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Gravitational collapse and black hole formation in a braneworld Wang, Daoyan 2015

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Gravitational Collapse and Black HoleFormation in a BraneworldbyDaoyan WangB.Sc., University of Science and Technology of China, Hefei, China, 2004M.Sc., The University of British Columbia, Vancouver, Canada, 2008A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2015c© Daoyan Wang 2015AbstractIn this thesis we present the first numerical study of gravitational collapse in braneworld withinthe framework of the single brane model by Randall-Sundrum (RSII). We directly show that theevolutions of sufficiently strong initial data configurations result in black holes (BHs) with finiteextension into the bulk. The extension changes from sphere to pancake (or cigar, as seen from adifferent perspective) as the size of BH increases. We find preliminary evidences that BHs of thesame size generated from distinct initial data profiles are geometrically indistinguishable. As such,a no-hair theorem of BH (uniqueness of BH solution) is suggested to hold in the RSII spacetimesstudied in this thesis—these spacetimes are axisymmetric without angular momentum and non-gravitational charges. In particular, the BHs we obtained as the results of the dynamical system,are consistent with the ones previously obtained from a static vacuum system by Figueras andWiseman. We also obtained some results in closed form without numerical computation such asthe equality of ADM mass of the brane with the total mass of the braneworld.The calculation within the braneworld requires advances in the formalism of numerical relativity(NR). The regularity problem in previous numerical calculations in axisymmetric (and sphericallysymmetric) spacetimes, is actually associated with neither coordinate systems nor the machine pre-cision. The numerical calculation is regular in any coordinates, provided the fundamental variables(used in numerical calculations) are regular, and their asymptotic behaviours at the vicinity of theaxis (or origin) are compatible with the finite difference scheme. The generalized harmonic (GH)formalism and the BSSN formalism for general relativity are developed to make them suitable forcalculations in non-Cartesian coordinates under non-flat background. A conformal function of themetric is included into the GH formalism to simulate the braneworld.iiPrefaceChapter 1 is the introduction and nothing in the chapter is original work. The author rederivedthe formulae in the literature while keeping d (the dimension of spacetime) and ǫ (the sign tocharacterize spacelike or timelike foliation of spacetime) general.All the works in the rest chapters (Chap. 2, 3, 4 and 5), except for the works properly cited,are original. Within these:— A significant part of the work regarding initial data (presented in Chap. 4) was conductedin the collaboration with Evgeny Sorkin. Specifically, Evgeny proposed “direct method”(defined in the chapter), taught the author to calculate the total energy in general relativityfollowing [87], and proposed to study the mass-area relation.— Eq. (5.34), the form of the relation among the circumstances of black holes, was proposedby Toby Wiseman.— All the remaining works were carried out by the author, with the guidance of Matthew W.Choptuik and William G. Unruh.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xNotations and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Overview of Our Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Spacetime Foliation and the Decomposition of Einstein’s Equations . . . . . . . . 71.2.1 The Decomposition of Einstein’s Equations . . . . . . . . . . . . . . . . . . 91.2.2 Israel’s Junction Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.3 Coordinate Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3 Randall-Sundrum Braneworld II (RSII) . . . . . . . . . . . . . . . . . . . . . . . . 121.3.1 Formalism for General Braneworld . . . . . . . . . . . . . . . . . . . . . . . 121.3.2 Randall-Sundrum Braneworld II . . . . . . . . . . . . . . . . . . . . . . . . 131.3.3 Parameter Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3.4 Vacuum Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3.5 Matter in RSII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15ivTable of Contents1.4 The Global Structures of AdS Spacetime and the Poincare´ Patch . . . . . . . . . . 161.4.1 AdS Spacetime and the Poincare´ Patch . . . . . . . . . . . . . . . . . . . . 161.4.2 The Penrose Diagram of the AdS2 Spacetime . . . . . . . . . . . . . . . . 181.4.3 The Infinities of the Poincare´ Patch . . . . . . . . . . . . . . . . . . . . . . 201.4.4 The Poincare´ Patch of AdS3 . . . . . . . . . . . . . . . . . . . . . . . . . . 211.4.5 The RSII Braneworld and Its Structure . . . . . . . . . . . . . . . . . . . . 261.5 Evolution Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.6 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.6.1 Finite Difference Approximation . . . . . . . . . . . . . . . . . . . . . . . . 301.6.2 Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.6.3 Tests for General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . 342 Characteristics in the Braneworld Spacetime . . . . . . . . . . . . . . . . . . . . 362.1 Boundary Conditions at the Brane . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.2 Apparent Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.2.1 The Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.2.2 Smoothness of the Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.2.3 Apparent Horizon on the Brane and in the Bulk . . . . . . . . . . . . . . . 402.3 Event Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.3.1 Event Horizon in the Braneworld . . . . . . . . . . . . . . . . . . . . . . . 412.3.2 Event Horizon on the Brane . . . . . . . . . . . . . . . . . . . . . . . . . . 422.3.3 Extrinsic Curvature as Geodesics Deviation . . . . . . . . . . . . . . . . . 432.3.4 The Relation between the Event Horizons . . . . . . . . . . . . . . . . . . 452.4 Energy in the Braneworld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.4.1 Total Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.4.2 Total Energy in the Braneworld with Axisymmetry . . . . . . . . . . . . . 492.4.3 Total Energy in Conformally Flat Space of the Braneworld . . . . . . . . . 542.5 ADM “Mass” and Hawking “Mass” of the Brane . . . . . . . . . . . . . . . . . . . 562.5.1 ADM Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.5.2 Hawking Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.5.3 The ADM “Mass” of the Conformally Flat Space . . . . . . . . . . . . . . 582.6 A Quest for Brane Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59vTable of Contents3 Axisymmetric Spacetime with Non-Flat Background . . . . . . . . . . . . . . . 623.1 Regularity Problem and Our Conjecture . . . . . . . . . . . . . . . . . . . . . . . 633.1.1 The Existing Analysis and The Existing Solutions . . . . . . . . . . . . . . 643.1.2 The Conjecture: Variables Rather Than Coordinates . . . . . . . . . . . . 663.1.3 Cartesian Components Method . . . . . . . . . . . . . . . . . . . . . . . . 683.1.4 Results for General Symmetric Tensors . . . . . . . . . . . . . . . . . . . . 713.1.5 Test of the Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.1.6 Validity and the Extension of the Conjecture . . . . . . . . . . . . . . . . . 733.1.7 Summary of Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.2 The Generalized Harmonic Formalism in Non-Cartesian Coordinates . . . . . . . . 773.2.1 An Introduction to the Generalized Harmonic Formalism . . . . . . . . . . 773.2.2 The Generalized Harmonic Formalism in Cylindrical Coordinates . . . . . 793.2.3 Background Removal in the Literature . . . . . . . . . . . . . . . . . . . . 803.3 The Generalized Harmonic Formalism in Non-Flat Backgrounds . . . . . . . . . . 823.3.1 Tensorial Source Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.3.2 Conformal Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.3.3 The Implementation of Generalized Harmonic Formalism . . . . . . . . . . 843.4 Evolution of Massless Scalar Field under Cylindrical Coordinates . . . . . . . . . 853.4.1 Initial Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.4.2 Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.4.3 Superficially Singular Metric Representation . . . . . . . . . . . . . . . . . 913.5 Remark about Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934 Initial Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.1.1 Laplacian Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.1.2 Direct Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.2 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.3 The Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.3.1 The Solution and Apparent Horizon . . . . . . . . . . . . . . . . . . . . . . 994.3.2 Brane Geometry as Seen by a Brane Observer . . . . . . . . . . . . . . . . 994.3.3 Asymptotic Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.4 The Total Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.4.1 The Relation with the Area of Apparent Horizon . . . . . . . . . . . . . . 104viTable of Contents4.5 Discussion: the Results of Laplacian Specification . . . . . . . . . . . . . . . . . . 1064.6 Prepare the Initial Data for Evolution . . . . . . . . . . . . . . . . . . . . . . . . . 1074.6.1 Prepare the Initial Data for Specific Initial Source Functions . . . . . . . . 1074.6.2 Total Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095 Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.1 The Evolution as an Initial Value Problem . . . . . . . . . . . . . . . . . . . . . . 1115.2 Details of the Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . 1145.3 Gauge Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.3.1 Fixing Gauges via Source Functions . . . . . . . . . . . . . . . . . . . . . . 1155.4 Constraint Violation and the Cure . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.4.1 Constraint Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.4.2 Imposing Constraints on the Brane . . . . . . . . . . . . . . . . . . . . . . 1185.5 The Evolution with an Apparent Horizon . . . . . . . . . . . . . . . . . . . . . . . 1205.5.1 Smoothness of Apparent Horizons in the Braneworld . . . . . . . . . . . . 1225.5.2 Apparent Horizon Finder . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1225.5.3 Dissipation at the Excision Boundary . . . . . . . . . . . . . . . . . . . . . 1245.6 Tests and the Validation of the Numerical Scheme . . . . . . . . . . . . . . . . . . 1245.7 The Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.7.1 The Evolution Process and Apparently Stationary State . . . . . . . . . . 1255.7.2 Black Holes as the Result of Gravitational Collapse . . . . . . . . . . . . . 1295.7.3 The Relation with Black Strings . . . . . . . . . . . . . . . . . . . . . . . . 1355.7.4 The No-hair Feature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385.7.5 The Comparison with Figueras-Wiseman Solution . . . . . . . . . . . . . . 1395.7.6 Brane Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406 Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1456.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1456.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148AppendicesA Generalized BSSN Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158viiTable of ContentsB Extrinsic Curvature as Geodesics Deviation: C ≥ 1 Case . . . . . . . . . . . . . 163viiiList of Tables5.1 BHs Produced from Different Initial Data Families . . . . . . . . . . . . . . . . . . 135ixList of Figures1.1 The Area of AH as a Function of the Radius on the Brane . . . . . . . . . . . . . . 41.2 The Embedding of a Hypersurface in Spacetime . . . . . . . . . . . . . . . . . . . . 71.3 Penrose Diagrams of AdS2 and its Poincare´ Patch . . . . . . . . . . . . . . . . . . 201.4 Penrose Diagrams of AdS3 and its Poincare´ Patch . . . . . . . . . . . . . . . . . . 251.5 r = const Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.6 t = const Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.7 z = const Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.1 Manifold M and its Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.2 The Closed Surfaces to Calculate Total Energy and Energy in the Bulk . . . . . . 502.3 The Embedding of a Closed Surface . . . . . . . . . . . . . . . . . . . . . . . . . . 513.1 The Demonstration of the Cartoon Method . . . . . . . . . . . . . . . . . . . . . . 643.2 The Preservation of Spherical Symmetry Monitored by Φ . . . . . . . . . . . . . . 893.3 The Convergence Tests of the Normal Simulation . . . . . . . . . . . . . . . . . . . 903.4 The Convergence Tests using the Original Einstein’s Equations . . . . . . . . . . . 913.5 The Convergence Test of the Superficially Singular Simulation . . . . . . . . . . . . 923.6 The Comparison of the Results from the Two Simulations . . . . . . . . . . . . . . 934.1 Initial Data: function A and function −B . . . . . . . . . . . . . . . . . . . . . . . 1004.2 The Apparent Horizons in Initial Data . . . . . . . . . . . . . . . . . . . . . . . . . 1004.3 Brane Masses in Initial Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.4 The Asymptotic Behaviour in Initial Data . . . . . . . . . . . . . . . . . . . . . . . 1034.5 Total Mass VS The Area of Apparent Horizon . . . . . . . . . . . . . . . . . . . . . 1054.6 The Asymptotic Behaviour in Initial Data Obtained via Laplacian Specification . . 1064.7 Initial Data: Function A and Function −B . . . . . . . . . . . . . . . . . . . . . . 1094.8 The Asymptotic Behaviour in Initial Data . . . . . . . . . . . . . . . . . . . . . . . 110xList of Figures5.1 The Evolution as an Initial Value Problem with Boundary Conditions . . . . . . . 1115.2 The Convergence Tests for The Enhanced Damping Method . . . . . . . . . . . . . 1195.3 The Convergence Tests for The Constraint Imposing Method . . . . . . . . . . . . 1215.4 The Snapshots of Four Evolutions Resulting in BHs with Different Sizes . . . . . . 1275.5 The Snapshots of Three Evolutions Resulting in BHs with Different Sizes . . . . . 1285.6 An Evolution that Produces a Medium BH . . . . . . . . . . . . . . . . . . . . . . 1305.7 An Evolution that Produces a Medium BH—continued. . . . . . . . . . . . . . . . 1315.8 An Evolution that Produces a Small BH . . . . . . . . . . . . . . . . . . . . . . . . 1325.9 An Evolution that Produces a Small BH . . . . . . . . . . . . . . . . . . . . . . . . 1335.10 An Evolution that Produces a Large BH . . . . . . . . . . . . . . . . . . . . . . . . 1345.11 The Shape of BHs with All Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365.12 The Shape of BHs via Embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . 1375.13 The Embedding of a Closed Surface . . . . . . . . . . . . . . . . . . . . . . . . . . 1405.14 The Comparison with the Figueras-Wiseman Solution via Embedding . . . . . . . 1415.15 Energies in 4D GR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1435.16 “Energies” on Brane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1435.17 Brane Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144xiNotations and AbbreviationsPlease pay particular attention to sign conventions: the sign of metric, the sign of the Christoffelsymbol, and the sign of the Riemann tensor. The conventions we employ in this thesis, are thesame as those in Baumgarte-Shapiro [34] (therefore the same as those in Wald [32]). The sign ofextrinsic curvature is the same as Baumgarte-Shapiro [34] (therefore the opposite of Wald [32]).GR General RelativityBH Black HoleBS Black StringPDE Partial Differential EquationFDA Finite Difference ApproximationGH Generalized HarmonicRSII Randall-Sundrum braneworld model II (the single brane model)gµν spacetime metricγµν spatial metrichµν brane metric∇ the covariant derivative associated with gD the covariant derivative associated with γD the covariant derivative associated with hd the dimension of the whole spacetimeMetric signature (−,+,+,+,+)α lapse functionβµ shift functionxiiNotations and AbbreviationsChristoffel symbols Γαβγ =12gαµ (gµβ,γ + gµγ,β − gβγ,µ),where ,µ ≡ ∂µRiemann Tensor Rγµαβ defined via ∇α∇βvγ −∇β∇αvγ = Rγµαβ vµ, ∀ vector field vµnµ unit normal vector of t = const hypersurfaceKµν ≡ −γ αµ γ βν ∇αnβ, extrinsic curvature of t = const hypersurfacenν unit normal vector of the braneKµν ≡ −h αµ h βν ∇αnβ , extrinsic curvature of the branetilde˜ quantities and operations associated with conformal metricbar¯ quantities and operations associated with background metricΛ cosmological constant in the bulkλ tension of the branekn = 8πGn where Gn is Newton’s constant in n dimensionxiiiAcknowledgementsFirst and foremost I would like to thank my supervisor, Matthew Choptuik. He taught me howsolid scientific work should be conducted. He encouraged me to pursue a challenging yet importantproject. Among other supports, I especially appreciate the freedom that he gave me during theseyears.I would like to thank William Unruh for his inspirations and his guidance, as well as hisinvolvement and support during my PhD studies.I thank Evgeny Sorkin for the collaboration on the initial value problem and discussions onvarious other topics. I thank Frans Pretorius for the PAMR/AMRD package and his continuousadvise, and his generous share of his code on binary black holes. The conversation with Frans,Evgeny and Matt at Matt’s home in August 2011, about regularity and Generalized Harmonicformalism in non-Cartesian coordinate, was crucial to help me realize the confront of the researchand motivated me to pursue further.I thank Toby Wiseman and Pau Figueras for their generosity to send me their final data tocompare, and the useful discussions.I would like to thank the members of my committee: Colin Gay and Mark Van Raamsdonk forthe questions and comments which helped me to improve my thesis.It was a pleasure being a member of the numerical relativity group at UBC. I am happy tothank my friends and collegues, Arman, Ben, Bruno, Daniel, Graham, Jason and Silvestre. Specialthanks go to Arman for his good questions which helped to improve the thesis. I also thank Arman,Graham and Silvestre for proofreading certain parts of my thesis.I want to thank Junqi Guo for the collaboration on f(R) study (which is not a part of thisthesis).I am grateful to my wife, Yuanyuan Gao, for her love, all kinds of support, understanding andcompanionship during these years, which enable me to do what I do.I would like to thank my parents and my sister for their love, sacrifice and the freedom theygave me, and thank my parents in law for their support and understanding.xivTo my family.xvChapter 1Introduction1.1 OverviewOur observable universe is 3+1 dimensional. The exploration on the possibility of the existenceof extra dimensions can trace back to the work by Kaluza and Klein in 1920s [1, 2], where theytried to unify electromagnetism with gravity by using the metric tensor in five dimensional (5D)spacetime in which our universe was a 3+1 dimensional hypersurface. The basic idea of braneworldscenarios is that our observable universe could be a 3+1 dimensional hypersurface (the brane)embedded in a higher dimensional spacetime (the bulk). Only gravity can propagate freely intothe bulk while all the matters are confined onto the brane. Among all the braneworld models, thesingle brane model constructed by Randall and Sundrum (also known as RSII) [3], is remarkable forits simplicity. It assumes one extra dimension with infinite size, a negative cosmological constantΛ in the bulk, tension λ on brane which makes the brane a gravitating object, and Z2 symmetry ofthe bulk with respect to the brane. λ is set to a value such that general relativity (GR) is recoveredon the brane at low energies [4]. However, the high energy behaviour, where the dynamics couldbe rather different from GR, is not understood as clearly.Black holes (BHs), among others, are objects that can illustrate the high energy behaviour. BHsare predicted by GR, and there is strong observational evidence for their existence [5], thereforeRSII needs to reproduce BHs in order to be a viable physical theory. In the absence of matter, thereexist a particular class of solutions in RSII constructed from vacuum solutions of four dimensional(4D) GR using the construction method in [6]. These solutions are called black strings whenthe corresponding 4D solutions are black holes. However, the black strings are unstable dueto the Gregrory-Laflamme instability [7, 8]. In fact, Lehner-Pretorius [69] showed that certaintype of black string (which is different from the black string in braneworld) evolves into a seriesof 3D spherical BHs connected by black strings and the black strings continue to evolve in thesame pattern, which yields a self-similar configuration. Within finite asymptotic time, a nakedsingularity is created, which violates the cosmic censorship hypothesis. Besides, the Kretschmann11.1. Overviewscalar RµναβRµναβ (where Rµναβ is the Riemann tensor) diverges at the AdS horizon [6], whichmeans the black string (in RSII) solution itself has pathology. Therefore, black strings can not beformed via natural processes such as gravitational collapse.What is the bulk counterpart of the 4D black hole? Noticing that the instability is most severeat the AdS horizon where black strings might “pinch off”, Chamblin-Hawking-Reall [6] proposedthat BH with finite extension into the bulk was a stable state and was actually the end state ofgravitational collapse. The investigation on this proposal started from studying one aspect of thisproposal: the existence of such black objects. Tanaka [9] and Emparan et al. [10] conjectured, viaholography, that static black holes with radii much greater than the AdS length, can not exist inRSII. Fitzpatrick et al. [11] and Gregory et al. [12] then argued that such a conjecture may notbe justified since the arguments in [9, 10] did not take into account the strongly coupled nature ofthe holographic theory.Several other groups investigated this proposal using numerical techniques and tried to findspecific solutions describing the BHs. As a first step, Shiromizu-Shibata [13] studied time sym-metric initial datasets for the Einstein’s equations, demanding that the data contained apparenthorizons, so that subsequent evolution would presumably settle down to black holes. Kudoh-Tanaka-Nakamura [14] carried out calculations on static BHs in the braneworld, finding solutionswith finite extension into the bulk, but only for radii that were small compared to the AdS ra-dius. Subsequent work by Kudoh [15] showed that corresponding solutions could also be found inthe 6D braneworld, and that the horizon sizes could be larger than the AdS radius in that case.Further numerical calculations in 5D by Tanahashi-Tanaka [16] found time symmetric initial datawith large apparent horizons. In 2008, Yoshino [17] repeated the calculation done in [14], usinga more accurate numerical scheme, and argued that due to certain systematic errors exhibited bythe solutions—and hence presumably affecting the previous computations—there was no evidencethat BHs with finite extension into the bulk could exist at all (i.e. not even small ones). In 2011,Kleihaus et al [19] also claimed the same.Finally, the debate was settled by the remarkable work carried out by Figueras-Wiseman [20] in2011. Via perturbing AdS/CFT solution (which itself was also numerically constructed [21]), theyobtained static black holes with a range of sizes, including ones large compared to the AdS scale.Recently (2014), Figueras presented an improved calculation [22] which obtained black holes withmuch wider range, thus presumably black holes can be of any size. In 2012, large BHs were alsoobtained by Abdolrahimi et al [24] using a different method, and there was preliminary evidencethat their solution was the same as the Figueras-Wiseman solution.21.1. OverviewWithout proving the stability and uniqueness of the solution however, it is not clear whether theFigueras-Wiseman BHs can be formed via natural processes such as gravitational collapse. Moregenerally, despite the success on the statics side, the full dynamics of RSII—a more important andinteresting aspect—remains poorly understood, especially in the high energy regime. The answersto fundamental questions, such as the end state of gravitational collapse, or the interaction betweenthe brane and bulk, remain unknown or vague.The main goal of this thesis work is to study the full dynamics in RSII via numerical calculation.Specifically, we will study the dynamical process of black hole formation as a result of gravitationalcollapse of massless scalar field (which lives only on the brane). To our knowledge, there is noprevious calculations of the full dynamics of (any) braneworld scenario. The study is crucial tounderstand the dynamics of braneworld models. Technically, as it turns out, the study deepensour understanding and also extends the formalism of numerical relativity.The calculation in the braneworld is significantly more difficult than that in GR. The confine-ment of matter to the brane and the interaction between the brane and the bulk due to branecontent including matter and brane tension, are the roots of the difficulties [25]. Gravitational col-lapse inevitably produces energies high enough to make the interaction between the brane and thebulk significant. This rules out any attempts to solve the problem using brane content only. Dueto this interaction, the brane equations are not closed, thus the dynamics of the whole spacetime(including both the brane and the bulk) needs to be studied.Here we successfully performed a numerical study, and we directly show that the results ofthe gravitational collapse of sufficiently strong initial configurations are BHs with finite extensioninto the bulk. The extension changes from sphere to pancake (or cigar, seen from a differentperspective) as the size increases. There are indications that BHs with the same size producedfrom different initial data, have the same shape, which means the details of initial data are lost inthe resulting BHs. Therefore a no-hair theorem (uniqueness of BH) is suggested to hold also forBHs in the RSII spacetimes that are studied in this thesis. In particular, the BHs we obtained asthe results of dynamical systems, are consistent with the ones previously obtained from a staticproblem by Figueras-Wiseman [20]. Please refer to Fig. 1.1 for a preview.There are gaps between the existing knowledge in numerical relativity and the knowledge neededfor the numerical calculation of braneworlds. We developed some of these to make our numericalcalculation possible. With these developments, we solved the regularity problem in numericalrelativity (the irregularity associated with simulations in axisymmetric or spherically symmetricspacetime which appeared in previous numerical relativity calculations) by studying it from a31.1. Overviewdifferent perspective than previous works did. Our work deepens the understanding of the problem,and can potentially completely solve the problem. We generalized evolution formalisms (generalizedharmonic and BSSN) of GR to make them suitable for simulations in non-Cartesian coordinateunder non-flat background. A conformal function of the metric is included in the formalism tosimulate the braneworld. The constraint violations found in the braneworld calculations, are curedby imposing the constraints at the brane.0.04 0.1 1 10 2010−210−1100101102103104ra/ℓAbulk/ℓ3  BS5D SchFW DataOur DataFigure 1.1: The area of bulk apparent horizon as a function of the areal radius of the horizonon the brane. The coupling strength of the brane tension is proportional to 1/ℓ (where ℓ is theAdS length), which is invisible to the BHs whose sizes (the areal radii on the brane) are muchsmaller than ℓ. Therefore small BHs behave as 5D Schwarzschild. When the size is much largerthan ℓ, the asymptotic relation is that of the corresponding black string. Our data obtained fromthe evolutionary system, is consistent with the data obtained by Figueras-Wiseman [20] from astatic system.1.1.1 Overview of Our ApproachThe system is formulated as an initial value problem (IVP), with Einstein’s equations in thebulk as the governing equation. The brane content, including brane tension and the matter, isimposed via Israel’s junction conditions [26], which serve as parts of the boundary conditions ofthe IVP. The matter on the brane evolves as ordinary 4D matter (which only “feels” the 4D metricon the brane).Given that calculations of this sort (or even published attempts) do not exist, and due to theprohibitive computational cost of performing the calculations in the fully 5D context, we restrict41.1. Overviewourselves to spherical symmetry on the brane, which makes the system axisymmetric in the bulk(i.e. functions depend on two spatial dimensions plus time).Even with this significant simplification, we are faced with a challenging task, not least sincethere are several features of the problem that have not been addressed, or fully resolved, in previousnumerical relativity studies. Apriori, there are the following challenges(1) The numerical treatment of delta-function (distributional) matter(2) The numerical treatment of a brane with tension(3) Regularity issues induced by axisymmetry(4) Appropriate evolution schemes for use in non-Cartesian coordinate systems, and under non-flatbackgrounds(5) Properly incorporating the AdS boundary conditions(6) Higher dimensional numerical relativityAmong the features mentioned above, some are more challenging and cause more severe prob-lems than others, and could be made into good projects by themselves. In fact, Oliver Rinne’sPhD thesis (at University of Cambridge, 2005) [71] was on regularity issues in axisymmetry. Hestudied a specific formalism (Z(2+1)+1 system) for axisymmetric system. On the other hand,what we are going to present in this thesis is a “direction change” of the study on this topic.We will conjecture that the regularity problems are caused by the the fundamental variables usedin numerical simulations, rather than coordinates. We further developed a few general methodsto produce formalisms that yield regular results. Hans Bantilan’s PhD thesis (at Princeton Uni-versity, 2013) [68] was on the AdS spacetime in 5D, where lightlike signals can propagate to thespatial infinity and come back to its departure location within a finite proper time as measuredby a timelike observer. In this thesis, by employing the formalisms we developed to simulatebraneworld spacetime (in Chap. 3), there are no issues associated with the higher dimensions. Thespacetime of the braneworld is asymptotically a part of the Poincare´ patch of the AdS spacetimewhose causal structure is similar to that of Minkowski spacetime. Therefore Cauchy surfaces existin the spacetime (so that Cauchy problem is well-defined), and the above mentioned problem as-sociated with lightlike signals will not occour. There will be complications related to the Killinghorizons of the Poincare patch, however, such as a timelike curve of finite length can reach theKilling horizons. Please refer to Sec. 1.4 for more details about AdS spacetime, the Poincare´ patchand the background of the RSII braneworld.51.1. OverviewDuring the investigation, we encountered peculiar properties and challenges which were notanticipated. In solving for time symmetric initial data, in most situations the problem can reduceto solving the Hamiltonian constraint for one unknown variable. In RSII, however, there have to beat least two unknown variables in the initial data to make Israel’s boundary conditions consistent;normally the shift and lapse functions can be arbitrarily specified utilizing the gauge freedomin choosing coordinates, while they have to satisfy certain conditions in the braneworld due toIsrael’s boundary conditions. In most situations, the constraint violation problem in free evolutionschemes, such as the generalized harmonic formalism and BSSN formalism, can be consideredsolved by constraint damping method [58, 61–63]. However, the same method does not control thesevere violation at the brane of the braneworld. We found that we could solve the violation byproperly imposing the constraints at the brane boundary.Therefore we add the following two additinal problems to the feature list(7) Peculiar properties in initial data problem(8) Constraint violation in braneworldIn the following chapters, we will show how we solve these challenges and achieve the simulationof the dynamical process of black hole formation as the result of gravitational collapse on the brane.The thesis is divided into three parts. In the remaining sections of Chap. 1, we will introduce thebraneworld in more detail plus some basic aspects of numerical computation. The second partis devoted to develop the machinery to perform simulations and to extract physical informationsin the braneworld. This part consists of Chap. 2 and Chap. 3. Chap. 2 develops the conceptualaspects, such as the smoothness of apparent horizon across the brane, the relation between branehorizons and bulk horizons, the coordinate gauge at the brane, and energy aspects of braneworld.In Chap. 3, our approach to the regularity problem will be presented. We also show how evolutionschemes of GR, such as the generalized harmonic formalism, can be developed and generalized innon-Cartesian coordinate under non-flat asymptotic spacetime background, to be made suitablefor simulations in the braneworld. The final part is the numerical simulation. Initial data isobtained in Chap. 4, which also provides numerical results to support the discussion on energyaspects in Chap. 2. In Chap. 5, we show how the constraint violations are solved, and also showthe simulation per se, and the physical results. Other than Chap. 1, all the works in this thesis,except for the ones properly cited, are original.61.2. Spacetime Foliation and the Decomposition of Einstein’s Equations1.2 Spacetime Foliation and the Decomposition ofEinstein’s EquationsIn this section, we will introduce the notations and conventions in the context of the (d−1)+1decomposition formalism of general relativity (GR), and Israel’s junction conditions, which areneeded to introduce the braneworld. Here d is the dimension of the whole spacetime. Metricsignature is (−,+, . . . ,+), which is the same as that in [29, 32, 34, 36].The (d−1)+1 decomposition of Einstein’s equations, is a generalization of the 3+1 formalismof GR [34, 36]. We start with a (d − 1) dimensional hypersurface Σ embedded in d dimensionalspacetime M . Σ can be thought as the brane in braneworld, or a “time”= constant hypersurfacein an evolutionary problem. I.e. the formalism introduced in this section applies to both cases.Especially, the parameter t in this section, is not reserved for “time”.Σ divides M into two parts M±. Please refer to Fig. 1.2.Σn+n−M+M−Figure 1.2: The embedding of a hypersurface in a higher dimensional spacetime. The wholespacetime M is composed of M+ (the half sphere) and M− (the plane with a big hole in themiddle to fit M+). The intersection of M+ and M− is hypersurface Σ (the circle). n± are unitnormal vectors of Σ, as seen in M±. The convention is that n+ points into M+, and n− pointsout of M−. Note: (i) the extrinsic curvature of Σ in M+ can be different from that in M−. Forexample, in this figure Σ is a curved line in M−, but a straight line in M+; (ii) In general n+ 6= n−(but a few authors incorrectly assumed n+ = n− when deriving junction conditions). For example,in the figure n+ is not defined inM− and n− is not defined inM+. Another example is n− = −n+in Randall-Sundrum braneworlds.Generally Σ can be a hypersurface within a hypersurface family that locally foliates the space-time. Let us use parameter t to characterize the foliation, where t = constant is a specific hy-persurface (Σt) within the family. t (M+) > t (M−) is the convention we adopt. Let n± be unitnormal vectors of Σ, as seen in M±. The convention is that the direction of n± is the same as theincreasing direction of t. Therefore n+ points intoM+, and n− points out ofM−. The super-index71.2. Spacetime Foliation and the Decomposition of Einstein’s Equations± is omitted whenever we express anything that is valid for both signs. nα is normalized asnαnα = ǫ, (1.1)where ǫ is +1 when Σ is the brane in the braneworld and −1 when Σ is a “time”=constanthypersurface. In terms of t, the normal vector nµ isnµ = ǫ α∇µt, (1.2)where α ≡(√ǫ∇µt∇µt)−1 is the lapse function. ∇ is the covariant derivative associated with themetric gµν . The greek index µ, ν = 0, . . . (d− 1), where 0 is the time coordinate. Definemµ ≡ αnµ, (1.3)which satisfies mµ∇µt = 1.The decomposition of Einstein’s equations is expressed in terms of the following quantities.The induced metric on Σγµν ≡ gµν − ǫ nµnν . (1.4)The extrinsic curvature is defined asKµν ≡ −γ αµ γ βν ∇αnβ = −γ αµ ∇αnν . (1.5)Note the geometrical meaning: the extrinsic curvature is defined as the “change rate” of the normalvector nα along the hypersurface, as seen in the bulk (M±). Since the intrinsic observer on Σ cannot even “feel” nα, the extrinsic curvature characterizes the extrinsic nature of the embedding. 1The definition of Riemann tensor is the same as that in [29, 32, 34, 36], which is defined via anarbitrary vector field vµ∇α∇βvγ −∇β∇αvγ = Rγµαβ vµ. (1.6)We also adopt the same sign convention of Christoffel symbols as that in [29, 32, 34, 36]Γαβγ =12gαµ (gµβ,γ + gµγ,β − gβγ,µ) , (1.7)1The extrinsic curvature can also be obtained as the measure of the deviation of the geodesics in Σ and M ,produced by the same vector lying within Σ. Please refer to section 2.3.3 and appendix B for the study from thisperspective.81.2. Spacetime Foliation and the Decomposition of Einstein’s Equationswhere ,µ ≡ ∂µ.1.2.1 The Decomposition of Einstein’s EquationsRepeating the Gauss-Codazzi-Ricci decomposition [29, 32, 34, 36] while keeping d and ǫ general,the Einstein tensor Gµν ≡ Rµν − 12gµνR is decomposed as(d)Gµνnµnν =12(−ǫ · (d−1)R+K2 −KµνKµν); (1.8)(d)Gµνnνγµα = DαK −DµKµα; (1.9)(d)Gµνγµαγνβ =ǫα (LmKαβ − γαβLmK) +(d−1)Gαβ+ ǫ(2KαµKµβ −KKαβ +12γαβ(K2 +KµνKµν))+ 1α (γαβDµDµα−DαDβα) , (1.10)where D is the covariant derivative associated with the hypersurace metric γµν . Lm is the Liederivative with respect to mµ. The super-indices (d) and (d−1) are here to characterize dimension.For example, (d−1)Gµν and (d−1)R are the intrinsic Einstein tensor and the intrinsic Ricci scalar ofΣ.Imposing Einstein’s equations in the d dimensional spacetime (which relates the geometry withmatter via matter’s stress tensor Tµν)(d)Gµν = kdTµν , (1.11)we obtain Hamiltonian constraint, momentum constraint and evolution equationkd ρ =12(−ǫ(d−1)R+K2 −KµνKµν); (1.12)ǫ kd Sα = DαK −DµKµα; (1.13)kd Sαβ =ǫα (LmKαβ − γαβLmK) +(d−1)Gαβ+ ǫ(2KαµKµβ −KKαβ +12γαβ(K2 +KµνKµν))+ 1α ((γαβDµDµα−DαDβα) , (1.14)where kd = 8πGd andGd is Newton’s constant in d dimension. We have defined ρ ≡ Tµνnµnν , Sα ≡ǫ Tµνnνγµα, Sαβ ≡ Tµνγµαγνβ , which yield the following decompostion: Tµν = ρnµnν + nµSν +91.2. Spacetime Foliation and the Decomposition of Einstein’s EquationsSµnν + Sµν . Taking trace gives T = S + ǫρ, where S ≡ Sµνγµν .The evolution equation (1.14) can be equivalently expressed as [34, 36]LmKαβ = ǫDαDβα+ α[−ǫ (d−1)Rαβ +KKαβ − 2KαµKµβ + ǫ kd(Sαβ − γαβS + ǫρd− 2)]. (1.15)The definition equation of extrinsic curvature (1.5), can be equivalently expressed asLmγαβ = −2αKαβ. (1.16)(1.15) or (1.14), together with (1.16) form a complete set of evolution equations [34, 36] using γµνand Kµν as fundamental variables. This formalism is called ADM-York formalism of GR.1.2.2 Israel’s Junction ConditionsIn classical electromagnetism, there are situations where electric charges are highly concentratedon two dimensional surfaces such as the interfaces of different materials. The distribution of theelectric charges is then singular in the 3D space. The electric fields appear to be discontinuousacross the singular layers, and the relation relating the discontinuities of the electric fields withthe singular distributions of the electric charges, is called a junction condition.Similarly, in GR, if the stress tensor is highly concentrated on Σ, it will cause discontinuities.The discontinuities are described by Israel’s junction conditions.Israel’s first junction condition [26] states that the intrinsic geometry of the hypersurface is welldefined. i.e. the induced metric of Σ obtained from M+ agrees with the induced metric obtainedfrom M−[ˆγµν ]ˆ = 0, (1.17)where the notation [ˆaˆ] ≡ a+ − a−.Integrating (1.14) over an infinitesimal layer across Σ, we get Israel’s second junction condition[26]kd Sµν = ǫ([ˆKµν ]ˆ− γµν [ˆK ]ˆ)or [ˆKµν ]ˆ = ǫkd(Sµν − γµνSd− 2), (1.18)where Sµν is the singular part of the projected energy-momentum tensor on Σ defined asSµν ≡∫ 0+0−Sµνdl, (1.19)where dl ≡ dt (∂t)µ nµ = αdt is the proper length/time across Σ, and l is (arbitrarily) set to be101.2. Spacetime Foliation and the Decomposition of Einstein’s Equationsl = 0 at Σ.Integrating (1.12) and (1.13) givesSµ ≡∫ 0+0−Sµdl = 0; ̺ ≡∫ 0+0−ρdl = 0, (1.20)which is because the right hand sides (RHSs) of (1.12) and (1.13), as well as the non-Lm terms onthe RHS of (1.14), are all well-defined and finite on both sides of the hypersurface.Eq. (1.13) and (1.18) giveDµS µν = −[ˆSν ]ˆ, (1.21)which is the singular matter’s conservation law on Σ.Because of the identity [ˆa2ˆ] = [ˆaˆ]{ˆa}ˆ, where {ˆa}ˆ ≡ a+ + a−, eq. (1.12) giveskd [ˆρˆ] = 12([ˆK ]ˆ{ˆK }ˆ − [ˆKµν ]ˆ{ˆKµν }ˆ). Combining with (1.18), we have[ˆρˆ] = ǫ2(S2− d {ˆK }ˆ −(Sµν − γµνSd− 2){ˆKµν }ˆ)= − ǫ2Sµν {ˆKµν }ˆ. (1.22)Eq. (1.21) and (1.22) are constraint/conservation of the singular matter.1.2.3 Coordinate DescriptionA coordinate system is needed to perform numerical relativity. A coordinate system using t asa coordinate, can be constructed as the following. On each Σt, a coordinate system xi is assignedand xi is set to be differentiable across Σt. Here the Latin index i runs from 1 to (d − 1). (t, xi)is then a coordinate system.Since (∂t)µ∇µt = 1 = mµ∇µt, the shift function βµ ≡ (∂t)µ − mµ is perpendicular to ∇µt,therefore in the tangent space of Σt. The evolution equations can be written in coordinate systemby using Lmµ = L(∂t)µ −Lβµ , where L(∂t)µ is simply ∂t in the coordinate system in which t servesas one of the coordinate [31]. In general βµ = βi (∂i)µ and γµν = γij(dxi)µ(dxj)ν . They canbe further reduced to βi and γij in this coordinate system by using (∂i)µ = (0, 0, ..., 1, ..., 0) and(dxi)µ = (0, 0, ..., 1, ..., 0), where only the i-th position is 1. An infinitesimal vector pointing from(t, xi) to (t+ dt, xi + dxi) is then dt (∂t)µ + dxi (∂i)µ, and its length is derived asds2 =(ǫ α2 + βiβi)dt2 + 2βidtdxi + γijdxidxj , (1.23)where βi ≡ γijβj .111.3. Randall-Sundrum Braneworld II (RSII)1.3 Randall-Sundrum Braneworld II (RSII)In this section we will introduce the single brane model by Randall-Sundrum (RSII). We willfirst introduce the results regarding the general ideas of braneworld (there is one extra dimension,and matter is confined on the brane), then introduce the setup that is unique in RSII.From now on, both the brane and “time”=constant hypersurface will be discussed. Thereforewe use different notations to distinguish between them. We will use Kµν , nν , hµν , Dµ, (d−1)Gµν ,(d−1)Rµν to denote the extrinsic curvature, unit normal vector, induced metric, covariant deriva-tive, Einstein tensor, Ricci tensor of the brane (therefore nµnµ = 1), and use Kµν , nµ, γµν ,Dµ, (d−1)Gµν , (d−1)Rµν to denote the corresponding quantities of the “time”=const hypersurfaces(therefore nµnµ = −1). Especially, the parameter t is reserved for “time”=const hypersurfaces.1.3.1 Formalism for General BraneworldA class of braneworlds are defined as a five dimensional spacetime where the matter is trappedon a (3+1) dimensional brane Σ, but gravity can propagate freely into the bulk. Since the braneΣ is our observed universe, we would expect that the induced equations on the brane becomeEinstein’s equations at a certain limit. Using Gauss’ equations and the d-dimensional Einstein’sequations, one gets [4](d−1)Gαβ =d− 3d− 2kd{Tµνhµαhνβ + hαβ(Tµνnµnν +T1− d)}+(KKαβ −KαγKhβ)− 12hαβ(K2 −KµνKµν)− Eαβ , (1.24)where Tαβ is the stress tensor in d dimensional spacetime, and T is the trace. Eµν is defined asEµν ≡ (d)Cαβγδ nαnγhβµhδν . (1.25)(d)Cµνγδ is the Weyl tensor defined by the following relation [4](d)Rµνγδ =(d)Cµνγδ +2d− 2(gµ[γ (d)Rδ]ν − gν[γ (d)Rδ]µ)− 2(d− 1)(d− 2)(d)R gµ[γgδ]ν , (1.26)where we used the notation A[αβ] ≡ 12 (Aαβ −Aβα).From (1.24) one sees that the matter content Tµν needs to be specified. The extrinsic curvatureand the projection of the Weyl tensor need to be addressed.121.3. Randall-Sundrum Braneworld II (RSII)1.3.2 Randall-Sundrum Braneworld IIIn RSII, the size of the extra dimension is infinite. The bulk is empty except for a negativecosmological constant. Therefore the stress tensor isTαβ = −Λgαβ + Sαβδ(l), (1.27)where Λ is the cosmological constant in the bulk, l is the proper length discussed in section 1.2.2.The brane is located at l = 0. In general, l is only defined in a neighbourhood of the brane. Thebrane content isSαβ = −λhαβ + ταβ , (1.28)where ταβ is the stress energy tensor of matter on brane, and λ is the tension of the brane, whichis required for the consistency of the theory (see below).There is a Z2 symmetry with respect to the brane [3] in Randall-Sundrum braneworlds, so thatn− = −n+. Z2 symmetry is mirror symmetry followed by an identification operation: a spacetimeM with Z2 symmetry can be obtained by taking a piece of spacetime with a boundary, thengenerating its image by parity transformation with respect to the boundary, and then gluing thesetwo pieces together by identifying the piece with its image (i.e. identifying point p with its imageas the same point, for all p ∈M). In another word, Randall-Sundrum braneworlds are “one-sided”spacetimes. This point will be made clearer in Sec. 1.3.4. As a consequence of n− = −n+, onecan obtain K+αβ = −K−αβ . Israel’s second junction condition (1.18) is then reduced toK+αβ = −K−αβ =12kd(Sαβ − hαβSd− 2)= 12kd(λ hαβd− 2 + ταβ − hαβτd− 2). (1.29)This equation relates Kαβ with the matter distribution on the brane, which can eliminate theextrinsic curvatures in (1.24). The Eαβ term in (1.24) is related to the geometry of the bulk andcan not be eliminated easily. However, as analysed in [4], this term is only important at highenergy regime.Now, (1.24) can be evaluated on either side of the brane, which may be realized by performingthe evaluation at l 6= 0 and then taking l → 0, because it does not make sense to do calculationexactly on the brane due to the delta-functions. One bonus relation obtained from this procedureis [ˆEαβ ]ˆ = 0, which is a consequence of the Z2 symmetry. The Einstein tensor on the brane istherefore [4](d−1)Gαβ = −Λd−1hαβ + kd−1ταβ + k2dπαβ − E±αβ , (1.30)131.3. Randall-Sundrum Braneworld II (RSII)whereΛd−1 = (d− 3)kd( 18(d− 2)λ2kd +1d− 1Λ), (1.31)kd−1 = k2dλd− 34(d− 2) , (1.32)παβ =2τταβ − hαβτ28(d− 2) +18(hαβτµντµν − 2ταγτ γβ). (1.33)This recovers Einstein’s equations on brane at low energy regime since Eµν is also negligible inthe low energy regime [4]. However, the behaviour could be quite different from GR at high energyscheme. Furthermore, due to the interaction between the brane and the bulk, the equations onthe brane do not form a close system. Therefore the attempts to solve the dynamics by evolvingonly the brane content, is ruled out.1.3.3 Parameter SettingIn order to let kd−1 have the correct sign, (1.32) requires that [4]λ > 0. (1.34)According to (1.31), Λd−1, the equivalent cosmological constant on the brane, can achieve anyvalue by tuning the value of λ. In the case Λd−1 is taken to be zero, (1.31) yieldsλ = 2(d− 2)ℓ√kd, (1.35)where ℓ is the AdS radius in the bulk, which relates to Λ asℓ =√− (d− 1)(d− 2)2Λ . (1.36)Eq. (1.32) and eq. (1.35) implyGd−1 =√(d− 3)2 · 2πGdℓ Gd, (1.37)where Gn ≡ kn/8π is Newton’s gravitational constant in n dimension. This relation clearly showsthat ℓ is related to the relation between the Newton’s constant on the brane with that in thebulk. In the theory of RSII, ℓ is a freely adjustable parameter, whose value can be determined by141.3. Randall-Sundrum Braneworld II (RSII)experiments. Taking d = 5, (1.37) reduces to G4 = G5 ·√8πG5/ℓ. Choosing the unit k5 = ℓ = 1,this equation implies k4 = Vacuum SolutionThere are a class of solutions when matter is absent [6, 42–44]ds2 = e−2|y|/ℓ (hµνdxµdxν) + dy2, (1.38)where y ∈ (−∞,∞) is the extra-dimension and the brane is located at y = 0. xµ is the coordinatein the 4D section. hµν does not depend on y, and can be any 4D vacuum solution of GR [6].Define z ≡ ℓey/ℓ when y ≥ 0, and z ≡ ℓe−y/ℓ when y ≤ 0. In coordinate space, this is a two-to-onemapping which maps ±y to z = ℓ exp(|y|/ℓ) ∈ [ℓ,∞). However, this “two-to-one” feature is onlysuperficial, because the (xµ, y) and (xµ,−y) are actually the same physical points, according tothe Z2 symmetry. Under coordinates (xµ, z), (1.38) is changed intods2 = ℓ2z2[(hµνdxµdxν) + dz2], (1.39)where z ≥ ℓ covers the whole physical spacetime, and the brane is located at z = ℓ.The simplest case is to let hµν be 4D Minkowski spacetime. The corresponding 5D space is apart of the Poincare´ patch of AdS (Anti-de Sitter) space, and can be regarded as the counter partof Minkowski spacetime in the braneworld (see Sec. 1.4). hµν can also take black hole solutions,and the corresponding solutions in the braneworld are called black strings.1.3.5 Matter in RSIIBecause of the specific form of the RS braneworld II, equation (1.21) and (1.22) now reduce toDµS µν = 0; (1.40)0 = 0, (1.41)where D is the covariant derivatives associated with hµν . Eq. (1.40) is important since it is theconservation law of the matter on brane. Since the tension part λhµν satisfies the conservationlaw, then eq. (1.40) requires that the matter part ταβ must satisfy the conservation law. This isconsistent with equation of motion of matter on brane, which takes its form in 4D GR since matteris strictly trapped on the brane and can not directly “feel” the extra dimension.151.4. The Global Structures of AdS Spacetime and the Poincare´ PatchWe will study the gravitational collapse of massless scalar field on the brane. For masslessscalar field (denoted by Φ), the conservation law is equivalent to its equation of motionDµDµΦ = 0. (1.42)The matter’s energy momentum tensor isτµν = DµΦDνΦ−12hµν(hαβDαΦDβΦ). (1.43)1.4 The Global Structures of AdS Spacetime and thePoincare´ PatchAs discussed in Sec. 1.3.4, in the context of the braneworld, the counter part of the Minkowskispacetime isds2 = ℓ2z2(− dt2 + dr2 + r2(dθ2 + sin2 θdφ2)+ dz2), where z ≥ ℓ. (1.44)The spacetime with this metric and z ≥ 0, is a Poincare´ patch [75–77] of the AdS spacetime withAdS radius ℓ. In this section we will discuss the global structure of AdS spacetime and its Poincare´patch. The discussion is necessary for us to examine whether the RSII braneworld can be definedas an initial value problem, and whether event horizon can be defined. These two aspects are goingto be discussed in Sec. AdS Spacetime and the Poincare´ PatchThe d dimensional homogeneous isotropic spacetime satisfying Einstein’s equations with apositive (or negative) cosmological constant, is called a de Sitter (or Anti-de Sitter) spacetime,and is denoted as dSd (or AdSd). In this section, we focus on the Anti-de Sitter spacetime. Asdiscussed in [75–77], AdSd is a hyperboloid of radius ℓ−X20 −X21 +d−1∑i=1Y 2i = −ℓ2, (1.45)161.4. The Global Structures of AdS Spacetime and the Poincare´ Patchembedded in the flat spacetime expanded by (X0, X1, Y1, ..., Yd−1) whose metric isds2 = −dX20 − dX21 +d−1∑i=1dY 2i . (1.46)One coordinate system of the AdS spacetime is defined asX0 = ℓ secχ cosT, (1.47)X1 = ℓ secχ sinT, (1.48)Yi = ℓ ωi tanχ, for i = 1, ..., d− 1. (1.49)Here ωi satisfyd−1∑i=1ω2i = 1.i.e. they form a (d− 2) dimensional unit sphere in (d− 1) dimensional Euclidean space. Under thiscoordinate system, the AdS spacetime isds2 = ℓ2cos2 χ(−dT 2 + dχ2 + sin2 χdΩ2d−2), (1.50)where dΩ2d−2 stands for the line element on a (d−2) dimensional unit sphere in (d−1) dimensionalEuclidean space. Eq. (1.50) with χ ∈ [0, π/2) and T ∈ (−∞,∞) defines the whole AdS spacetime,and is therefore a global cover of the AdS spacetime [75–77]. 2Alternatively, another coordinate system isX0 =ℓ2z(d−2∑i=1x2i − t2 + z2 + 1), (1.51)X1 =ℓ tz , (1.52)Y1 =ℓ2z(d−2∑i=1x2i − t2 + z2 − 1), (1.53)Yi+1 =ℓ xiz , for i = 1, ..., d− 2. (1.54)The range of the variables are z ∈ (0,∞), t ∈ (−∞,∞). x1, x2, ..., xd−2 form a (d− 2) dimensional2In eq. (1.47) and (1.48), T is periodic with the period 2π, therefore there exist closed timelike curves. Theglobal AdS spacetime is to unfold these closed curves by removing the identification of T with T + 2π.171.4. The Global Structures of AdS Spacetime and the Poincare´ PatchEuclidean space. The metric of the AdS spacetime is nowds2 = ℓ2z2(−dt2 + dz2 +d−2∑i=1dx2i). (1.55)Expressing the space expanded by xi’s under spherical coordinates, this metric isds2 = ℓ2z2(−dt2 + dz2 + dr2 + r2dΩ2d−3), (1.56)where dΩ2d−3 stands for the line element on a (d − 3) dimensional unit sphere in (d − 2) dimen-sional Euclidean space. This spacetime is called a Poincare´ patch of the AdS spacetime, and thecoordinates (1.55) or (1.56) are called Poincare´ coordinates. 31.4.2 The Penrose Diagram of the AdS2 SpacetimeIn this section we mainly focus on the causal structure of the spacetimes, which can be con-veniently studied via the tool known as the Penrose diagram. For simplicity, let us start withAdS2—the 2D AdS spacetime. The metric of the global cover isds2 = ℓ2cos2 χ(−dT 2 + dχ2), (1.57)and the ranges of the two variables are T ∈ (−∞,∞) and χ ∈ (−π/2, π/2). 4 The Penrose diagramis shown in Fig. 1.3(a), where the T = const and χ = const curves are simply horizontal and verticallines. Eq. (1.57) shows that the null curves are straight lines with the slope dT/dχ = ±1 (e.g. theorange lines in the figure).An interesting feature of the AdS spacetime is that the lightlike signals can propagate to spatialinfinities (χ = ±π/2), and then come back to its departure location within a finite proper time(measured at the departure location). Or precisely speaking, there exist closed causal curves withfinite proper length, which connect “local” regions with spatial infinities. As an example, let usconsider a lightlike signal departing at point A in Fig. 1.3(a) and propagating along the orangeline AB to point B (spatial infinity). It then propagates back from B to C along the orange lineBC. For an observer sitting at χ = 0, its proper time lapses π, which is finite. On the other hand,3(1.56) remains unchanged by the scaling: (t, z, r) → (ct, cz, cr) where c is an arbitrary positive constant, andit can be either dimensionful or dimensionless. In particular, when taking c = 1/ℓ, the (t, z, r) can be regarded asparameters with length dimension measured by unit ℓ. In this section (Sec. 1.4) these coordinates will be treatedas dimensionless parameters since it is easier to relate to the global cover. In all the other parts of the thesis, wewill treat them as parameters with length dimension measured by unit ℓ.4Or equivalently but more closely to relate to (1.50) and higher dimensional AdS spacetime, we can also takeχ ∈ [0, π/2) with ψ = 0, π, where ψ is to parametrize ω1 in (1.49) as ω1 = cosψ.181.4. The Global Structures of AdS Spacetime and the Poincare´ Patchthe proper distance from the observer to χ = π/2 in a T = const hypersurface is∫ π/201cosχdχ = 2 arctanh[tan(χ2)]∣∣∣π/20= 2 arctanh(1) = ∞. (1.58)Actually the lengths of all the spacelike curves connecting a point at χ = π/2 with a point atχ 6= π/2 are infinite. i.e. χ = π/2 is truly a spatial infinity. For higher dimensional AdS spacetimeswith d > 2, the above analysis is also valid.The metric of the Poincare´ patch under the Poincare´ coordinates is nowds2 = ℓ2z2(−dt2 + dz2), (1.59)where t ∈ (−∞,+∞) and z ∈ (0,+∞). To get the Penrose diagram of the Poincare´ patch, werelate the coordinates (T, χ, ...) with the coordinates (t, z, r, ...) by the equality of eq. (1.47∼1.49)and eq. (1.51∼1.54). The coordinates (T, χ, ...) can then be expressed in terms of (t, z, r, ...). i.e. forevery (t, z, r, ...), there is a corresponding (T, χ, ...), which is a point on the Penrose diagram of theglobal AdS spacetime. The Penrose diagram of the Poincare´ patch, is then a subset of the Penrosediagram of the global AdS spacetime. The Penrose diagram for the Poincare´ patch is obtained asFig. 1.3(b,c). In the figures, the blue lines are t = const, and the black lines are z = const. Forthe Penrose diagram, the boundary is divided into(1) Point i0 represents z = ∞, the spatial infinity.(2) Point i+ represents t = +∞, the future timelike infinity.(3) Point i− represents t = −∞, the past timelike infinity.(4) The line j+z represents the future Poincare´ horizon (see, e.g. [39]), which is a Killing horizonassociated with the Killing vector ∂t. As explained below in Sec. 1.4.3, this boundary can bereached by timelike curves of finite proper length, therefore these are not infinities. However, ifthe future oriented null curves are parametrized by the Killing parameter t, the null curves willend at this boundary when taking t→∞. For example, the curve defined by (t, z) = (t, t+1)is a null curve. When t→ +∞, this curve ends at j+z . i.e. this boundary appears to be infinityas measured by the Killing parameter t. In this sense, this boundary is an analogy of the nullinfinity of Minkowski spacetime [32]. We will put the word “infinity” into double quotes if itis not a true infinity but appears to be infinity as measured by t.(5) Similarly, j−z is the past Poincare´ horizon.191.4. The Global Structures of AdS Spacetime and the Poincare´ Patch(6) z = 0 (which is also χ = −π/2), the conformal boundary (see, e.g. [40]) of the Poincare´ patch,which is a part of the timelike spatial infinity of the global cover.Χ = 0 Χ = Π  2Χ =-Π  2T = 0T = ΠT =-ΠACBO(a)i 0i+i-z = 0z = 1t = 0jz+jz-(b)OABCDEFGΧ = 0 Χ = Π  2Χ =-Π  2T = ΠT =-ΠHI(c)Figure 1.3: Penrose diagrams of AdS2 and its Poincare´ patch. (a) is the Penrose diagram of thewhole AdS2. The T = const lines are horizontal, and χ = const lines are vertical. The range ofthe two variables are T ∈ (−∞,∞) and χ ∈ (−π/2, π/2). Lightlike curves are locally straight lineswith slope dT/dχ = ±1. (b) is the Penrose diagram of the Poincare´ patch, while (c) emphasizesthat it is embedded into the global AdS spacetime.Fig. 1.3(b,c) show that in the Poincare´ patch there still exist lightlike signals going to spatialinfinity (at the conformal boundary) and coming back whose trajactory can be connected by atimelike curve with finite proper time. e.g. the trajectory C → A→ B in Fig. 1.3(c) is connectedby the purple line BC whose proper time is finite.1.4.3 The Infinities of the Poincare´ PatchIn the last subsection, we show that at the i0, i± and j±z , at least one of t = ∞ and z = ∞ istaken. Now we examine whether these are “true” infinities, by evaluating the proper lengths. Theproper length of the t = 0 curve is∫ ∞0dzz =∫ 10dzz +∫ ∞1dzz , (1.60)201.4. The Global Structures of AdS Spacetime and the Poincare´ Patchwhere both terms on the right hand side are +∞. Actually the lengths of all the spacelike curvesconnecting a point at z = ∞ with a point at z 6= ∞ are infinite. Thus i0 is a true spatial infinity.Similarly, i± are true timelike infinities.For the Poincare´ horizons which appear to be null “infinities” measured in terms of the Killingparameter t, it is easier to perform the calculation under (T, χ) coordinates. One can see thatthe proper time of FG—a curve connecting the past Poincare´ horizon with the future Poincare´horizon as shown in Fig. 1.3(c)—is finite, which also means the lightlike signal propagating alongF → O → G enters from the past Poincare´ horizon and escapes via the future Poincare´ horizon,within a finite proper time measured by an observer (whose world-line is FG). This means theinformation or entities in the space can “vanish” within a finite time. Also, the proper length ofspacelike curve ED is finite. Therefore j±z are not true infinities.On the other hand, however, the Poincare´ patch (1.55) has a timelike Killing vector ∂t, andj±z (the Poincare´ horizons [39]) are the Killing horizons associated with this Killing vector. In theliterature, z = ∞ is often referred as AdS horizon [6]. When approaching j±z , the time parameterassociated with the Killing vector (which is t) approaches infinity. An observer whose world-lineis the integral curve of the Killing vector (i.e. his spatial coordinates of the Poincare´ coordinatesremain constants), is called a Killing observer. For a Killing observer, his proper time takes(−∞,∞). j+z is the future boundary of the spacetime region that can possibly be observed by anyKilling observer, and j−z is the past boundary of the spacetime region that can possibly be affectedby any Killing observer.1.4.4 The Poincare´ Patch of AdS3AdS2 catches the most important features of the more general AdSd. Also, AdSd has a transla-tional symmetry in the space expanded by xi’s. Therefore, any finite xi’s can be brought to xi = 0using this symmetry, where the above discussions regarding AdS2 still apply. On the other hand,however, the behaviour by “turning on” the space expanded by xi’s, might introduce some newfeatures, because the xi can take infinities. Therefore AdS3 is a better model for the more generalAdSd with d ≥ 3.For AdS3, the metric of the global cover isds2 = ℓ2cos2 χ(−dT 2 + dχ2 + sin2 χdψ2), (1.61)211.4. The Global Structures of AdS Spacetime and the Poincare´ Patchwhere χ ∈ [0, π/2), ψ ∈ [0, 2π) and T ∈ (−∞,∞). The metric of the Poincare´ patch isds2 = ℓ2z2(−dt2 + dz2 + dr2), (1.62)where z ∈ (0,∞), t ∈ (−∞,∞) and r ∈ (−∞,∞). 5 Eq. (1.47-1.49) and eq. (1.51-1.54) now readsX0 = ℓ secχ cosT =ℓ2z(d−2∑i=1x2i − t2 + z2 + 1), (1.63)X1 = ℓ secχ sinT =ℓ tz , (1.64)Y1 = ℓ ω1 tanχ = ℓ cosψ tanχ =ℓ2z(d−2∑i=1x2i − t2 + z2 − 1), (1.65)Y2 = ℓ ω2 tanχ = ℓ sinψ tanχ =ℓ rz , (1.66)where we have parametrized ωi’s by ω1 = cosψ and ω2 = sinψ.The Penrose diagram of the global cover is a cylinder, where the radius of the cylinder is χ, theangle is ψ and the vertical axis is T . Similar to AdS2, the Penrose diagram of the Poincare´ patchcan be embedded into that of the global cover. Expressing (T, χ, ψ) by (t, z, r) via eq. (1.63-1.66),and then plotting in terms of (t, z, r), gives the Penrose diagram of the Poincare´ patch embedded inthe Penrose diagram of the global cover. Please refer to Fig. 1.4. Fig. 1.4(a) is the Penrose diagramof AdS3 and the Penrose diagram of its Poincare´ patch with some special lines. The vertical blackline is the line of (χ, ψ) = (π/2, π) which represents (t, z, r) = (t, 0, 0) with t ∈ (−∞,∞). The othertwo straight lines (which are half blue half black), correspond to the null “infinities” for r = consthypersurfaces. These three lines and the plane surrounded by them, form the Penrose digram ofthe Poincare´ patch of the AdS2 discussed in Sec. 1.4.2. The three vertices of this triangle are i0and i± which represent the spatial infinity and timelike infinities, respectively. The boundary ofAdS3 is divided into (see Fig. 1.4)(1) i0, point (T, χ, ψ) = (0, π/2, 0), the spatial infinity of z = ∞ and r = ±∞.(2) i+, point (T, χ, ψ) = (π, π/2, π), the future timelike infinity.(3) i−, point (T, χ, ψ) = (−π, π/2, π), the past timelike infinity.(4) j+z , straight line described by (ψ = 0 or π, T = −χ cosψ+π/2), the future null “infinity” withfinite r.5Again, equivalently but more closely related to higher dimensional spacetime, we can also take r ∈ [0,∞) withθ = 0, π where ω3 is parametrized as ω3 = cos θ.221.4. The Global Structures of AdS Spacetime and the Poincare´ Patch(5) j−z , straight line described by (ψ = 0 or π, T = χ cosψ − π/2), the past null “infinity” withfinite r.The subscript z is to label the fact that the null “infinities” are within the t − z plane (sincer = const). Similarly, the two crossing lines (half red half black) on the surface of the cylinder, arethe null infinities for z = const surfaces, and are denoted as j±r , which are true infinities. i.e.(6) j+r , straight line on the side surface of the cylinder described by (χ = π/2, T = ±ψ withT ≥ 0), the future null infinity with finite z.(7) j−r , straight line on the side surface of the cylinder described by (χ = π/2, T = ±ψ withT ≤ 0), the past null infinity with finite z.(8) z = 0, the wedge portion of the side surface of the cylinder, the conformal boundary of thePoincare´ patch, which is a part of the spatial infinity (χ = π/2) of the global cover.The side surface of the cylinder corresonds to χ = π/2. Its gray portion (on Fig. 1.4(a)) is thez = 0 hypersurface in the Poincare´ patch. Unfolding the cylinder along ψ = 0, we get Fig. 1.4(b).The figure shows that the two crossing lines on the surface of the cylinder are actually straightlines with slope dT/dψ = ±1, which are null curves: taking χ = π/2, eq. (1.61) implies straightlines with dT/dψ = ±1 are null. Fig. 1.4(c) emphasizes the null “infinities” of the Poincare´ patch.We take null curves parametrized by the Killing parameter t, and these null curves end at the null“infinities” when t → ∞. The orange lines are null curves with finite r, which end at j+z . Thegreen lines are null curves with finite z, which end at j+r (and start from j−r ). At j±r (or j±z ), z (orr) is finite. On the other hand, the purple lines are null curves which end at the surface where allof t, r and z are infinite. Therefore these null “infinities” with all of t, r and z approaching theirinfinities are denoted as(9) j+rz, the future null “infinity” with infinite r and infinite z.(10) j−rz, the past null “infinity” with infinite r and infinite z.Within these null “infinities”, j±z are not true infinities, as discussed above for AdS2. Similarly,j±rz are not true infinities either. On the other hand, i0, i± and j±r are located at the surface ofthe cylinder (where χ = π/2), and are true infinities. The boundaries that can be reached by nullcurves are defined as the future/past Poincare´ horizons [39]j± ≡ j±r⋃j±z⋃j±rz. (1.67)231.4. The Global Structures of AdS Spacetime and the Poincare´ Patchj± are the upper/lower boundary surfaces of the Penrose diagram (of the Poincare´ patch). Theexpression for these two surfaces can be obtained as follows [78]. Using eq. (1.63-1.66), (t, z, r) canbe expressed in terms of (T, χ, ψ) as [78]t = sinTcosT − sinχ cosψ , (1.68)z = cosχcosT − sinχ cosψ , (1.69)r = sinχ sinψcosT − sinχ cosψ . (1.70)Because z = ∞ at j±, the equation for surfaces j± is described by requiring the denominator of(1.69) to be zero, which iscosT − sinχ cosψ = 0. (1.71)The upper/lower surfaces of Fig. 1.4(a,c) are generated using this equation. The upper surface isconfirmed to be the future null “infinity” by Fig. 1.4(c) where the future null curves end at theupper surface. By the time reversal symmetry of the spacetime, the lower surface is also confirmedto be the past null “infinity”.Now we prove that the Poincare´ horizons described by (1.71) are indeed the Killing horizonsfor Killing vector ∂t. We definef ≡ cosT − sinχ cosψ, (1.72)then the Poincare´ horizons are described by f = 0. For these hypersurfaces to be the Killinghorizons, we need to prove [38] (1) the Killing vectors ∂t are orthogonal to these hypersurfaces;(2) the Killing vectors are null on these hypersurfaces.In general, the hypersurfaces described by f = const is orthogonal to the vector field ∂αf . ∂αfis evaluated as∂αf = (− sinT,− cosχ cosψ, sinχ sinψ), (1.73)under the coordinates (T, χ, ψ). A direct evaluation shows that ∂αf become null at the hypersur-faces specified by f = 0. On the other hand, a long but otherwise direct calculation shows that atthe hypersurfaces specified by f = 0(∂t)α =(sinT/ cos2 χ)· ∂αf. (1.74)i.e. at the Poincare´ horizons, ∂t is proportional to ∂αf . Therefore the two conditions for thePoincare´ horizons being Killing horizons are met.241.4. The Global Structures of AdS Spacetime and the Poincare´ Patch(a)HΨ = 2 Π Li 0i 0i+i-HΨ = 0 L HΨ = Π Ljr+jr+jr-jr-r= 0r= 1 r=-1t= 0t= 1t=-1(b) (c)Figure 1.4: Penrose diagrams of AdS3 and its Poincare´ patch. In (a), the cylinder is the Penrosediagram of the whole AdS3, while the portion surrounded by the gray surfaces (with differentdarkness) is the Penrose diagram for the Poincare´ patch. For the notations on the diagram: i0 andi± are for points; j±r and j±z are for lines; j±rz are for surfaces; and CB stands for the conformalboundary which is the wedge portion of the side surface of the cylinder. Note: j±rz appearing on theside of the vertical black line, are actually for the surfaces rather than for the vertical black line.(b) is the conformal boundary (the z = 0 hypersurface of the Poincare´ patch), which is the part ofthe Penrose diagram on the side surface of the cylinder. (c) is to emphasize the null “infinities”,where the orange lines are the null curves with finite r, the green lines are the null curves withfinite z and the purple lines are the null curves with all of t, r, z approaching their infinities. Whentaking t→ +∞, the orange lines end at j+z , the green lines end at j+r and the purple lines end atj+rz.251.4. The Global Structures of AdS Spacetime and the Poincare´ PatchThe upper/lower boundary surfaces have another interesting property. We can put a secondpatch on top of the existing patch, by requiring that the i0 of the new patch is the i+ of theold patch. The new patch can be regarded as a transformation from the old patch, where thetransformation is composed of the following two steps: (1) the patch is rotated by π with χ = 0 asthe rotational axis. This operation is ψ → ψ+ π. (2) the patch is moved up vertically by π, whichis T → T + π. After these two operations, eq. (1.71) remains unchanged. i.e. it also describes theupper/lower boundaries of the new patch. Therefore, there is no gap between the two patches.1.4.5 The RSII Braneworld and Its StructureTo get a better idea about the structure of the Poincare´ patch and the brane, we show thefollowing hypersurfaces with one of the coordinates being constants. Fig. 1.5 shows the r = consthypersurfaces, where the blue lines are t = const. Or in another word, these blue lines are theintegral curves of ∂z, and the black lines are that of ∂t. Fig. 1.6 shows the t = const hypersurfaces,where the blue lines are the integral curves of ∂z, and the red lines are that of ∂r. Fig. 1.7 showsthe z = const hypersurfaces, where the red lines are the integral curves of ∂r, and the black linesare that of ∂t. In particular, the brane in the RSII braneworld is the z = 1 hypersurface, which isshown in Fig. 1.7(b).The following discussion applies to general d case.As introduced in Sec.1.3.4, the spacetime background of RSII braneworld is to take the z ≥ 1portion of the Poincare´ patch (which is called the bulk). Therefore the z = 0 boundary is eliminatedfrom the Penrose diagram, and the global causal structure is the same as that of Minkowskispacetime.In fact, any z = z0 (where the constant z0 ∈ (0,∞)) can serve as the brane, because theextrinsic curvature of the z = z0 hypersurface can be calculated as Kµν = hµν , which satisfiesthe Israel’s junction condition (1.29) applied to the vacuum case. Here hµν is the reduced metricon the hypersurface, and the extrinsic curvature is calculated based on the unit normal vector ofthe hypersurface, nµ, pointing into the bulk (larger z direction). Within the AdS spacetime, thez = z0 hypersurfaces are not geodesic surfaces (in the sense that the extrinsic curvatures of thehypersurfaces do not vanish, see Sec. 2.3.3), but are surfaces with constant acceleration, in thesense that every freely moving massive particle within a z = z0 surface has a constant accelerationas measured by the co-moving inertial observer in the AdS spacetime. Let d be the dimension ofthe AdS spacetime, and let vµ be the d-velocity of a massive particle freely moving within a z = z0hypersurface. The path of the particle is then described by the timelike geodesics generated by the261.4. The Global Structures of AdS Spacetime and the Poincare´ Patchtangent vector vµ based on the intrinsic metric of the z = z0 hypersurface, which is vαDαvµ = 0,where D is the covariant derivative associated with hµν . Let uµ be the d-velocity of an observer,then the acceleration of the particle observed by this observer isaµ = uα∇α (vµ − uµ) . (1.75)If the observer is an inertial observer in the AdS spacetime, his trajectory is then described bya timelike geodesics as uα∇αuµ = 0. Furthermore, if the observer’s instant velocity is vµ, he isa co-moving inertial observer. The acceleration of the massive particle moving along a timelikegeodesics in the z = z0 hypersurface, as measured by a co-moving observer in the AdS spacetime,is then vα∇αvµ, which is described by eq. (2.27) (see Sec. 2.3.4)vα∇αvµ = nµvαvβKαβ = nµvαvβhαβ = −nµ. (1.76)This acceleration is a constant vector (as measured by the comoving inertial observer in the AdSspacetime) pointing out of the bulk.(a) (b) (c)Figure 1.5: r = const hypersurfaces, where (a) is r = 0, which is the AdS2 diagram. (b) is r = −1and (c) is r = −8. The blue lines are the integral curves of ∂z, and the black lines are that of ∂t.271.4. The Global Structures of AdS Spacetime and the Poincare´ Patch(a) (b) (c)Figure 1.6: t = const hypersurfaces, where (a) is t = 0. (b) is t = 2 and (c) is t = 8. The bluelines are the integral curves of ∂z , and the red lines are that of ∂r.(a) (b) (c)Figure 1.7: z = const hypersurfaces, where (a) is z = 0, which is opened as Fig. 1.4(b). (b) isz = 1, which is the brane in the RSII braneworld. (c) is z = 7. The red lines are the integralcurves of ∂r, and the black lines are that of ∂t.281.5. Evolution Schemes1.5 Evolution SchemesThere exist various formalisms of GR, among which only the ones that are strongly hyperbolic(refer to, for example [34]), can be used as a well-defined formalism of an initial value problem.In this section we only (very) briefly sketch the generalized harmonic formalism since furtherdevelopments will be present in the next chapter with more details.The generalized harmonic (GH) formalism [61] uses the gauge source functionsHα ≃ ∇β∇βxα = −Γαµνgµν ≡ −Γα, (1.77)as fundamental variables. The notation ≃ means the equation is a constraint relation. Einstein’sequations can now be written as−12gαβgµν,αβ − gαβ(,µgν)β,α −H(µ,ν) +HβΓβµν − ΓανβΓβµα = kd(Tµν −1d− 2gµνT). (1.78)A coordinate gauge choice can now be realized via specifying the Hµ’s. As long as Hµ does notinclude derivative of metric functions, the principal part of the above equation − 12gαβgµν,αβ ismanifestly strongly hyperbolic.Both the generalized harmonic formalism and the BSSN formalism [34, 45, 46] are widely usedin the literature, yet none of them is sufficient to simulate braneworld and we have to developthem further. In this thesis, the generalized harmonic formalism will be employed to evolve thebranewrold spacetime. Thus we put the introduction and the development of the BSSN formalismto appendix A.1.6 Numerical MethodsThe equations of motion of gravitational theory are non-linearly coupled partial differentialequations (PDEs). Due to the non-linearity and the complexity, it is not very realistic to studythe full dynamics in closed form, especially the behaviour at high energy regime where the fieldsare so strong that perturbation methods do not apply. We here use a numerical approach. In thissection we introduce finite difference approximation (FDA) methods to solve the PDEs. The focusis on the various tests to distinguish numerical solutions from numerical artifacts.291.6. Numerical Methods1.6.1 Finite Difference ApproximationTo demonstrate the concepts in a less abstract way, let us consider the following model problem,which is non-linear wave equation in flat 3+1 dimension spacetime under axisymmetry with sourceterm f (which does not depend on the wave function Φ). This model problem includes a fewfeatures that are important for numerical calculation in braneworlds. The equation is assumed tobe (−∂tt + ∂xx + ∂yy + ∂zz)Φ + Φ2 = f in Cartesian coordinates, or(−∂tt + ∂ρρ +1ρ∂ρ + ∂zz)Φ+ Φ2 = f, (1.79)in cylindrical coordinates (t, ρ, φ, z) that are adapted to the axisymmetry. Therefore the axisym-metry implies ∂φΦ = 0, which has been applied in (1.79). Let us assume the spatial domain isρ ∈ [0, ρmax], z ∈ [0, zmax].The whole domain, both spatial and temporal, is divided into discrete grids (or meshes). Inprinciple this division can be arbitrary, as long as the grid/mesh elements are small. The meaningof “small” is going to be clear by the discussion in section 1.6.2. To be more specific and to honorsimplicity, here let us employ uniform grid. Therefore the spatial domain can beρi = (i− 1)∆ρ, i = 1, 2, ..., nρ where ∆ρ =ρmaxnρ − 1; (1.80)zj = (j − 1)∆z, j = 1, 2, ..., nz where ∆z =zmaxnz − 1. (1.81)For simplicity, let us choose ∆ρ = ∆z = h. The time domain is also discretized and the timeinterval between two subsequent discretized time levels can be expressed as ∆t. ∆t/h is called theCourant factor.We use notationΦni,j ≡ Φ(tn, ρi, zj) ≡ Φ((n− 1)∆t, (i − 1)∆ρ, (j − 1)∆z), (1.82)and similar notation for function f . We replace the differential operators by their FDA operators301.6. Numerical Methodswith second order accuracy:∂ρρΦ →Φni+1,j − 2Φni,j +Φni−1,jh2 , (1.83a)∂ρΦ →Φni+1,j − Φni−1,j2h , (1.83b)∂zzΦ →Φni,j+1 − 2Φni,j +Φni,j−1h2 , (1.83c)∂ttΦ →Φn+1i,j − 2Φni,j +Φn−1i,j(λh)2 . (1.83d)The FDA operators are obtained by Taylor expansions such asΦni+1,j = Φni,j + hΦ,ρ +h22! Φ,ρρ +h33! Φ,ρρρ +h44! Φ,ρρρρ +O(h6),which yieldΦni+1,j − 2Φni,j +Φni−1,jh2 = ∂ρρΦ+h212Φ,ρρρρ +O(h4). (1.84)The term h212Φ,ρρρρ + O(h4) = O(h2) is the difference between the exact operator and the FDAoperator, which is called truncation error. When h is small (so that the truncation error is notsignificant), the differential operators can be replaced by their FDA counter parts. Other FDAoperators in (1.83) can be obtained similarly. The discretized PDE reads−Φn+1i,j − 2Φni,j +Φn−1i,j(λh)2 +Φni+1,j − 2Φni,j +Φni−1,jh2 +1ρiΦni+1,j − Φni−1,j2h+Φni,j+1 − 2Φni,j +Φni,j−1h2 +(Φni,j)2 = fni,j. (1.85)Now we are ready to introduce the general notations to make the discussion clearer. A set ofPDEs, such as equation (1.79), can be collectively denoted asLu = f, (1.86)where L stands for differential operators and all other operations, u stands for the fundamentalvariables (the unknown functions) to solve for, and f stands for the terms in the equations thatdo not include u. In equation (1.79), u = Φ, and Lu = LΦ =(−∂tt + ∂ρρ + 1ρ∂ρ + ∂zz)Φ + Φ2.The discrete FDA operators, such as equation (1.84), can be collectively denoted asAΦ = EΦ+ hp ·EΦ, (1.87)311.6. Numerical Methodswhere A stands for the FDA version of the exact operator E . hp means that the approximationlevel is of p-th order in h, E stands for the error operator — more specifically, hpEΦ is the error.Using (1.87), we can discretize (1.86) asLhuh = fh, (1.88)where h is to label resolution. An example of (1.88) is (1.85).In (1.85), the approximation is of second order in h. Generally the approximation order of Lhis p, which can be formally expressed asLh = L+ hpE. (1.89)From the discussion above, one sees that the validity of FDA needs to be built upon thefollowing two assumptions: (1) the funtion Φ is smooth; (2) h is small, so that the truncation erroris not significant.However, these two conditions are not sufficient to guarantee that the numerical result uh isactually a approximation of the exact solution u. Therefore systematic test mechanisms need tobe developed to distinguish numerical solutions from numerical artifacts.1.6.2 TestsFirst, often it is neither practical nor necessary to let equation (1.88) be satisfied exactly.Instead, (1.88) is considered to be satisfied when residual rh ≡ Lhuh − fh is sufficiently small.Again, “small” does not have any measurable meaning yet.Multiplying equation (1.85) by ρi, we get the following equation−ρi(Φn+1i,j − 2Φni,j +Φn−1i,j)(λh)2 +ρi(Φni+1,j − 2Φni,j +Φni−1,j)h2 +Φni+1,j − Φni−1,j2h+ρi(Φni,j+1 − 2Φni,j +Φni,j−1)h2 + ρi(Φni,j)2 = ρifni,j . (1.90)(1.85) and (1.90) share exactly the same numerical properties, such as convergence, smoothness,regularity, etc. But the two residuals have different numerical values. Therefore, the residual being“small”, has no absolute meaning.This feature can be expressed in a more abstract way as: Lu = f and g · Lu = g · f have thesame numerical properties. Here g is a non-zero, smooth function over the domain. For example g321.6. Numerical Methodscan be an arbitrary non-zero constant to make the residual take any value. Therefore, the absolutevalue of residual does not have any meaning. So how to distinguish between a numerical solutionand a numerical artifact? And how small is the residual to be considered sufficiently small? Thesequestions will be answered by the following analysis.Assume the numerical result uh that satisfies Lhuh = fh + rh is obtained, where rh is theresidual. Generically, uh is a numerical solution, if the following equation is satisfied when uh issubstituted back into equation (1.86)limh→0Luh − fh = 0. (1.91)Let us see what it meansLuh − fh = Lhuh − hpEuh − fh = rh − hpEuh = rh +O(hp). (1.92)Therefore (1.91) is satisfied, if rh is negligible compared to hpEuh (in this sense rh is small).However, technically it is impossible to apply a continuous operation L to discrete functionuh, and then eq. (1.91) can only be understood formally. Instead, uh is considered a numericalsolution, iflimh→0rhI = 0,where rhI ≡ LhI uh − fh,where LhI 6= Lh that satisfies limh→0LhI = L. (1.93)Since LhI is independent of Lh (a different discretization), rhI is called independent residual.For the model problem, we can use the following discretization as the independent discretizedoperators∂rrΦ →2Φni,j − 5Φni+1,j + 4Φni+2,j − Φni+3,jh2 , (1.94a)∂rΦ → −3Φni,j − 4Φni+1,j +Φni+2,j2h , (1.94b)∂zzΦ →2Φni,j − 5Φni,j+1 + 4Φni,j+2 − Φni,j+3h2 , (1.94c)∂ttΦ →2Φni,j − 5Φn−1i,j + 4Φn−2i,j − Φn−3i,j(λh)2 . (1.94d)This discretization is different from (1.83), and is also of the second order accuracy.331.6. Numerical MethodsIn general, the approximation order of LhI is denoted as m, thereforeLhI = L+ hmEI = Lh − hpE + hmEI, (1.95)rhI = LhI uh − fh = (Lh − hpE + hmEI)uh − fh = rh − hpEuh + hmEIuh. (1.96)Again, here it is required that∥∥rh∥∥ is negligible compared to min(∥∥hpEuh∥∥ ,∥∥hmEIuh∥∥), thereforethe independent residual rhI converges to zero at min(p,m)-th order. Here ‖u‖ is the norm of u.For the model problem, p = m = 2, therefore the independent residual behaves as a secondorder quantity: when h decreases to h/2, the independent residual rhI decreases to r(h/2)I = 14rhI .Note, the independently discretized operators LhI can be very different from the discretizedoperators Lh used to obtain the solution. LhI and Lh do not need to be of the same method. Forexample, one can use finite element method or spectrum method to obtain the solution, but useFDA as independent operators to evaluate the independent residual.1.6.3 Tests for General RelativityFor a numerical problem, often there are a certain number of equations to solve, for an equalnumber of fundamental variables (the unknown functions). If the number of equations is less thanthe number of unknown functions, in principle there are no unique solutions. On the other hand,in GR, the number of equations is greater than the number of unknown functions. In this case theredundant equations are called constraints.As an example, in 3 + 1 formalism of GR, there are six functions γij to be solved for, bysolving the six evolutionary equations. The other four equations are the Hamiltonian constraintand momentum constraints. Analytically, if the constraints are satisfied initially, the consistency(Bianchi identity) guarantees them to be satisfied at all times, as long as the evolutionary equationsare satisfied during the evolution. However, numerically there are always small violations tothe constraints, and there is no guarantee the violations are controllable. Therefore, for generalrelativity, the constraints need to be tested as well. i.e. in order to make sure all the componentsof Einstein’s equations are satisfied, both the independent residual test and the convergence testfor constraints are needed.Equivalently, in the case a certain formalism of GR is employed to obtain the numerical results,the results can be substituted into another formalism of GR to produce residuals, and the residualsshould converge at the expected order. For example, one can use generalized harmonic formalismto obtain the solution, and then substitute the solution into original Einstein’s equations to get341.6. Numerical Methodsresiduals, and check whether the residuals converge as expected.35Chapter 2Characteristics in the BraneworldSpacetimeThe presence of the brane imposes interesting new physics. This chapter is devoted to developthe formalisms to describe the following topics associated with the brane.Israel’s junction conditions impose cusps in some metric components, which serve as boundaryconditions for these metric components. In this chapter we will discuss the boundary conditionsof the remaining metric components. We will also discuss other effects of Israel’s conditions suchas the smoothness of the apparent horizon across the brane.The main goal of our study is to numerically simulate the process of black hole formation. Thedefinition of a black hole relies on the global causal structure of the spacetime. We will discusshow a black hole can be defined in the braneworld. In the braneworld, the causal structure ofthe spacetime is determined by the spacetime geometry in 5D, therefore the 4D apparent horizonand event horizon on the brane should play no direct role in braneworlds. However, since the 4Dbrane is all one can observe, we will study the 4D quantities as well, and compare them with theresults in GR to see the observable difference from GR. We will also study the relation betweenthe horizons on the brane and the horizons in the bulk.Energy in GR is not a locally defined quantity since the energy “density” can be of any value[94]. However, if the system presents asymptotic translational symmetry in time, in certain casesquasi-local energy can be defined, such as ADM energy in asymptotically flat spacetime. A moregeneral formalism of energy obtained from Hamilton-Jacobi analysis [95] [87], will be directlyemployed in the braneworld to obtain the total energy of the system.There is also energy exchange between the brane and the bulk. In this chapter we will presentour preliminary study on this topic.362.1. Boundary Conditions at the Brane2.1 Boundary Conditions at the BraneIn this section we discuss the properties and the gauge freedom in the boundary conditions atthe brane.The vacuum solution of the braneworld is eq. (1.38)ds2 = e−2|y|/ℓ (hµνdxµdxν) + dy2, (2.1)where y ∈ (−∞,∞), and the brane is located at y = 0. For the general case (non-vacuum), wesetup the coordinate system (xa, y), where y is the extra dimension. Latin indices (a, b, . . . ) are forthe coordinates on y = constant surfaces, and their values take 0, 1, 2, 3. Greek indices (µ, ν, . . . )take 0, 1, 2, 3, 4, and are used for the coordinates of the whole spacetime. Therefore the metric isds2 = gabdxadxb + 2gaydxady + gyydydy.The coordinate y is set to inherit the following properties: y = 0 is where the brane is located,and y is adapted according to the Z2 symmetry, such that the metric components:gab(xa,−y) = gab(xa, y), (2.2)gyy(xa,−y) = gyy(xa, y), (2.3)and gay(xa,−y) = −gay(xa, y). (2.4)Under this coordinate choice, Israel’s first junction condition is simply gab|y=0+ = gab|y=0− = hab,where hab is the intrinsic metric on the brane (expressed in the coordinates xa), induced fromthe bulk metric on either sides of the brane. Israel’s second junction condition imposes conditionson the extrinsic curvature Kab (of the brane embedding in the bulk). These conditions can betranslated into conditions on ∂gab/∂y, which will serve as the boundary conditions for gab.gyy and gay are not related to Israel’s conditions. Rather, since the braneworld spacetimes are“one-sided” (see Sec. 1.3.2 and Sec. 1.3.4), in general there is no generic reasons to require they-coordinate lines (the intersection of the xa = constant surfaces) to be perpendicular with they = 0 surface (the brane), which means gay∣∣y=0+ 6= 0. Taking y → 0 in (2.3) and (2.4), we getgyy∣∣y=0− = gyy∣∣y=0+ , (2.5)and gay∣∣y=0− = −gay∣∣y=0+ . (2.6)372.2. Apparent HorizonHowever, since gay∣∣y=0+ 6= 0 in general, it means gay∣∣y=0 are not defined. This is because only theinduced metric on the brane and the extrinsic curvature of the brane are important, while gay arenot needed in defining the induced metric and the extrinsic curvature of the brane. Similarly, gyyis not needed in defining the brane geometry either, which means generically gyy is not defined,although it could be defined as gyy∣∣y=0 = gyy∣∣y=0− = gyy∣∣y=0+ . We havegenerically, gµy are not defined on the brane.Although generally gµy are not defined, it is convenient to choose the coordinates at the brane suchthat the y-coordinate lines are perpendicular with the brane. We call this gauge as perpendiculargauge. Under this coordinate gauge, we can then definegay∣∣y=0 = 0. (2.7)This coordinate gauge has desirable properties such as the smoothness of apparent horizon that isgoing to be in Sec. 2.2.Note that gyy∣∣y=0 is still unconstrained.2.2 Apparent HorizonAn apparent horizon is needed to monitor the evolution of spacetime if a black hole is presentduring the evolution. This section is devoted to studies on apparent horizons in braneworlds.2.2.1 The DefinitionThe definition of apparent horizon can be found in standard texts [34, 35, 37]. Let nα be futuredirected timelike unit vector normal to t = constant hypersurfaces. Let S denote a closed (d− 2)dimensional spatial surface within a t = constant hypersurface, and sα is its unit normal vectorpointing towards the outgoing direction, which is within the same t = constant hypersurface. Theinduced metric on S is then (not to be confused with the mα defined in (1.3))mαβ ≡ gαβ + nαnβ − sαsβ . (2.8)382.2. Apparent HorizonLet vα be an arbitrary vector field, the relative change rate in the area elements of S along vα is[37]Θ(v) ≡ 1√mLv√m = 12mµνLvmµν = mµν∇µvν , (2.9)where m is the determinant of mµν . The final expression is the expansion of vα along S.For the (d − 2) dimensional spacelike surface S, there exist two null curves orthogonal to thissurface. Let us denote the two future directed null vectors tangent to these two null curves as ±lα.The relative change rate of the area elements of S along the null geodesics congruences producedby ±lµ are thenΘ± ≡ mαβ∇α ±lβ . (2.10)A trapped surface is an S whose Θ± < 0, which means the null geodesics congruences producedby both +lµ and −lµ drag S to the surfaces with smaller area elements. An S withΘ+ = 0 (2.11)is called a marginally outer trapped horizon (MOTH), whose area elements stay the same underthe Lie-dragging of +lµ. The MOTH is not unique in a given spacetime since there can be otherMOTHs within a MOTH. An apparent horizon is defined as the outermost MOTH.Now let us construct ±lα. Because timelike normal vector nα is orthogonal to the spacelikenormal vector sα, the two future directed null vectors orthogonal to S are±lα ≡ nα ± sα, (2.12)where +lα is outgoing and −lα is ingoing. Substituting this into (2.10), we obtainΘ± = mαβ∇α (nβ ± sβ) = ±Dαsα −K + sαsβKαβ , (2.13)where Dα denotes the covariant derivative associated with γαβ , and Kαβ is the extrinsic curvatureof the t = constant hypersurface. Therefore, the follow equation is satisfied at the apparent horizonΘ+ = Dαsα −K + sαsβKαβ = 0. (2.14)The above discussion shows that the definition of apparent horizon relies on the choice oft = constant hypersurfaces. For a given spacetime, different slicing conditions can result in drasti-cally different apparent horizons. Taking Schwarzschild spacetime as an example. In Schwarzschild392.2. Apparent Horizoncoordinates or the isotropic coordinates, the apparent horizon coincides with the event hori-zon, where the physical singularity is inside of the apparent horizon. On the other hand, theSchwarzschild spacetime can be sliced in such a way that there is no apparent horizon [33].2.2.2 Smoothness of the HorizonIsrael’s conditions impose cusps to some components of the metric. In this section we will studywhether this affects the smoothness of apparent horizon. The smoothness of an apparent horizoncan be studied via sα by asking whether sα is continuous across the brane. Rewriting Θ+ = 0 asDαsα = K − sαsβKαβ, (2.15)we then apply Gauss’ theorem (in curved space) on the t = constant hypersurface. By the sameprocedure to derive the junction condition of electric field across a surface 6, we can find the junc-tion condition for sα across the brane. If the right hand side of (2.15) is finite (by the Z2 symmetrywith respect to the brane, this condition is true if the t = constant hypersurfaces are chosen tobe perpendicular with the brane), then the integration over an infinitesimal layer across the branevanishes. Therefore the component of sα that is perpendicular to the brane, is continous acrossthe brane. This continuity of the perpendicular component, together with the Z2 symmetry withrespect to the brane, imply that the perpendicular component of sα must be zero. Therefore sα iscontinuous. i.e.the direction of an apparent horizon is continuous across the brane, if the slicing condition issuch that the t = constant hypersurfaces are perpendicular to the brane.Under the coordinates setting in Sec. 2.1, the slicing condition is expressed as gty = 0. Inparticular, the perpendicular gauge (2.7) satisfies the slicing condition.2.2.3 Apparent Horizon on the Brane and in the BulkGenerically the causal structure is determined by 5D geometry. However, only the branequantities are directly observable, therefore we study the relation between 4D and 5D quantities.6The procedure to derive E+ − E− ∝ σ from ∇ ·E ∝ ρ, where ρ is volume charge density and σ is areal chargedensity of the singular layer. E is the electric field, and E+ is the electric field on one side of the singular layer, andE− is the electric field on the other side.402.3. Event HorizonThe question we try to address in this subsection is whether the apparent horizon seen on the brane(which is calculated based on the brane geometry only), and the bulk apparent horizon (which iscalculated based on the bulk geometry), agree with each other on the brane. This can be studiedvia the expansions of the outgoing null geodesics congruences on the brane and in the bulkΘbrane =(hαβ + (r)nα (r)nβ − (r)sα (r)sβ)Dα((r)nβ + (r)sβ)= (r)mαβ∇α((r)nβ + (r)sβ)(a)= (r)mαβ∇α (nβ + sβ) (b)= (mαβ − nαnβ)∇α (nβ + sβ)= Θbulk − nαnβ∇α(nβ + sβ). (2.16)where (r)mαβ = hαβ + (r)nα (r)nβ − (r)sα (r)sβ is the projection operator that projects to the (d− 3)-surface on the brane, and D is the covariant derivative associated with the brane metric hαβ .Anything with a superscript (r) is a quantity defined only on the brane. The vector nα is theunit normal vector that is perpendicular to the brane. Assuming perpendicular gauge (2.7), wehave sβ = (r)sβ and nβ = (r)nβ on the brane, which are used in deriving eq. (b) from eq. (a).The difference between the two Θ’s, even at the apparent horizon where Θbulk = 0, is generallynon-zero. i.e. generally these two apparent horizons do not agree. Therefore, we will study therelation between event horizons in the next section.2.3 Event Horizon2.3.1 Event Horizon in the BraneworldIn this subsection we examine whether event horizon is well-defined in the spacetime of theRSII braneworld, and discuss how to define an evolution problem.The global causal structure of AdS spacetime and its Poincare´ patch was introduced in Sec. 1.4.The spacetime background of RSII braneworld is to take the z ≥ 1 portion of the Poincare´ patch.Therefore the z = 0 boundary is eliminated from the Penrose diagram, and the global causalstructure is the same as that of Minkowski spacetime. In particular, similar to Minkowski space-time, there is no signal travelling to spatial infinity and then coming back within a finite localproper time, and Cauchy surfaces exist [6]. To discuss the Cauchy surfaces, we define the future/-past horizons as i0⋃ i±⋃ j±r⋃ j±z⋃ j±rz (see Sec. 1.4.4). Considering a t = constant hypersurface412.3. Event Horizon(which is spacelike), any future causal curves coming from the past horizon will either go acrossthis hypersurface, or hit the brane and then are reflected to travel back into the bulk (due tothe Z2 symmetry with respect to the brane) which eventually go across this hypersurface. Simi-larly, all the past causal curves of the future horizon go across this hypersurface. Therefore, anyt = constant hypersurface is a Cauchy surface, because all the developments of any past event arecaptured by the surface, and all the future events can be predicted by the data on this surface. Ormore precisely, all the inextendible future causal curves of the past horizon and all the inextendiblepast causal curves of the future horizon, go across the Cauchy surface [32]. Therefore, a Cauchyproblem (an initial value problem) is well-defined.The z ≥ 1 branch has the horizons which are not true infinities. Accordingly, the definitionof event horizon is modified as below. In fact, the definition of event horizon in asymptoticallyMinkowski spacetime can be directly carried over, while the only change is to replace the notionof null infinities in the definitions by the future/past Poincare´ horizons defined by eq. (1.67). Theevent horizon is the boundary of the spacetime region which can not be connected to the externalworld by future oriented null geodesics. i.e. the event horizon is the collection of “the points ofno return”. The “external world” needs to be defined. Similar to the case of asympototicallyMinknowski spacetime, the external world is defined as the past of the future Poincare´ horizon,therefore the event horizon is the boundary of the spacetime separating the region that can beconnected to the future Poincare´ horizon by future null geodesics from the region that can not.The future Poincare´ horizon, on the other hand, are defined as the “infinities” (as measured bythe Killing parameter t) of future null curves departing from the external world. i.e. there is acircular argument in these definitions. The circle can be ended by physically identifying certainspacetime region(s) as the external world. Therefore, as long as the external world can be physicallyidentified, the event horizon can be defined, and this argument applies to any spacetime (i.e. it isnot limited to the case of the Poincare´ patch of the AdS spacetime).2.3.2 Event Horizon on the BraneThe event horizon is defined in Sec. 2.3.1. In the following we will study whether the eventhorizon based on the brane geometry is well-defined. The causal structure of the braneworld isdetermined by the 5D geometry, rather than the 4D geometry of the brane. The null geodesics gen-erated by the outgoing, future oriented null vectors on the bulk event horizon, form the boundary(the event horizon) separating the spacetime region that can be connected to the future Poincare´horizon by future oriented null geodesics, from the region that can not. Let us consider the out-422.3. Event Horizongoing future oriented null vectors within the brane originated from the intersect of the bulk eventhorizon and the brane. If the future oriented null geodesics generated by these null vectors willstay on the brane forever, then these null geodesics are the boundaries separating the spacetimeregions that can be connected (by future oriented null geodesics) with the future Poincare´ hori-zon from the regions that can not. i.e. (1) these null geodesics stay on the brane forever; (2)these null geodesics are the boundaries separating the spacetime regions that can arrive the futurePoincare´ horizon from the regions that can not. i.e. these null geodesics form the event horizonon the brane. Therefore, the key for the well-definedness of the event horizon on the brane, isto study the relation between the null geodesics produced based on the brane geometry and thenull geodesics produced based on the bulk geometry, generated from the same null vector whichinitially lies within the brane. This relation can be described by the extrinsic curvature of thebrane.2.3.3 Extrinsic Curvature as Geodesics DeviationThis subsection (Sec. 2.3.3) is the foundation of the study on the relation of event horizons.The work in this subsection is a “re-discovery” of the Gaussian curvature (see, e.g. [80]) and theGauss-Weingarten equation [81] in differential geometry.The motivation is as follows. Let us examine how to measure the extrinsic nature of theembedding of a hypersurface Σ into a higher dimensional spaceM . Σ is considered flatly embeddedinto M , if Σ appears to be flat in M , in the sense that an arbitrary straight line as seen in Σ isalso a straight line as seen in M . Since straight lines are geodesics, the above statement can bemore precisely rephrased as: if the hypersurface is flatly embedded, an arbitrary geodesics in Σ(consistent with the hypersurface metric γαβ) will also be a geodesics in M (consistent with thebulk metric gαβ). For non-flat embedding, these two types of geodesics are not the same in general7. Therefore, this motivates us to describe the extrinsic curvature as the deviation of the geodesicsdefined in Σ from the geodesics defined in M for a vector initially lying on Σ. This point of viewto describe the extrinsic curvature, is referred as geodesics point of view (GEP for short).On the other hand, the embedding described via the covariant derivative of the unit normalvector along Σ, as what has been done in eq. (1.5), is referred as normal vector point of view (NVPfor short).Generally if a d−C dimensional sub-manifold Σ is embedded in a d dimensional manifold M ,7As an example, we can think of a sphere S2 : x2 + (y − 1)2 + z2 = 1 embedded in R3. Take a vector lying onS2 : (∂x)µ at (0, 0, 0). The geodesics produced by it on S2 is the equator, while the geodesics produced by it in R3is the x-axis.432.3. Event HorizonC is called the co-dimension. In this subsection, we will prove NVP and GEP are equivalent inthe C = 1 case, in the sense that the embedding studied from GEP (shown below) will also leadto the same definition of the extrinsic curvature as eq. (1.5), which is defined from NVP. In theRSII braneworld, there is only one extra dimension, thus the co-dimension is C = 1. For C > 1case, please refer to appendix B.The basic idea is to study the two geodesics generated by an arbitrary vector Tα ∈ Σ, viaequations T βDβTα = 0 and T β∇βTα = 0, where D is the covariant derivative in Σ and ∇ is thecovariant derivative in M . Then we compare the two geodesics to get the difference, which candescribe the embedding nature of Σ inM . We adopt the following approach: rewrite T βDβTα = 0as T β∇βTα = leftover, then the leftover is the deviation of the two geodesics. Let us denote theunit normal vector of Σ as nα whose length square is ǫ = nαnα = ±1 where ǫ = 1 if the extradimension is spacelike and ǫ = −1 if the extra dimension is timelike. The reduced metric of thehypersurface isγαβ = gαβ − ǫ nαnβ. (2.17)For a general tensor Tα1...αkβ1...βl ∈ Σ, the covariant derivative associated with the metric γαβis [34, 36]DγTα1...αkβ1...βl = γα1δ1 ...γǫlβl γνγ ∇νT δ1...δkǫ1...ǫl . (2.18)Therefore ∀ Tα ∈ Σ, we haveDαT µ = γ βα γµγ∇βT γ , (2.19)which tells us that the geodesics generated by Tα in Σ is just0 = TαDαT µ = Tαγ βα γµγ∇βT γ = γµγTα∇αT γ , (2.20)orγµνTα∇αT ν = 0. (2.21)On the other hand, we have the following identity which can be obtained by the fact that nαTα = 0nµnνTα∇αT ν = −nµTαT ν∇αnν . (2.22)From (2.21) and (2.22), we obtainTα∇αT µ = (γµν + ǫ nµnν)Tα∇αT ν = −ǫnµTαT β∇αnβ . (2.23)442.3. Event HorizonWe can now define the deviation of the two types of geodesics equation as the right hand side ofequation (2.23). It is then clear that the deviation vanishes, if and only ifTαT β∇αnβ = 0. (2.24)This result is for arbitrary Tα defined on Σ, and only the contraction with Tα appears in thisexpression, which means we can use the following quantity to describe the embedding of Σ in MKαβ ≡ −γ µα γ νβ ∇µnν . (2.25)i.e. NVP and GEP are using the same quantity (Kαβ) to describe the embedding. The deviationequation (2.23) in terms of the extrinsic curvature is now rewritten asTα∇αT µ = ǫ nµTαT βKαβ, (2.26)from which one see that the deviation is in the perpendicular direction (nµ direction).2.3.4 The Relation between the Event HorizonsThe work in this subsection (Sec. 2.3.4) was first independently carried out in [81].For the braneworld, the 3 + 1 brane is embedded into the 4 + 1 dimensional bulk. Using thenotations in braneworld, the geodesics deviation equation (2.26) is nowvα∇αvµ = nµvαvβKαβ , (2.27)where vα is the tangent vector of an arbitrary geodesics within the brane. n is the unit normalvector of the brane, and ǫ = nµnµ = 1 has been applied.To study the event horizon relations, we focus on the case where vα is a null vector. In RSII,Kαβ is related to brane content by Israel’s junction condition (1.29)Kαβ =12kd(λ hαβd− 2 + ταβ − hαβτd− 2), (2.28)which implies the deviation of null geodesics isvα∇αvµ =12kdnµvαvβταβ . (2.29)452.4. Energy in the BraneworldTherefore the deviation of the two geodesics amounts to whether vαvβταβ vanishes. Generallythe right hand side of eq. (2.29) does not vanish (otherwise it is a new energy condition for thematter. The discussion of energy condition is beyond the scope of our project). However, in caseτµν = 0, when the matter on the brane vanishes, the two geodesics coincide. In another word,when the matter is absent, at the intersect of the bulk event horizon and the brane, the futureoriented null geodesics that are produced by the null vectors lying in Σ, will stay on the braneforever. Since these null geodesics are on the event horizon of the bulk, they are the boundariesseparating the spacetime regions that can be connected (by future oriented null geodesics) withthe future Poincare´ horizon from the regions that can not. i.e. (1) these null geodesics stay on thebrane forever; (2) these null geodesics are the boundaries separating the spacetime regions thatcan arrive the future Poincare´ horizon from the regions that can not. Therefore, from the branepoint of view, they form the event horizon on the brane. i.e.an event horizon on the brane is well-defined when there is no matter around the horizon.For gravitational collapse processes, if the systems eventually reach the stationary states thatthe matters either go into the black holes, or get bounced back and travel towards spatial infinity,then there are no matters at the horizons and the event horizons on the brane are well-defined.2.4 Energy in the BraneworldIn order to quantitatively describe the spacetime evolution, and gravitational interaction be-tween the brane and the bulk, we need to introduce certain quantities, such as energy. However,there is no local definition of energy in general relativity. Instead, there have been many attemptsto define quasilocal energy in general relativity, and many of these definitions are only well-definedin a certain background. In this section, the definition developed by Brown and York obtainedfrom a Hamiltonian-Jacobi analysis of the gravitational action [95] (Hawking and Horowitz alsogave a similar derivation [87]), is applied to the braneworld.2.4.1 Total EnergyIn this subsection, we introduce the energy defined in [87, 95]. In the next subsection, we willapply this definition to the braneworld spacetime.462.4. Energy in the BraneworldIn general the action of a gravitational system isI(g,Φ) =∫M[ R16πGd+ Lm(g,Φ)]− 18πGd∮∂MK , (2.30)where M is the spacetime manifold, K is the extrinsic curvature of the boundary ∂M embeddingintoM . Lm is the Lagrangian for all the matter fields and the matter fields are collectively denotedas Φ.Let us choose the boundary ∂M = Σt1⋃Σt2⋃BQ, where BQ ≡ ⋃t∈[t1,t2] SQt as shown inFig. 2.1. SQt is the closed (d − 2) dimensional spatial surface embedded in each Σt, and Q is thesingle parameter to characterize the family of the enclosed (d− 2) dimensional surfaces in each Σt.When Q → ∞, SQt goes to the spatial infinity boundaries. BQ is setup such that its normal unitvector Qµ is perpendicular to nµ (so that nµ lies within BQ).QSQttΣt2Σt1ΣtBQFigure 2.1: Manifold M and its boundaries. The boundary ∂M is composed of Σt1 , Σt2 , and BQ.On the diagram, the boundaries are shown to be the upper/lower surface of the cylinder, and theside surface of the cylinder. Here BQ ≡⋃t∈[t1,t2] SQt , where SQt is the enclosed (d− 2) dimensionalspatial surface embedded in each Σt, and Q is the single parameter to characterize the family ofthe closed surfaces in each Σt. Choose BQ such that its normal unit vector Qµ is perpendicular tonµ (so that nµ lies in BQ).The Hamiltonian is then [87, 95]H =∫Σt(αH + βνMν) +18πGd∮SQt[−ǫ Qµ(Kγµν −Kµν)βν + α · (d−2)k], (2.31)where ǫ ≡ nµnµ = −1. (d−2)kαβ is the extrinsic curvature of SQt embedded in Σt, and (d−2)k is its472.4. Energy in the Braneworldtrace. H and Mν are Hamiltonian constraint and momentum constraint defined asH ≡ −ǫ16πGd(−K2 +KµνKµν + ǫ · (d−1)R+ 16πGdρ), (2.32)Mν ≡ ǫ8πGdDµ(Kγµν −Kµν)− Sν . (2.33)The Hamiltonian constraint H and momentum constraints M are implied by Einstein’s equations,therefore vanish for physical configurations, and should be dropped in (2.31).Hamiltonian (2.31) diverges in general. However, it is the physical Hamiltonian that matters[87]. The physical Hamiltonian is H − H¯, where H¯ is the Hamiltonian of a background spacetime.We denote background quantities by a bar ( ¯ ). The background is a static solution, then itscontribution is only− 18πGd∮SQtα (d−2)¯k,where the integration is over a closed surface in the background spacetime that is isometric to theSQt (that is the closed surface chosen in the physical spacetime).The physical energy is the physical HamiltonianE = H − H¯ = limQ→∞18πGd∮SQt[−ǫ Qµ(Kγµν −Kµν)βν + α((d−2)k − (d−2)¯k)], (2.34)where the Hamiltonian constraint H and momentum constraint M are dropped, since they vanishat physical configurations. Also, at spatial infinities where there is asymptotic time translationalsymmetry, the lapse function and the shift functions go to the form of the background spacetime.However, as proved by Shi-Tam [90], the definition does not yield a positive definite energyexcept for the time symmetric case (Kµν = 0). Also, the definition is gauge dependent. Yau andLiu [91–93] defined another formula for energy, which is gauge independent, and can be positivedefinite under certain conditions. However, our initial data is time symmetric which means thedefinition by Brown-York and Hawking-Horowitz is sufficient. Also, we can still use the definitionby Brown-York and Hawking-Horowitz during evolution since the energy is characterized by theasymptotic behaviour at spatial infinities, which is not affected by finite time evolution (i.e., thelocal behaviour is not able to propagate to spatial infinities within a finite time evolution).482.4. Energy in the Braneworld2.4.2 Total Energy in the Braneworld with AxisymmetryWe assume the energy (2.34) is well-defined in the braneworld. In the braneworld, the spacetimebackground isds¯2 = ℓ2z¯2(− dt¯2 + dr¯2 + dz¯2 + r¯2(dθ¯2 + sin2 θ¯dφ¯2)), where z¯ ≥ ℓ. (2.35)A bar is used to denote the quantities associated with the background. For this background, thelapse α¯ = 1/z¯ and the shift β¯ν = 0. Therefore, (2.34) reduces toE = limQ→∞18πGd∮SQtα¯((d−2)k − (d−2)¯k), (2.36)which will be the definition of the energy of the braneworld.To calculate the energy, we need to set a closed surface family SQ that goes to spatial infinityas Q → ∞. The two requirements on defining the family are: (i) SQ goes to spatial infinity asQ → ∞, and (ii) the closed surface is smooth to a certain degree so that the extrinsic curvature(d−2)kαβ is well-defined at any point on SQ. In our case where the system has axisymmetry(spherical symmetry on the brane) with coordinates (t, r, θ, φ, z) adapted to the symmetry, we maychoose the closed surface as, for example, Fig. 2.2(a). Quantities Q, u and v are the parametersfor defining the closed surfaces. Please refer to the capture of the figure for the details.Without loss of generality, the spatial metric of any t = constant slice can bedl2 = ℓ2z2[e2A+2B(dr2 + dz2)+ e2A−2Br2(dθ2 + sin2 θdφ2)]. (2.37)The brane is located at z = ℓ. This is the most general spatial metric for the axisymmetricconfiguration because, by taking the symmetry into account, the most general form can takedl2 = ℓ2z2[η˜rrdr2 + η˜zzdz2 + 2η˜rzdrdz + r2η˜θθ(dθ2 + sin2 θdφ2)]. (2.38)For a given t = constant slice, everything only depends on r and z, therefore η˜rrdr2 + η˜zzdz2 +2η˜rzdrdz can transform into a conformally flat form. Lastly, the freedom in the conformal function,can be used to fix the brane at z = ℓ [14, 47].To calculate the energy, we embed SQt into the background spacetime (2.35). In general, it isnot guaranteed that such embedding is possible for an arbitrarily chosen closed surface, although itturns out all the closed surfaces considerred in this thesis could be embedded into the background492.4. Energy in the Braneworldrz(r = 0, z = 1)(I)(II)Q(III)rz(r = 0, z = 1)(a) (b)(I)Q(II)(III)Figure 2.2: The closed surfaces to calculate total energy and energy in the bulk. Diagram (a) isused to calculate the total energy. z = 1 is where the brane is located. The system is sphericallysymmetric on the brane and r is the radius of the spherical coordinate. The closed surface iscomposed of three segments: segment (I) starts at r = Q and its coordinate length (measured bythe coordinate z) is v ·Q. The length of segment (III) is u ·Q. Here u ∈ (0, 1) and v ∈ (0, 1) andtheir values are fixed for a specific closed surface family. Segment (II) is an arc (a quarter of a circlewhose radius is (1−u) ·Q)) to connect these two segments smoothly. The small segment below thebrane is to show that there is another part below the brane which is related to the said part by Z2symmetry. Note, the closed surface should goes smoothly across the brane (i.e. the closed surfaceis perpendicular to the brane). In general case this perpendicular surface is not represented byr = constant. When the coordinate gauge at the brane is taken to be perpendicular gauge (2.7)so that grz∣∣z=ℓ = 0, this surface is represented by r = constant. Diagram (b) is to add two dashedlines (along the brane) onto diagram (a). The use of (b) is going to be explained in Sec. 2.6.502.4. Energy in the Braneworldspacetime. The embedding is a mapping from SQt in the physical spacetime to its image in thebackground spacetime by keeping the intrinsic geometry of the closed surface. i.e. the intrinsicgeometry of the image of SQt in the background spacetime is the same as the intrinsic geometryof SQt in the physical spacetime. There is a freedom in this mapping (which is “where we put theimage”), and we fix this freedom by naturally mapping the intersection of the closed surface withthe brane in the physical spacetime to the brane of the background spacetimez¯|z=ℓ = ℓ. (2.39)For a closed surface in the physical space with metric (2.38), the embedding into the background0.2 0.411.21.41.6BDCArz(a)0.2 0.4 0.611.¯D¯C¯A¯r¯z¯(b)Figure 2.3: The embedding of a closed surface. Fig. (a) is the closed surface in the physicalspacetime (2.38), and Fig. (b) shows its embedding into the background spacetime (2.35). Thefreedom of the embedding is fixed by mapping A to A¯.space (2.35) is demonstrated by Fig. 2.3. The embedding boils down into the following two condi-tions:(1) On the r − z plane, an point B (see Fig. 2.3) on the closed surface, represents a 2-sphereexpanded by coordinates (θ, φ), whose proper area is 4πr2η˜θθ/z2. The area of the 2-sphere at512.4. Energy in the BraneworldB¯ (the image of B in the background spacetime) is 4πr¯2/z¯2, thereforer2z2 η˜θθ =r¯2z¯2 . (2.40)(2) C is a point on the closed surface that is infinitesimally close to B. Let us denote the coor-dinates of B by (r, z), and denote the coordinates of C by (r + dr, z + dz). The square ofthe length of the infinitesimal line is(η˜rrdr2 + η˜zzdz2 + 2η˜rzdrdz)/z2. Correspondingly, thecoordinates of their images (B¯ and C¯) are (r¯, z¯) and (r¯ + dr¯, z¯ + dz¯), and the square of thelength of the infinitesimal line is(dr¯2 + dz¯2)/z¯2. The equality of these two lengths reads1z2(η˜rrdr2 + η˜zzdz2 + 2η˜rzdrdz)= 1z¯2(dr¯2 + dz¯2). (2.41)Here we emphasize that the infinitesimal line is taken along the closed surfaces. The condition“along the closed surface” defines how dr and dz are related (also defines how dr¯ and dz¯ arerelated).The freedom related to “where to put the images” is fixed by mapping A (the point of the closedsurface on the brane) to A¯ (the point of the closed surface on the brane).For metric (2.37), these two conditions reduce tor2z2 e2A−2B = r¯2z¯2 , (2.42)1z2 e2A+2B (dr2 + dz2)= 1z¯2(dr¯2 + dz¯2). (2.43)For background metric (2.35), the lapse funtion to evaluate (2.34) is α¯ = ℓ/z¯ and the shiftfunction is β¯µ = 0.∮SQt(α · (d−2)k)diverges and the divergence is cancelled by the divergence in∮SQt(α · (d−2)¯k). Noticing the ℓ2/z2 factor, we examine whether the contribution from segment (II)and segment (III) in Fig. 2.2(a), vanishes as Q → ∞. In fact, one can show by direct calculationthat∮SQt(α · (d−2)k)from segment (II) (the arc) converges to zero as 1/Q, as long as A and B con-verge to zero (at any rate) as Q→ ∞, so does∮SQt(α · (d−2)¯k). Therefore∮SQtα((d−2)k − (d−2)¯k),the contribution to the total energy, converge to zero at least as fast as 1/Q. The same argumentalso applies to segment (III). Therefore, only the r = Q segment contributes to the total energy. Orwithin the bulk, only the region r ≫ z, contributes to the total energy.522.4. Energy in the BraneworldNow we complete the proof by proving the claim “∮SQt(α · (d−2)k)from segment (II) and seg-ment (III) converges to zero as 1/Q”. The calculation on segment (III) is similar to the calculationon segment (II). Furthermore, acually segment (III) does not need to exist to define the closedsurface, if we take u = 0. For conciseness, here we only present the calculation on segment (II)(the arc).The metric is (2.37), where r and z along the arc can be parametrized as r = uQ+ ρ cosχ andz = 1 + vQ+ ρ sinχ, where ρ = (1− u)Q,χ ∈ [0, π/2]. A direct calculation giveslimQ→∞∮SQt(α · (d−2)k)= 8π limQ→∞∫ π/20dχ e2A−2B 1z3 (ξ1 + ξ2)= 8π limQ→∞∫ π/20dχ 1z3 (ξ1 + ξ2) . (2.44)Here ξ1 ≡ −r2 − 2rρ cosχ+ (3r2ρ sinχ)/z and it is easy to numerically check∫ π/20dχ (ξ1/z3) ∝ 1/Q.ξ2 ≡ r2ρ (−3A,ρ +B,ρ), and A,ρ and B,ρ converge to zero at least as fast as 1/ρ since this cor-responds to A ∼ log ρ and B ∼ log ρ, whilst the reality is A → 0 and B → 0 when ρ → ∞.Therefore∫ π/20dχ (ξ2/z3)converges to zero at least as fast as∼∫ π/20dχ (r2/z3) ∝ 1/Q.This completes the proof.Another potential contribution to the total energy is the “cusp” of the closed surface across thebrane, if there is any. Within the physical spacetime, we can properly choose the closed surfacesuch that it crosses the brane smoothly. Under the coordinate gauge grz∣∣z=ℓ = 0, the smoothnesscondition is equivalent to dr/dz∣∣z=ℓ = 0, where the derivative is taken along the trajectory of theclosed surface in the r − z plane. This condition is satisfied by the surface specified by r = Q.Within the background spacetime, the smoothness condition is equivalent to dr¯/dz¯∣∣z¯=ℓ = 0. If weuse z as the parameter of the trajectory in the r¯ − z¯ plane of the background spacetime, we havedr¯/dz¯ = dr¯/dzdz¯/dz ,532.4. Energy in the Braneworldwhere the derivatives are taken along the trajectory of the closed surface in the r¯ − z¯ plane ofthe background spacetime. Therefore the condition dr¯/dz¯∣∣z¯=ℓ = 0 is just dr¯/dz∣∣z¯=ℓ = 0. If thiscondition is met, there is no cusp (therefore no contribution to the total energy) across the brane.In fact, since we will take Q → ∞ eventually, it is sufficient to examine whether this conditionis satisfied by only keeping the lowest order of A and B in the expressions. However, conditiondr¯/dz∣∣z¯=ℓ = 0 is generally not gauranteed. A direct calculation from eq. (2.42) and eq. (2.43)yieldsdr¯/dz∣∣z¯=ℓ ≈ r · (A+A,z +B −B,z)∣∣z=ℓ, (2.45)where ≈ means this relation is true up to the lowest order in A and B. In general, it is hard tosay whether this term vanishes. However, in the conformally flat space that we are going to studyin the next subsection (Sec. 2.4.3), the smoothness condition is indeed satisfied (see eq. (2.53)).Furthermore, we will obtain a relation between the total energy of the whole braneworld, and theADM energy calculated based on the brane geometry.2.4.3 Total Energy in Conformally Flat Space of the BraneworldA general way to realize the embedding is to encode the two conditions (eq. (2.42) and (2.43))into numerical methods. However, if we take into account the asymptotic behaviours of A and Bin the following special case, the embedding and the calculation of total energy can be obtainedanalytically without using numerical methods.We consider a special case by assuming B ≪ A in r ≫ z region 8. In this region, the spatialmetric reduces todl2 = ℓ2z2 e2A (dr2 + dz2 + dθ2 + sin2 θdφ2), (2.46)for which the Hamiltonian constraint is (under the unit ℓ = 8πG5 = 1, which implies 8πG4 = 1via (1.37))(∂rr + ∂zz)A+2z2(1− e2A)+ 2A,rr −2A,zz + (A,z)2 + (A,r)2 = 0, (2.47)and Israel’s junction condition isA,z∣∣z=ℓ = 1− eA. (2.48)In the r≫ z region (which is the only region that contributes to the total energy as proved above),8This condition was observed (i.e. a posterior result, rather than a prior assumption) in static star solutions [47].This condition holds in a solution for small static BHs [14]. This condition is also compatible with Hamiltonianconstraint [48]. This condition will also turn out to hold in one version of our initial data (see Chap. 4).542.4. Energy in the Braneworldeq. (2.47) reduces to the following by taking only the linear term in A(∂rr + ∂zz)A−4Az2 +2A,rr −2A,zz = 0, (2.49)with linearized Israel’s condition asA,z∣∣z=ℓ = −A. (2.50)The Hamiltonian equation (2.49) subject to boundary condition (2.50) can then have an solutionwith closed formA ≈ α1rz , (2.51)here we use ≈ to emphasize that this solution is only true at r ≫ z region.Remark: this solution satisfies the linearized Hamiltonian constraint subject to the linearizedIsrael’s boundary condition, and the boundary condition at r → ∞. However, there is no proofregarding uniqueness. Therefore, we should justify (2.51) is indeed a solution via the result obtainedby numerical calculation, which is going to be carried out in Sec. 4.3.3.In physical spacetime, the coordinates of the r = Q surface are now (z, θ, φ). We will findthe metric of the corresponding surface in the background spacetime expressed under coordinates(z, θ, φ) as well. The two conditions of the embedding are nowr¯z¯ =rz eA; (2.52a)(r¯′)2 + (z¯′)2z¯2 =e2Az2 . (2.52b)where r = Q is a fixed value, and ′ denotes d/dz. Eq. (2.52) is the ODE set that defines r¯(z)and z¯(z) for fixed r, subject to the “initial” value z¯∣∣z=1 = 1. By construction, the r = Q surfacein physical space and r = Q surface in background space have the same intrinsic metric, and thecoordinates of the surfaces are the same one: (z, θ, φ). It is then straightforward to carry out(2.34). Note the shift functions are asymptotically zeros and the lapse function is asymptotically1/z¯ for background (2.35).The solution of the ODE set by keeping the lowest order in A isz¯ = z · exp[α1(z − 1)rz]; r¯ = r · exp(α1r). (2.53)It is then a routine work to carry out the calculation of total energy using (2.34). Here we only552.5. ADM “Mass” and Hawking “Mass” of the Branelist some important intermediate resultsMtotal =18πG5limQ→∞∫SQtα((d−2)k − (d−2)¯k)= 18πG5· 2 · 4π · limQ→∞∫ vQ1dz Q α((d−2)k − (d−2)¯k)= 18πG5· 24πα1∫ ∞1dzz4 = α1/G5. (2.54)The factor 2 in the second line is due to the Z2 symmetry with respect to the brane in RSII;4π is from the integration over (θ, φ); Q is the determinant of the intrinsic metric of the surface:Q = e3Ar2/z3; and we have used (d−2)k = −2z/r + 5α1/r2, (d−2)¯k = −2z/r + 2α1/r2; r = Q;α = eA/z.Also, from the expressions one see that, the result would have diverged as∼ Q, if the backgroundterm (d−2)¯k is absent.2.5 ADM “Mass” and Hawking “Mass” of the BraneDue to the equivalence between mass and energy, we use these two words interchangeably.As will be explained in Sec. 2.6, the masses calculated based on the brane geometry, are notreally energies on the brane, from braneworld point of view. Actually they play no direct roles inbraneworld, therefore we put “mass” in quotes in the title of this section. The purpose of studyingthese quantities is to compare the braneworld with GR to examine whether there are observationaldifferences. In previous sections, the dimension of the spacetime is arbitrary. In this section, weonly consider 3+1 dimensional spacetime (the brane).We will first introduce ADM mass and Hawking mass in Sec. 2.5.1 and Sec. 2.5.2, then we willderive the ADM mass in the conformally flat space (2.46).2.5.1 ADM MassThe ADM mass is introduced in standard texts [29, 32, 36]. One way to obtain it is to use(2.34) by setting the lapse α = 1 and the shift βµ = 0 [36]. Note the ADM mass is only definedwhen the background is asymptotically flat. Applying (2.34) to a asymptotically flat spacetime,the ADM mass reduces to [36]MADM =116πG4limQ→∞∮SQt[D¯ iqij − D¯i(q¯klqkl)]si, (2.55)562.5. ADM “Mass” and Hawking “Mass” of the Branewhere qij is the spatial metric, and i, j = 1, 2, 3 is the spatial coordinate index. q¯ij is the (flat)background metric, and D¯ is the covariant derivative associated with q¯ij . si is the unit normalvector of SQt pointing outwards, which was denoted as Qα in (2.34).Note, only the spatial metric is needed in the definition.For the spherical symmetric case, the metric can always be rewritten asdl2 = grrdr2 + r2(dθ2 + sin2 θdφ2), (2.56)in spherical coordinate (r, θ, φ), where r is the areal radius. In this situation the ADM mass canbe derived to beMADM = limr→∞Madm, (2.57)whereMadm =r2G4(grr − 1)√grr, (2.58)which is defined as the integrand in (2.55). In the special case of Schwarzschild metric wheregrr = (1− 2mG4/r)−1, Madm = m/√1− 2mG4/r, which has r dependence. Within the horizon,2mG4/r > 1, therefore this quantity is not well-defined within the horizon.2.5.2 Hawking MassThe Hawking mass is defined on two dimensional surfaces S with spherical topology, and itcharacterizes the mass in the space enclosed by the surface. The Hawking mass is described in termsof spin-coefficient formalism, for which please refer to [83, 85, 86]. In terms of spin-coefficients, theHawking mass is defined asMH =√A16πG24(1 + 12π∮Sρρ′dS), (2.59)where A is the area of S. The ρ and ρ′ are two of the spin-coefficientsρ = 12√2mαβ∇α (nβ + sβ) =12√2Θ+, (2.60a)ρ′ = 12√2mαβ∇α (nβ − sβ) =12√2Θ−, (2.60b)572.5. ADM “Mass” and Hawking “Mass” of the Branewhere Θ± are defined in (2.13). Substituting (2.60a) and (2.60b) back into (2.59), we getMH =√A16πG24(1 + 116π∮SΘ+Θ−dS). (2.61)At an apparent horizon Θ+ = 0, therefore the following mass-area relation holds at the apparenthorizonMH =√A16πG24. (2.62)In the case of spherical symmetric metric (2.56), the Hawking mass is calculated asMH =r2G4(1− (grr)−1), (2.63)where we have usedΘ+ = −Θ− =2r√(grr)−1,which are directly calculated by applying the definitions (2.13) to metric (2.56). However, thecalculation of Θ’s depends on the time components of the metric too. This result is only valid incase the configuration is time symmetric (where the extrinsic curvature of t = constant hypersurfaceis zero). This motivates us to give (2.63) a new notation Mh, which agrees with MH in the timesymmetric case.In the special case of Schwarzschild metric where grr = (1− 2mG4/r)−1, MH = m and has nor dependence.2.5.3 The ADM “Mass” of the Conformally Flat SpaceLet us now apply the ADM mass or Hawking mass for the conformally flat space (2.46). Firstlet us rewrite it in terms of the areal radius r˜, which is defined as the radius associated with thearea of the r = constant surface. Thereforer˜ = reA. (2.64)Rewrite (2.46) in terms of r˜, we havedl2brane = (1 + rdA/dr)−2 dr˜2 + r˜2(dθ2 + sin2 θdφ2), (2.65)582.6. A Quest for Brane Energywhich is the spatial metric on the brane. Substitute in the asymptotic behaviour of A at z =1, r≫ 1: A ≈ α1/rz = α1/r, we havedl2brane = (1− α1/r)−2 dr˜2 + r˜2(dθ2 + sin2 θdφ2).Substitute this into (2.58) or (2.63), we obtain the brane ADM mass asMbraneADM = α1/G4. (2.66)Comparing this with eq. (2.54), we obtain a somewhat surprising resultMtotal = (G4/G5) ·MbraneADM. (2.67)2.6 A Quest for Brane EnergyThere is energy exchange between the bulk and the brane. To quantitatively describe theenergy exchange, we need to define the energy in the bulk, and the energy on the brane. Basicallyonly the energy in the bulk, or the energy on the brane, is sufficient since the other can be definedby subtraction from the total energy.To serve as the energy of the brane, an expression should satisfy:(1) its value is a part of the total energy. This rules out the ADM mass defined based on thebrane geometry since it is equal to the total energy for a class of space configurations. Pleaserefer to eq. (2.67).(2) it is not conserved during the evolutions, because of the energy exchange between the braneand the bulk. This requirement rules out any quasi-local definition on the brane, such as theADM mass and Hawking mass evaluated from brane geometry. This is because, the quasi-localenergies are defined as the limit at spatial infinities utilizing the time translational symmetryat spatial infinities, therefore conserved.The definition of an energy, especially that for the brane, is subtle and this section is to requestthe study of energy on the brane, rather than providing a solution. The reader can skip thissection from here since the following is an attempt (that has conceptual issue) rather than a result,although the attempt turns out to have surprisingly good features exhibiting in the simulations wecarried out.592.6. A Quest for Brane EnergyWe would like to search the definition from the Hamiltonian-Jacobi analysis described above.Since the energy of a certain region is defined as an integration over the boundary enclosing theregion, if this concept could be generalized (which actually can not), we can tentatively definethe energy of the bulk as the integration of (2.34) over the closed surface of Fig. 2.2(b). i.e. weenclose it along the brane. The difference, which is the integration of (2.34) along the brane, canbe accordingly defined as the energy on the brane.However, without going into the details, we know this tentative definition is problematic for thefollowing reasons. There is a sharp corner in the surface, where the extrinsic curvature is infiniteand the integration of the extrinsic curvature over the corner can be indefinite. In the backgroundspacetime, integrating along the brane makes sense physically. There is a requirement, however,that the surface is embedded into the background so the intrinsic metric of the surface stays thesame, which can not be met for any non-trivial brane. Or alternatively, we keep the “embedding”requirement, but then we have to give up the “integrating along the brane” in background spacetime(which also means we give up the freedom fixing condition (2.39).). Also, since this is a definitionregarding an integration on the brane, which will fail when there is a physical singularity on thebrane. Therefore this definition can not be treated seriously. It is only a (very) rough description.We hope the resulting quantity changes monotonically with the amount of the energy exchangedbetween the brane and the bulk.In the simulation (presented in Chap. 5), we will choose the brane to be the surface in thebackground spacetime. i.e. we give up the requirement that the intrinsic metric of the surfacesin physical spacetime and background spacetime are the same. (2.42) and (2.43) are the tworequirements for embedding, and we have to keep one of these to make the definition unambiguous.Here we choose to keep (2.42) because it is simpler and also because the areal radius still makessense when there is a physical singularity (where condition (2.43) fails).Considering requirement (2.42), it is natural to express the energy in terms of spherical coordi-nate (r˜, θ, φ) with r˜ as the areal radius, where the physical metric is dl2 = gr˜r˜dr˜2+r˜2(dθ2 + sin2 θdφ2).Since this definition only includes the integration on the brane, we may rewrite it asEbrane =18πG5∫Bα((d−2)k − (d−2)¯k), (2.68)= limQ→∞18πG5∫ Q0dr˜∫ π0dθ∫ 2π0dφ√det[q] α((d−2)k − (d−2)¯k). (2.69)where r˜ = r¯ is the areal radius and r¯ is the areal radius in the background space. q is the spatialmetric of the brane (in physical spacetime) and we have√det[q] = r˜2 sin θ√gr˜r˜. B stands for the602.6. A Quest for Brane Energyspatial part of brane manifold (in physical spacetime). Please note it is G5 (rather than G4) inthe expression, and α is the lapse in 5D restricted on the brane, rather than the lapse functioncalculated from brane geometry. (d−2)k is the extrinsic curvature of the 3-brane embedded in 4Dspatial bulk. Note a crucial feature of this energy is that, it is an integration over B, and itsmeaning is the energy over B. i.e. it might be possible to think of energy density, which is actually(spatial) coordinate independent on the brane. Noticing the second requirement of the energy onthe brane (that the definition should not be semi-local), this definition might have captured a keyfeature.61Chapter 3Axisymmetric Spacetime withNon-Flat BackgroundThe cost of numerical computation increases dramatically with the number of dimensions. Thisis usually called the curse of dimensionality. To date, it is impractical to directly perform numericalcalculations in 4 + 1 dimension at a reasonable resolution, or study problems with the demand ofhigh resolution (such as critical phenomena) in more than (effectively) 2+1 dimension. Therefore,one of the first steps to study a physical system in higher dimensional spacetime, is to considerthe system with some kind of symmetry, such as spherical symmetry and axisymmetry, to reducethe effective dimension.However, in many situations, singularities, instabilities, unavoidable noises or other irregulari-ties, arise in the numerical calculations performed under the coordinates adapted to the symmetry,either at the origin or at the axis of the symmetry ([50] and references therein). On the other hand,this kind of issue does not occur in the simulations of the same system performed under Cartesiancoordinates. This kind of behavior is called the regularity problem in non-Cartesian coordinates.To make the discussion and presentation smoother, we will only mention axisymmetry below,which also applies to spherical symmetry.Where does the irregularity come from? Since the irregularity appears in cylindrical coordinatesor spherical coordinates rather than Cartesian coordinates, it is widely believed that the irregularitycomes from coordinate system choice ([50] and references therein). The terms involving 1/rn (wherer symbolically stands for the radius coordinate in spherical coordinates or cylindrical coordinates)often appears in the equations, and it is widely believed that, with or without machine precisionplaying a role, these terms are responsible for the irregularity [50].What we do differently in this thesis is to make distinction between the coordinate system, andthe fundamental variables—the unknown functions to be solved for—used in simulation (which areusually the components of tensors and pseudo-tensors), and reveal that the fundamental variables(rather than the coordinate system, or the operators in the coordinate system) are responsible623.1. Regularity Problem and Our Conjecturefor irregularities. In particular, neither 1/rn-terms nor machine precision, is relevant to regularityproblem. Therefore the regular results obtained from simulations performed in Cartesian coordi-nates, are actually due to the fact that the tensor (and pseudo-tensor) components are regular inCartesian coordinates, rather than the regularity of Cartesian coordinates itself. As one will see inthis chapter, actually it does not make sense to talk about the regularity of a coordinate system.The key to avoid irregularity issue in any coordinate system, is to construct tensors (and pseudo-tensors) in terms of regular variables that are compatible with the finite difference approximation(FDA) scheme at the vicinity of the axis (or the origin). There are many ways to express tensors(and pseudo-tensors) in terms of regular variables. To embody the construction, we present ageneral method to construct regular variables out of Cartesian components (which play the roleof regular variables). The method can, for example, enable the generalized harmonic formalismto be used in non-Cartesian coordinates. Then we analyze why certain other formalisms in theliterature can avoid regularity issue as well.Utilizing the knowledge obtained in studying regularity problem, the evolution schemes suchas the generalized harmonic formalism and the BSSN formalism of GR can be rewritten undergeneral coordinates which overcome the irregularity issue. On the other hand, there is anotherproblem associated with the braneworld: the asymptotic spacetime background of braneworld isnot flat, while the existing knowledge in the literature are only for asymptotically flat spacetimes.It is not clear how to setup the source functions (of the GH formalism) in the braneworld. Thesecond part of this chapter is devoted to solving the non-flat background problem.3.1 Regularity Problem and Our ConjectureVery often (but not always), the numerical calculations performed in cylindrical coordinatesadapted to axisymmetry, generate irregularities in the vicinity of the symmetry axis, but the samecalculations performed in Cartesian coordinates do not generate irregularities. This phenomenais called the regularity problem. The irregularity, can appear as: (i) singularity in certain funda-mental variables used in numerical calculations in which situation the code would crash and thefundamental variables would not converge, or (ii) non-smoothness of certain fundamental variables,or (iii) smooth fundamental variables which do not converge at the expected order — i.e. they cannot pass the independent residual tests.In this section, we will first introduce the existing analysis of regularity problem, then identifysome of the existing methods that can overcome regularity problem, then we will declare a conjec-633.1. Regularity Problem and Our Conjectureture regarding the true source of the problem. We back up the conjecture by a detailed analysisof the deviation between the numerical results and the analytical results, which reveals the secondkey element to yield regular results.3.1.1 The Existing Analysis and The Existing SolutionsLet us take a specific example as the carrier of the existing analysis: the wave equation of ascalar field Φ in 3+1 (flat) spacetime with axisymmetry. (x, y, z) are the Cartesian coordinates,and (ρ, φ, z) are the cylindrical coordinates. The equation of motion is(∇2 − ∂tt)Φ = (∂xx + ∂yy + ∂zz − ∂tt)Φ =(∂ρρ +1ρ∂ρ + ∂zz − ∂tt)Φ, (3.1)where we have applied the fact that the space is axisymmetric so that the φ-derivatives are zeros.It is “obvious” that the term (1/ρ)∂ρΦ was singular and it was widely believed ([50] and referencestherein) that terms like this were responsible for regularity problems.Based on the analysis, the crucial step to cure irregularity was to modify the differential op-erators so that terms like 1/rn did not appears. There are quite a few widely-used techniques tocure the regularity problem. Among these techniques, a class of methods called Cartoon meth-ods [50, 61], utilize the observation that “there is no regularity issue in Cartesian coordinates” tomodify the operators in a way so that the operators are effectively Cartesian.xyFigure 3.1: The demonstration of the cartoon method. The system is axisymmetric with respectto z axis. For stencils which involve three grid points, only three slices are needed: the y = 0 slice,the one above it and the one below it. Note, this diagram is from [50].The Original Cartoon MethodThe original cartoon method [50] was proposed by Alcubirre et al in 1999. Here I will again usea scalar field Φ to demonstrate the basic ideas. The physical space is three dimensional space, with643.1. Regularity Problem and Our Conjecture(x, y, z) being the Cartesian coordinates, and the system is axisymmetric with respect to z axis.Analytically, only one slice, say the y = 0 slice, is needed to include all the information because ofthe axisymmetry. Numerically, the cartoon method is trying to use only y = 0 slice as well. Themethod can be implemented via the following four steps.(1) Write the code using equations expressed in Cartesian coordinates.(2) Use only the y = 0 slice. Of course, this is impossible in Cartesian coordinates, since thediscretization of derivatives with respect to y needs more than one slices in the y direction. Tobe specific, let us assume the equations consist of second order derivatives at most, and thediscretization is of the second order accuracy. In this situation, the stencil only needs threegrid points to do the finite difference approximation for the differential operators (Fig. 3.1).However, the function values on the upper and the lower slices are not known.(3) Obtain the function values on the upper and the lower slices by numerical interpolation con-structed from the values within y = 0 slice, utilizing the axisymmetry. The process is shownin Fig. 3.1.(4) Replace the function values on the upper and the lower slices, by the interpolated valuesobtained in step (3), and then substitute into the discretization.From these four steps, operators such as 1/rn are avoided, and only Cartesian operators are directlyused.The Lie Derivative Cartoon MethodIn the third step of the original cartoon method, in order to get derivatives with respect toy, numerical interpolation is used. The Lie derivative cartoon method [61] improves this, byanalytically replacing these derivatives by those within the y = 0 slice. For the particular example,using the fact x∂y − y∂x being the Killing vector, we have∂yΦ =yx∂xΦ.Taking derivatives with respect to y on both sides, we have∂yyΦ =1x∂xΦ +yx∂xyΦ.653.1. Regularity Problem and Our ConjectureThe laplacian ∇2Φ can then be evaluated at the y = 0 plane as∇2Φ = (∂xx + ∂yy + ∂zz)Φ =(∂xx +1x∂x +yx∂xy + ∂zz)Φ∣∣∣∣y=0=(∂xx +1x∂x + ∂zz)Φ. (3.2)The derivatives in y direction are replaced by the derivatives within the y = 0 slice, on the analyticallevel (rather than numerical level in the original cartoon method).Background Removal MethodThe cartoon method is a general approach to solve regularity problem, which can be used inany formalism of GR. On the other hand, Brown [97] and Gourgoulhon [36] developed a method torewrite the BSSN and generalized harmonic formalisms, by a background removal method, whichwill be described in detail in the following sections. The same method was also applied in [21]to solve static problems using Ricci-DeTurck flow methods. The simulation using this rewrittenBSSN formalism in spherical coordinates “turned out to be” regular [56]. However, the authorsdid not analyze why this method is regular. Sorkin-Choptuik [54, 55] used a different methodto remove the background effect, which again “turned out to be” regular, without analyzing thereason.3.1.2 The Conjecture: Variables Rather Than CoordinatesVarious other attempts on the solution to regularity problem had been proposed (see [50] andreferences therein) and they were not very successful until the cartoon method appeared. Thecartoon method is a general method that solves the regularity problem, and there exist othermethods which overcome the regularity problem as well[36, 71, 97]. Here we will take a closer lookand provide a deeper understanding of the topic. The understanding can serve as a criteria anda guideline to develop regularized formalisms, and can potentially completely solve the regularityproblem associated with coordinates.We start from the facts rather than speculations:(1) The Lie derivative cartoon method works well [61–63].(2) 1/x still appears in eq. (3.2) which was produced based on Lie derivative cartoon method.(3) Eq. (3.2) is the same as (3.1) (equation in cylindrical coordinates), if x is identified with ρ.(4) Certain background removal methods (performed in non-Cartesian coordinates) [36, 54, 55, 97]are free from regularity issues.663.1. Regularity Problem and Our Conjecture(5) There exist many successful simulations with 1/rn terms (many references, e.g: [59, 61–63, 71]).(6) All in all, a multiplication by 1/rn terms does not change the numerical feature. (ρ · ∂ρρ + ∂ρ)Φis not problematic, therefore (∂ρρ + ∂ρ/ρ)Φ should be as safe.These facts lead me to ask whether the spherical/cylindrical coordinates were the sources ofthe problem, and ask the question: does the regularity issue come from the terms (1/rn) inthe operators associated with the coordinate system, or somewhere else such as the fundamentalvariables being used in numerical simulation?In fact, if we take a closer look at the difference between the past simulations in cylindricalcoordinates and those in Cartesian coordinates, the situation is either to use cylindrical components(of tensors and pseudo-tensors) under cylindrical coordinate, or to use Cartesian componentsunder Cartesian coordinates. i.e. whenever one switches from cylindrical coordinates to Cartesiancoordinates, he also “naturally” switches the fundamental variables from cylindrical componentsto Cartesian components. i.e. he has performs two changes: change of coordinates, and change offundamental variables. It is not clear whether the regular results obtained from simulations underCartesian coordinates, are due to the change of coordinates, or due to the change of fundamentalvariables. All the previous studies, including those yielding regular results, did not make thedistinction between the effects from coordinates and the effects from fundamental varibles. Here wewill make the distinction and make the conjecture: The regularity issue come from the fundamentalvariables used in simulation, rather than the operators associated with coordinate systems.According to the conjecture, the fact that simulations in Cartesian coordinates with Cartesiancomponents are regular, is because the Cartesian components (of tensors and pseudo-tensors) areregular, rather than “Cartesian coordinates are regular”, or “operators in Cartesian coordinatesare regular” 9. According to the conjecture, the key to avoid regularity issue, is to express varioustensors and pseudo-tensors in the equations in terms of regular functions10. In the following, wewill first construct regular functions out of Cartesian components, then construct an example as adirect test to our conjecture.9According to the conjecture, the fact that the cartoon methods produce regular simulations, is because theCartesian components (of tensors and pseudo-tensors) are used as fundamental variables, rather than that theoperators are made effectively Cartesian.10According to the conjecture, eq. (3.1) should be free from regularity problem. The simulation confirms thisclaim.673.1. Regularity Problem and Our Conjecture3.1.3 Cartesian Components MethodIt is fairly easy to express a tensor in terms of regular components and there are multiple waysto achieve this. As a specific example, here we adopt a “canonical” approach: since the Cartesiancomponents are regular, one way to express tensors in terms of regular components, is to rewritetensors in cylindrical coordinates in terms of the (regular) Cartesian components, using the basictensor transformation relation. Since there is a geometrical relation between Cartesian coordinatesand cylindrical coordinates, we will do the coordinate transformation at certain region of the space,such that the radius coordinate ρ (or r) coincides with one of the Cartesian coordinates (referto Fact 3 in Lie derivative cartoon method above), and the functional forms in the cylindricalcoordinates, are the same as that in the Cartesian coordinates. Let us call this procedure asCartesian component method.To clarify the concept and demonstrate the usage, let us take metric function in 3+1 dimensionas an example. We use indices “cylin” and “Cart” to indicate cylindrical and Cartesian, respec-tively. The procedure to express tensors in terms of their Cartesian components in cylindricalcoordinates, comprises the following four steps.(1) write down the most general form of the metric (according to axisymmetry) in cylindricalcoordinates (t, ρ, φ, z)g(cylin) =g(cylin)tt g(cylin)tρ 0 g(cylin)tzg(cylin)tρ g(cylin)ρρ 0 g(cylin)ρz0 0 g(cylin)φφ 0g(cylin)tz g(cylin)ρz 0 g(cylin)zz, (3.3)(2) apply coordinate transformationg(y)µν =∂xα∂yµ∂xβ∂yν g(x)αβ , (3.4)to transform this metric into its Cartesian coordinates (t, x, y, z) at the location (y = 0, x ≥ 0),which is φ = 0 half plane in cylindrical coordinates, where the positive half of the x axis683.1. Regularity Problem and Our Conjecturecoincides with ρ. Now the metric in Cartesian coordinates at (y = 0, x ≥ 0) isg(Cart) ≡ηtt ηtx 0 ηtzηtx ηxx 0 ηxz0 0 ηyy 0ηtz ηxz 0 ηzz(3.5)=g(cylin)tt g(cylin)tρ 0 g(cylin)tzg(cylin)tρ g(cylin)ρρ 0 g(cylin)ρz0 0 g(cylin)φφ /x2 0g(cylin)tz g(cylin)ρz 0 g(cylin)zz, (3.6)where the first matrix is defined byηµν(t, x, z) ≡ g(Cart)µν (t, x, y, z)∣∣∣y=0,x≥0,which is merely the restriction of Cartesian components onto (y = 0, x ≥ 0) half plane. Thesecond matrix is the calculation result of (3.3) via coordinate transformation relation (3.4).(3) compare the components in (3.5) and (3.6), and rewrite the tensor in terms of Cartesiancomponents. Take the tt component as an example: the above relation tells us that ηtt(t, x, z) =g(cylin)tt (t, ρ, z) for all ρ = x ≥ 0, therefore ηtt and g(cylin)tt have the same value at every (t, ρ, z),which means they have the same function form in terms of x (and ρ). Since g(Cart)µν (t, x, y, z) isregular, its restriction to (y = 0, x ≥ 0), ηtt(t, x, z), must be regular, which means g(cylin)tt (t, ρ, z)is regular in cylindrical coordinates. Actually all functions are of the same form as the Cartesiancomponents, except for φφ component, for which we have g(cylin)φφ /x2 = ηyy (and remember,x = ρ), and this relation suggests to rewrite the φφ component as ρ2ηyy.(4) Finally we assemble components together to express the metric in cylindrical coordinates interms of Cartesian components. The metric in cylindrical coordinates can be written asg(cylin) =ηtt ηtρ 0 ηtzηtρ ηρρ 0 ηρz0 0 ρ2ηφφ 0ηtz ηρz 0 ηzz, (3.7)693.1. Regularity Problem and Our Conjecturewhich is the same asηtt ηtx 0 ηtzηtx ηxx 0 ηxz0 0 ρ2ηyy 0ηtz ηxz 0 ηzz, (3.8)with the subindices renamed. In the future we will denote this process roughly asηµν(t, ρ, z) ≡ g(Cart)µν∣∣∣y=0=φ,x=ρ(t, ρ, 0, z).To this point the procedure of “expressing (metric) tensors in terms of their Cartesian compo-nents” is complete. Afterwards, Einstein’s equations (and other tensorial equations) are expressedin cylindrical coordinates to perform numerical calculations, which are regular.Let us study how this method is related to Lie derivative cartoon method. The residuals ofEinstein’s equations are ℜµν ≡ Gµν − kdTµν . The following two equations are obtained by directcalculation, via Lie derivative cartoon method and our Cartesian components method, respectively.ℜLDC =ℜLDCtt ℜLDCtx 0 ℜLDCtzℜLDCtx ℜLDCxx 0 ℜLDCxz0 0 ℜLDCyy 0ℜLDCtz ℜLDCxz 0 ℜLDCzz, (3.9)ℜCC =ℜCCtt ℜCCtρ 0 ℜCCtzℜCCtρ ℜCCρρ 0 ℜCCρz0 0 ℜCCφφ 0ℜCCtz ℜCCρz 0 ℜCCzz=ℜLDCtt ℜLDCtx 0 ℜLDCtzℜLDCtx ℜLDCxx 0 ℜLDCxz0 0 ρ2ℜLDCyy 0ℜLDCtz ℜLDCxz 0 ℜLDCzz, (3.10)where LDC stands for Lie derivative cartoon method, and CC stands for our Cartesian componentsmethod. LDC is expressed under Cartesian coordinates, while CC is expressed under cylindricalcoordinates. The result means, with identifying x = ρ, the two residuals are the same, exceptfor a multiplication of ρ2 in the φφ components in cylindrical coordinates. However, as explainedin fact (6), or Sec. 1.6, a multiplication of a smooth, non-zero function onto a residual equation,does not change the numerical properties of the numerical calculation. Therefore, our Cartesiancomponent method is the same as Lie derivative cartoon method via a different approach underdifferent philosophy.This agreement is not surprising. We can actually “prove” it as the following: both the Lie703.1. Regularity Problem and Our Conjecturederivative cartoon method and our Cartesian components method, are expressing the same tensor(the Einstein tensor) under the same coordinates (with identifying x with ρ), at the same physicallocation (y = 0 = φ, x = ρ ≥ 0), using the same variables (Cartesian components), therefore theresults from the two methods must be the same.In the end, we reiterate that there are multiple ways to express tensors in terms of their regularcomponents (such as the example with a superficially singular metric in Sec. 3.4, or backgroundremoval method in generalized harmonic formalism and BSSN formalism to be introduced below).Here expressing the metric (and other tensors or pseudo-tensors) in terms of their Cartesian com-ponents using the transformation relation of tensors and pseudo-tensors, is just a specific exampleto obtain regular components.Also, the above example can actually be improved: using local flatness [52] at ρ = 0, the aboveηφφ can be replaced by ηρρ + ρW with W |ρ=0 = 0 (or replaced by ηρρ + ρ2W with W,ρ|ρ=0 = 0).The local flatness condition can also be obtained by the procedure in Sec. Results for General Symmetric TensorsIn GR, the governing equations Gµν = kdTµν are symmetric tensors of (0, 2) type (i.e. tensorswith two “downstairs” indices). As long as we know how to deal with symmetric tensors of (0, 2)type, we know how to deal with Einstein’s equations — we know how to deal with GR.Let us consider a general symmetric tensor of (0, 2) type which in general has the followingexpression in Cartesian coordinates (t, x, y, z)T (Cart) =Ttt Ttx Tty TtzTtx Txx Txy TxzTty Txy Tyy TyzTtz Txz Tyz Tzz. (3.11)Now we will express T in cylindrical coordinates. Since the final expression should be independentof φ, one can express T at any value of φ. e.g. let us choose φ = 0. At this “location” we havex = ρ, y = 0. 11The axisymmetry is expressed asL(−y∂x+x∂y)T = 0. (3.12)11Again, we emphasize that the expression in cylindrical coordinates does not depend on the value of φ, and herewe just take advantage of this fact and do the calculation at φ = y = 0.713.1. Regularity Problem and Our ConjectureOpening up this expression, we obtain the following relations at y = 0Tty = Txy = Tyz = 0, (3.13)Ttt,y = Ttx,y = Ttz,y = Txz,y = Txx,y = Tzz,y = Tyy,y = 0, (3.14)Ttx = x · U, Txz = x · V, Tyy − Txx = x ·W, where W |x=0 = 0, (3.15)where (U, V,W ) are regular expressions in terms of the Tµν , which specific forms are irrelevant atthis point.Performing a coordinate transformation from Cartesian coordinates (t, x, y, z) to cylindricalcoordinates (t, ρ, φ, z) and using (3.13) (and take φ = 0), we obtainT (cylin) =τtt τtρ 0 τtzτtρ τρρ 0 τρz0 0 ρ2 · τφφ 0τtz τρz 0 τzz, (3.16)where τµν(t, ρ, z) ≡ Tµν |y=φ=0,x=ρ (t, ρ, 0, z), which are guaranteed regular if the Cartesian com-ponents are regular.The condition (3.15) now readsτtρ = ρ · U, (3.17)τρz = ρ · V, (3.18)τφφ = τρρ + ρ ·W such that W |ρ=0 = 0. (3.19)The first two can be alternatively expressed as τtρ|ρ=0 = τρz |ρ=0 = 0 12, which are parity conditions,and the last condition is called local flatness. To complete the parity conditions at ρ = 0, similarto the derivation of (3.14) at y = 0, now opening up (3.12) at x = 0, and then renaming the indicesand notations, we getτtt,ρ∣∣ρ=0 = τtz,ρ∣∣ρ=0 = τρρ,ρ∣∣ρ=0 = τφφ,ρ∣∣ρ=0 = τzz,ρ∣∣ρ=0 = 0. (3.20)i.e. the local flatness condition and the parity conditions about the metric tensor obtained inSec. 3.1.3, actually apply to any symmetric tensors of (0,2) type.12For functions that can be expressed as Taylor expansion f =∑∞i=0 firi, the condition f |r=0 = 0 implies f0 = 0.Therefore f = ∑∞i=1 firi = r ·∑i=0 fi+1ri ≡ r · V .723.1. Regularity Problem and Our Conjecture3.1.5 Test of the ConjectureHere we study a testing problem: massless scalar field evolution in 5D. By the same procedureas what is used in 4D, the metric in cylindrical coordinates (t, r, θ, φ, z) isg(4+1)cylin =ηtt ηtr 0 0 ηtzηtr ηrr 0 0 ηrz0 0 ηθθr2 0 00 0 0 ηθθr2 sin2 θ 0ηtz ηrz 0 0 ηzz. (3.21)Einstein’s equations in terms of (3.21) are equivalent to that from Lie derivative cartoon method,and the simulation is, without surprise, regular.Now, to directly test our conjecture, we also perform the simulation using the following metricrepresentationg(4+1)singular =ηtt ηtr 0 0 ηtzηtr ξ · r+1r 0 0 ηrz0 0 ηθθr2 0 00 0 0 ηθθr2 sin2 θ 0ηtz ηrz 0 0 ηzz. (3.22)i.e. we purposely introduce a “singular” term ξ · (r+1)/r (or, we have defined ξ = rr+1 ·ηrr), whichwould be troublesome from the conventional point of view. However, if our conjecture is correct,then the simulation in terms of (3.22) would be just as good as the simulation in terms of (3.21),since ξ = rr+1 · ηrr is regular.To test the claim, we performed the simulation in terms of both metric representations. It turnsout that the results of the two simulations are the same (which means the superficially singularterm ξ · r+1r does not cause trouble). Therefore, this simulation supports our conjecture.For the details of the simulation, please refer to Sec. Validity and the Extension of the ConjectureThe conjecture is simple, and its usage and validity have been demonstrated though examples.However, the demonstration is not a proof. In this section we try to give a proof (in a physicist’ssense) and we will address the following three questions:(1) Why terms like 1/rn do not cause singularity problems?733.1. Regularity Problem and Our Conjecture(2) When taking only regular functions, are there still any other kind of regularity issues (otherthan the singularity issue)? And if yes, how to cure it?(3) What role does machine precision play in the regularity problem?To answer these questions, we study at what degree the finite difference approximation (FDA)can represent a derivative operator. It is clear to use an example. For a function f(x) with h asthe spacing of the grids, let us consider the following FDAfi+1 − fi−12h = f′(x) + h2C2f ′′′ + h4C4f ′′′′′ + · · · =∞∑n=0h2nC2nf (2n+1),where Cn are constants which specific values are irrelevant here, and f (n) is the n-th derivativeof f . The expression shows the FDA of a first order derivative, discretized at a finite differencescheme that is of the second order accuracy. Generally, the FDA (A) of a d-th derivative operator,discretized at a finite difference scheme that is of a-th approximation order, isAf (d) = f (d) + haCaf (a+d) + h2aC2af (2a+d) + · · · =∞∑n=0hnaCnaf (na+d), (3.23)where h is the grid spacing.In GR, when r is small, the leading orders of the Cartesian components of symmetric tensorsof (0,2) type (denoted as f) are asymptotically either linear or quadratic in r (refer to Sec. 3.1.4).Therefore, as long as a+ d ≥ 3, we have f (na+d) = 0 (n = 1, 2, ...) in (3.23), thereforeAf (d) = f (d). (3.24)i.e. the FDA approximation with a+ d ≥ 3, is exact when f is linear or quadratic in r. In general,when the FDA is exact in the vicinity of the symmetry axis, we say that the FDA is compatiblewith the asymptotic behaviour of fundamental variables. Therefore, as long as the equations arewell-behaved at the continuous limit, they are also well-behaved at the discrete level. In particular,terms with multiplications of 1/rn are well-behaved since the discretization results of the functionsare exact, and multiplication operation is (almost) exact as well.The above equation explained why our solution to regularity issue works, and it can also predictthe failure of FDA. From equation (3.23) one can see that if f (a+d) 6= 0, then f (d) and haCaf (a+d)are comparable at small r (because (a+d)−a = d), then the FDA is a bad approximation. i.e. theFDA has to be exact, otherwise the error (haCaf (a+d)) is of the same order as the value (f (d)),743.1. Regularity Problem and Our Conjecturewhich invalidates the FDA. For example, when f = r3 and a = 2, then the FDA for f ′ has trouble.Given the asymptotic behaviour of f being f ∼ r3, one can find ways to cure, such as(1) Use higher order FDA.(2) f ′ → diff(f, r3) × 3r2, then only discretize the “diff”. Here the notation “diff” is the partialderivative: diff(A,B) ≡ ∂A/∂B.(3) Rewrite f as f = r ·F and let F be the fundamental variable instead. The criterion to rewritef in terms of F is that the asymptotic behaviour of F is compatible with the FDA in thevicinity of the symmetric axis.These method can be easily generalized to other asymptotic behaviours (for example, using thethird method, a function with asymptotic behaviour as f ∼ r5/2 can be rewritten as f = r1/2 ·F ).The first two methods have been used in the literature and proved to be successful. Overall, ouropinion is that, as long as the asymptotic behaviours are known, there is always a way to easilyfix these issues. 13We end this section by analyzing machine precision and show that it is not related to the irreg-ularity issues. In the above analysis, we only analyzed truncation error (caused by the finitenessof h) with machine precision ε (∼ 10−16 in double precision) being ignored and we concluded thatboth 1/rn terms and non-(1/rn) terms are exact, as long as the FDA scheme is compatible withthe asymptotic behaviours of the fundamental variables in the vicinity of r = 0. Now we restore εand analyze its effects. The concern regarding ε is that the 1/rn terms would amplify the effect ofmachine precision so that the errors in 1/rn terms might be significantly larger than the errors innon-(1/rn) terms. In the following, we will show that the errors in 1/rn terms are not larger thanthe errors in non-(1/rn) terms.For specificity, let us consider the FDA off,rr + f/r2 − 1/r2,13However, in the literature there is yet another type of “cure” to regularity problem. Let us take f = r3 usingsecond order FDA as the example. The basic idea of the cure is to still discretize f ′ as f ′ → (fi+1 − fi−1)/2h,which is actually wrong according to our analysis above since the truncation error is comparable to the value itself.But it is easy to prove that, for any fixed r, the truncation error convergences to zero as the resolution increases.However, for any resolution, there is always a region in the vicinity of r = 0, where the error is still comparable tothe value itself. In particular, the values at the first few grids in the vicinity of r = 0 will always be wrong. Since themistake only happens in a finite region (which size shrinks as the resolution increases), the result in other regions isstill correct, as long as the simulation is stable. Therefore the focus of this kind of “cure” is to make the simulationstable. Our opinion is that, these methods start from a mistake and try to do something to cover the mistake sothat it is controllable within a shrinking region (as the resolution increases), and even if the mistake is controllable,the behaviour at r = 0 is always problematic (for example, the independent residual tests would not be passed atthe point). Instead of searching for any “cure” of this kind, our suggestion is to develop methods based on our twocriteria (see Sec. 3.1.7) on regularity, such as the three ways presented above.753.1. Regularity Problem and Our Conjecturewhere the asymptotic behaviour of f(r) in the vicinity of r = 0 is f ∼ 1 + r2. The concern is thatf,rr ∼ f ∼ 1, but f/r2 would be of order 1/ε when r is as small as ∼√ε, therefore the additionoperation in f,rr + f/r2 is not accurate. Symbolically, we denote this concern of non-accuracy asf,rr + f/r2 ∼ 1 + 1/ε = ε−1 · (1 + ε), (3.25)whose accuracy is the same as 1 + ε since multiplication operation does not lose accuracy. Wewill analyze what it takes to let this non-accuracy in f,rr + f/r2 show up. For the expressionf,rr + f/r2 − 1/r2, the discretized equation at the second order FDA is (fi+1 − 2fi + fi−1)/h2 +(f − 1)/r2. Representing f,rr as (fi+1 − 2fi + fi−1) /h2, relies on the validity of Taylor expansionssuch as fi+1 = fi + hf,r + (h2/2)f,rr + (h3/6)f,rrr + .... It is then crucial to examine whetherthe higher orders in these expansions can be expressed accurately by floating point numbers.To let (fi+1 − 2fi + fi−1) /h2 be of second order accuracy, it is required that (h3/6)f,rrr in theexpansion must be representable by floating point number. Even if we give up the desire for secondorder accuracy, it is required at least (h2/2)f,rr is representable in the expansion by floating pointnumber. i.e. it is required the value of (h2/2)f,rr is not lost in fi + hf,r + (h2/2)f,rr, which canbe symbolically expressed as ∼ 1 +√ε+ ε. Therefore in (fi+1 − 2fi + fi−1) /h2, the operation issymbolically (1 + √ε + ε − 2 + 1 − √ε + ε)/ε. The accuracy level is determined by the largestand the smallest values involved in the addition (subtraction) operations, which are 1/ε and ε/ε.i.e. the accuracy level of this FDA isε−1 ·(1 + ε). (3.26)i.e. in order to let the FDA (fi+1 − 2fi + fi−1) /h2 represent f,rr at the lowest order accuracy, theε is required to be representable in the operation 1 + ε. Comparing (3.26) with (3.25) one seesthat, the accuracy level (due to machine precision) of non-(1/rn) terms is the same as that of the1/rn terms. If the operations involved 1/rn terms (such as f,rr + f/r2) lose their accuracy due tomachine precision, then the FDAs of non-(1/rn) terms can not represent the derivative operatorsany more (such as (fi+1 − 2fi + fi−1) /h2 fails to represent f,rr at any grid point, rather thanmerely a few grids in the vicinity of r = 0). i.e. what it takes to fail f,rr + f/r2, is the failure ofFDA method.Or equivalently, provided the FDA is still valid (so that finite difference method can be used),then f,rr + f/r2 does not lose its accuracy (therefore machine precision can still be ignored).i.e. for the aspect of machine precision, the 1/rn terms are not worse than the non-(1/rn) terms.Therefore machine precision plays no role in regularity. The discussion above reveals that the763.2. The Generalized Harmonic Formalism in Non-Cartesian Coordinatestruncation error (due to the finiteness of h) is fundamentally different from the machine error.The former is a generic feature associated with the FDA method which has nothing to do withmachines, while the latter is due to the limited ability of machines. One can improve the latter byusing more accurate numbers such as quadruple-precision floating point numbers with the machineprecision ε ∼ 10−34, while the truncation error can not be improved in this way.3.1.7 Summary of RegularityThis subsection is the summary of the regularity discussion, which should be treated as arephrase of our conjecture: the irregularity has nothing to do with coordinate system (and operatorsin the coordinate system) or machine precision. The simulation in cylindrical coordinates (or anyother coordinates) is regular, provided the following two conditions are met:(1) The fundamental variables being used in numerical calculations, are regular functions;(2) The FDA scheme is compatible with the asymptotic behaviours of the fundamental variablesat the symmetry axis (or the origin or any other special locations in the coordinate system).3.2 The Generalized Harmonic Formalism inNon-Cartesian CoordinatesIn the previous sections, we described how to apply the Cartesian components method toexpress the metric functions in terms of regular functions. In this section, we apply the samemethod to the generalized harmonic formalism (and to BSSN in appendix A).3.2.1 An Introduction to the Generalized Harmonic FormalismNow we briefly introduce the generalized harmonic formalism [61] of GR. The source functionsHµ are defined asHα ≡ ∇β∇βxα = −Γαµνgµν , (3.27)and Hµ ≡ gµνHν . In terms of Hµ, Einstein’s equationsRµν = kd(Tµν −1d− 2gµνT),773.2. The Generalized Harmonic Formalism in Non-Cartesian Coordinatesreduce to [61]−12gαβgµν,αβ − gαβ(,µgν)β,α −H(µ,ν) +HβΓβµν − ΓανβΓβµα = kd(Tµν −1d− 2gµνT). (3.28)The generalized harmonic formalism [61] is to lift Hµ as fundamental variables, then the definingequations (3.27) are not defining equations any more. Instead, they are now constraintsCα ≡ Hα − Γαµνgµν ≃ 0, (3.29)where ≃ means the equations are constraint relations. i.e. in the generalized harmonic formalism,the fundamental variables are the metric functions gµν and the source functions Hµ, where Hµ aresubject to the constraints (3.29). Since Hµ are now fundamental variables, the principal parts of(3.28) are now − 12gαβgµν,αβ , which are manifestly strongly hyperbolic.Using the Bianchi identities, it can be proved [61] that the constraints (3.29) will always besatisfied during an evolution, as long as they are satisfied at any instant of the evolution and theHamiltonian constraint together with the momentum constraints are also satisfied at the instant.In numerical simulations, however, there are always modes violating the constraints, and theviolation modes might grow with time (even exponentially), which will produce unphysical results.Gundlach et al. then suggested [58] to add constraint damping terms to the left hand side ofeq. (3.28), which can beZµν ≡ κ(n(µCν) −1 + p2 gµνnβCβ), (3.30)where κ > 0, −1 ≤ p ≤ 0. If the constraints are satisfied, the damping terms vanish. If smallviolation modes develop, the damping terms can damp out the modes [58] and drive the numericalresults back to physical configurations.Since Hµ are now fundamental variables, their equations of motion need to be imposed. Theconstraints Hα ≃ −Γαµνgµν = ∇β∇βxα show that the Hµ are related to coordinate gauge choices.There are a lot of freedom to impose these equations of motion. For example, Hµ can even befunctions of coordinates and metric functions such as Hµ = gµνxν . But generally it is required thatHµ do not include the derivatives of metric functions (which means the equations of motion suchas Hµ = gµν,αgνα are generally not recommended), so that the principal parts of (3.28) are notaffected. We do not intend to extensively discuss the gauge choices in this section. Please refer toSec. 5.3 for the details of some popular gauges being used in the literature.783.2. The Generalized Harmonic Formalism in Non-Cartesian CoordinatesTo simplify future discussions, we defineΓα ≡ Γαµνgµν .3.2.2 The Generalized Harmonic Formalism in Cylindrical CoordinatesNumerical evolutions using GH in cylindrical coordinates in terms of Hµ, are unsuccessful dueto the fact that the Hµ’s are singular. Instead, our approach is again to use Cartesian componentsmethod: employ the coordinate transformation relation for Christoffel symbols(y)Γκαβ =∂xµ∂yα∂xν∂yβ(x)Γγµν∂yκ∂yγ +∂yκ∂xµ∂2xµ∂yα∂yβ , (3.31)to express Hµ’s in terms of their Cartesian components. By the same procedure, we obtainHtHρHφHz=ht + (1/ρ) (ηtρ/ηφφ)hρ + (1/ρ) (ηρρ/ηφφ)0hz + (1/ρ) (ηρz/ηφφ), (3.32)where hα(t, ρ, z) ≡ (Cart)Hα(t, x = ρ, y = 0, z) with sub-index renamed, therefore are regular.Specifying the gauge amounts to choosing appropriate hα (rather than Hα). e.g. the harmonicgauge now reads hµ = 0, which is indeed the harmonic gauge in Cartesian coordinates. From(3.32), one can see why the simulations with Hµ as fundamental variable fail—because Hr issingular, which violates our first criterion for regularity.By the same procedure, the metric in spherical coordinates (t, r, θ, φ), with spherical symmetry,isg(3+1)sphere =ηtt ηtr 0 0ηtr ηrr 0 00 0 ηθθr2 00 0 0 ηθθr2 sin2 θ. (3.33)793.2. The Generalized Harmonic Formalism in Non-Cartesian CoordinatesAgain, ηθθ is replaced by ηrr + rW , considering local flatness. The source functions areHtHrHθHφ=ht + (2/r) (ηtr/ηθθ)hr + (2/r) (ηrr/ηθθ)cot θ0. (3.34)For 5D, the metric in cylindrical coordinates (t, r, θ, φ, z) isg(4+1)cylin =ηtt ηtr 0 0 ηtzηtr ηrr 0 0 ηrz0 0 ηθθr2 0 00 0 0 ηθθr2 sin2 θ 0ηtz ηrz 0 0 ηzz, (3.35)where ηθθ is replaced by ηrr + rW , considering local flatness. And the source functions areHtHrHθHφHz=ht + (2/r) (ηtr/ηθθ)hr + (2/r) (ηrr/ηθθ)cot θ0hz + (2/r) (ηrz/ηθθ). (3.36)3.2.3 Background Removal in the LiteratureIn the literature, there exist a class of methods for solving the regularity issue of the Hµ’sthrough the use of the source functions with a background term subtracted, which can all becalled background removal methods.In [54, 55], the authors used (for example in the axisymmetric case in 5D)Hr = Hˆr +2r ,Ht = Hˆt, Hz = Hˆz,where 2/r is the value of Hr in flat spacetime background, and the variables with a hat (ˆ ) arethe variables the authors used as fundamental variables. From the discussion above, we obtainHˆr = hr +2r(ηrrηθθ− 1)= hr +2r(ηθθ − rWηθθ− 1)= hr −2Wηθθ,803.2. The Generalized Harmonic Formalism in Non-Cartesian Coordinateswhich is related to hr by a regular term, therefore is regular. Similarly, theirHˆt = ht +2rηtrηθθand Hˆz = hz +2rηrzηθθ,are regular too (because ηtr and ηrz are asymptotically linear in r as discussed in section 3.1.4).On the other hand, there is another way to remove the background [36, 97]. Instead of usingHµ’s, the authors developed the formalisms to useHˆµ ≃ −gµν(Γναβ − Γ¯ναβ)gαβas fundamental variables, where the “bar” means that the associated quantities are for the back-ground. Hˆµ is actually a vector (while Hµ is not). Although no simulations using this method havebeen reported14, this method should be able to generate regularized simulations if the backgroundis chosen correctly. The reason is as follows: if the background is chosen to be flat, in Cartesian co-ordinates (where Γ¯ναβ = 0) we have Hˆµ = −gµνΓναβgαβ = hµ, where hµ is the Cartesian componentof the source function, which is what is used in our Cartesian components method. Since Hˆµ is avector, its transformation from Cartesian coordinates to cylindrical coordinates at (y = 0, x ≥ 0) issimply Hˆ(cylin)t = Hˆ(Cart)t = ht, Hˆ(cylin)r = Hˆ(Cart)x = hr, and Hˆ(cylin)z = Hˆ(Cart)z = hz, while othercomponents are zeros. Therefore Hˆµ = hµ holds in cylindrical coordinates as well. i.e. when thebackground is flat, the fundamental variables used by the background removal method in [36, 97],are the same as the ones used in Cartesian components method. Therefore, by our conjecture, thismethod has no regularity issues, as long as the Cartesian components are regular and the FDAscheme is compatible with the asymptotic behaviour of the fundamental variables.This formalism can solve the regularity issue (by choosing the background properly), and thesource function is now a vector. Furthermore, it can remove any background (especially non-flatbackgrounds) 15. We will develop it further in Sec. 3.3.14We will use a generalization of this method in our simulation of the braneworld.15However, when the background is not flat, the regularity of the source functions is not guaranteed, in whichcase the background needs to be analyzed in a case-by-case basis.813.3. The Generalized Harmonic Formalism in Non-Flat Backgrounds3.3 The Generalized Harmonic Formalism in Non-FlatBackgroundsThe spacetime background for RSII isds2 = ℓ2z2(− dt2 + dr2 + r2(dθ2 + sin2 θdφ2)+ dz2). (3.37)For this background, the h’s from eq. (3.36) are(ht, hr, hz) =(0, 0,−3z). (3.38)Normally setting up the gauge choices is to let hµ take specific values or satisfy specific conditions.For example, in the case for the harmonic gauge, the hµ are zeros. However, the spacetimebackground is now non-trivial, in the sense that hz is not zero for the background, and it is notclear how to let gauges, such as harmonic gauge, make sense. In other words, it is not clear howto setup hµ.Generally, in the current literature, the gauge choices for GH are all in Cartesian coordinateswith a flat spacetime as the background. In the last section, we have developed the GH in non-Cartesian coordinates, where we have used the assumption that Cartesian components are perfect.For the braneworld, however, Cartesian components might not be sufficient in the sense that itis not clear how to easily employ the existing gauge choices used in the literature. i.e. actuallyCartesian components are not perfect and we need to find a way to “get rid of” the background.3.3.1 Tensorial Source FunctionsThe first way is to use the “background removal” methods mentioned above. For example,following [97], we employ the following functions to serve as the fundamental variableshµ ≃− gµν(Γναβ − Γ¯ναβ)gαβ, (3.39)where the bar ( ¯ ) stands for quantities and operations associated with the background. Asdiscussed in section 3.2.3, when the background is taken to be flat, hµ in (3.39) is the same as thehµ used in “Cartesian components” methods. Yet, (3.39) can be applied to non-flat background.Another good feature is that hµ is now a tensor (vector), which geometrically makes more sense. Forthe braneworld, if (3.37) is taken as the background in (3.39), we have hα = 0 for the background.823.3. The Generalized Harmonic Formalism in Non-Flat BackgroundsHowever, still this can not be easily linked with the existing work in the literature where thebackground is flat. i.e. it is not clear what gauge should be given to the hα.3.3.2 Conformal TransformationNoticing the background metric (3.37) is conformally flat, we can do a conformal transformationg˜µν ≡ (z2/ℓ2)gµν to obtain g˜µν whose background is flat. Actually we can use a general conformalfactor Ψ in the conformal transformation as g˜µν ≡ Ψ−2gµν . The only requirements on this Ψ are:(a) Ψ = ℓ/z when the metric is the background metric; and (b) Ψ goes to ℓ/z asymptotically atspatial infinities. Or even more generally, we define conformal transformationg˜µν ≡ Ψ−qgµν , (3.40)where q is a numerical factor that can be set as any value for convenience. A tilde (˜ ) is used forthe quantities and operations associated with g˜µν . For example, the Christoffel symbols areΓ˜αµν =12 g˜αβ (g˜βν,µ + g˜µβ,ν − g˜µν,β) . (3.41)Repeating the derivations in [32] or [36] while keeping q and d general, we obtainRµν = R˜µν +R(Ψ)µν , (3.42)whereR(Ψ)µν ≡−q(d− 2)2 ∇˜µ∇˜ν lnΨ−q2 g˜µν ∇˜α∇˜α lnΨ+ q2(d− 2)4(∇˜µ lnΨ∇˜ν lnΨ− g˜µν ∇˜α lnΨ∇˜α lnΨ). (3.43)Substituting this into Einstein’s equationsRµν −12gµνR = kdTµν (3.44)⇔ Rµν = kd(Tµν −1d− 2gµνT). (3.45)we haveR˜µν = kd(Tµν −1d− 2 g˜µν T˜)−R(Ψ)µν , (3.46)833.3. The Generalized Harmonic Formalism in Non-Flat Backgroundswhere T˜ ≡ g˜µνTµν . The equations are Einstein’s equations with a modified right hand side.Depending on how the left hand side is rewritten, one can proceed with either one of thefollowing two approaches. The first approach is to define the source function asH˜β ≡ −Γ˜βαδ g˜αδ, (3.47)which results in the following GH formalism with a conformal function−12 g˜αβ g˜µν,αβ − g˜αβ(,µg˜ν)β,α − H˜(µ,ν) + H˜βΓ˜βµν − Γ˜ανβΓ˜βµα = kd(Tµν −g˜µν T˜d− 2)−R(Ψ)µν . (3.48)For braneworld simulation, this formalism is sufficient since the conformal transformation let thebackground of g˜µν be flat which enables us to borrow the existing results regarding gauge spec-ification in the literature. Eq. (3.48) can be solved using (3.35) as the metric and (3.36) as thesource function (every quantity needs to have a tilde though).Alternatively, R˜µν can be rewritten using tensorial source functions. This approach does betterin the following aspects: it is not limited to the case where the background of g˜µν is flat, and thesource function forms a tensor. The background spacetime of g˜µν is denoted as g¯µν . We furtherdefineC˜αµν ≡ Γ˜αµν − Γ¯αµν , (3.49)∆Γ˜β ≡ C˜βαδg˜αδ, (3.50)h˜β ≡ −∆Γ˜β = −(Γ˜βαδ − Γ¯βαδ)g˜αδ. (3.51)Repeating the derivation in [36] (for the BSSN derivation though), while keeping d general andwithout requiring g˜ = g¯, we obtain the final equation as− 12 g˜αβ∇¯α∇¯β g˜µν − ∇¯αg˜β(µ∇¯ν)g˜αβ − ∇¯(µh˜ν) + h˜αC˜αµν − C˜αµβC˜βαν= kd(Tµν −g˜µν T˜d− 2)− R¯µν −R(Ψ)µν . (3.52)3.3.3 The Implementation of Generalized Harmonic FormalismThe discussions in this subsection regarding the implementation apply to all the GH formalisms(3.28), (3.48) and (3.52). Here we use (3.52) as a specific example.The (t, µ) components of equation (3.52) are the Hamiltonian constraint and momentum con-843.4. Evolution of Massless Scalar Field under Cylindrical Coordinatesstraints. Eq. (3.51) are extra constraints due to the introduction of h˜. In this sense, performingthe evolution using all the components in (3.52), is a full-evolution scheme with Hamiltonian andmomentum constraints being satisfied. Constraints (3.51) are driven to be satisfied by adding theconstraint violation damping terms [58] into (3.52). Therefore we give it the name full-evolutionwith source driving gauge. This is what has been adopted in the literature. However, it is not veryclear how the coordinates condition is imposed via h˜µ (or even Hµ).Alternatively, we can adopt the same strategy as that in BSSN (therefore we give it the nameBSSN-like method): the lapse and shift functions are specified in the “ordinary way” (such as max-imal slicing, 1 + log slicing, Gamma freezing, etc), while the Hamiltonian/momentum constraintsare left un-evolved. Also, an evolution equation for h˜µ will be derived, rather than using h˜µ to drivelapse and shift. Using Γ¯ββα = 12∂α ln g¯ (and Γ˜ββα = 12∂α ln g˜), we obtain ∇¯αg˜αβ = h˜β − g˜αβ∇¯αF ,where F ≡ 12 ln (g˜/g¯), which is a scalar. Taking derivative with respect to t, we have∇¯th˜µ = −∇¯tg˜αβ∇¯αg˜βµ − g˜αβ∇¯t∇¯αg˜βµ + ∇¯t∇¯µF, (3.53)within which there are terms ∇¯t∇¯tg˜αβ, which will be replaced by quantities with first-order andzeroth-order in ∇¯t, via eq. (3.52). As for the Hamiltonian/momentum constraints, the dampingterms for their violation modes could be constructed. Beyond these, h˜µ + ∆Γ˜µ ≃ 0 are still con-straints (which are similar to the constraints Γ˜i ≃ γ˜jk((d−1)˜Γi jk − (d−1)¯Γi jk)in BSSN formalism).Note, we have not implemented a code using BSSN-like methods yet. Performing simulationsusing this method is part of our future plans.The BSSN method is widely used in numerical relativity, and we have simply borrowed itsspirit, in deriving the generalized harmonic formalism in non-Cartesian coordinates in non-flatbackground. For completeness, we also generalized the BSSN formalism to non-flat backgroundwith non-Cartesian coordinates. However, since we will not use the BSSN in simulating thebraneworld, the development of the BSSN method is put in appendix A.3.4 Evolution of Massless Scalar Field under CylindricalCoordinatesIn this section, we study a model problem: the massless scalar field collapse in 5D underaxisymmetry. The reason for choosing the collapse in 5D under axisymmetry is that this system isclose to our main project — the massless scalar field collapse in the braneworld, and is therefore an853.4. Evolution of Massless Scalar Field under Cylindrical Coordinatesinstructive model problem. We will use this section to demonstrate the use of the various methodsdeveloped in this chapter. Namely: we will test the Cartesian components methods of generalizedharmonic formalism. Then we will test our conjecture by introducing a seemingly singular terminto the metric, while the fundamental variables are known to be regular.3.4.1 Initial DataAt the initial instant, we assume the metric takes the following formds2 = e2A(− dt2 + dr2 + r2(dθ2 + dφ2 sin2 θ)+ dz2), (3.54)under cylindrical coordinates (t, r, θ, φ, z), with r = 0 being the symmetry axis. The range of thecoordinates are of course r ≥ 0, 0 ≤ θ ≤ π, 0 ≤ φ < 2π,−∞ < z < ∞. In this simulation we willkeep the spacetime symmetric about z = 0 plane, therefore the range of z (in the code) is [0,∞).The initial data is obtained by solving the momentum constraints and the Hamiltonian con-straint. We take time symmetric initial data, therefore the momentum constraints are satisfiedautomatically, which leaves only the Hamiltonian constraint to be solved. The Hamiltonian con-straint readsA,rr +A,zz +2A,rr + (A,r)2 + (A,z)2 +16((Φ,r)2 +(Φ,z2))= 0, (3.55)where Φ is the massless scalar field. We choose its initial configuration to beΦ(0, r, z) = a · exp(−(√z2 + r2 − r0)2σ2), (3.56)which is symmetric about z = 0. Here we purposely choose the spherically symmetric initialconfiguration, to test the quality of our axisymmetric code by examining whether a sphericallysymmetric initial data will remain spherically symmetric. For the testing, we take a = 0.1, σ =0.25, r0 = 1.The time derivatives of spatial components of the metric are taken as zeros since the initial dataconfiguration is time symmetric. The other metric components are taken trivially gtr = gtz = 0,but the time derivatives of gtt, gtr, gtz are chosen such that ht = hr = hz = 0 (ref. eq. (3.58)) atthe initial instant [62, 63].863.4. Evolution of Massless Scalar Field under Cylindrical Coordinates3.4.2 EvolutionThe evolution is performed using our Cartesian components method applied to the generalizedharmonic formalism (eq. (3.28)), therefore the metric representation is (3.35)g =ηtt ηtr 0 0 ηtzηtr ηrr 0 0 ηrz0 0 ηθθr2 0 00 0 0 ηθθr2 sin2 θ 0ηtz ηrz 0 0 ηzz, (3.57)where ηθθ is replaced by ηrr + rW , considering local flatness condition at r = 0, as discussed inSec. 3.1.3. The gauge sources are (3.36)HtHrHθHφHz=ht + (2/r) (ηtr/ηθθ)hr + (2/r) (ηrr/ηθθ)cot θ0hz + (2/r) (ηrz/ηθθ). (3.58)During the evolution, the gauge is chosen to be ht = hr = hz = 0, which is equivalent to harmonicgauge in Cartesian coordinates.A constraint damping term [58] is added to the left hand side of eq. (3.28)Zµν ≡ κ(n(µCν) −1 + p2 gµνnβCβ), (3.59)where κ > 0, −1 ≤ p ≤ 0. κ = 6 and p = 0 were chosen during the simulation.The simulation is performed in compactified coordinates Rˆ ≡ r/(r + 1) and Zˆ ≡ z/(z + 1). Inthis way the spatial infinities, r = ∞ and z = ∞ are mapped to Rˆ = 1 and Zˆ = 1. Since we focuson the behaviour in the region that is close to r = z = 0, where the resolution in terms of (Rˆ, Zˆ)and (r, z) are comparable, the compactification does not cause loss in resolution. The boundaryconditions at r = ∞ or z = ∞ are trivial Dirichlet condition such that the metric is asymptoticallyflat. At z = 0, the spacetime remains symmetric about z = 0. The boundary conditions atr = 0 are parity conditions and local flatness condition which are consequences of axisymmetry:∂rηtt|r=0 = ∂rηzz|r=0 = ∂rηtz|r=0 = ∂rηrr|r=0 = ηtr|r=0 = ηrz|r=0 = W |r=0 = ∂rΦ|r=0 = 0.873.4. Evolution of Massless Scalar Field under Cylindrical CoordinatesThe simulations are performed by “standard” FDA of second order accuracy∂tf →fn+1i,j − fn−1i,j2 ·∆t , (3.60)∂Rˆf →fni+1,j − fni−1,j2 ·∆Rˆ, (3.61)∂Zˆf →fni,j+1 − fni,j−12 ·∆Zˆ, (3.62)∂ttf →fn+1i,j − 2fni,j + fn−1i,j(∆t)2 , (3.63)∂RˆRˆf →fni+1,j − 2fni,j + fni−1,j(∆Rˆ)2 , (3.64)∂ZˆZˆf →fni,j+1 − 2fni,j + fni,j−1(∆Zˆ)2 , (3.65)∂tRˆf →14 ·∆t ·∆Rˆ(fn+1i+1,j − fn+1i−1,j + fn−1i−1,j − fn−1i+1,j), (3.66)∂tZˆf →14 ·∆t ·∆Zˆ(fn+1i,j+1 − fn+1i,j−1 + fn−1i,j−1 − fn−1i,j+1), (3.67)∂RˆZˆf →14 ·∆Rˆ ·∆Zˆ(fni+1,j+1 − fni−1,j+1 + fni−1,j−1 − fni+1,j−1). (3.68)where the index i (or j) is to characterize the grid position in Rˆ (or Zˆ) direction, and ∆Rˆ (or ∆Zˆ)is the (uniform) spacing of the grids in Rˆ (or Zˆ) direction. In the simulation, we set ∆Rˆ = ∆Zˆ.The superscript n, n+1, n−1 is to characterize the discretized time levels, while ∆t is the spacing.Since compactified coordinates are used, the Courant factor is defined as ∆t/min(∆r) = ∆t/∆Rˆ,where min is taken over all ∆r. The Courant factor is set to be 0.5 in the simulation.The residual equations are the discretized equation (3.28) (the GH formalism of GR). Theupdate scheme is to obtain quantities at time level n+1 from given quantities at level n and n−1,by solving the residual equations. We solve the residual equations by pointwise Newton-Gauss-Seidel iteration, until residuals are smaller than a “small” threshold.A Kreiss-Oliger [73] style numerical dissipation is adopted to control high frequency numericalnoises. Since we are using second order FDA, a fourth order KO dissipation is sufficient. Beforeobtaining the advanced time level n+ 1, the dissipation is applied to both n and n− 1 time levels[61].883.4. Evolution of Massless Scalar Field under Cylindrical CoordinatesTests and Validation of the Numerical SchemeThe first test we perform is to examine whether spherical initial data evolved under axisym-metric code, can remain spherically symmetric. Fig. 3.2 shows instants during the process whenthe majority of Φ field is being bounced back at the centre. The figure shows that the sphericalsymmetry is indeed preserved during the evolution.(a) (b) (c) (d)Figure 3.2: The preservation of spherical symmetry monitored by Φ. The horizontal axis is r,and the vertical axis is z. The simulation was carried out by 16 cores, and these graphs showthe results at the r = z = 0 corner. Since compactified coordinates are used, the sphericalconfiguration will appear to be non-spherical. However, when r and z are small, the distortion dueto the compactification is small. This is why we choose to show the r = z = 0 corner. The fourgraphs show four instants when the pulse travels towards the center, and then gets bounced backand travels outwards. The graphs show that the spherical symmetry is indeed preserved duringthe evolution.By the general theory of numerical solutions, the numerical result is a numerical solution onlyif the independent residual tests and the convergence tests of the constraints are passed. The inde-pendent residuals are obtained by evaluating the residuals resulting from a different discretizationwhich is to discretize the Rˆ and Zˆ derivatives by forward finite difference approximation of secondorder accuracy, while the time derivative is discretized by backwards finite difference approxima-tion with second order accuracy. Fig. 3.3 shows the convergence of independent residual of thett component of (3.28), and the convergence of the t component of the constraint equation. Theresults of other independent residual tests and convergence tests are similar, therefore omitted.These tests show that the simulation produces valid physical results, which confirms that ourCartesian components method does not have the regularity issue.Another way to justify that the solution is indeed a physical solution, is to directly evaluatethe residuals of original Einstein’s equations Rµν = kd(Tµν − gµν Td−2), without referring to thesource functions at all. i.e. we use the generalized harmonic formalism to obtain the solutions, thensubstitute them into original Einstein’s equations to get the residuals. Fig. 3.4 shows the tt and893.4. Evolution of Massless Scalar Field under Cylindrical Coordinates0 0.5 1 1.5 2 2.500. Res. of tt EquationtimeL 2-norm  coarse resolution4 × fine resolution(a)0 0.5 1 1.5 2 2.5012345678 x 10−4 Convergence of Ht + ΓttimeL 2-norm  coarse resolution4 × fine resolution(b)(c) (d)Figure 3.3: The independent residual tests and the constraint tests. Results from two resolutionsare needed to perform the tests. Here the finer resolution’s grid spacing is one half of that inthe coarser resolution. If the numerical results are actually numerical solutions, the independentresiduals obtained at finer level are one-fourth of the independent residuals obtained at the coarserresolution, since these results are obtained from the FDA with second order accuracy, and theindependent residuals are also discretized at second order accuracy. Fig. (a) is the independentresidual test of the (tt) component of Einstein’s equations (3.28). Fig. (b) is the test of the t-component of the constraint equations Hµ+Γµ ≃ 0. Fig. (c) is the snapshot of the (tt) componentof the independent residual at the (r = 0, z = 0) corner, at some instant. The function with zerovalue is shown as a reference, which appears as a plane with uniform colour (light blue). The onetaking larger value is obtained from the coarser grid, which value should be 4 times of the valueobtained from the finer grid at all grid points at all instants. We purposely show the (r = 0, z = 0)corner which is the location suffering the most severe irregularity (if there is). This figure showsthat there is no irregularity in our simulation, and the independent residual indeed convergencesas expected at every grid point. Fig. (d) is the snapshot of the residual of constraint Ht + Γt atthe (r = 0, z = 0) corner, at some instant.903.4. Evolution of Massless Scalar Field under Cylindrical Coordinatesrz components respectively, as two examples. Indeed they converge as second order quantities.0 0.5 1 1.5 2 2.500.0050.010.0150.02Res. of tt Comp. of Einstein Eq.timeL 2-norm  coarse resolution4 × fine resolution(a)0 0.5 1 1.5 2 2.500.511.522.533.54 x 10−3 Res. of rz Einstein Eq.timeL 2-norm  coarse resolution4 × fine resolution(b)Figure 3.4: The convergence tests using the residuals obtained from the original Einstein’s equa-tions. Fig. (a) is the convergence of the residual obtained from evaluating the (tt) componentof the original Einstein’s equations Rµν = kd(Tµν − gµν Td−2). Fig. (b) is the test from the (rz)component.3.4.3 Superficially Singular Metric RepresentationTo test our conjecture, instead of using (3.57), here we perform the evolution in terms of thefollowing representation of metric which includes a superficially singular termgsingular =ηtt ηtr 0 0 ηtzηtr ξ · r+1r 0 0 ηrz0 0 ηθθr2 0 00 0 0 ηθθr2 sin2 θ 0ηtz ηrz 0 0 ηzz, (3.69)where ηθθ is replaced by ξ · r+1r + rW . The boundary conditions of ξ are ξ|r=0 = 0, ξ|r=∞ =1, ∂zξ|z=0 = 0, ξ|z=∞ = r/(r + 1).913.4. Evolution of Massless Scalar Field under Cylindrical CoordinatesThe source functions are of courseHtHrHθHφHz=ht + (2/r) (ηtr/ηθθ)hr +(2/r2)· ξ · (r + 1)/ηθθcot θ0hz + (2/r) (ηrz/ηθθ). (3.70)The metric representation gsingular should be problematic due to ξ/r terms, according to theconventional point of view in [50]. However, according to our conjecture, as long as ξ is regular,and its asymptotic behaviour is compatible with the relevant FDA (in the sense that the FDA isable to accurately represent the derivatives of ξ at small r), then the result would be regular (i.e.not problematic). Since ηrr ∼ const + r2 at small r, one should expect ξ ∼ r · ηrr ∼ r at small r,and the second order finite difference approximation to its derivatives should be exact. Thereforethe simulation in terms of ξ should be regular, if our conjecture holds.The independent residual test and constraint convergence test of the simulation in terms of(3.69) and (3.70) are shown in Fig. 3.5. It shows the simulation produces valid physical results aswell. The superficially singular term ξ/r does not cause problem.0 0.5 1 1.5 2 2.500. Res. of tt EquationtimeL 2-norm  coarse resolution4 × fine resolution(a)0 0.5 1 1.5 2 2.500. x 10−3 Convergence of Ht + ΓttimeL 2-norm  coarse resolution4 × fine resolution(b)Figure 3.5: The independent residual tests and constraint tests for the simulation using the su-pervifially singular metric (3.69). Fig. (a) is the independent residual test of the (tt) componentof Einstein’s equations (3.28). Fig. (b) is the test of the t-component of the constraint equationsHµ + Γµ ≃ 0. These two tests show that the simulation using (3.69) and (3.70) indeed satisfiesEinstein’s equations.Now we examine whether these two simulations produce the same result. We perform this test923.5. Remark about Regularityby comparing the same quantities resulting from two simulations. Here we show the L2-norms of(α − 1) (where α is the lapse function) and Φ in Fig. 3.6, which shows that the two simulationsindeed produce the same results.Therefore, our conjecture passed this challenging test.0 0.5 1 1.5 2 2.500. from SimulationstimeL 2-normof(α−1)  normalsingular(a)0 0.5 1 1.5 2 2.500.Φ from SimulationstimeL 2-normofΦ  normalsingular(b)Figure 3.6: The comparison of the results from two the simulations. In the legend, “normal” meansthe result obtained from the simulation using metric representation (3.57) and “singular” meansthat obtained from (3.69) where there is a superficially singular term in the metric representation.(a) shows the L2-norms of (α − 1) from the two simulations, and (b) shows the L2-norms of Φfrom the two simulations. These two graphs show that the two simulations indeed produces thesame results.3.5 Remark about RegularityWe have a conjecture regarding the regularity issue: provided the fundamental variables usedin a numerical simulation are regular functions, and the FDA scheme is compatible with theasymptotic behaviours of the fundamental variables at the symmetric axis (or the origin), there isno irregularity issue.According to the conjecture, the key to overcome irregularity issues is to express tensors (andpseudo tensors) in terms of regular functions that are compatible with the FDA. One way todo it is to carry the regular Cartesian components into cylindrical coordinates (by coordinatetransformation relations) and let them serve as the fundamental variables.There exist formalisms in the literature that are able to generate simulations without regular-ity issue, according to our conjecture. Among these specific methods, our Cartesian componentsmethod and Lie derivative cartoon method are equivalent (although the philosophies underlying933.5. Remark about Regularitythem are different). Both methods use Cartesian components, and are general ways to rewriteequations in non-Cartesian coordinates using Cartesian components, which apply to any formal-ism of GR. The background removal methods with tensorial source functions [36, 97], however,only apply to the formalisms where Christoffel symbols (or other connection coefficients) playfundamental roles, such as generalized harmonic formalism and BSSN formalism. In these for-malisms, when the background is chosen to be flat, this background removal method generates thesame results as those of our Cartesian components method (and Lie derivative cartoon method).One advantage of the background removal method is that it can be applied in non-flat background,where our Cartesian components method (and Lie derivative cartoon method) need to be modified.The gauge choice in terms of the tensorial source functions needs further study. The backgroundremoval method used in [54, 55], however, is different from any of the above. It is also regular,although it is not clear how to setup coordinate gauges in this formalism.Above all, I emphasize the following: for the aspect of regularity, our conjecture is morefundamental than any specific formalisms in this chapter. For example, Cartesian componentsare not guaranteed to be singularity free—an example is the behaviour in the vicinity of thephysical singularities. In this case the Lie derivative cartoon or Cartesian component method willnot help. In background removal methods, the modified source function serving as fundamentalvariable, is the subtraction of two terms. In principle it is possible that both terms are singularbut the function resulting from the subtraction is regular, and in this case the formalism canstill generate regular results. All in all, before using any formalisms, one need to check the twoaspects of the conjecture: regular functions being used as fundamental variables, and FDA beingcompatible with the asymptotic behaviours of the fundamental variables in the vicinity of theaxis/origin/other special points.94Chapter 4Initial DataThe evolution of the braneworld is specified by the evolution equation(s) with the initial dataand the boundary conditions. The governing equations of the braneworld are Einstein’s equationsin the 5D bulk: Rµν − 12gµνR = −Λgµν . We will use the formalism of Einstein’s equations withthe conformal factor (3.46). For the conformal factor Ψ, we simply choose Ψ = ℓ/z and q = 2.Therefore the conformal metric isg˜αβ = Ψ−qgαβ =z2ℓ2 gαβ . (4.1)According to the knowledge we obtained from the regularity discussion in Chap. 3, the most generalyet regular metric can beg˜ =η˜tt η˜tr 0 0 η˜tzη˜tr η˜rr 0 0 η˜rz0 0 η˜θθr2 0 00 0 0 η˜θθr2 sin2 θ 0η˜tz η˜rz 0 0 η˜zz, (4.2)where η˜θθ is replaced by η˜rr + rW˜ .In this chapter and the next, we choose the unit ℓ = k5 = 1 (which implies 8πG4 = 8πG5 = 1).In this chapter, the initial data is obtained by solving the Hamiltonian constraint, subject to theboundary conditions imposed by Israel’s junction condition.The initial data is specified on a spacelike hypersurface. Because the evolution equations arepartial differential equations with second order time derivatives, the initial data is specified by thevalues of the evolving quantities (to be specified below) and their first order time derivatives. Thelapse and the shift functions are related to the gauge freedom in choosing the coordinates whichcan be chosen arbitrarily, therefore the initial data is the combination of spatial (d− 1)-metric gij ,matter distribution, and their first order time derivatives that satisfy the Hamiltonian constraint954.1. Formulationand momentum constraints. For time symmetric initial data, the momentum constraints aresatisfied automatically, which leaves the Hamiltonian constraint to be solved. In most situationsof GR, we can then choose a spatial metric ansatz with only one unknown function such asdl2 = A(dr2 + r2(dθ2 + sin2 θdφ2) + dz2). The Hamiltonian constraint is then solved for A. Wewill examine whether this situation will still be true in the braneworld.4.1 FormulationIn the braneworld, Israel’s boundary condition can not be met if there is only one unknownvariable in the spatial metric. To “absorb” this “inconsistency”, there must be at least two unknownvariables in the spatial metric. The spatial metric can be taken to have a conformal form in (r, z)coordinatesdl2 = ℓ2z2[e2A+2B(dr2 + dz2)+ e2A−2Br2(dθ2 + sin2 θdφ2)]. (4.3)The brane is located at z = ℓ, while z = ∞ is the spatial infinity. Eq. (4.3) is actually the mostgeneral spatial metric for the axisymmetric (in the bulk) case, as discussed in Sec. 2.4.2. Thebackground spacetime (vacuum) corresponds to A = B = 0.The Hamiltonian constraint is32(∇2A)− 12(∇2B)+ 1− e4B2r2 +3z2(1− e2(A+B))+ 3U,rr −3U,zz +32 (U,z)2 + 32 (U,r)2 = 0, (4.4)where ∇2 ≡ ∂rr + ∂zz, and U ≡ A − B. The domain within which to solve the equations is(r, z) ∈ [0,∞) × [1,∞). Let Φ be the massless scalar field that lives on brane, Israel’s junctioncondition (1.29) is then translated intoA,z = 1− eA+B −16e−A−B (Φ,r)2 , (4.5a)B,z = −14e−A−B (Φ,r)2 , (4.5b)from which one see that a vanishing B is only possible when there is no matter (Φ,r = 0) on thebrane.The boundary conditions at the symmetric axis r = 0 are simply the parity condition andthe local flatness condition which translate into A,r∣∣r=0 = B,r∣∣r=0 = B∣∣r=0 = Φ,r∣∣r=0 = 0.When z → ∞, the spacetime should approach the background: A|z→∞ = B|z→∞ = 0. If the964.1. Formulationmatter is localized to finite r, then the spacetime should approach the background as r → ∞:A|r→∞ = B|r→∞ = 0 = Φ|r→∞.There is only one equation (the Hamiltonian constraint) for two variables A and B, thereforethere remains freedom in the initial data. We fix the freedom by imposing the form of B. Based onthe specific forms of B, we used two specifications: Laplacian specification and direct specification.4.1.1 Laplacian SpecificationSince there is no other requirement on B except for the boundary conditions, we will impose anequation for B, for example, ∇2B = 0. However, it turns out that there are numerical instabilityissues at spatial infinities, which suggests us to carry the asymptotic behaviour at these boundariesby some factors. I.e. we define Y as B ≡ fBY where the asymptotic behaviour at spatial infinitiesis carried by fB. Since we do not know the asymptotic behaviour yet, we will try to choose fB inan experimental manner. There is actually a guidance we can start with. A similar problem is thestatic star configuration studied in [47], where the boundary condition at z →∞ is asymptoticallyB ∼ constz3 . (4.6)If the matter is localized at finite r, the boundary condition at r→∞ is asymptotically [47]B ∼ constr . (4.7)As discussed in Chap. 2, the asymptotic bahaviour of A at r ≫ z is supposed to be A ∼ 1/rz forconformally flat space. We can then introduce factors likeA = X(r + 1)z , (4.8a)B = Y r2(r + z)3 , (4.8b)where the r2 in equation (4.8b) is to let the boundary conditions B∣∣r=0 = B,r∣∣r=0 = 0 be satisfiedautomatically. In the numerical method, we will directly solve for X and Y instead of A and B.The equation of B can be imposed via Y as∇2Y − aY b = 0. (4.9)974.2. Numerical MethodsThe principle part ∇2 plays the role as a smoother, and the second part is to serve as an amplitudedamper (to be explained below) when both a and b are set to be positive values, such as a = b = 3.The functionality of the damping term is to prevent the amplitude of Y from getting too large.The damping term is needed because instabilities arise at the brane when the amplitude of Ybecomes too large.4.1.2 Direct SpecificationSince the only requirements on B are the boundary conditions among which only the boundarycondition at the brane is non-trivial, we may directly set B to take the following simple form sothat (4.5b) holdsB(r, z) = ∂B∂z∣∣∣∣z=1· f(z), (4.10)where f(z) satisfies the condition: f,z∣∣z=1 = 1. In case we want the magnitude of B to be small,we can also let f(z) satisfy a second condition: f |z=1 = 0. There is still a lot of freedom to choosef . e.g. we can choose it to be f(z) = (z − 1)/zpz , where pz = 4 if we want B to satisfy (4.6) (notnecessary though). Or we can let B die off faster by choosingf(z) = (z − 1) · exp[− (z − 1)2 /σz2], (4.11)where σz can be chosen arbitrarily. In all the numerical results presented in this chapter, σz = 2σris chosen, where σr is going to be defined in (4.14). Because (4.10) and (4.11) yield B|z=1 = 0, wehaveB(r, z) = −14e−A0 (Φ,r)2 · f, (4.12)where A0(r, z) = A(r, z)|z=1. It is easy to check all the boundary conditions of B are satisfied.It turns out the numerical behaviour of the direct specification is better than that of theLaplacian specification, so we will mainly focus on the direct specification method.4.2 Numerical MethodsIn order to let the spatial infinities be a part of our computational domain, compactifiedcoordinates (Rˆ, Zˆ) are usedRˆ ≡ rr + r0; Zˆ ≡ z − ℓz − ℓ+ z0, (4.13)984.3. The Numerical Solutionwhere r0 and z0 are parameters to control the compactification. r0 = z0 = 1 is used for thenumerical results that are going to be shown in this chapter. The discretized grids are uniformin Rˆ and Zˆ, and the equations are discretized by finite difference approximation with secondorder central stencil, which are shown by eq. (3.60) to eq. (3.68). Then Newton-Gauss method isemployed iteratively to solve for the numerical solution. After the numerical solutions are obtained,we validate the solutions by the mean of independent residual test.4.3 The Numerical Solution4.3.1 The Solution and Apparent HorizonWe used the following profile for the matter fieldΦ = A · exp[− (r − x0)2σ2r]. (4.14)The condition Φ,r∣∣r=0 = 0 needs to be satisfied in the initial data, therefore we either choosex0 = 0, or choose x0 to be at least several σr.The independent residual, calculated as the residual of equation (4.4) under a different dis-cretization scheme (other than eq. (3.60) to eq. (3.68)) that is of second order accuracy, shouldbehave like a second order quantity everywhere in the domain if a numerical solution is obtained.To show the independent residuals compactly, we use their L2 norms. As an example, we per-form the independent residual test to the numerical results obtained from the calculation with(A , σr, x0) = (2.8, 0.15, 0) using the direct specification. The L2 norms of the independent resid-ual at resolutions 64 × 64 (the domain in Rˆ direction and Zˆ direction are uniformly divided into64 intervals), 128× 128 and 256× 256, are 0.0799, 0.0194 and 0.00498, respectively. The L2 normshrinks by a factor of 3.9-4.1 when the grid spacing decreases by a factor of 2. The independentresidual indeed converges to zero at second order, therefore the numerical scheme is justified.The results of the numerical calculation with (A , σr , x0) = (3.2, 0.3, 0) using the direct speci-fication, are shown in Fig. 4.1. The scalar field is strong enough to produce an apparent horizonwhich is shown in Fig. Brane Geometry as Seen by a Brane ObserverIn this subsection we focus on the geometry of the brane, since the bulk is not directly observ-able. In the next subsection, we will study the geometry of the bulk. Again, here we reiterate994.3. The Numerical Solutionr/(1 + r)(z−1)/zFunction A  0 0.2 0.4 0.6 0.8 function Ar/(1 + r)(z−1)/zFunction −B  0 0.2 0.4 0.6 0.8 function −BFigure 4.1: Function A and −B in the initial data metric (4.3) obtained from the parameters(A , σr, x0) = (3.2, 0.3, 0)0.1 0.2 0.3 0.4 Apparent Horizonrz  AH in BulkAH on BraneFigure 4.2: The apparent horizons for the configuration with parameters (A , σr, x0) = (3.2, 0.3, 0).The red x in the figure is the apparent horizon calculated using the brane geometry only. It doesnot coincide with the intersect of the brane and the bulk apparent horizon.1004.3. The Numerical Solutionthat the masses of the spacetime obtained from the brane geometry, play no direct roles in thebraneworld. Especially, they do not really represent the masses of the brane. The purpose ofstudying these masses, is to compare with GR and obtain some observable difference from GR.We will study two different masses defined on the brane—brane ADM mass and the Hawkingmass (see Sec. 2.5.1 and 2.5.2). The proper area radius r˜ on the brane isr˜ = eA−Br, (4.15)so that the metric on the brane is nowds2brane = gr˜r˜dr˜2 + r˜2dΩ2, (4.16)where gr˜r˜ = e4B [1 + r (dA/dr − dB/dr)]−2.In terms of gr˜r˜ and r˜, the Hawking mass isMH =r˜2G4(1− (gr˜r˜)−1), (4.17)which has r˜ dependence. Then the r˜ dependence of the Hawking mass is an observable differencefrom 4D GR. One example is shown in Fig. 4.3.The Hawking mass goes to ADM mass as r˜ →∞MADM = limr˜→∞r˜2G4(1− (gr˜r˜)−1)= α1 + β1G4. (4.18)α1 and β1 are defined via the asymptotic behaviour of A and B at large r, which are assumed tobe A∣∣z=1 ≈ α1/r and B∣∣z=1 ≈ β1/r. As it is shown in Sec. 4.3.3, the asymptotic behaviour of Bactually implies β1 = 0.The masses for the results of the configuration with (A , σr, x0) = (3.2, 0.3, 0), are shown inFig. 4.3. In the next section we will study the total mass in the bulk. To distinguish differentmasses, the ADM mass on brane will be referred as MbraneADM.4.3.3 Asymptotic BehaviourThe asymptotic behaviour at r ≫ z is crucial for the calculation of total energy (see Sec. 2.4.2).The construction of B in the direct specification according to eq. (4.12), directly implies |B| ≪ |A|at the r ≫ z region. When B is negligible, in Sec. 2.4.2 we showed that a solution at r ≫ z region1014.3. The Numerical Solution0 0.2 0.4 0.6 0.8 1051015202530r1+rMassesMasses on Brane  MHMadmMADMapparent horizonFigure 4.3: Brane Masses from the initial data configuration with parameters (A , σr, x0) =(3.2, 0.3, 0). The result is MADM = 22.67. The graph shows that the Hawking mass is not aconstant in the braneworld, in contrast to 4D GR. Hawking mass agrees with ADM mass only atr →∞. Madm (defined in equation (2.58)) still blows up around the apparent horizon.1024.3. The Numerical SolutionwasA ≈ α1rz when r ≫ z. (4.19)However, as discussed in that section, there is no proof of the uniqueness of this solution. Thereforein this section we need to test whether the solution is indeed given by eq. (4.19). Please refer toFig. 4.4 for the results of a series of simulations from the family (A , σr, x0) = (A , 0.3, 0) usingthe direct specification. The graphs show that the asymptotic behaviour is indeed described byeq. (4.19).      α1 =0.000357zr·A  α1/zr = 24.60r = 31.00r = 41.67r = 63.00r = 127.00      α1 =0.0007683zr·A      α1 =0.001663zr·A      α1 =0.003624zr·A100 101 102      α1 =0.007985zr·A      α1 =0.01798zr·A  α1/zr = 24.60r = 31.00r = 41.67r = 63.00r = 127.00      α1 =0.04215zr·A      α1 =0.1051zr·A      α1 =0.2833zr·A100 101 102      α1 =0.8936zr·AFigure 4.4: The asymptotic behaviour of A at r ≫ z region, shown by the simulations from thefamily (A , σr, x0) = (A , 0.3, 0). In the first diagram, we plot r ×A versus z in log-log scale. Thered line, is plotted assuming r×A = α1/z. If A ≈ α1/rz, one would expect the plots for differentr should follow the red line. Other diagrams are plotted in the same way, but with different α1resulted from different A . The diagrams look almost identical, despite that α1 has changed overthree orders of magnitude. It means that the asymptotic behaviour of A at r ≫ z region is indeedA ≈ α1/(rz).1034.4. The Total Energy4.4 The Total EnergyThe asymptotic behaviour of eq. (4.19) is confirmed by Fig. 4.4, therefore the calculation inSec. 2.4.2 applies. The calculation shows that the total energy (Mtotal) in the whole spacetime isMtotal = α1/G5 = 8πα1. (4.20)Consequently, as shown by eq. (2.67), the relation between the brane “energy” and the total energyis then simplyMbraneADM = Mtotal. (4.21)4.4.1 The Relation with the Area of Apparent HorizonIn 4D GR case, there is a simple relation between the Hawking energy of the sphericallysymmetric vacuum spacetime and the area of the horizon of the black hole sitting at the symmetriccenter, which is eq. (2.62). In the braneworld, we would like to examine whether the total energycan also be characterized by the area of apparent horizon in 5D braneworld. To study this relation,we choose the configurations where the matter outside of the apparent horizon are negligible.Fig. 4.5 shows the relation between the total energy Mtotal, and√4πAbulk, where Abulk is thearea of the apparent horizon in the bulk. The figure shows these two quantities are equal. Thereforewe haveMtotal =√4πAbulk. (4.22)How to understand this relation? Eq. (4.21) tells us that the mass can also be understood asthe ADM mass calculated on the brane. The size of BH—the areal radius of the intersect of thehorizon with the brane—is ra ∼ (0.7, 2.0) according to the area-versus-radius relation shown byFig. 1.1, where ra is the areal radius of the horizon on the brane. These BHs are then consideredto be medium to large. For large BHs, the area of the horizon in the bulk reduces to the areaof black string, which is equal to the area of the horizon on the brane, and the brane geometry(without matter) reduces to 4D GR. Eq. (4.22) might be justMbraneADM =√4πAbrane. (4.23)On the other hand, given there are many BHs of medium size in this data set, the knowledgeof large BHs (that the areas reduces to those for black strings) might not apply. Therefore therelation might be generically about total energy and the area of apparent horizon in the bulk.1044.4. The Total Energy8 10 12 14 16 18 20 22 2481012141618202224Total Mass√4piAinBulkMass-Area Relation  Datay = xFigure 4.5: The mass-area relation of apparent horizon. This plot is produced from simulationswith various parameters (A , σr, x0), using both direct method and Laplacian method. The onlycriteria is that the configurations defined by the parameters are able to produce apparent horizonsin the bulk.1054.5. Discussion: the Results of Laplacian SpecificationAt the end, we emphasize that the range of the size of the BHs is merely ra ∼ (0.7, 2.0), andFig. 4.5 exhibits a small but systematic deviation between the data and the line of a linear fit.Therefore further study is needed.4.5 Discussion: the Results of Laplacian SpecificationThe total energy is obtained in eq. (4.20), which was derived based on the condition that|B| ≪ |A| in r ≫ z region. By the construction of the “direct specifiction”, |B| ≪ |A| is metautomatically. On the other hand, without pre-setting this condition, this condition emerged inthe star solution [47] and the small black hole solution [14]. In our initial value problem, there isno such a pre-set condition in Laplacian specification method. It is then interesting to see whetherthis condition can still emerge. Fig. 4.6 shows the results of a series of simulations from the family(A , σr, x0) = (A , 0.15, 0.5) using Laplacian specification, where we see that indeed A ≈ α1/rzemerges. The validity of the discussion leading to Mtotal = α1/G5 is based on the assumption      α1 =0.005473zr·A  α1/zr = 24.60r = 31.00r = 41.67r = 63.00r = 127.00100 101 102      α1 =0.5847zr·A                    α1 =0.005473c =0.01267zr2.3·B  cr = 24.60r = 31.00r = 41.67r = 63.00r = 127.00100 101 102                    α1 =0.5847c =0.1232zr2.3·BFigure 4.6: The asymptotic behaviour of A at r ≫ z region obtained via Laplacian Specification,shown by simulations from the family (A , σr, x0) = (A , 0.15, 0.5). This diagram shows that theasymptotic behaviour of A is indeed A ≈ α1/(rz). Instead of almost identical configurations asin Fig. 4.4, this diagram shows interesting pattern: 1) the asymptotic behaviour of stronger datais more like A ≈ α1/rz; 2) disregard the trend with the strength of the data, as long as r is bigenough, there is always A ≈ α1/rz. The asymptotic behaviour of B is observed to be B ≈ c/r2.3.The fact that c/α1 decreases with α1 explains that A become more like α1/(rz) with the increaseof α1.A ≈ α1/rz and |B| ≪ |A|. Since B ≈ c/r2.3 and A ≈ α1/rz, ∀ (small) ε > 0, ∃ Q = (cv/α1ε)10/3such that |B/A| ≤ ε for r = Q, z ∈ [1, v · Q] (which is the region to calculate the total energy,as shown in Fig. 2.2(a) and discussed in Sec. 2.4.2). Therefore the assumptions hold, and we stillhave Mtotal = α1/G5.1064.6. Prepare the Initial Data for Evolution4.6 Prepare the Initial Data for EvolutionOften the lapse α˜ and the shifts η˜ti are freely specifiable in initial data. However, for thebraneworld, the Israel’s junction condition imposes non-trivial boundary conditions on the lapsefunction and the shift functions. For our specific initial data ansatz with time symmetric initialconfiguration, the boundary conditions for the shift functions are automatically satisfied by settingη˜ti = 0. The lapse function, on the other hand, has a non-trivial boundary condition. We denotethe tt component of the metric as gtt = −(L2/z2)e2a, then Israel’s boundary condition for a isa,z = 1− eA+B +112e−A−B (Φ,r)2 . (4.24)Using the direct specification, a can takea = f ·(1− eA + 112e−A (Φ,r)2), (4.25)where f is the same f used in eq. (4.10) to specify B.4.6.1 Prepare the Initial Data for Specific Initial Source FunctionsGiven the values of the metric functions and their time derivatives, the constraints for sourcefunctions h˜µ + ∆Γ˜µ ≃ 0 (which is satisfied at initial instant), can give us the initial values of thesource functions. Alternatively, as used by Pretorius [63], if one would like to set a certain gaugeat the initial instant by setting the value of h˜µ, then one can use the constraint equations to workbackwards to get the values of ∂tη˜tµ. In this way the initial values of ∂tη˜tµ is expressed in terms ofthe value of the metric components and their derivatives. Let us denote the obtained expressionas η˜tµ,t(η˜, ∂η˜).This procedure works well in ordinary spacetime (such as the 5D massless scalar field studiedin section 3.4). However, in the braneworld, generally η˜tz,t(η˜, ∂η˜) is not continuous at the braneunder the perpendicular gauge (2.7). This is because, ∂tη˜tz should be zero at the brane, due tothe boundary condition η˜tz∣∣z=1 = 0 at all time. On the other hand, η˜tz,t(η˜, ∂η˜) will include zderivatives of other metric functions (i.e. η˜tt,z, η˜rr,z, η˜zz,z, η˜θθ,z), which generally results in a non-zero η˜tz,t(η˜, ∂η˜) at the brane, due to Israel’s junction condition. i.e. η˜tz,t is not continuous at thebrane.This discontinuity problem can be solved by properly adjusting η˜zz . If we do not chooseη˜zz = η˜rr at the initial instant, there is the freedom to setup the value of η˜zz since neither Israel’s1074.6. Prepare the Initial Data for Evolutionjunction condition nor parity condition impose any requirement on η˜zz. We can use this freedomto adjust η˜zz such that η˜tz,t(η˜, ∂η˜) = 0 at the brane. In this case the simplest ansatz for the initialmetric isds2 = ℓ2z2(− e2adt2 + e2A+2Bdr2 + e2A−2B(dθ2 + sin2 θdφ2)+ e2bdz2), (4.26)for which the Hamiltonian constraint reads− −1 + e4B2r2 −3e2(A+B)z2 +12z2 e2(A−b+B)(6 + 6z2 (A,z)2 + 3zB,z + 2z2 (B,z)2+ zb,z (3 + zB,z)− zA,z (9 + 3zb,z + 4zB,z) + 3z2A,zz − z2B,zz)+ 12 (A,r)2 + 12A,rb,r +12 (b,r)2 + 2A,r + b,r − 4B,rr− 3A,rB,r −32b,rB,r +52 (B,r)2 +A,rr +12b,rr −B,rr = 0. (4.27)The boundary conditions of a,A,B are nowa,z =112e−2A−2B (Φ,r)2 , A,z = −16e−2A−2B (Φ,r)2 , B,z = −14e−2A−2B (Φ,r)2 . (4.28)The only requirement on b (from solving η˜tz,t(η˜, ∂η˜) = 0) is its derivative at the braneb,z = −(h˜z +16e−2A−2B (Φ,z)2), (4.29)so that ∂tη˜tz is continuous on the brane. The h˜z in (4.29) is the target initial value of h˜z that wetry to achieve at the initial instant. For example, h˜z = 0 if the harmonic gauge is chosen to beimposed at the initial instant.Similar to the direct specification method applied to B above, we can adopt direct specificationmethod to specify (a,B, b)a = f ·( 112e−2A (Φ,r)2), B = f ·(−14e−2A (Φ,r)2), b = f ·(−(h˜z +16e−2A (Φ,z)2)). (4.30)Ora = f ·( 112e−2A0 (Φ,r)2), B = f ·(−14e−2A0 (Φ,r)2), b = f ·(−(h˜z +16e−2A0 (Φ,z)2)),(4.31)where A0 ≡ A|z=1, is the value of A on the brane. While both of (4.30) and (4.31) work, the1084.6. Prepare the Initial Data for Evolutionnumerical experiments show that (4.31) enables the Hamiltonian constraint equation (which is anelliptic equation in term of A) converges faster when performing numerical relaxation. Thereforewe adopt (4.31).For an example of the initial data, please refer to Fig. 4.7.r/(1 + r)(z−1)/zFunction A  0 0.2 0.4 0.6 0.8 Initial Data: function Ar/(1 + r)(z−1)/zFunction −B  0 0.2 0.4 0.6 0.8 Initial Data: function −BFigure 4.7: Function A and −B in initial data metric (4.26) from the initial data configurationwith parameters (A , σr, x0) = (2.5, 0.2, 0)4.6.2 Total EnergyThe total energy from the metric (4.26) can be obtained by the same method presented inSec. 2.4.2. For this metric, according to equation (4.30) or (4.31), we have b ∼ 0 and B ∼ 0 atlarge r (or large z) region. Therefore the embedding conditions (2.40) and (2.41) reduce tor¯z¯ =rz eA; (4.32a)(r¯′)2 + (z¯′)2z¯2 =1z2 , (4.32b)where ′ ≡ d/dz.Only the asymptotic behaviour of A at r ≫ z region contributes to the calculation of totalenergy. The asymptotic behaviour is (shown by Fig. 4.8)A ≈ α1r at r≫ z. (4.33)1094.6. Prepare the Initial Data for EvolutionThis asymptotic behaviour gives the solution to (4.32) at the lowest order of A asz¯ ≈ z, r¯ ≈ r · exp(α1r), (4.34)which yield the total energyMtotal = α1/G5 = 8πα1. (4.35)Therefore, the equality of the total energy with the ADM “mass” calculated on the brane, alsoholds for metric (4.26).100 101 102  α1 =0.0002199zr·A  α1r = 24.60r = 31.00r = 41.67r = 63.00r = 127.00100 101 102  α1 =1.08zr·A  α1r = 24.60r = 31.00r = 41.67r = 63.00r = 127.00Figure 4.8: The asymptotic behaviour of A at r ≫ z region, shown by numerical solutions forthe family (A , σr, x0) = (A , 0.15, 0.5). The diagrams are almost identical, and here we only showthe one with the smallest α1 and the one with the largest α1. These diagrams show that theasymptotic behaviour of A is A ≈ α1/r for α1 over a wide range of magnitude. Therefore, theconclusion of A ≈ α1r at r ≫ z, is robust.110Chapter 5Evolution5.1 The Evolution as an Initial Value Problem✲ r✻zBulk: (d)Gµν + Λgµν = 0.AdSAdSAxisBrane (z = ℓ): Israel’s BC and matter’s equations of motion.Figure 5.1: The evolution as an initial value problem with boundary conditions. z ∈ [ℓ,∞) isthe extra dimension, while the brane is located at z = ℓ (expressed by the shadowed line in thediagram). r ∈ [0,∞) is the radius coordinate, and r = 0 (z axis) is the symmetry axis of the SO(3)symmetry.The evolution of the braneworld is specified by the evolution equation(s) with initial data andthe boundary conditions. The initial data was discussed in Chap. 4. Here we present the evolutionequation and the boundary conditions.The governing equations are composed of the 5D Einstein’s equations in the bulk and thematter’s equation of motion on the brane. The 5D Einstein’s equations areGαβ = −Λgαβ =: kdTαβ ⇔ Rαβ = kd(Tαβ − gαβTd− 2). (5.1)1115.1. The Evolution as an Initial Value ProblemIts form in terms of the conformal metric g˜αβ isR˜αβ = kd(Tαβ − g˜αβT˜d− 2)−R(Ψ)αβ , (5.2)where g˜, Ψ, R(Ψ)αβ and R¯αβ were introduced in section 3.3. For the conformal function and theconformal parameter, we take q = 2 and Ψ = 1/z in our calculations.The most general conformal metric in terms of regular functions η˜, is (see eq. (3.35))g˜αβ =η˜tt η˜tr 0 0 η˜tzη˜tr η˜rr 0 0 η˜rz0 0 η˜θθr2 0 00 0 0 η˜θθr2 sin2 θ 0η˜tz η˜rz 0 0 η˜zz; (5.3)where η˜θθ is taken as equal to η˜rr+rW˜ , which makes the local flatness condition η˜θθ∣∣r=0 = η˜rr∣∣r=0be satisfied automatically.The evolution equation (5.2) can be solved in two ways. The first way, is to use (3.48), in termsof the following source functions (see eq. (3.36))H˜tH˜rH˜θH˜φH˜z=h˜t + (2/r) (η˜tr/η˜θθ)h˜x + (2/r) (η˜rr/η˜θθ)cot θ0h˜z + (2/r) (η˜rz/η˜θθ). (5.4)Alternatively, a second way, is to use the generalized harmonic formalism with conformal functionand tensorial source functions, which is equation (3.52)− 12 g˜αβ∇¯α∇¯β g˜µν − ∇¯αg˜β(µ∇¯ν)g˜αβ − ∇¯(µh˜ν) + h˜αC˜αµν − C˜αµβC˜βαν= kd(Tµν −1d− 2 g˜µν T˜)− R¯µν −R(Ψ)µν . (5.5)When the background in (5.5) is taken to be flat spacetime (which is what we adopted in thenumerical calculations presented in this chapter), the two ways are equivalent. For discussion andpresentation purpose, we will adopt the second, eq. (5.5).1125.1. The Evolution as an Initial Value ProblemA second part of the governing equations is the equation of motion of the matter on the brane,which only “feels” the 4D metric. For massless scalar field, the equation of motion isDαDαΦ = 0, (5.6)where D is the covariant derivative associated with the brane metric hαβ .Next we look at boundary conditions. The boundary conditions at the spatial infinities (r →∞or z →∞) are Dirichlet conditions such that the metric takes the form of the backgroundds2 = ℓ2z2(− dt2 + dr2 + r2(dθ2 + sin2 θdφ2)+ dz2), (5.7)which is a Poincare´ patch of the AdS spacetime (see Sec. 1.4). The background of RSII braneworldis the z ≥ ℓ portion. In terms of the conformal metric components, the boundary conditions atr → ∞ and z → ∞ are η˜rr = 1, η˜tt = −1, η˜zz = 1, η˜tr = 0, η˜tz = 0, η˜rz = 0, W˜ = 0, and Φ = 0 forthe matter at r →∞.The boundary conditions on the symmetry axis are parity conditions, and the local flatnesscondition, which are η˜tt,r|r=0 = 0, η˜rr,r|r=0 = 0, η˜tz,r|r=0 = 0, η˜zz,r|r=0 = 0, η˜tr|r=0 = 0, η˜rz|r=0 =0, W˜ |r=0 = 0, Φ,r|r=0 = 0.The boundary conditions at the brane are Israel’s junction condition (1.29)K+αβ = −K−αβ =12kd(λ hαβd− 2 + ταβ − hαβτd− 2), (5.8)which translates into conditions on η˜tt,z, η˜tr,z , η˜rr,z, W˜,z . The expressions are long and the specificforms do not matter at this stage, and are thus not written here. The expressions are in terms ofmetric functions, their first order derivatives with respect to r and t, and Φ,r, Φ,t. As discussed inSec. 2.1, generically there is no boundary condition for η˜µz, yet we need their boundary conditionsin the numerical calculation. These conditions are more subtle and will be discussed below.The massless scalar field Φ only exists at z = ℓ. Therefore it is not defined in the bulk.1135.2. Details of the Numerical Implementation5.2 Details of the Numerical ImplementationFor spatial coordinates, we use “compactified coordinates” to include spatial infinities into thecalculation domain, and numerical grid is uniform in Rˆ and Zˆ:Rˆ ≡ rr + r0; Zˆ ≡ z − ℓz − ℓ+ z0, (5.9)where r0 and z0 are parameters to control the scale of the compactification. The derivatives inthe equations need to be changed accordingly. For example, f,r = f,Rˆ · ∂Rˆ∂r , etc. The numericalcalculations are performed using the central stencils of finite difference approximation operatorsof second order accuracy, whose explicit forms are∂tf →fn+1i,j − fn−1i,j2 ·∆t , (5.10)∂Rˆf →fni+1,j − fni−1,j2 ·∆Rˆ, (5.11)∂Zˆf →fni,j+1 − fni,j−12 ·∆Zˆ, (5.12)∂ttf →fn+1i,j − 2fni,j + fn−1i,j(∆t)2 , (5.13)∂RˆRˆf →fni+1,j − 2fni,j + fni−1,j(∆Rˆ)2 , (5.14)∂ZˆZˆf →fni,j+1 − 2fni,j + fni,j−1(∆Zˆ)2 , (5.15)∂tRˆf →14 ·∆t ·∆Rˆ(fn+1i+1,j − fn+1i−1,j + fn−1i−1,j − fn−1i+1,j), (5.16)∂tZˆf →14 ·∆t ·∆Zˆ(fn+1i,j+1 − fn+1i,j−1 + fn−1i,j−1 − fn−1i,j+1), (5.17)∂RˆZˆf →14 ·∆Rˆ ·∆Zˆ(fni+1,j+1 − fni−1,j+1 + fni−1,j−1 − fni+1,j−1). (5.18)where the index i and j are grid indices in Rˆ and Zˆ directions, and ∆Rˆ and ∆Zˆ are the (uniform)spacing of the grids in Rˆ and Zˆ directions. The superscripts n, n+1, n− 1 are the discretized timelevels, where ∆t is the spacing. In the simulations performed in this chapter, we set r0 = z0 andthe resolution in Rˆ is the same as that in Zˆ. Courant factor ∆t/min(∆r) = ∆t/(r0 ·∆Rˆ) is set tobe 0.5.The residual equations are the discretized eq. (5.5) and eq. (5.6). Let n denote the current timelevel. The update scheme is to obtain time level n+1 from given quantities at level n and n−1, by1145.3. Gauge Freedomsolving the residual equations. We solve the residual equations by pointwise Newton-Gauss-Seideliteration in a black-red manner (see, e.g. [73]) until residuals are smaller than a “small” threshold.A Kreiss-Oliger (KO) [73] style numerical dissipation is added to control high frequency numer-ical noises. Since we use second order FDA to discretize eq. (5.5) and eq. (5.6), a fourth order KOdissipation (see [61] for the specific form) is employed. Following [61], the dissipation is applied toboth n and n− 1 time levels before solving for the advanced time level n+ 1.Adaptive Mesh Refinement (AMR) is used to reach high resolution (when needed). We usedPAMR/AMRD [65] as the tool to realize the parallelization of the code. Both of the KO styledissipation and AMR are built into PAMR/AMRD. The simulations producing BHs in this chapter,however, were obtained using only one CPU with uniform grid structure, because a shootingmethod is employed to locate apparent horizons, which is not parallelized, nor adapted to AMR.Our plan for the future is to upgrade the code to use flow method to locate apparent horizons,which can be parellelized and adapted to AMR [62].5.3 Gauge FreedomAs discussed in section 3.3, there are two ways to perform evolutions using GH formalism: theBSSN-like method and the source function driven method. Here we adopt the source functiondriven method. Therefore we need to consider how to impose gauges via the source functions,and how to make sure the h˜µ + ∆Γ˜µ ≃ 0 constraints are preserved during the evolutions. In thissection we focus on imposing gauges. Please refer to section 5.4 for the study of the constraintpreservation.5.3.1 Fixing Gauges via Source FunctionsThere are gauge freedoms in gravitational theories, which are about the coordinate choices. Itis important to properly specify coordinates in numerical relativity to avoid coordinate pathology,to evolve spacetime with strong fields, and to deal with physical singularities. For the generalizedharmonic formalism, the following gauges are generally adopted in the simulations in the literature.(1) The simplest gauge is the harmonic gaugeh˜µ = 0. (5.19)1155.3. Gauge Freedom(2) The lapse driving gauge used by Pretorius [63]:˜h˜t + c1α˜− α˜0α˜s − c2h˜t,µn˜µ = 0, (5.20)where s ≥ 0, c1 > 0, c2 > 0. It is generalized from the damped harmonic equation x,tt+ c1 ·x+c2 ·x,t = 0. With ˜, the equation has the functionality as a smoother. The effect of this gaugeis to drive the lapse function towards its desired value α˜0. For example, Pretorius found thatthe instability tends to happen at the apparent horizon excision boundary when the value ofthe lapse function is too small. He then chose α˜0 = 1, the value of the lapse function in flatspacetime.One can also apply this method to the spatial components h˜i, to achieve the desired gauges.(3) Damped wave gauge [60, 98]. In our case, the gauge readsh˜µ = c1 log( η˜Pα˜)n˜µ − c2 η˜µiβ˜i/α˜, (5.21)where η˜ =[η˜rr η˜zz − (η˜rz)2](η˜θθ)2 is the determinant of the spatial (conformal) metric inCartesian coordinate. P, c1, c2 are positive parameters. The effect is to damp out the dynamicsin spatial coordinates on the time scale 1/c2, and to suppress the growth in√η˜/α˜ (whenP = 1/2) on time scale 1/c1.In ours simulations, (3) was adopted for the simulations that produce small black holes. Forsmall black holes, it is crucial to set both r0 and z0 much less than 1 to let the black hole boundariesinclude many grid points, so that the small black holes are well-resolved. (2) was adopted forthe simulations that produce large black holes. Pretorius’ group successfully performed manysimulations [62, 66, 67] using (5.20) as the slicing condition, and h˜i = 0 for spatial coordinates,which are also the conditions we use. α˜0 = 1 is chosen. For large BHs, (1) also works well. In anycase, after BHs are obtained and the evolution is stablized, we gradually change the gauge into (3)to drive the value of α˜ towards α˜0 = 1 so that the lapse rate of the coordinate time is comparableto that of the proper time, which enables us to define and to monitor the apparently stationarystate that is going to be introduced in Sec. Constraint Violation and the Cure5.4 Constraint Violation and the Cure5.4.1 Constraint DampingThe generalized harmonic formalism, has the constraints Cµ ≡ Hµ+Γµ ≃ 0 (or C˜ ≡ h˜µ+∆Γ˜µ ≃0). During the evolutions, there are always numerically errors that violate the constraints, andoften the modes of the deviation from the constraint equations grow with time, and drive theevolution away from Einstein’s equations. The phenomena, numerical violation from constraintequations growing with time, is a very common challenge for numerical relativity (see, e.g. [58]).One way to improve the performance, is to add the following constraint damping terms [58] tothe left hand side of the evolution equations (3.28)Zµν ≡ κ(n(µCν) −1 + p2 gµνnβCβ), (5.22)where κ > 0 and p > −1 are constant parameters. For evolution equations (5.5), the dampingterms are changed toZ˜µν ≡ κ(n˜(µC˜ν) −1 + p2 g˜µν n˜βC˜β), (5.23)where C˜µ ≡ h˜µ + ∆Γ˜µ. These damping terms have been proven to be useful in keeping theconstraints satisfied for simulations in “ordinary” spacetimes (e.g. the simulation in 5D carried outin Sec. 3.4). However, for the braneworld, it turns out that these damping terms are not sufficient— the constraints do not converge to zero as the resolution increases. In fact, the residuals of theconstraints almost stay the same when the resolution increases.We experimented with many changes to the damping term—some worked better than others.One version was to make κ spacetime dependent. We tried a few choices and it turns out thefollowing choice worked to a certain degreeκ→ κ zn(z − c ℓ)n , (5.24)where c was chosen to be, for example, 0.9 or 0.95, which effectively puts more damping nearthe brane. This change enhanced (a little) the convergence of the constraints. However, theconvergence criterion is still not perfectly satisfied.It turns out the following choice is more successful, although we can not explain why at this1175.4. Constraint Violation and the Curemoment. More work is needed.Z˜µν = κ(n˜(µC˜ν) − F ·1 + p2 g˜µν n˜β C˜β), (5.25)where F may depend on spacetime. i.e. only the second term of Z˜µν is multiplied by F , whichis a spacetime dependent function. We experimented with different forms of F , and the followingfamily worked the best so far (among all our trials)F = znzn − c ℓn , (5.26)where c is close to 1, which again adds more damping on the brane. A choice was n = 4, c = 0.99(refer to Fig. 5.2). This damping was successful, since the constraints converged at the expectedorder.The success of this method means that the second term in constraint (5.23) and the first termmay have very different effects.The constraint damping is very important. We do not have specific guidance to give the formsof such damping parameter settings. A survey over the parameters showed that the effect ofdamping is very sensitive to parameters. Also, even for the most successful result (Fig. 5.2), theindependent residuals (and the residual of constraint equations) increase rapidly after certain time.i.e. very long term simulations seems to be problematic.5.4.2 Imposing Constraints on the BraneFrom the result of Fig. 5.2, we learn that 1) the gauge choice h˜µ = 0 works; 2) the dampingshould be larger at the brane, which motivates us to exactly impose constraints h˜µ +∆Γ˜µ = 0 onthe brane.How to impose constraints h˜µ +∆Γ˜µ = 0? What one usually does in successful simulations inordinary spacetimes (non-braneworld, such as the 5D simulation in Sec. 3.4), is to set h˜’s to certainpre-set values (e.g. in the case of the harmonic gauges, we have h˜ = 0). In this situation, the effectof constraint damping is to drive the metric components to the values that satisfy the constraintequations, rather than doing something to the source functions h˜µ. i.e. to impose constraints, itis the metric (rather than h˜’s) that should be guided.How does this guidance happen? Let us define should˜hµ ≡ −g˜µν(Γ˜ναβ − Γ¯ναβ)g˜αβ . i.e. theyare what the h˜’s should be, if the constraints are exactly satisfied. The should˜h’s are functions ofthe metric components and their derivatives. One can set the value of metric components or their1185.4. Constraint Violation and the Cure0 0.5 1 1.5 2 2.500.0020.0040.0060.0080.01Res. of tt Comp. of Einstein Eq.timeL 2-norm  coarse resolution4 × fine resolution(a)0 0.5 1 1.5 2 2.500.511.522.53 x 10−3 Res. of rz Einstein Eq.timeL 2-norm  coarse resolution4 × fine resolution(b)0 0.5 1 1.5 2 2.501234 x 10−4 Convergence of h˜t +∆Γ˜ttimeL 2-norm  coarse resolution4 × fine resolution(c)0 0.5 1 1.5 2 2.5012345678 x 10−4 Convergence of h˜r +∆Γ˜rtimeL 2-norm  coarse resolution4 × fine resolution(d)0 0.5 1 1.5 2 2.500. x 10−3 Convergence of h˜z +∆Γ˜ztimeL 2-norm  coarse resolution4 × fine resolution(e)Figure 5.2: This convergence test is for the results obtained from the simulation using harmonicgauge h˜µ = 0, and constraint damping Z˜µν = κ(n˜(µC˜ν)− 1+p2 g˜µν n˜αC˜α · z4z4−0.99). The initial datais Φ = A exp[−(r − x0)2/σ2r]with (A , r0, σr) = (0.05, 1, 0.25). After the results are obtainedby the generalized harmonic formalism with a conformal function, the results are substituted intothe original Einstein’s equations in terms of the original metric functions without the conformalfunction, to get the residuals. The residuals should converge to zero as second order quantities,if the results are numerical solutions. Fig. (a) is the convergence of the residual obtained fromthe (tt) component of Einstein’s equations Rµν = kd(Tµν − gµν Td−2). Fig. (b) is the test fromthe (rz) component. Fig. (c,d,e) show the convergence of constraints: h˜µ + ∆Γ˜µ with µ = t, r, z,respectively. The residuals converge at the expected order. However, this is the result obtainedfrom very extreme damping, and all the residuals have up-climbing tails.1195.5. The Evolution with an Apparent Horizonderivatives at the brane to let should˜hµ = h˜µ (therefore imposing constraints), which can be donein multiple ways. Here we adopt the following. should˜hµ = h˜µ can be equivalently expressed bysetting the values of η˜tz,z, η˜rz,z and η˜zz,z , which in principle can serve as the boundary conditionsfor η˜µz—as shown in Sec. 2.1, generically there is no boundary condition for η˜µz.In addition, we impose the perpendicular gauge at the brane (η˜tz∣∣z=ℓ = η˜rz∣∣z=ℓ = 0) since thisgauge gives the smoothness of the apparent horizon across the brane, as discussed in Sec. 2.2.2.Now we have two conditions for η˜tz (and two conditions for η˜rz): one is constraint imposingcondition which is a condition on the value of η˜tz,z (and η˜rz,z), the other is the perpendiculargauge condition which is a condition on the value of η˜tz (and η˜rz). Imposing two boundaryconditions on one function is achieved by the following trick: let one of the conditions be satisfiedautomatically by choosing the forms of the functions:η˜tz ≡ (z − ℓ) · ξ˜t, (5.27)η˜rz ≡ (z − ℓ) · ξ˜r. (5.28)In this way, the η˜tz |z=ℓ = η˜rz|z=ℓ = 0 are automatically satisfied. The conditions on η˜tz,z andη˜rz,z are now converted into the conditions on the values of ξ˜t and ξ˜r. We then use ξ˜t and ξ˜r asfundamental variables instead of η˜tz and η˜rz.It turns out that this method works well, and there is no need to use special damping.Beyond the constraints imposed on the brane, the “normal and plain” damping term Z˜µν =κ(n˜(µC˜ν) − 1+p2 g˜µν n˜αC˜α)is still employed to control “ordinary” violation modes in the bulk. Thetests of the method are shown in Fig. The Evolution with an Apparent HorizonDuring the evolution, sometimes singularities are formed and the code crashes. There are atleast two types of singularities. The first is the coordinate singularity due to pathological coordinategauges, which can be avoided by properly choosing coordinate gauges (which is non-trivial). Thesecond is the physical singularity. However, Penrose’s cosmic censorship hypothesis [84] statesthat, the physical singularity is always hidden behind an event horizon. While there is no proof ofthis hypothesis, it does seem to be satisfied in many cases.Event horizon is the boundary in spacetimes that separates the events which can be causallyconnected with future Poincare´ horizon by future oriented null geodesics, from the events which cannot. The interior surrounded by event horizons, by definition, can not affect the regions outside of1205.5. The Evolution with an Apparent Horizon0 0.5 1 1.5 2 2.50123456 x 10−3 Res. of tt Einstein Eq.timeL 2-norm  coarse resolution4 × fine resolution(a)0 0.5 1 1.5 2 2.5012345678 x 10−4 Res. of rz Einstein Eq.timeL 2-norm  coarse resolution4 × fine resolution(b)0 0.5 1 1.5 2 2.500.511.522.533.5 x 10−4 Convergence of h˜t +∆Γ˜ttimeL 2-norm  coarse resolution4 × fine resolution(c)0 0.5 1 1.5 2 2.501x 10−4 Convergence of h˜r +∆Γ˜rtimeL 2-norm  coarse resolution4 × fine resolution(d)0 0.5 1 1.5 2 2.500.511.5 x 10−4 Convergence of h˜z +∆Γ˜ztimeL 2-norm  coarse resolution4 × fine resolution(e)Figure 5.3: The convergence Tests for the simulation using gauge h˜µ = 0. The initial data isΦ = A exp[−(r − x0)2/σ2r]with (A , r0, σr) = (0.05, 1, 0.25). After the results are obtained bythe generalized harmonic formalism with a conformal function, the results are substituted intothe original Einstein’s equations in terms of the original metric functions without the conformalfunction, to get the residuals. The residuals should converge to zero as second order quantities,if the results are numerical solutions. Fig. (a) is the convergence of the residual from the (tt)component of Einstein’s equations Rµν = kd(Tµν − gµν Td−2). Fig. (b) is the test for the (rz)component. Fig. (c,d,e) show the convergence of constraints: h˜µ+∆Γ˜µ with µ = t, r, z, respectively.In all the tests shown in the figures, the spacing of the coarser grid is ∆Rˆ = ∆Zˆ = 1/256 and thespacing of the finer grid is ∆Rˆ = ∆Zˆ = 1/512. The simulations were performed using 16 CPUsand the test is performed on the result obtained by the CPU at the Rˆ = Zˆ = 0 corner, which isthe region that would suffer from the most severe problems (if there was). The convergences areshown to be good. After t ∼ 2, the residuals suddenly decreased, because the interesting dynamicspropagated out of the region where we evaluate the residuals.1215.5. The Evolution with an Apparent Horizonthe event horizons. Therefore, one way to avoid the physical singularity in the calculation domainis to perform the evolution without referring to the interior of the event horizons (so that theinterior of the event horizons is excised from the calculation domain).Event horizon, however, is a concept based on the global structure of the whole spacetime,which is therefore not quite useful during the evolution since one can not tell the event horizonuntil the full evolution is completed. But the full evolution is not able to be obtained withoutknowing the event horizon to excise the physical singularities. Fortunately there is the concept ofapparent horizon which is locally (in time) defined. Apparent horizons are not guaranteed to existin a certain evolution. But, if they do exist, they are inside of the event horizons. Therefore theinterior of an apparent horizon can not affect the exterior of the event horizon. Since the apparenthorizon lies inside the event horizon, and sometimes by a long way, the Penrose hypothesis doesnot guarantee that singularities can not exist outside the apparent horizon. But again, oftenthe physical singularities (to be formed in a future instant) are inside of the apparent horizon.Therefore, if an apparent horizon appears, we may excise the interior of the apparent horizon toget rid of the physical singularities from the calculation domain. This idea (black hole excision)was proposed by W. G. Unruh [70], and had become a common practice to deal with physicalsingularities in numerical relativity.To perform black hole excision, one needs to locate the apparent horizon.5.5.1 Smoothness of Apparent Horizons in the BraneworldIn the braneworld, the Israel’s junction condition at the brane essentially imposes cusp con-ditions to certain metric functions. This raises the question of whether apparent horizons willbe non-smooth across the brane. By the discussion of the smoothness of apparent horizons inSec. 2.2.2, the apparent horizons in the braneworld is smooth under the perpendicular gauge(2.7). Under gauge grz∣∣z=1 = 0 (perpendicular gauge), the smoothness can be simply expressed as(dr/dz)∣∣z=1 = 0, where the derivative is evaluated along the apparent horizons.5.5.2 Apparent Horizon FinderIn this subsection we introduce a method to locate apparent horizons [64].We introduce polar coordinates (ρ, χ) via r = ρ sinχ, z = 1 + ρ cosχ (length dimensions are inthe unit of ℓ). In this coordinate system, the symmetric axis r = 0 is χ = 0, and the brane z = 11225.5. The Evolution with an Apparent Horizonis χ = π/2. On a t = constant hypersurface, we define functionf ≡ ρ− ̺(χ), (5.29)and let the apparent horizon be the one with f = 0. The form of ̺ is going to be determined by theapparent horizon equation (2.14). The spacelike outpointing unit vector normal to f = constantsurfaces is nowsα = pα√pβpβ, where pα ≡ γ βα ∂βf. (5.30)Substituting sα into (2.14), we get a second order ordinary differential equation (ODE) of ̺ withrespect to χ. The boundary condition at χ = 0 (the symmetric axis r = 0) is d̺/dχ = 0; theboundary condition at χ = π/2 (the brane) is d̺/dχ = 0 under the perpendicular gauge at thebrane (η˜rz = η˜tz = 0), because of the Z2 symmetry of the braneworld and the smoothness of theapparent horizon. Apparent horizon is obtained via solving this second order ODE subject to theboundary conditions. Numerically, we use a version of the shooting method to obtain the apparenthorizon. There are multiple ways to implement shooting methods. The basic idea is to start thetrajectory (implied by the ODE) at certain initial point (χ = 0), and then solving the ODE subjectto the initial conditions yields a value of d̺/dχ at the final point (χ = π/2) of the trajectory. Wethen adjust the inital point accordingly until d̺/dχ = 0 at final point. One version is implementedas the following. At χ = 0, we pick up certain value of z(0)p (which is ̺(χ = 0)) as the initial valueof z and solve the ODE to obtain the trajectory to χ = π/2. Not losing generality, let us assumethe value of d̺/dχ at χ = π/2 is positive. Then we try different initial value of z(0) until we findan initial value z(0)n such that d̺/dχ is negative at χ = π/2. Now we have found a bracket of theinitial guesses: (z(0)p, z(0)n). Then we can use the following binary search to find the apparenthorizon. We use z p for z(0)p, z n for z(0)n and Rp for d̺/dχ in the pseudo code.eps = pre-set small valueRp = 10 * epsdo while ( abs(Rp) > eps )z_m = (z_p + z_n) / 2solve ODE to get Rp at chi = pi/2, by initial value z = z_mif (Rp > 0) thenz_p = z_melse if (Rp < 0) thenz_n = z_m1235.6. Tests and the Validation of the Numerical Schemeend ifend doz(0) = z_mIn performing shooting method above, we need to solve the ODE, for which we used the Runge-Kutta method of second order accuracy.5.5.3 Dissipation at the Excision BoundaryTo perform the excision, which is to ignore the interior of the apparent horizon during numericalevolutions, is conceptually simple and neat. However, technically it is tricky and probably messyto deal with the excision boundary. To ignore the interior of the apparent horizon, we only evolvethe exterior, and the boundary condition at the excision boundary is “no boundary condition”.Instabilities and noises tend to happen at the excision boundary, which should be removed bycertain numerical dissipation. One way to do it, is to reconstruct the interior from the exterior,such that the reconstructed functions are smooth across the excision boundary. Then we applyKreiss-Oliger dissipation to remove the noises. This process, while sounding trivial, is very difficultto realize. There is no good way to reconstruct smooth functions across the excision boundary.This method is able to evolve the spacetime with excision for “a while”, so its success is limited.Another way, which is the best among the methods we tried, is to use the one-side dissipation at theexcision boundary, which was developed by Calabrese et al. [72], and adopted in PAMR/AMRD[65]. This method significantly improves the performance.5.6 Tests and the Validation of the Numerical SchemeAs pointed out in Sec. 1.6.2, the tests after the numerical results are obtained, are essential tomake sure the results are actually numerical solutions rather than numerical artifacts. To this end,the independent residual tests need to be performed. For systems with constraints, convergencetests for constraint residuals need to be performed as well.For GR, one can either perform independent residual tests together with constraint convergencetests, or perform convergence tests for residuals obtained from a different formalism of GR. Givengeneralized harmonic formalism (5.5) was used to obtain the solutions, where the conformallytransformed metric and source functions were fundamental variables, now we perform the conver-gence tests of the residuals obtained from the original Einstein’s equations (5.1). i.e. the numericalsolutions are obtained via the generalized harmonic formalism with the conformally transformed1245.7. The Numerical Solutionmetric as the fundamental variables. Then the solutions are transformed back to obtain the phys-ical metric, and are then substituted into the original Einstein’s equations to get residuals. Theseresiduals should converge to zero at the expected order (which is of order 2 in our simulations), ifthe numerical solutions are obtained. The tests are shown in Fig. 5.3.Beyond this, we also performed independent residuals, which are perfect thus omitted here.Beyond these, we also performed convergence tests for the residuals of the constraints, which areshown in Fig. The Numerical SolutionThe scheme can be used to study a wide range of dynamical processes, such as critical phe-nomena, the evolution problem in cosmology, gravitational wave from collapse, etc. But we willlimit our attention to the end states of gravitational collapse at this time.We performed a series of simulations from the initial data metric (4.26) with the initial matterfield asΦ = A · exp[−(r − x0)2/σ2r], (5.31)from different initial data families. Within each family, only amplitude A changes, while Gaussianparameters (σr and x0), σz (the parameter in the “direct specification” function f in (4.11), whichis used in eq. (4.31)) and the compactification parameters (r0 and z0), are fixed.5.7.1 The Evolution Process and Apparently Stationary StateThe initial data represents a localized Gaussian pulse. Since the initial data is time symmetric,the pulse evolves into two pulses: the ingoing pulse and the outgoing pulse. For weak data, theingoing pulse is bounced back from r = 0 to travel outwards, which is the same phenomena as thatin GR. The unique phenomena in RSII, is the interaction between the brane and the bulk, whichmainly appears as the energy leaking into the bulk from the brane. Please refer to Fig. 5.4 and5.5.Sufficiently strong data will lead to BHs. The spacetime with BH can continue to evolve usingBH excision techniques. The properties of the BHs are studied via apparent horizons in the bulk.Apparent horizon is generally different from event horizon. However, at the end of an evolution,the system reaches its stationary state and its apparent horizon coincides with its event horizon[89]. In the braneworld, when matter is absent around the horizon, the intersection of the bulk1255.7. The Numerical Solutionevent horizon with the brane, is actually a well-defined event horizon on the brane as proved inSec. 2.3, therefore observable, in principle.We monitor ra, Abulk and C5 during an evolution, where ra is the areal radius of the intersectionof the apparent horizon with the brane, Abulk is the area of the apparent horizon in the bulk, andC5 is the length of the circumstance of the horizon (restricted on the r− z plane) in the bulk. Forthe simulations that can reach their end states or their apparently stationary state (to be definedbelow), the quantities reach the values that are almost constants. Please refer to Fig. 5.6 andFig. 5.7 for an evolution which produces a BH of medium size, Fig. 5.8 for an evolution whichproduces a BH at a smaller size, Fig. 5.9 for an evolution which produces a small BH and Fig. 5.10for an evolution which produces a large BH. Fig. 5.4 and 5.5 show the evolutions from two initialdata families, monitored by the Kretschmann scalar RµναβRµναβ .To obtain the end state of a system, it is necessary to let the evolution continue sufficientlylong so that the system settles down. Even so, it is non-trivial to recognize the end state in a givensimulation, due to coordinate effects. When a system reaches its end state, the system has a Killingvector that is asymptotically timelike, which corresponds to the time translational symmetry. Ifthe slicing condition is such that the t = constant slices are Lie-dragged by the Killing vector, it iseasy to overcome the spatial coordinate effects, by embedding the apparent horizons into a fixedspace, such as the background space. During a particular evolution, the apparent horizon evolveswith time, but its embedding in the fixed space will evolve into a time-independent state, if theslicing condition is adapted according to the Killing vector.However, it is non-trivial to impose such slicing conditions. Such a slicing condition is currentlynot imposed in our numerical schemes 16. As a result, we do not have a time-independent state atthe end of the evolutions. In the simulations, instead, we can obtain apparently stationary states—the states with BHs whose horizons (embedded into a fixed space) appear to be stationary for afinite time. By this definition, the apparently stationary state that stays stationary for infinitetime is actually the end state.To recognize the apparently stationary state of a system, we monitor the evolution of theapparent horizon by its embedding into the “vacuum” backgroundds2 = ℓ2z¯2(− dt¯2 + dr¯2 + r¯2(dθ¯2 + sin2 θ¯dφ¯2)+ dz¯2). (5.32)16In the literature, there are the so-called symmetry seeking coordinate conditions [118–120] towards the conditionswhich can evolve into the coordinate configurations such that the time coordinate t is adapted to the Killing vectorassociated with the time translational symmetry of a stationary system. It will be our future work to implementsuch slicing conditions in our code.1265.7. The Numerical SolutionFigure 5.4: The snapshots of four evolutions resulting in BHs with different sizes, monitored viaKretschmann scalar RµναβRµναβ . Each small panel represents an instant of the space, where thehorizontal and vertical axes are r and z axes expressed in compactified coordinates r/(r+ r0) and(z − ℓ)/(z − ℓ + z0), therefore the bottom of each panel is z = ℓ, the brane. The complete spacein r direction is shown. In z direction, only the part with interesting dynamics is shown. Theevolutions are from the family with x0 = 2, σr = 0.2, σz = 0.4, and r0 = z0 = 2 are chosen. Theyproduce no-BH, BH with ra = 0.29ℓ, 0.61ℓ, 3.78ℓ from A = 0.04, 0.15, 0.24, 0.49 respectively. Thetwo smaller BHs are apparently stationary states, and the largest BH is not since the configurationhas not settled down to stationary state after evolving for a long time, and eventually the codecrashes. In general it is harder for an evolution to settle down if the excision surface is going acrossthe “wiggling” regions. This appears to be a technical issue related to the black hole excision atrelatively coarse resolutions. The black ellipses in the figures are the excision surfaces inside theapparent horizons. The evolutions clearly show how the energy flows from the brane into the bulk,and flow from the exterior towards the symmetric axis. Part of the energy is captured by blackholes, the remaining energy continue to flow towards the “far end” of the bulk.1275.7. The Numerical SolutionFigure 5.5: The snapshots of three evolutions resulting in BHs with different sizes, monitoredvia Kretschmann scalar RµναβRµναβ . The evolutions are from the family with x0 = 1, σr =0.1, σz = 0.2, and r0 = z0 = 1 are chosen. They produce no-BH, BH with ra = 0.22ℓ, 2.9ℓ fromA = 0.03, 0.16, 0.54 respectively. The smaller BH is a apparently stationary state, and the largerBH is not since the configuration is not settled down to stationary state after evolving for a longtime. Again, presumbly this is a technical issue related to the black hole excision.1285.7. The Numerical SolutionHere a bar (¯ ) is used to emphasize that this spacetime is fixed, and the z¯ ≥ 1 portion is actuallythe background of the braneworld spacetime. The embedding is demonstrated by Fig. 2.3 (alsoeq. (2.40) and eq. (2.41)). The apparently stationary state appears as an “accumulating” curve inthe embedding plot. Please refer to Fig. 5.6, Fig. 5.7, Fig. 5.8 and Fig. 5.9 as examples where theapparently stationary states are obtained. The processes of the settlement into time-independentstates, show that the portion of the apparent horizons that is close to the brane, gets settled first,when the remaining portion might be still dynamical.Similar to apparent horizon, the existance of apparently stationary state is not guaranteed, andits relation with end state is not clear. In our simulations, as it turns out, apparently stationarystates can be easily obtained by long term evolutions, as long as the apparent horizons do not crossthe regions with interesting dynamics (the “wiggling” regions), which can be realized by properlychoosing the initial data such that the wiggles either finally travel away from the apparent horizons,or are captured by the black holes. Furthermore, the plots of the quantities of apparently stationarystates (such as the plots of Abulk-versus-ra and C5-versus-ra shown by Fig. 5.11), generated fromthe evolutions of the initial data profiles from different families, exhibite certain trends, whilstthe same plots with horizons that are not apparently stationary state do not have trends. Also,the Abulk-versus-ra plot (the upper panel of Fig. 5.11) agrees perfectly with that obtained in thestatic system studied by Figueras-Wiseman in [20]. Therefore, we conjecture that the apparentlystationary state is close to the end state, and we use apparently stationary state to approximatethe end state.5.7.2 Black Holes as the Result of Gravitational CollapseFor the BHs as apparently stationary states of gravitational collapse, we focus on the followingaspects: the topology, the size and the shape.For the topology, one can see that the BHs appear to be localized on the brane with finiteextension into the bulk.For the sizes, please refer to the results of all the simulations we performed, which are shown byFig. 5.11, and table 5.1 for the results from some selected simulations. We did not try extremelyhard to find the largest/smallest BHs possible. But within the simulations we performed, weobtained BHs withra ∈ (0.04ℓ, 19.6ℓ). (5.33)At the end of an evolution, the matter has either fallen into the BH, or escaped to infinity,which makes the brane tension be the only content associated with the brane. The strength of the1295.7. The Numerical Solution0.1 0.2 0.3 0.4 by Coordinatesrz(a) AH evolution by coordinates0.2 0.4 by Embeddingr¯z¯(b) AH evolution by embedding0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1− + r)Φ  t = 0t = 1t = 2.28t = 2.5t = 3t = 8.88(c) The evolution of massless scalar field ΦFigure 5.6: This figure and Fig. 5.7 show an evolution that produces apparently stationary state asmedium BH with (ra,Abulk, C5) = (0.601, 2.785, 3.164), using parameters (A , x0, σr, σz , r0, z0) =(0.24, 2, 0.2, 0.4, 2, 2) which defines the initial data and the compactification. (a) is the evolutionof the apparent horizon, where r and z are coordinates. The black dashed line is the first apparenthorizon that appears during the evolution, and the red “-.” line is the apparently stationary stateof this specific evolution of gravitational collapse. Other lines are apparent horizons at intermediateinstants, which color changes continuously from black to red. To better study the evolution withoutcoordinate distortion effects, we embed each apparent horizon into the background spacetime:ds2 = ℓ2z¯2(− dt¯2 +dr¯2 + r¯2(dθ¯2 + sin2 θ¯dφ¯2)+dz¯2). The embedding is demonstrated by Fig. 2.3.The embedding plot is shown in (b). This graph shows more clearly that the apparently stationarystate is obtained: the shape of the apparent horizon has settled down, since the coordinate timet ∼ 5.5. (c) shows the evolution of the massless scalar field that only lives on the brane. The initialprofile is a Gaussian pulse, which splits into ingoing and outgoing branches. The ingoing branchis gradually captured by the BH.1305.7. The Numerical Solution2.5 3 3.5 4 4.5 5 5.5 6 6.50.570.580.590.60.610.62r a/ℓt(a) The evolution of ra2.5 3 3.5 4 4.5 5 5.5 6 6.522.ℓ3t(b) The evolution of Abulk2.5 3 3.5 4 4.5 5 5.5 6 5/ℓt(c) The evolution of C52.5 3 3.5 4 4.5 5 5.5 6 6.500.α˜)t(d) The minimum of α˜Figure 5.7: This is the continuation of Fig. 5.6. Here we show the evolution of ra, Abulk, C5 andthe minimum of lapse function α˜ over the whole calculation domain. One can see the apparentlystationary state is obtained by sub-figure (b) in Fig. 5.6, since t ∼ 5.5. The quantities approachesquasi-constant since then. However, the quantities do not stay strictly at constants. Combiningthe plot of α˜, finally the values at t = 5.8 are recorded as the data for the apparently stationarystate. min(α˜) ∼ 0.35 means the lapsing rate of coordinate time is comparable with that of propertime.1315.7. The Numerical Solution0.05 0.1 0.15 0.2 by Coordinatesrz(a) AH evolution by coordinates0.05 0.1 0.15 by Embeddingr¯z¯(b) AH evolution by embedding1 1.5 2 2.5 3 3.5 4 4.5 a/ℓt(c) The evolution of ra1 1.5 2 2.5 3 3.5 4 4.5 500.ℓ3t(d) The evolution of Abulk1 1.5 2 2.5 3 3.5 4 4.5 5/ℓt(e) The evolution of C51 1.5 2 2.5 3 3.5 4 4.5 500.α˜)t(f) The minimum of α˜Figure 5.8: This figure shows an evolution that produces apparently stationary state assmall BH with (ra,Abulk, C5) = (0.221, 0.184, 1.311), using parameters (A , x0, σr, σz , r0, z0) =(0.16, 1, 0.1, 0.2, 1, 1).1325.7. The Numerical Solution0.005 0.01 0.015 0.02 0.025 0.03 0.0351.0051.011.0151.021.0251.031.035AH by Coordinatesrz(a) AH evolution by coordinates0.01 0.02 0.03 0.041.0051.011.0151.021.0251.031.0351.041.045AH by Embeddingr¯z¯(b) AH evolution by embedding0.6 0.65 0.7 0.75 0.8 0.850.020.0250.030.0350.040.045r a/ℓt(c) The evolution of ra0.6 0.65 0.7 0.75 0.8 0.8500. x 10−3Abulk/ℓ3t(d) The evolution of Abulk0.6 0.65 0.7 0.75 0.8 0.850. 5/ℓt(e) The evolution of C50.6 0.65 0.7 0.75 0.8 0.850.α˜)t(f) The minimum of α˜Figure 5.9: This figure shows an evolution that produces apparently stationary state as smallBH with (ra,Abulk, C5) = (0.0432, 0.00155, 0.269), using parameters (A , x0, σr, σz , r0, z0) =(0.08, 0.5, 0.1, 0.2, 0.5, 0.5).1335.7. The Numerical Solution0 1 2 3 4 551015202530AH by Coordinatesrz(a) AH evolution by coordinates1 2 3 4 5 6 7 8 9 1011121314152468101214161820222426283032AH by Embeddingr¯z¯0 0.5 129.53030.53131.532(b) AH evolution by embedding1 2 3 4 515.12215.12415.12615.12815.1315.13215.13415.136r a/ℓt(c) The evolution of ra1 2 3 4 5286428652866286728682869Abulk/ℓ3t(d) The evolution of Abulk1 2 3 4 514.614.6514.714.7514.814.8514.914.95C 5/ℓt(e) The evolution of C51 2 3 4 50.650.70.750.80.850.90.951min(α˜)t(f) The minimum of α˜Figure 5.10: This figure shows an evolution that produces large BH as apparently station-ary state with (ra,Abulk, C5) = (15.1, 2866, 14.9), using parameters (A , x0, σr, σz, r0, z0) =(1.05, 2, 0.5, 1, 2, 2). This is the case that data is so strong that all the interesting dynamics iscaptured by the BH. Note the plots of ra and Abulk are noisy, which we can not explain. Thismight be caused by the KO dissipation—the dissipation appears to have stronger effects at largerspatial coordinates, and its perturbing effect on larger horizon is also stronger. As a result, theenergy of the BH gradually decreases, which causes the decrease in Abulk.1345.7. The Numerical Solutionr0 z0 σz x0 σr A ra/ℓ Abulk/ℓ3 C5/ℓ0.5 0.5 0.2 0.5 0.1 0.08 0.0432 0.00155 0.2691 1 0.3 0 0.3 2.05 0.600 2.80 3.181 1 0.2 1 0.1 0.16 0.221 0.18 1.312 2 0.4 2 0.2 0.24 0.601 2.79 3.162 2 1 2 0.5 1.05 15.1 2866 14.9Table 5.1: BHs produced from different initial data from different families. The initial data profilefor the massless scalar field on the brane is Φ = A · exp[−(r − x0)2/σ2r]. σz is the parameter toset up metric functions via “direct specification” eq. (4.11), which is then substituted into (4.31).The spatial metric for initial data is eq. (4.26). r0 and z0 are simply compactification parametersdefined in eq. (5.9).brane tension is proportional to 1/ℓ, therefore invisible to small BHs whose size ra ≪ ℓ. TheseBHs will be asymptotically 5D Schwarzschild, therefore Abulk = 2π2r3a and C5 = 2πra. Pleaserefer to Fig. 5.11.For the shape of large BHs, please refer to Sec. The Relation with Black StringsThe black string solution is ds2 = ℓ2z2(habdxadxb + dz2), where a = 0, 1, 2, 3 and xa standsfor a coordinate other than z, and hab is a BH solution of vacuum Einstein’s equations in 4D,which does not depend on coordinate z. This could be called black cone instead of black string,if we had considered the intrinsic geometry of the horizon of black strings. It is named string,in the sense that the(habdxadxb + dz2)part is a string. The “string” shape can be revealed byembedding the horizon into the background spacetime (5.32). Therefore, the embedding of theBHs into this background, can give a direct comparison with the black strings. The embedding ofthe BHs (represented by the apparently stationary states) of different sizes is shown in Fig. 5.12.Please refer to the caption of the figure for more details.Fig. 5.12 shows that small BHs are almost spherical. The BHs gradually change into a cigar-shape as the size increases, which suggests to call those BHs, black cigars, as first suggested in [6].As the size of BH increases, the portion of the horizon that is close to the brane gradually changesinto a black string. Also, the ℓ2/z¯2 factor in the background metric (5.32), contributes to Abulk asℓ3/z¯3, which means the contribution from large z¯ region become relatively negligible. As a result,for large BHs, only the portion that is close to the brane actually contributes to Abulk, regardlessthe behaviours in the large z¯ region. This explains why the relation Abulk-versus-ra for large BHs(shown in the upper panel of Fig. 5.11), behaves as that of black strings. On the other hand, this1355.7. The Numerical Solution10−210−1100101102103104ra/ℓAbulk/ℓ3  Black String5D SchFW DataOur Data0.04 0.1 1 10 20100101ra/ℓC 5/ℓ  5D Sch4 log(ζ · ra)Our DataFigure 5.11: The shape of BHs with all sizes. Small BHs are asymptotically 5D Schwarzschild. Thearea-radius relation for large BHs is that of black string, which is also 4D Schwarzschild accordingto AdS5/CFT4 [20, 22]. The area-radius relation is consistent with the one obtained from a staticproblem by Figueras-Wiseman. C5 for large BHs follows C5 = 4 log(ζ · ra) [23], where ζ = 2.71 isthe best fit of our data.1365.7. The Numerical Solution0.2 0.4 0.6¯/(c + ra)(z¯−1)/(c+r a)  (i)(ii)(iii)(iv)(v)0.2 0.4 0.6 0.8¯/ra(z¯−1)/ra  (i)(ii)(iii)(iv)(v)Figure 5.12: The shape of BHs with all sizes shown by embedding into the background spacetime(5.32). The horizons are scaled by a factor of(c + ra). Note both coordinates r¯ and (z¯ − 1) arescaled by the same factor, therefore the shape is not affected by the scaling. The scaling is tobring BHs with very different sizes (here ra ∼ (0.04ℓ, 15ℓ)) onto comparable plotting scale. Herethe scaling parameter on the left panel is c = 0.3, and the right panel corresponds to c = 0. Theleft panel still carries the information of sizes, which emphasizes on the uniqueness feature. Theright panel emphasizes on how the shape changes with size, which can also be read-off from theleft panel.Small BHs are asymptotically 5D Schwarzschild, which are almost spherical. They then changeinto cigars (seen from the embedding point of view) as the size increases, which suggest to callthem black cigars as firstly suggested in [6].More importantly, the BHs are apparently stationary states from different families. For family(i), the parameters specifying the family are (x0, σr, σz , r0, z0) = (0.5, 0.1, 0.2, 0.5, 0.5). Theseparameters for other families are: (ii) (0, 0.3, 0.3, 1, 1); (iii) (1, 0.1, 0.2, 1, 1); (iv) (2, 0.2, 0.4, 2, 2);(v) (2, 0.5, 1, 2, 2). From the left panel we see that BHs obtained from different families with thesame size, almost agree with one another. This indicates the detail of initial data is lost, and thesolution is unique. The right panel shows the relative extension into the bulk increases with thesize, and the portion (of the horizons) that is close to the brane looks more and more like blackstring as the size increases.1375.7. The Numerical Solutionalso means that the relation of Abulk-versus-ra can not reveal the difference between the BHs andblack strings.Therefore, we study C5 (the length of the circumstance of the horizon restricted on the r − zplane) versus ra, since C5 is infinite for black strings, but finite for BHs. Furthermore, Fig. 5.12shows that the relative extension into the bulk (as seen from the point of view of the backgroundmetric (5.32)), increases with the size on the brane. It is not very clear whether the relativeextension has an upper limit. On the other hand, by the uniqueness of the BH solutions that isgoing to be studied in Sec. 5.7.4, and the comparison with Figueras-Wiseman solution that is goingto be shown in Sec. 5.7.5, there is strong indication that our solutions generated from evolutionsystems, are the same as those were obtained by Figueras-Wiseman in their static systems. Forthe Figueras-Wiseman solution, as shown in the upper figure of Fig. 5.14, the large BHs has alimiting shape which is the AdS5/CFT4 solution [20, 21]. Generally, for configurations with fixedshape in the space (5.32), there is the following property for large raC5 = 4 log(ζ · ra), (5.34)regardless what the shape is, as long as the shape is fixed. This statement can be justified by directnumerical experiments for a few shapes. ζ is determined by the specific shape. C5 versus ra isplotted as the lower panel of Fig. 5.11, which supports the assumption that large BHs have the sameshape, and the best fit of ζ is 2.71. ζ ∼ 2.8 was first independently found by Figueras-Wiseman[23], and eq. (5.34) was proposed by Toby Wiseman in [23].Large BHs have the same shape (black cigar) in the background space (5.32), and black stringsalso appear as strings (rather than cones) in this space, whilst C5/ra merely appears as log(ra)/ra(black pancake) which is not as great as a fixed shape (personal tastes), therefore it makes moresense to name BHs as black cigars as first suggested in [6], rather than black pancakes.5.7.4 The No-hair FeatureWe purposely performed the simulations from distinct initial data families. Because the resultof a well-posed numerical simulation depends smoothly on its initial data, if only one family isconsidered (only one parameter A in the initial data profile is changed), the quantities (such asAbulk, C5 and ra) will depend smoothly on A . Therefore, for a given family, relations betweenquantities such as Abulk-versus-ra, will emerge. For a different family, in principle the relation ofAbulk-versus-ra might be different from the relation obtained from the previous family. Should1385.7. The Numerical Solutionthis happens, the BH solutions are not unique. In reality, however, the relations of Abulk-versus-raobtained from different families are the same: the Abulk-versus-ra relation plotted using the dataobtained from the evolutions from different initial data families is shown in the upper panel ofFig. 5.11, which appears to be one single curve. Similarly, C5-versus-ra is plotted as the lowerpanel of Fig. 5.11, which also appears to be one single curve. i.e. Fig. 5.11 shows that the BHswith the same sizes have the same areas and the same circumferences, regardless which familiesthe results are generated from. This indicates that the shape of the horizon is solely determined bythe size ra, regardless which initial data family the BHs are generated from. Therefore a no-hairtheorem of the BH solution in RSII is suggested. In general, however, the BH solutions in AdSspacetimes may not be unique (see, e.g. [117]). Therefore, the uniqueness of the BHs is limitedto the RSII spacetimes studied in this thesis—these spacetimes are axisymmetric without angularmomentum and non-gravitational charges. In this situation, the shapes of the horizons are directlystudied in Fig. 5.12, where one can see that the BHs with the same size produced from differentfamilies actually have almost the same shape, which is an indication that the detail of the initialdata is lost in the final state, and the geometry of a BH is solely determined by its size. Thereforea no-hair theorem (the uniqueness of BH solution) is suggested to hold for BHs in RSII.5.7.5 The Comparison with Figueras-Wiseman SolutionIf the BH solutions in RSII are unique in axisymmetric spacetimes without angular momentumand non-gravitational charges, then the BH solutions should be the same, regardless how the BHsolution is obtained. In particular, the BHs produced as the end states of evolutionary systemsshould agree with the ones obtained from the static problem studied in [20, 22]. Here we still useapparently stationary states to approximate end states. Following [20, 22], we plot our data ofAbulk versus ra on top of the Figueras-Wiseman data in the upper panel of Fig. 5.11. The figureshows the agreement with Figueras-Wiseman solution as illustrated by Abulk versus ra is perfect.To better compare with their solution, we also embed the BHs into the space (5.32) with z¯ ≤ 1,which is what Figueras-Wiseman did in [20]. Following [20], the freedom of the embedding is fixedby mapping the r = 0 ends (i.e., the axis ends) of the horizons to the point (r¯, z¯) = (0, 1) in (5.32),instead of mapping the z = 1 ends (i.e., the brane ends) to z¯ = 1 in (5.32) as what we did inthe above sections. i.e. instead of performing the embedding as Fig. 2.3, here we perform theembedding as Fig. 5.13. Please refer to Fig. 5.14 for the results. Fig. 5.14 shows the two resultsqualitatively agree. However, there are a few differences between these two figures: the largest BHmeets the vertical axis at r¯ ≈ 0.468 for our data, but at r¯ ≈ 0.457 for Figueras-Wiseman data; there1395.7. The Numerical Solution0.2 0.411.21.41.6BDCArz(a)0 0.1 0.2¯D¯C¯A¯r¯z¯(b)Figure 5.13: The embedding of a closed surface. Fig. (a) is the closed surface in the physicalspacetime, and Fig. (b) shows its embedding into the background spacetime (5.32) with z¯ ≤ 1.The freedom of the embedding is fixed by mapping D to D¯.exist line crossings (can be seen by zooming in) in our data but there is not in Figueras-Wisemandata. Together with the line crossings in Fig. 5.12, it implies that generally apparently stationarystates are, close to but distinct from, end states. Here we emphasize that our solution for largeBHs were obtained at a resolution that is effectively coarse at the horizon, thus the solution shouldbe less reliable. Furthermore, as one can see from the processes of the settlements to apparentlystationary states (Fig. 5.6, Fig. 5.7, Fig. 5.8 and Fig. 5.9), the portion of the apparent horizonthat is close to the brane gets settled first, while the remaining portion (the portion that is farfrom the brane, and close to the symmetric axis) might be still dynamical. This makes the portionthat is close to the brane more reliable than the portion that is far from the brane. Therefore,the embedding plot by fixing the brane end of the apparent horizon as done in Fig. 5.12, is morereliable than the embedding plot by fixing the axis end of the apparent horizon as done in Fig. Brane EnergyIn this section we focus on the physics on the brane, by studying the quantities obtained fromthe reduced metric (hµν) on the brane. The rationale is that the brane is all one can directlyobserve (while the bulk is invisible). We compare quantities obtained on the brane, with the same1405.7. The Numerical Solution0.0 0.2 0.4 0.6 0.8 Figueras-Wiseman solution0 0.2 0.4 0.6 0.8¯r¯(b) Our solutionFigure 5.14: The comparison with the Figueras-Wiseman solution via embedding into (5.32),with z¯ ≤ 1. The freedom of the embedding is fixed by mapping the r¯ = 0 ends of the horizons toz¯ = 1 in (5.32). Note, figure (a) is from [20], and the labels (r, z) should be understood as (r¯, z¯).The large BHs have a limiting shape which is the AdS5/CFT4 solution [20, 21]. In (b), the largestBH is the one with ra = 15.1.1415.7. The Numerical Solutionquantities obtained from 4D GR. We will study ADM mass and Hawking mass. As explained inSec. 2.5, these masses are only defined at asymptotic spatial infinities. However, let us first studythe ADM mass and Hawking mass in 4D GR, to obtain some insights.To compare with the braneworld where a spherical symmetry is present on the brane, we firstconsider gravitational collapse under spherically symmetric configuration in 4D GR. The initialdata is time symmetric Gaussian (massless scalar field), centred at r = 1. The pulse splits intoingoing and outgoing branches. For data that is not strong enough to produce BH, the ingoingbranch is bounced back and then travel outwards. For the ADM mass and Hawking mass in 4DGR, please refer to Fig. 5.15. The figure shows that:(1) the energy is conserved (remains to be constant at spatial infinity);(2) the energy is monotonic with radius r, which means the energy “density” (not well-defined ingeneral) is positive;(3) there is a “stair” between two separate matter pulses—this is due to Birkhoff’s theorem:vacuum solutions in spherical symmetry are unique (which are Schwarzschild’s solutions).Therefore, in the vacuum gap between the two pulses, the solution is should be Schwarzschild,where the ADM/Hawking energy is well-defined as well. Thus Hawking mass is constant withinthis gap, while ADM mass is approximately constant.For the ADM mass and Hawking mass on the brane, please refer to Fig. 5.16, where one cansee that the above mentioned features (2) and (3) are lost. Feature (1) still holds—this is becausethe masses are defined as the limit at the spatial infinity utilizing the time translational symmetry.Or in another word, this is because the dynamics happening locally can not propagate to thespatial infinity within finite time. However, it is expected that feature (1) does not to hold in thebraneworld. This is because, there is energy exchange between the bulk and the brane, and thesimulations show that the “wiggles” are moving from the brane into the bulk. Therefore we needa quantity that can describe this phenomena.It turns out the energy defined by equation (2.69) can indeed fulfill this task. Please refer toFig. 5.17. This energy has feature (2) and feature (3) mentioned above. The energy at spatialinfinity, on the other hand, changes with time. Fig. 5.17(b) shows the change with time: thebrane loses energy in majority of time (which agrees with the phenomena we observed from thesimulations we performed), but it gains some energy when the incoming pulse gets bounced backat the origin.1425.7. The Numerical Solution0 0.2 0.4 0.6 0.8 1−0.0200. ADM Energyr/(1 + r)ADMEnergy0 0.2 0.4 0.6 0.8 1−0.0200. Hawking Energyr/(1 + r)HawkingEnergyFigure 5.15: Energies in 4D GR. Each line is plotted at an instant (i.e., constant coordinate time).The black dashed line is at the earliest instant, and the red “-.” line is at the last instant. Otherlines’ colour changes gradually from black to red, with respect to coordinate time. Each line onlyhas one colour since it stands for one instant.0 0.2 0.4 0.6 0.8 1−0.1−0.0500. ADM “Energy”r/(1 + r)ADM“Energy”0 0.2 0.4 0.6 0.8 1−0.1−0.0500. Hawking “Energy”r/(1 + r)Hawking“Energy”Figure 5.16: “Energies” on Brane. Each line is plotted at an instant (i.e., constant coordinatetime). The black dashed line is at the earliest instant, and the red “-.” line is at the last instant.Other lines’ colour changes gradually from black to red, with respect to coordinate time. Each lineonly has one colour since it stands for one instant.1435.7. The Numerical Solution0 0.2 0.4 0.6 0.8 Energyr/(1 + r)BraneEnergy0 0.5 1 1.5 2 Energycoordinate timeBraneEnergyFigure 5.17: This figure shows the brane energy defined by equation (2.69). In (a), each lineis plotted at an instant (i.e., constant coordinate time). The black dashed line is at the earliestinstant, and the red “-.” line is at the last instant. Other lines’ colour changes gradually fromblack to red, with respect to coordinate time. Each line only has one colour since it stands for oneinstant. (b) shows the energy at spatial infinity changing with coordinate time.144Chapter 6Conclusion and Future WorkThe basic idea of braneworld scenarios is that our observable universe could be a 3+1 dimen-sional surface (the “brane”) embedded in a higher dimensional spacetime (the “bulk”). The singlebrane scenario constructed by Randall and Sundrum (also known as RS-braneworld II), is whatwe study in this thesis project. The basic setup is as follows: the single brane (the observableuniverse) is embedded in the bulk with one extra dimension of infinite size. The matter is strictlytrapped on the brane while gravity is free to access the bulk. The bulk is therefore “empty”, buthas an assumed negative cosmological constant. The bulk has a Z2 symmetry with respect tothe brane, and, the brane has a tension which enable fine-tuning to any equivalent cosmologicalconstant on the brane. General relativity (GR) is recovered on brane at low energies, but thebrane dynamics can be rather different from GR at high energies. This latter regime is the focusof our research. Specifically, we study the dynamical process of black hole formation as a result ofgravitational collapse of massless scalar field.6.1 ConclusionWe have achieved the following: in terms of developing the machinery, we discover/develop/in-vent several novel facts, formalisms and techniques regarding NR and braneworld. The regularityproblem in previous NR simulations in axisymmetric (and spherically symmetric) spacetime, isactually associated with neither coordinate systems nor the machine precision. The numerical cal-culation is regular in any coordinates, provided the fundamental variables (used in simulation) areregular, and their asymptotic behaviour at the vicinity of the axis (or origin) is compatible with thefinite difference scheme. Generalized harmonic (GH) and BSSN formalisms for general relativityare developed to make them suitable for simulations in non-Cartesian coordinate under non-flatbackground. A conformal function of the metric is included into the formalism to simulate thebraneworld. The usual constraint damping term used in GH, can not control the severe constraintviolation in braneworld. The violation is cured by imposing the constraints properly. In solving1456.2. Future Workelliptic equations (Hamiltonian constraint, for instance), using functions to carry the asymptoticbehaviour at spatial infinities could be crucial. The delta-function matter can be simulated by“integrating out” the delta function and the brane content can be encoded in Israel’s junctioncondition.On the physics side, we perform the first numerical study of gravitational collapse in braneworldwithin the framework of the single brane model by Randall-Sundrum (RSII). The scheme is capableof obtaining apparently stationary states as the results of gravitational collapse, which are BHslocalized on the brane, with finite extension into the bulk. The extension changes from sphereto flattened pancake (or from sphere to cigar, from the embedding point of view) as the sizeof BH increases. There is strong evidence that the detail of initial data is lost in the resultingBH, therefore no-hair theorem of BH (uniqueness of BH solution) is suggested to hold in RSIIspacetimes that are studied in this thesis—these spacetimes are axisymmetric without angularmomentum and non-gravitational charges. In particular, the BHs we obtained as the apparentlystationary states from the dynamical system, are consistent with the ones previously obtained froma static problem by Figueras and Wiseman. We also obtained some results in closed form withoutnumerical computation: the smoothness of apparent horizon across the brane under perpendiculargauge, the well-definedness of event horizon on the brane, and the equality of ADM mass of thebrane with the total mass of braneworld.6.2 Future WorkThere are many potential research directions/projects suggested by this project, some of whichwill become our future work. One such direction is to derive the mass-area relation of the (apparent)horizons in closed form. Another project is to look at the examples related to holographic principle:the equality of bulk energy and the ADM/Hawking energy calculated from the brane geometrywhich we have proved to hold for two classes of spacetimes. We would like to examine whetherthe relation holds in general space, and study the asymptotical configurations at spatial infinities.As for this work per se, it can be expended/upgraded as follows(1) Apparently stationary states were obtained, but it is not clear whether end states were ob-tained. The method to identify end state needs to be developed. Some attempts would be toimplement the slicing conditions in [118–120].(2) The current slicing condition and coordinate gauges can perform a wide range of initial dataand yield apparently stationary states. However, for certain initial data, the gauges we used1466.2. Future Work(Harmonic gauge, lapse driven gauge, damping wave gauge) can not yield a simulation thatlast “forever”. Therefore we need to improve the performance by studying coordinate gauges.(3) There is slight inconsistency in the initial stages of the evolution (which fade out duringevolution), which might be due to the truncation error, or inconsistency in initial data. Furtherstudy is needed.(4) The energy of the brane is needed to describe the interaction between the brane and thebulk. The brane energy we obtained is not conceptually satisfying, although the qualitativebehaviour is surprisingly good. The energy aspect needs more study.(5) The current code can run in a parallel environment since it is based on AMRD/PAMR. How-ever, we used a single CPU to do all the simulations since shooting method were employed tosearch for the apparent horizon. One improvement is to use flow method to search for apparenthorizon, which can be used in a parallel environment.For the same reason, although the code is capable of using the adaptive mesh refinement(AMR) that is built-in in AMRD/PAMR, because of the shooting method used in findingapparent horizon, we performed the simulations under unigrid. Hopefully we can turn on theAMR feature when the shooting method is replaced by flow method. In this way we can alsoimprove the accuracy of the result. 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Randall, “Linearized gravity in brane backgrounds,” JHEP0003, 023 (2000) [hep-th/0002091].[117] D. Sudarsky and J. A. Gonzalez, “On black hole scalar hair in asymptotically anti-de Sitterspace-times,” Phys. Rev. D 67, 024038 (2003) [gr-qc/0207069].[118] D. Garfinkle and C. Gundlach, “Symmetry seeking space-time coordinates,” Class. Quant.Grav. 16, 4111 (1999) [gr-qc/9908016].[119] C. Bona, J. Carot and C. Palenzuela-Luque, “Almost-stationary motions and gauge condi-tions in general relativity,” Phys. Rev. D 72, 124010 (2005) [gr-qc/0509015].[120] C. Bona, L. Lehner and C. Palenzuela-Luque, “Geometrically motivated hyperbolic coordi-nate conditions for numerical relativity: Analysis, issues and implementations,” Phys. Rev. D72, 104009 (2005) [gr-qc/0509092].157Appendix AGeneralized BSSN FormalismThe original BSSN [45, 46] formalism is only applicable to 4D spacetime with Cartesian co-ordinate. It was generalized into non-Cartesian coordinate in [36, 97] for 4D spacetimes withflat backgrounds. We will generalize BSSN into arbitrary dimension, general background, generalcoordinate and general conformal function (without requiring γ˜ = γ¯, see below), yet free fromirregularity issue. This work is essential to make BSSN applicable to higher dimensional space-time with non-trivial background, such as braneworld simulation. The derivation in [36] is largelyborrowed in this section.Fundamental variables to evolve: ϕ, K, γ˜ij , A˜ij , Γ˜i, which are defined asγ˜ij ≡ e−qϕγij , (A.1)Aij ≡(Kij −1d− 1γijK), (A.2)A˜ij ≡ e−qϕAij , (A.3)Γ˜i ≡ γ˜jk((d−1)˜Γi jk − (d−1)¯Γi jk), (A.4)where γ˜, the determinant of γ˜ij , is set to a time-independent function, such as γ¯ (the determinantof the background spatial metric). The bar-ed quantities are associated with the time-independentbackground metric.Let’s derive the evolution equations for these quantities.The starting points are the evolution equations for Kαβ and γαβ in ADM-York formalism:eq. (1.15) and (1.16). They are written below asLmKαβ = ǫDαDβα+ α[−ǫ (d−1)Rαβ +KKαβ − 2KαµKµβ + ǫ kd(Sαβ − γαβS + ǫρd− 2)], (A.5)Lmγαβ = −2αKαβ, (A.6)158Appendix A. Generalized BSSN Formalismfrom which we can deriveLm ln γ1/2 =12γijLmγij = −αK. (A.7)The equation of motion for K isLmK = γijLmKij +KijLmγij= ǫDiDiα+ α[A˜ijA˜ij +1d− 1K2 + kd(d− 3d− 2ρ−ǫSd− 2)], (A.8)where the Hamiltonian constraint is used. The equations of motion for ϕ, γ˜ij are derived to beLmϕ =2q(d− 1)[Lm ln γ1/2 − Lm ln γ˜1/2]= 2q(d− 1)(−αK − Lm ln γ˜1/2). (A.9)Lmγ˜ij = e−qϕLmγij − qe−qϕγijLmϕ = −2αA˜ij +2d− 1 γ˜ijLm ln γ˜1/2. (A.10)For Γ˜i, we haveΓ˜i = γ˜jk((d−1)˜Γi jk − (d−1)¯Γi jk)= −D¯j γ˜ij + γ˜ij∂j[ln(γ¯/γ˜)1/2]. (A.11)⇒ ∂tΓ˜i = −D¯j(∂tγ˜ij)+ ∂tγ˜ij∂j[ln(γ¯/γ˜)1/2]. (A.12)where we have applied (d−1)¯Γj jk = 12∂k ln γ¯ (and (d−1)˜Γjjk = 12∂k ln γ˜), and ∂tD¯j− D¯j∂t = 0 whichis because ∂t(d−1)¯Γi jk = 0. ∂tγ˜ij can be easily evaluated from (A.10) via ∂tγ˜ij = −γ˜ikγ˜jl∂tγ˜kl.The evolution of A˜ij needs the result of the evolution of Aij , which isLmAij =LmKij −1d− 1 (KLmγij + γijLmK)= (LmKij)TF +2αd− 1KKij −2αd− 1γijKlmKlm, (A.13)where TF means trace free. ThereforeLmA˜ij = e−qϕLmAij − qe−qϕAijLmϕ= e−qϕ[ǫDiDjα− αǫ(d−1)Rij + ǫαkd(Sij − γijS + ǫρd− 2)]TF+ αKA˜ij − 2αA˜ilA˜l j +2d− 1 A˜ijLm ln γ˜1/2. (A.14)(d−1)Rij needs to be rewritten in terms of the Γ˜i defined in (A.11). By the same procedure for the159Appendix A. Generalized BSSN Formalismconformal transformation carried out in [32, 36], we have(d−1)Rij = (d−1)R˜ij −q(d− 3)2 D˜iD˜jϕ−q2 γ˜ij D˜kD˜kϕ+ q2(d− 3)4(D˜iϕ D˜jϕ− γ˜ij D˜kϕ D˜kϕ).(A.15)Therefore(d−1)R = e−qϕ((d−1)R˜− q(d− 2)D˜kD˜kϕ−q2(d− 3)(d− 2)4 D˜kϕD˜kϕ). (A.16)Expressing (d−1)R˜ij and (d−1)R˜ in terms of Γ˜i, has been done in section 3.3 (or [32, 36])(d−1)R˜ij = (d−1)R¯ij −12 γ˜klD¯kD¯lγ˜ij − D¯kγ˜l(iD¯j)γ˜kl − D¯(iΓ˜j) + Γ˜kC˜kij − C˜kilC˜l kj . (A.17)Here we have defined C˜i jk ≡ (d−1)˜Γi jk − (d−1)¯Γi jk, and Γ˜i ≡ γ˜ij Γ˜j . Contracting (A.17) with γ˜ij ,we can obtain the expression of R˜ in terms of Γ˜i.In (A.14), DiDjα needs to be expressed by its counter partsDiDjα = DiD˜jα = D˜iD˜jα− CkijD˜kα= D˜iD˜jα−q2(D˜iαD˜jϕ+ D˜jαD˜iϕ− γ˜ij D˜kϕD˜kα), (A.18)where Ckij ≡ (d−1)Γkij − (d−1)˜Γkij , and Ckij = q2(δkiD˜jϕ+ δkjD˜iϕ− γ˜ijD˜kϕ)are used, and thelatter can be obtained by repeating the relevant derivations in [36].To perform numerical relativity, all the equations above need to be rewritten as partial deriva-tives with respect to t, which are related to Lm by Lm = L∂t − Lβ . We must be careful with theLie derivatives of tensor densities with respect to β. An object X is a tensor density of weight w,if X = tensor× γw/2. Its Lie derivative isLβX =[LβX]w=0+ wX∂iβi. (A.19)Let’s now figure out the weight of fundamental variables. Because of (A.1), we have eϕ =(γ/γ˜)1q(d−1) , therefore the weight of eϕ is 2−w˜q(d−1) , where w˜ is the weight of γ˜. Similarly, the weightof γ˜ij and A˜ij is w˜−2d−1 , the weight of upstairs γ˜ij and A˜ij is 2−w˜d−1 . The value of w˜, can actuallybe determined as follows. γ˜/γ is a scalar, because its value does not change under coordinatetransformation. Therefore, γ˜ = (γ · tensor), which implies w˜ = 2. Therefore, eϕ, γ˜ij and A˜ij are160Appendix A. Generalized BSSN Formalismall tensors17. Knowing w˜ = 2, it is then easy to deriveLβ ln γ˜1/2 =12γ˜Lβγ˜ = · · · = βi∂i ln γ˜1/2 + ∂iβi. (A.20)The formulae can then be rewritten in terms of coordinate derivatives by opening Lβ ln γ˜1/2. Let’stake (A.9) as an example. Using m = ∂t − β and ∂tγ˜ = 0, we have∂tϕ =2q(d− 1)(−αK + βi∂i ln γ˜1/2)+ βi∂iϕ+2q(d− 1)∂iβi, (A.21)where we have applied L∂tT = ∂tT in a coordinate system where t is a coordinate, for tensor T .The Hamiltonian constraint is nowkd ρ ≃12(−ǫ(d−1)R+ d− 2d− 1K2 − A˜ijA˜ij), (A.22)where (d−1)R is of course replaced by its expression in equation (A.16). ≃ means the equation isa constraint relation. The momentum constraint readsǫkdSi ≃ D˜iK −(D˜jA˜j i +q(d− 1)2 D˜jϕA˜ji +1d− 1D˜iK)= d− 2d− 1 D˜iK −(D˜jA˜j i +q(d− 1)2 D˜jϕA˜ji). (A.23)SummaryThe fundamental variables are ϕ, K, γ˜ij , A˜ij , Γ˜i. The equations of motion are (A.8, A.9, A.10,A.12, A.14). Constraints are the Hamiltonian constraint (A.22), momentum constraints (A.23),and the definition equations (A.1-A.4), which readγ˜ ≃ the pre-set time-independent function such as γ¯, (A.24)γ˜ijA˜ij ≃ 0, (A.25)Γ˜i ≃ γ˜jk(Γ˜i jk − Γ¯i jk). (A.26)17Quite a few authors counted the weights incorrectly—especially under Cartesian coordinate with flat backgroundwhere γ˜ was set to 1, where these authors incorrectly assumed w˜ = 0. Fortunately, the values of the weights per sedo not matter in the final expressions in terms of partial derivatives. It is the relative weight that matters. e.g. onecan keep w˜ general (without substituting w˜ = 2), therefore the weight of eϕ is 2−w˜q(d−1) . One can still obtain (A.21)correctly.161Appendix A. Generalized BSSN FormalismAxisymmetryFor simulations in axisymmetry—take 4D as an example—under cylindrical coordinates (t, ρ, φ, z),as shown in Chap. 3, the such defined Γ˜i reduces to the results obtained by our Cartesian com-ponents method if the background γ¯ij is flat, therefore regular. When the background is not flat,the behaviour needs to be analyzed in a case-by-case basis. The conformal metric and conformaltraceless extrinsic curvature areγ˜ij =γ˜ρρ 0 γ˜ρz0 ρ2(γ˜ρρ + ρW ) 0γ˜ρz 0 γ˜zz, A˜ij =A˜ρρ 0 A˜ρz0 ρ2(A˜ρρ + ρV ) 0A˜ρz 0 A˜zz, (A.27)where local flatness has been applied. All the fundamental variables depend on (t, ρ, z) only.162Appendix BExtrinsic Curvature as GeodesicsDeviation: C ≥ 1 CaseIn this section, we consider the case that a (d − C) dimensional object Σ embeds in the ddimensional space M . Therefore C is the codimension.Let (1)nµ, ... , (C)nµ be C continuous normal vector fields of Σ, with unit length (1)ǫ, ... , (C)ǫ,and (I)nµ’s are set to be mutually orthogonal, where the index I = 1, 2, ..., C. Each (I)ǫ = ±1,depending on the spacelike/timelike nature of the corresponding dimension.The procedure to obtain equation (2.23) can be straightforwardly generalized to C ≥ 1 case.For example, generalize equation (2.17) toγαβ = gαβ − (1)ǫ (1)nα(1)nβ − ...− (C)ǫ (C)nα(C)nβ . (B.1)Eventually the deviation of two geodesics produced by Tα ∈ Σ is derived to beC∑I=1(I)ǫ (I)nµTαT β∇α((I)nβ)≡ TαT βC∑I=1(I)ǫ (I)nµ (I)Kαβ. (B.2)i.e. we have defined C tensors(I)Kαβ = γ µα γ νβ ∇µ((I)nν), where I = 1, ..., C. (B.3)On the other hand, NVP takes the “change rate” of the direction of unit normal vector alongΣ to serve as extrinsic curvature. When C > 1, the direction of C dimensional orthogonal spaceis characterized by the wedge form(1)nµ ∧ ... ∧ (C)nν . (B.4)163Appendix B. Extrinsic Curvature as Geodesics Deviation: C ≥ 1 CaseTherefore, under NVP, the extrinsic curvature is18Kδµ...ν = γ αδ ∇α((1)nµ ∧ ... ∧ (C)nν), (B.5)which is a tensor of rank C + 1. As the form implies, Kδµ...ν does not lie within Σ. However, wecan prove that NVP is still equivalent to GEP.First, we define the notationdδ ≡ γ αδ ∇α. (B.6)Then it is easy to show thatKδµβ...ν = dδ((1)nµ ∧ (2)nβ ∧ ... ∧ (C)nν)=(d(δ)(1)nµ)∧ (2)nβ ∧ ... ∧ (C)nν+ (1)nµ ∧(d(δ)(2)nβ)∧ ... ∧ (C)nν + (1)nµ ∧ (2)nβ ∧ ... ∧(d(δ)(C)nν). (B.7)i.e. Leibniz rule. And of course the operation dδ should be excluded from the wedge’s. We use abracket on δ as d(δ) to express this fact explicitly.Before we continue, let us introduce some notationsNµν = gµν − γµν =C∑I=1(I)ǫ (I)nµ(I)nν . (B.8)(I)Nµν = Nµν − (I)ǫ (I)nµ(I)nν =C∑J=1,J 6=I(J)ǫ (J)nµ(J)nν . (B.9)Noticing that (I)nν(I)nν = (I)ǫ = constant along Σ, we have(dδ(I)nν)(I)nν = 0, (B.10)which impliesdδ(I)nν = γ αδ ∇α(I)nν =(γ αδ ∇α(I)nµ)g µν=(γ αδ ∇α(I)nµ)(γ µν +N µν ) =(γ αδ ∇α(I)nµ)(γ µν + (I)N µν), (B.11)18Or define the extrinsic curvature via the derivative of the wedge of the D − C tangent vectors of Σ.164Appendix B. Extrinsic Curvature as Geodesics Deviation: C ≥ 1 Casewhere we have applied equation (B.10) in deriving the last equal sign. The above equation is justdδ(I)nν = (I)Kδν +C∑J=1,J 6=I(J)ǫ(J)nν (IJ)Bδ, (B.12)where we have defined(IJ)Bδ ≡((J)nµdδ(I)nµ). (B.13)Overall, the above procedure is nothing but projecting dδ(I)nν into the tangent space of Σ and theorthogonal space of Σ.Now we can use the fact... ∧ (J)nν ∧ ... ∧ (J)nµ ∧ ... = 0, (B.14)and use equation (B.12) to rewrite equation (B.7) asKδµβ...ν = (1)K(δ)µ ∧ (2)nβ ∧ ... ∧ (C)nν+ (1)nµ ∧ (2)K(δ)β ∧ ... ∧ (C)nν + · · ·+ (1)nµ ∧ (2)nβ ∧ ... ∧ (C)K(δ)ν . (B.15)Again, when doing wedge, δ is not affected. The terms above are linearly independent, thereforeKδµβ...ν is a tensor whose coefficients of linearly independent tensors are (I)Kαβ. In another word,there is a one-to-one correspondence betweenKδµβ...ν and a set of (I)Kαβ. Therefore, NVP extrinsiccurvatureKδµβ...ν is equivalent to C quantities (I)Kµν , which characterize extrinsic curvature underGEP. i.e. GEP and NVP are equivalent for any codimension.165


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