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Dimensional reduction and spacetime pathologies Slobodov, Sergei 2010

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Dimensional Reduction and Spacetime Pathologies by Sergei Slobodov M.Sc., The St. Petersburg Technical University, 1985 M.Sc., The University of British Columbia, 2003 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Physics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) December 2010 c© Sergei Slobodov 2010 Abstract Dimensional reduction is a well known technique in general relativity. It has been used to resolve certain singularities, to generate new solutions, and to reduce the computational complexity of numerical evolution. These ad- vantages, however, often prove costly, as the reduced spacetime may have various pathologies, such as singularities, poor asymptotics, negative en- ergy, and even superluminal matter flows. The first two parts of this thesis investigate when and how these pathologies arise. After considering several simple examples, we first prove, using pertur- bative techniques, that under certain reasonable assumptions any asymptot- ically flat reduction of an asymptotically flat spacetime results in negative energy seen by timelike observers. The next part describes the topological rigidity theorem and its consequences for certain reductions to three dimen- sions, confirming and generalizing the results of the perturbative approach. The last part of the thesis is an investigation of the claim that closed timelike curves generically appearing in general relativity are a mathematical arti- fact of periodic coordinate identifications, using, in part, the dimensional reduction techniques. We show that removing these periodic identifications results in naked quasi-regular singularities and is not even guaranteed to get rid of the closed timelike curves. ii Statement of Collaboration The research covered by this thesis is a collection of projects done either alone or in collaboration with or under supervision of Kristin Schleich, Don Witt and Johan Brannlund. Specifically, chapter 3 is largely an unassisted effort, most of the chapter 4 is a presentation of the work done in collab- oration with Kristin Schleich, Don Witt and Johan Brannlund. Chapter 5 describes and extends the results first obtained by Kristin Schleich and Don Witt. Chapter 6 describes my original work, with guidance and feedback provided by Kristin Schleich. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Statement of Collaboration . . . . . . . . . . . . . . . . . . . . . iii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . x Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1 Causality and Hyperbolicity . . . . . . . . . . . . . . . . . . 7 2.2 Energy Conditions . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.1 Weak Energy Condition . . . . . . . . . . . . . . . . . 9 2.2.2 Null Energy Condition . . . . . . . . . . . . . . . . . 9 2.2.3 Average Null Energy Condition . . . . . . . . . . . . 10 2.2.4 Dominant Energy Condition . . . . . . . . . . . . . . 11 2.2.5 Timelike Convergence and the Strong Energy Condi- tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Asymptotic Flatness . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Total Energy and Angular Momentum . . . . . . . . . . . . 15 2.5 Axisymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.6 Dimensional Reduction . . . . . . . . . . . . . . . . . . . . . 17 3 Examples of Dimensional Reduction . . . . . . . . . . . . . . 22 3.1 Schwarzschild Blackhole . . . . . . . . . . . . . . . . . . . . . 22 3.2 Charged Blackhole . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3 Spinning Blackhole . . . . . . . . . . . . . . . . . . . . . . . 29 3.4 Axisymmetric Gravitational Radiation . . . . . . . . . . . . 31 iv Table of Contents 3.5 Five-dimensional Schwarzschild-like Spacetime . . . . . . . . 33 3.6 Higher-dimensional Schwarzschild Spacetime . . . . . . . . . 33 3.7 Effects of the Cosmological Constant . . . . . . . . . . . . . 34 3.7.1 De Sitter and Anti-de Sitter Spacetimes . . . . . . . . 35 3.7.2 Schwarzschild-(Anti-)de Sitter . . . . . . . . . . . . . 35 3.7.3 Four-dimensional AdS Soliton . . . . . . . . . . . . . 36 3.8 Doubling Reduced Spacetime . . . . . . . . . . . . . . . . . . 38 3.8.1 Doubling Reduced Minkowski Spacetime . . . . . . . 38 3.9 Axisymmetric Conformal Reduction . . . . . . . . . . . . . . 40 4 Violations of Energy Conditions in Dimensionally Reduced Spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.1 Perturbative Calculation . . . . . . . . . . . . . . . . . . . . 43 4.2 Violation of the Dominant Energy Condition in Axisymmetric Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5 Topological Restrictions on the Properties of Dimensionally Reduced Spacetimes . . . . . . . . . . . . . . . . . . . . . . . . 54 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.2 Topological Censorship . . . . . . . . . . . . . . . . . . . . . 55 5.3 PTC Corollary for Three-dimensional Spacetimes . . . . . . 56 5.4 Topological Rigidity of U(1) Reduction to Three Dimensions 58 5.5 No AF U(1) reduction from Four to Three Dimensions . . . 60 5.6 Examples of Reduction . . . . . . . . . . . . . . . . . . . . . 61 5.7 No Blackhole Horizons in Reduction to Three Dimensions . . 62 5.8 Considerations for Axisymmetric Numerical Schemes . . . . 63 6 Unwrapping Closed Timelike Curves . . . . . . . . . . . . . 66 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 6.2 Spacetimes with CTCs . . . . . . . . . . . . . . . . . . . . . 68 6.3 Unwrapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 6.3.1 Singular Spacetimes . . . . . . . . . . . . . . . . . . . 73 6.3.2 Quasi-regular Singularities in a Flat Spacetime . . . . 76 6.3.3 Singularities Created by Unwrapping . . . . . . . . . 79 6.4 Unwrapped Gott Spacetime . . . . . . . . . . . . . . . . . . 80 6.4.1 Construction . . . . . . . . . . . . . . . . . . . . . . . 81 6.4.2 CTCs of the Gott Spacetime . . . . . . . . . . . . . . 83 6.4.3 Unwrapping Gott Spacetime . . . . . . . . . . . . . . 84 6.5 Unwrapping Gödel Spacetime . . . . . . . . . . . . . . . . . 86 6.5.1 Unwrapping Procedure and Circular CTCs . . . . . . 87 v Table of Contents 6.5.2 Remaining Circular CTCs . . . . . . . . . . . . . . . 88 6.6 Multiple Snwrapping of the Gödel Spacetime . . . . . . . . . 89 6.6.1 Double Unwrapping . . . . . . . . . . . . . . . . . . . 91 6.6.2 Multiple Unwrapping . . . . . . . . . . . . . . . . . . 92 6.6.3 Sector-like CTCs in the Gödel Spacetime . . . . . . . 94 6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 vi List of Figures 2.1 Conformal diagram of an infinite Minkowski universe. Cour- tesy of Wikimedia Commons. . . . . . . . . . . . . . . . . . . 8 3.1 A schematic representation of axisymmetric dimensional re- duction and doubling of the Schwarzschild blackhole. One spatial slice is shown. Arrows show the orbits of the Killing vector φ. Spherical horizon r = 2M in three dimensions be- comes a half-circle after reduction, connected to the boundary at sin θ = 0. After doubling, the spacetime has no boundary and there are non-contractible causal curves starting and end- ing on I in this spacetime. One such curve is shown. . . . . 25 6.1 Unwrapping Minkowski space: a) pick a cylindrical chart, b) remove the fixed points of φ at r = 0, c) go to the universal cover by extending the range of φ to all reals. A t = z = 0 slice is shown as a collection of Cartesian charts with y = 0, x ≥ 0 removed and the surfaces φ = 2pi of one chart identified with the surface φ = 0 of the next chart. φ̄ is the global angular coordinate, φ̄ ∈ R. . . . . . . . . . . . . . . . . . . . . . . . . 77 6.2 A visualization of a two-dimensional spatial slice of the un- wrapped Minkowski space. . . . . . . . . . . . . . . . . . . . . 77 6.3 Conical wedge identification choices in the Gott spacetime. Wedge fill lines indicate the identified points. The strings are shown at the moment of the closest approach. Left: Gott identification, where CTCs are manifest (one CTC is shown). Center: Cutler identification, used to prove the existence of points not lying on any CTCs. Right: Carroll identification, used to visualize the existence of CTC-free spacelike hyper- surfaces, as the opposite sides of the wedge are identified at equal times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 vii List of Figures 6.4 Unwrapping Gott spacetime. a) The x = y = 0 subspace is removed and the universal cover is constructed by patching multiple Cartesian charts together. The former CTCs (in- dicated by the vertical arrows) are now open curves passing from chart to chart. The identifications between charts are in- dicated by the wavy arrows. b) An alternative way to unwrap: a single Gott chart is matched to a collection of Minkowski charts. The former CTCs wrap around just the two strings in the single Gott chart. . . . . . . . . . . . . . . . . . . . . . 85 6.5 Normalized concentric CTCs of radius R illustrated in the co- ordinates ( x 2R , t 2 √ 2 tan−1 sinhR ) . The CTCs lie inside a square with the side equal to two, but outside of a circle inscribed into it. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.6 Constructing the homotopy equivalence of the Gödel space after double unwrapping. We start with the (t, x, y = 0) sub- space with two points removed (left), then retract the space onto first two circles connected by a line segment, then the “figure 8”, and finally construct the universal cover of the “fig- ure 8” (only the first four levels of nodes are shown). Each edge corresponds to a circle wrapping around one of the two singularities in the original space, so the unwrapped space contains infinitely many singularities. . . . . . . . . . . . . . . 92 6.7 A tessellation of the two-dimensional quotient space of the Gödel spacetime. Each vertex corresponds to a timelike line in the full spacetime and each circle is an image of a closed null curve R = Rmin. The tessellation is dense enough to make any circular CTCs wrap around at least one such line, as shown. Once the lines are removed and the resulting non- simply connected spacetime is lifted into the full spacetime and then into its universal cover, no circular CTCs are present in this “multiply unwrapped” spacetime. . . . . . . . . . . . . 93 6.8 A visualization of the two-dimensional quotient space of the Gödel spacetime unwrapped at three points at once. Only a small patch of this space is shown, as unwrapping around two or more points results in a countable infinity of singular boundaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 viii List of Figures 6.9 Images of sector-like closed null curves in the (x, t) coordi- nates for a range of values R. Arrows indicate the future null directions. For large R the curve tends to a triangle with a fixed minimum angle. The angle increases as R goes down, and for R close to the minimum value of sinhRmin = 1 the null curve has to wrap around the origin several times to compensate for the time lost along the radial paths. . . . . . 95 ix Acknowledgements I am grateful to my thesis supervisor Dr. Kristin Schleich for firmly guiding me through the innumerable highs and lows of research and thesis writing and for her patience and encouragement at those times when the goal of finishing the job seemed unattainable. I am thankful to Dr. Don Witt for patiently explaining even simple things and giving me praise even when I hardly merited any. I enjoyed working together with Dr. Johan Brannlund, the closest to a genuine peer I had in over 15 years. I have received help and encouragement from a number of graduate students in the department, especially Kory Stevens, Henry Ling and Brian Shieh. I am thankful to Dr. Mark Van Raamsdonk for his advice to tough it out when I had all but lost any hope of finishing my degree, and to the rest of my supervisory commit- tee, Dr. Ingrid Stairs and Dr. Douglas Scott, for many helpful comments. I have received a lot of help over the years from the Freenode IRC chan- nels #math and #not-math, especially from the brilliant Eric Peterson, a graduate student at UCB as of this writing. I thank Robert Smith for his help with final editing and typesetting. I thank my son, Ilia, who not only provided some tricky illustrations for this thesis, but also served as a sounding board for my semi-coherent attempts to explain my work, and my daughter Alexandra, for her uncondi- tional love and for imagining me to be much smarter than I have any hope of being. There are no words to express my love and gratitude to my wife Maria, who bore the brunt of my decision to go to school for what ended up to be over nine years and has kept our family together through all this time. Research described in this thesis was supported, in part, by NSERC PGS B and UBC UGF scholarships. x Dedication To my mother, who believed in me even when I didn’t. xi Chapter 1 Introduction Einstein’s general relativity relates curvature of the spacetime to its matter content: the Einstein equation states that the Einstein curvature tensor Gab is proportional to the stress-energy tensor Tab of the matter, i.e., Gab ∝ Tab. This relation can be interpreted in two ways: one can calculate the stress- energy tensor if the spacetime metric is given, or one can specify the desired stress-energy tensor and solve for the spacetime metric. The former ap- proach is easy and determines the stress energy tensor uniquely, but the stress-energy tensor corresponding to an arbitrary metric is not guaranteed to describe anything like the “normal” matter we are accustomed to, such as stars, planets, electromagnetic radiation etc. It might, for example, cor- respond to negative energy, or to superluminal matter flows, often described informally as “tachyons”. A more useful problem is, in a sense, an opposite one: one specifies the desired matter content and the curvature in space at a given point in time (known as the initial surface) and uses it to calculate what happens to the matter and the space as it evolves in time. This is known as the initial value problem or Cauchy problem in general relativ- ity. Due to the non-linearity of the Einstein equation, the Cauchy problem is impossible to solve in closed form in all but trivial cases, so numerical methods must be used instead. In fact, the situation is even worse than that. The Cauchy problem has only been proven to be well-posed — to have a unique solution, at least for some finite time, and to have continuous dependence on initial conditions — only when the stress-energy tensor on the initial surface is well behaved. Specifically, it must describe subluminal matter sources with non-negative energy density. This is known as the domi- nant energy condition. The initial surface must not have naked singularities, those singularities not inside a blackhole. If any of these restrictions do not hold, the spacetime in question and its matter content are often described as pathological. While it is not out of the question that some pathological matter sources can be used to construct a well-posed Cauchy problem, the issue has to be investigated on a case-by-case basis. 1 Chapter 1. Introduction The motivation for this thesis was initially provided by the need to rec- oncile the 2 + 1 + 1 numerical formalism (see below) with the topological censorship theorems, discussed in section 5.2. The 2 + 1 + 1 numerical schemes aim to exploit the axisymmetry of some initial data sets in order to reduce the computational complexity of constructing a four-dimensional spacetime from an initial data set (see e.g., [1]). The standard approach to the Cauchy problem in general relativity (GR) is known as the 3 + 1 split [2], where 3 is the dimension of the spatial slice being evolved along a (one-dimensional) time coordinate. In the 2+1+1 approach one first dimen- sionally reduces the Einstein equation from 4 to 3 dimensions, by trading the dimension corresponding to the axisymmetry for additional terms in the stress-energy tensor. The resulting three-dimensional Einstein equation is then subjected to a 2 + 1 split, where now a two-dimensional spatial slice is evolved in time. The four-dimensional spacetime can then be reconstructed from the evolved three-dimensional one, if desired. An example of a problem where the 2 + 1 + 1 approach was hoped to produce interesting results is the head-on collision of two static or co-axially rotating blackholes. However, it follows from the topological censorship the- orems originally proven in the 1990s that the Cauchy surface in a globally hyperbolic (2 + 1)-dimensional spacetime without negative energy must ei- ther be simply connected — any loop can be contracted into a point — or have topology of a circle [3]. This condition is violated by axisymmetric re- duction of a two-blackhole spacetime. To avoid a contradiction, one is then led to suspect that the stress-energy tensor induced by the reduction in the reduced spacetime results in a negative energy density. The first step was to check this suspicion on some simple axisymmetric spacetimes. This investigation led at first to a surprising (or, in the hind- sight, maybe not so surprising) result, that even a spacetime as simple as the spherically symmetric vacuum blackhole in four dimensions, when di- mensionally reduced with respect to one of its rotational symmetries, yields a spacetime where the stress-energy tensor has some unusual properties. In particular, it has negative energy density. It also appears superluminal to some timelike observers. Adding electric charge, angular momentum or even gravitational radiation does not cure the problem, as the pathologies of the stress energy tensor of the reduced spacetime persist, at least asymptotically. Given that all the asymptotically flat reductions of various asymptot- ically flat spacetimes resulted in a pathological stress energy tensor, it is reasonable to suspect that this is a general feature of such reductions. To that end, we next investigated the conditions under which the presence of pathologies can be proven. One approach is to consider any asymptotically 2 Chapter 1. Introduction flat spacetime metric to be asymptotically a perturbation on the Schwarz- schild metric, and estimate the contribution of such perturbations to the induced stress-energy tensor of the reduced spacetime. We have proven that the weak energy condition (WEC) is violated, subject to some reasonable as- sumptions, for the axisymmetric dimensional reduction from any dimension D > 3 (see chapter 4). Since an axisymmetric reduction creates a boundary in the reduced spacetime where the symmetry axis used to be, care must be taken when calculating quantities such as the total energy of the reduced spacetime, as there is no closed surface ((D − 2)-sphere) over which the correspond- ing Komar integral can be calculated (see section 2.4). Fortunately, for the standard dimensional reduction the boundary is regular — the curvature of the reduced spacetime remains bounded near the boundary. This allows one to “double” the spacetime and remove the boundary by gluing another copy of it along the boundary. Now the total energy of a spacetime (also known as the ADM mass or ADM energy, after Arnowitt, Deser, and Mis- ner, who originally derived a Hamiltonian formulation of general relativity [4]) and other asymptotic quantities can be computed, and, surprisingly, the total energy vanishes identically after the reduction. This suggests that if one were to apply a suitable version of the positive energy theorem to the reduced spacetime (decribed in section 4.2), it follows that the reduction breaks the dominant energy condition, unless the reduced spacetime is flat. The perturbative calculation of the (pathological) induced stress-energy tensor has motivated a search for an underlying reason for the violations, which resulted in the topological rigidity theorem, formulated and proven in [5]. The theorem states that any globally hyperbolic bundle reduction to an asymptotically flat (AF) or an asymptotically locally (ALADS) space- time that satisfies the null energy condition (NEC) restricts the topology of the domain of outer communications of both the original and the re- duced spacetimes to just two possible cases when the reduced spacetime is three-dimensional: that of a disk or an annulus (a disk with a hole, no matter how small, cut in it). A corollary to the theorem is that there is no U(1) reduction of a four-dimensional AF globally hyperbolic spacetime to a three-dimensional AF or ALADS spacetime that satisfies NEC. The domains of applicability of the perturbative argument and the topo- logical rigidity theorem overlap, but are not identical. The perturbative cal- culation shows that any axisymmetric reduction of a (D > 3)-dimensional asymptotically flat axisymmetric spacetime to a (D−1)-dimensional asymp- totically flat spacetime results in a pathological stress-energy tensor. The topological rigidity theorem implies that a U(1) reduction of a four-dimensional 3 Chapter 1. Introduction AF spacetime with a U(1) Killing vector to a three-dimensional AF space- time must include negative energy. The above conclusion has significant implications for any 2+1+1 numer- ical scheme, as the initial value formulation in this case is not guaranteed to be well posed or even to exist. The proof of existence, uniqueness and con- tinuous dependence of evolution on initial data assumes subluminal matter flows with everywhere non-negative energy density [2]. A numerical im- plementation of the 2 + 1 + 1 approach that does not address these issues explicitly is likely to face serious obstacles. The 2+1+1 approach has been attempted by different groups [1, 6], with only a modest success, and it has been all but superceded by more general (and now more successful) fully (3+1)-dimensional techniques, where the amount of calculations is reduced when the axisymmetry is taken into consideration [7–9]. It is possible that some of the problems encountered by the pioneers of the 2+1+1 schemes are at least in part due to the pathological properties of the reduced spacetimes. While axisymmetric asymptotically flat reductions of asymptotically flat spacetimes necessarily result in negative energy and superluminal matter flows, other reductions do not have to. For example, one can avoid neg- ative energy by conformally rescaling the reduced metric. We analyze a well-known conformal rescaling of the reduced metric by the norm of the axisymmetric Killing vector in section 3.9. In this case there is a trade- off for keeping the energy density positive everywhere: the rescaled metric has a naked timelike singularity on the axis of symmetry and is no longer asymptotically flat, making it problematic to define asymptotic quantities, such as mass and charge. There are also reductions that do not create any (additional) pathologies. Some of them are described in section 5.6. For example, an axisymmetric reduction of a four-dimensional black string results in a well-behaved three- dimensional spacetime. This is not a counterexample to the corollary, as this spacetime is not asymptotically flat to begin with. Another example is the dimensional reduction of the anti-de Sitter (AdS) solution. There both the original and the reduced spacetimes are asymptotically AdS, not asymptotically flat. While analyzing the properties of dimensional reductions of different exact solutions, I have come across the fact that a simple reduction of the Gödel spacetime [10] with respect to a timelike Killing vector with open orbits produces a flat Euclidean space. This provided a motivation to inquire how the well-known Gödel closed timelike curves (CTC) [10, 11] arise from such a simple spacetime, given that there exists a nowhere singular timelike Killing vector field with a trivial topology (R), and whether the existence 4 Chapter 1. Introduction of CTCs in the total spacetime can be inferred from the properties of the reduced spacetime. While this question remains open, a related claim was investigated, namely that of Cooperstock and Tieu [12], that the CTCs of the Gödel spacetime, as well as those of the Gott spacetime [13], are nothing but an artifact of a bad coordinate identification. To that end, I have recast their suggestion to remove the “artificial” periodic identification of the angular coordinate in the polar chart into the formal “unwrapping” procedure. This procedure is investigated in detail using the simple examples of Minkowski, Gott and Gödel spacetimes. I have shown that unwrapping necessarily produces a naked quasi-regular singularity in a previously regular spacetime, thus confirming the accepted point of view that CTCs are not a mathematical artifact, but a necessary consequence of general relativity, contrary to the claims of Cooperstock and Tieu. While unwrapping the Gott spacetime does indeed remove all CTCs from it, the unwrapped Gödel space still had CTCs through every point. I next asked whether there is any way to remove the remaining CTCs from the Gödel spacetime, without creating an artificial boundary or matching it to a CTC-free spacetime with a different metric. As a result, I have constructed a “multiple unwrapping” procedure, where the Gödel space is first tessellated with a regular lattice of timelike lines, such that any CTC has to wrap around at least one of them, then the timelike lines are removed from the spacetime, and finally the resulting multiply connected spacetime is replaced with its universal cover, which no longer contains CTCs, but instead has a countable infinity of naked quasi-regular singularities. However, at any regular point the spacetime metric is still that of Gödel. While I have not proven that the suggested tessellation removes all CTCs from Gödel, this conjecture is supported by analyzing a different kind of a CTCs, a sector-like CTCs, which are also removed by this multiple unwrapping. As the results presented in this thesis show, dimensional reduction, while a useful tool, is also a dangerous one, as it more often than not results in pathologies of one kind of another, even if the original spacetime is per- fectly regular. On the flip side, various dimensional reductions can also cure (or mask) such pathologies as singularities, CTCs and “bad” stress-energy tensor. The thesis is structured as follows: chapter 2 describes the standard definitions, results and techniques that are extensively used in the rest of the thesis. Another important concept referred to in this thesis many times is that of singular spacetimes. The discussion of singularities is discussed in section 6.3.1. 5 Chapter 1. Introduction Chapter 3 analyzes the properties of several examples of axisymmetric reduction and the resulting pathologies, if any. Chapter 4 describes the perturbative arguments for the violations of the energy conditions. The topological reasons are presented in chapter 5. Finally, the investigation of CTC unwrapping is the subject of chapter 6. Throughout the thesis, we assume that the Einstein equation holds in any spacetime under consideration, and we use the units where 8piG/c2 = 1 so that the Einstein equation in the presence of the cosmological constant reads Gab+Λgab = Tab. Thus some of the expressions may differ from those given in other places by a factor of 8pi. 6 Chapter 2 Background We make repeated use of a number of concepts in general relativity, such as global hyperbolicity, energy conditions, asymptotic flatness and axisymme- try. While these are described in many standard texts on the subject, some of the definitions often differ slightly between them. This section reviews the necessary concepts and results. The formalism of dimensional reduction of a spacetime admitting a Killing vector is also reviewed. In the following we use a concept of a spacetime-with-boundary. It is obtained from a “usual” non-compact spacetime by attaching a boundary to it. The metric of such a spacetime can be conformally rescaled in a way that brings the boundary at infinity, usually denoted as I , to a finite distance, making its structure easy to see on a diagram. For AF spacetimes one makes a distinction between past and the future infinities, I −0 and I + 0 respectively, (see section 2.3). One such diagram is shown on Fig. 2.1. 2.1 Causality and Hyperbolicity A spacetime M is strongly causal if for any point p ∈ M there exists a neighborhood of any point p ∈ M through which no timelike curve passes more than once. A spacetime with boundary M ′ is globally hyperbolic if it is strongly causal and the sets J+(p,M ′)∩J−(q,M ′) (i.e., the intersection of the causal future1 of p and the causal past of q) are compact for all points p, q ∈ M ′. A Cauchy surface V ′ is a spacelike hypersurface such that every non-spacelike curve intersects this surface exactly once. A partial Cauchy surface is a surface that satisfies the weaker condition that each non-spacelike curve intersects the surface at most once2. Note V ′ for a manifold with boundary 1The timelike future (causal future) of a set S relative to U , I+(S,U) (J+(S,U)), is the set of all points that can be reached from S by a future directed timelike curve (causal curve) in U . The interchange of the past with future in the previous definition yields I−(S,U) (J−(S,U)). 2These definitions are the usual extensions of those for manifolds without boundary used in proofs of topological censorship for locally asymptotically anti-de Sitter spacetimes. 7 2.1. Causality and Hyperbolicity Figure 2.1: Conformal diagram of an infinite Minkowski universe. Courtesy of Wikimedia Commons. will have a boundary on I . The domain of outer communications (DOC) is the portion of a space- time M which is exterior to event horizons. Precisely D = I−(I +0 ) ∩ I+(I −0 ) for a connected component I0 for an AF spacetime and D = I−(I0) ∩ I+(I0) for an ALADS spacetime is the subset of M that is in causal contact with I . Note that D is the interior of an (n+1)-dimensional D′ = D ∪ I and that D′ is itself a globally hyperbolic spacetime with boundary. A second-order linear partial differential equation (PDE) for a field φ is said to be hyperbolic if and only if it can be expressed in the form ∇2φ + Aa∇aφ+ Bφ+ C = 0, where Aa is a smooth vector field [14, Eq. 10.1.19]. A second order quasi-linear3 PDE for a field φ is said to be hyperbolic if in can be expressed in the form gab(x, φ,∇cφ)∇a∇bφ = F (x, φ,∇aφ) [14, Eq. 10.1.21]. Here x represents the coordinates on the manifold and φ may consist of multiple fields. A concept important for numerical evolution is strong hyperbolicity, for it limits the propagation speed of high-frequency perturbations. See [15]. The precise definition of the term is discussed in [16], the important part for our purposes is that the ADM formulation of the initial value problem can be cast into a strongly hyperbolic form. Causality and hyperbolicity are essential if one wants to reconstruct the full spacetime from a given initial surface. Some of the spacetimes considered 3Linear in second derivatives. 8 2.2. Energy Conditions in this thesis, most notably the Gödel one, contain CTCs, and so are neither globally hyperbolic, nor causal. 2.2 Energy Conditions To be physically relevant, a matter source must have a stress-energy tensor that obeys a number of restrictions. Since most of this thesis revolves around violations of energy conditions, we review these conditions on the stress- energy tensor and the implications of their violations in some detail. Further discussion can be found in [2, §4.3]. 2.2.1 Weak Energy Condition In most cases observed classical physical matter has non-negative energy density. Energy density is observer-dependent, and for a timelike test ob- server with a velocity ua (normalized to gabu aub = −1) moving through the matter source with the stress-energy tensor Tab it is ρ = Tabu aub. (2.1) The weak energy condition (WEC) can then be formulated as ρ ≥ 0 (2.2) for all timelike observers. It is usually assumed that the vector ua is future- pointing, but since ρ is quadratic in ua, this restriction is not necessary. If the Einstein equation (without the cosmological constant written separately, Gab = Tab) is presumed to hold, then the energy density can be inferred from the spacetime curvature, ρ = Gabu aub. Often there exist an orthonormal frame at each point where the stress en- ergy tensor can be written in a canonical form T ab = diag{ρ0, P1, . . . , Pn−1}, where ρ0 is the energy density and (P1, . . . , Pn−1) are the components of the spatial momentum “vector” in this frame. This type of the stress energy ten- sor is called Type I in [2]. In this case, while the magnitude of the observed energy density varies between observers, its sign remains non-negative as long as ρ0 ≥ 0 and ρ0 ≥ −Pi for all i. 2.2.2 Null Energy Condition The null energy condition (NEC) is a variation of the WEC, but applied to null, rather than timelike, vectors: Tabk akb ≥ 0, (2.3) 9 2.2. Energy Conditions where gabk akb = 0. For a diagonal stress-energy tensor the NEC holds as long as ρ0 ≥ −Pi for all i. This condition is weaker than the weak energy condition (for example, negative energy density is allowed, as long as the pressure is strong enough to compensate). The NEC is necessary for the proofs of the singularity theorems, because it ensures focusing of null geodesics. Indeed, according to the Raychaudhuri equation [2] the null geodesic expansion θ is ka∇aθ = −Rabkakb + ωabωab − σabσab − 1 3 θ2, (2.4) and, given that vorticity of null geodesics ωab is zero, it becomes (using the fact that for null vectors Rabk akb = Tabk akb, whether or not the cosmological constant is present) ka∇aθ = −Tabkakb − σabσab − 1 3 θ2, (2.5) which is manifestly negative. So in this case the NEC is equivalent to the null convergence condition. 2.2.3 Average Null Energy Condition Both weak and null energy conditions can be violated in isolated regions of space. The well known Casimir effect [17], an attractive force between two parallel plates in vacuum, is one such example. Other examples include certain kinds of electromagnetic radiation states [18] and in certain entangled states in quantum mechanics [19]. In all these examples the localized regions of negative energy are surrounded by the regions of positive energy. If there is enough of it to “outweigh” the negative energy, then at least some of the results that assume NEC can still go through. To that end one can formulate the average null energy condition (ANEC), which holds when the average energy density measured along all complete null geodesics is non-negative: Add the mathematical formulation of ANEC:∫ ∞ −∞ Rabk akb(η′, η′) ds ≥ 0 (2.6) along any complete null geodesic s 7→ η(s) with tangent ka. This energy condition is sufficient to rule out a number of exotic phe- nomena in GR. In particular, the ANEC restriction on the stress-energy tensor is sufficient to prove the topological censorship theorems [20]. It is, however, possible to construct examples of spacetimes where ANEC does not hold [21]. 10 2.2. Energy Conditions 2.2.4 Dominant Energy Condition Another reasonable restriction on physical matter is that it does not travel faster than light or non-tachyonic. Mathematically this can be expressed as follows: the momentum density pa of the matter with the stress energy tensor Tab seen by a future-pointing timelike observer with velocity u a is pa = T ab u b. (2.7) This vector is non-spacelike when papb ≤ 0. (2.8) If this condition holds for all physical (i.e., future-pointing and non-spacelike) observers, and theWEC also holds, then the dominant energy condition (DEC) is said to hold. The DEC is necessary to prove the existence of the Cauchy problem in general relativity [2], as it establishes a lightcone and prevents matter from coming from infinity into the future domain of dependence. If this condition is not imposed, matter may essentially pop in and out of existence at will. Specifically, the conservation theorem [2, §4.3] states that Theorem. If the energy-momentum tensor obeys the dominant energy con- dition and is zero on a timelike boundary of a compact region, and on the initial surface of the region, then it is zero everywhere in the region. The DEC is only a sufficient condition for the Cauchy development to exist. It may still be possible to have a Cauchy development in some cases where the DEC is violated, however each specific case has to be analyzed separately. 2.2.5 Timelike Convergence and the Strong Energy Condition As discussed in section 2.2.2, the NEC implies the null convergence condi- tion. For a similar condition to hold for the timelike vectors, the following restriction on the stress energy tensor is required: Rabu aub = Tabu aub + 1 2 T − Λ ≥ 0. (2.9) This timelike convergence condition is known as the strong energy con- dition SEC when Λ = 0. When this condition holds, gravity appears as 11 2.3. Asymptotic Flatness an attractive, rather than a repulsive force, as the geodesic expansion is necessarily negative for a hypersurface-orthogonal geodesic congruence. Out of all the energy conditions described here, the NEC and DEC are of special interest in this thesis. In particular, the NEC is a necessary condition for the topological rigidity theorem, and the DEC is the one most commonly violated by Killing reductions. 2.3 Asymptotic Flatness An asymptotically flat spacetime is a representation of an isolated system in an “otherwise flat” spacetime (see [14]). There are different ways to define an asymptotically flat spacetime. The main idea is that the curvature (and the matter density) should fall off quickly enough to appear in the leading order as a “point mass” for an observer “at infinity”. One should be able to take a weak field (Newtonian) limit and obtain the total mass, total charge and total angular momentum of such a system, given by a suitable generalization of the Gauss law. A covariant way to do this was first introduced by Penrose [22], where a physical spacetime M is conformally embedded into an unphysical spacetime M̃ with a boundary ∂M = M̃ \ M . See [2, 14] for the detailed discussion and illuminating diagrams. The boundary describes the infinity of the original spacetime, and one can then talk about observers in the neighborhood of infinity. The properties of the spatial infinity in four dimensions were investigated in detail by Ashtekar and Hansen [23, 24]. It is worth noting the well known fact that the four-dimensional case is rather special as far as timelike geodesics in a spacetime produced by a point mass (described by a generalization of a Schwarzschild metric to an arbitrary number of dimensions) are concerned. Only in four dimensions do bound and stable geodesics exist. In three dimensions the spacetime is locally flat, and the total deficit angle is below4 2pi, so the timelike geodesics necessarily start and end at infinity, except for the radial ones, which start or end in the singularity. In five or more dimensions all non-circular geodesics end either at infinity or in the singularity, and the circular geodesics are unstable against small perturbations. Thus, higher dimensional asymptoti- cally flat spacetimes are not associated with gravitationally bound orbits of compact bodies in vacuum, and, in that sense, are not as “natural” as the four-dimensional asymptotically flat spacetime. In contrast, when a nega- tive cosmological constant is introduced, such as in the Schwarzschild-AdS 4Otherwise the spacetime is closed [25]. 12 2.3. Asymptotic Flatness spacetime, stable bound orbits exist in any dimension. See e.g., [26] for the details. We start with a coordinate-independent definition of AF and ALADS spacetimes. A spacetime (M , ) is asymptotically flat (AF) if it can be conformally included into a spacetime-with-boundary M ′ = M ∪ I , with metric g′ab, such that (a) for some conformal factor Ω ∈ C1(M ′), g′ab = Ω2gab on M and Ω vanishes on I but has null gradient which is non-vanishing point-wise on I , and (b) the boundary ∂M ′ = M ′ \ M = I is a disjoint union of past and future parts I +∪I−, each having topology S2×R with R’s complete null generators. Strictly speaking, this only defines null asymptotic flatness, as the spatial asymptotic flatness requires extra conditions to be imposed [14]. Since null asymptotic flatness is all that will be required, we do not make this distinc- tion. A spacetime (M , gab) is asymptotically locally anti-de Sitter (ALADS) if it can be conformally included into a spacetime-with-boundary M ′ = M ∪I , with metric g′ab, such that ∂M ′ = I is timelike (i.e., is a Lorentzian hypersurface in the induced metric) and M = M ′ \I . The conformal factor Ω ∈ C1(M ′) satisfies (a) Ω > 0 and g′ab = Ω 2gab on M , and (b) Ω = 0 and dΩ 6= 0 point-wise on I . I may in general have multiple components. Another way to define the asymptotically flat (3+1)-dimensional space- time is through the initial value formulation [14]: Fix a positive-definite 3-metric hab and extrinsic curvature tensor Kab on the initial surface. The spacetime can only be asymptotically flat if the metric approaches the flat metric fast enough (as O(1/r)) and the extrinsic curvature falls off suffi- ciently quickly (as O(1/r2)) as r →∞. More precisely the asymptotic flat- ness conditions can be stated as follows [27]: an initial data set (Σ3, hab,Kab) is asymptotically flat if there exists a diffeomorphism (a differentiable map with a differentiable inverse) F from R3 minus a ball B1 to the initial data 13 2.3. Asymptotic Flatness surface Σ3 minus a ball B2 such that F∗hab = ( 1 + M 2r )4 δab +O ( 1 r2 ) F∗Kab = O ( 1 r2 ) , (2.10) where δab is the flat Riemannian metric and r 2 = xax a in asymptotically Cartesian coordinates xa. The pullback F∗ will be omitted for notational simplicity. We note that the conformally flat metric (1 +m/2r)4δab is spherically- symmetric — it is in fact the spatial part of the Schwarzschild metric (dis- cussed in section 3.1) written in isotropic coordinates and so possesses a U(1) Killing vector ξS . An equivalent definition exists that uses a pertur- bation on the flat background [27]: Let h̃ab = hab − δab, where hab is again the metric of an achronal initial data surface Σ3, δab is the flat Euclidean 3-metric, and h̃ab satisfies the following fall-off conditions: h̃ab = O(1/r), ∇ch̃ab = O(1/r2) and ∇c∇dh̃ab = O(1/r3). The derivatives are taken with respect to the “background” flat metric δab. The definition (2.10) can be generalized to an arbitrary number of space- time dimensions D > 3 as hab = f(r)δab +O ( 1 rD−2 ) , (2.11) with the conformal factor f(r) = ( 1 + 16piM (D − 2)SD−2rD−3 )4/(D−3) , (2.12) where Sn = 2pi(n+1)/2 Γ(n+12 ) is the surface area of an n-dimensional sphere of unit radius. Three dimen- sions is a special case: there the spacetime is always flat outside of matter sources, and the concept of asymptotic flatness loses meaning. We this definition in chapter 4 to prove that an asymptotically flat di- mensional reduction of an asymptotically flat spacetime results in a patho- logical induced stress-energy tensor. 14 2.4. Total Energy and Angular Momentum 2.4 Total Energy and Angular Momentum Owing to being asymptotically Minkowski, the asymptotically flat space- times can be ascribed a finite total energy and angular momentum. These quantities are calculated as Komar integrals of the corresponding asymptotic symmetry: time translation and rotation [14]. Specifically, total energy E of an AF spacetime can be calculated as E = − 1 8pi lim S→I0 ∫ S ∗∇atb, (2.13) where S is a (D − 2)-dimensional sphere on the initial surface Σ, ta is the asymptotically Killing time translation vector in M satisfying ∇ata = 0 in the neighborhood of I + and ∗∇atb is the (D− 2)-form dual to ∇atb (it can be written using the Levi-Civita tensor as ∗∇atb = εab..cd∇ctd). In a coordinate form this becomes E = − 1 8pi lim S→I0 ∫ S tanb − tbna |tctc| ∇atbdA, (2.14) where na is a normal to the surface and dA denotes integration over the surface. If an initial slice is given, the total energy can be expressed using the ADM mass. The ADM mass M of an asymptotically flat spacetime can be calculated from the spatial metric hab of an asymptotically flat initial data set [28] (Σ, hab,Kab) as a limit at infinity of the surface term in the ADM Hamiltonian (generalized from [14, Eq. 11.2.4] for an arbitrary D ≥ 4): MADM = 1 16pi(D − 3) limr→∞ ∮ dAa ( ∂bhab − ∂ahbb ) , (2.15) where the integral is taken over a sphere of constant r and hab is expressed in asymptotically Cartesian coordinates. It is worth noting that total energy of the spacetime as a whole is not the sum of energies of objects inside it. For example, for a spacetime consisting of a blackhole surrounded by a perfect fluid ring analyzed in [29], the “partial” Komar mass for each of the two objects is negative, even though the total energy of the spacetime is positive, as required by the positive energy theorem (described in section 4.2). The total angular momentum of an AF spacetime is J = lim S→I0 ∫ S ∗∇aξb. (2.16) Here ξb is one of the asymptotic rotational Killing vectors. 15 2.5. Axisymmetry 2.5 Axisymmetry The definitions of axisymmetry given in literature vary somewhat and are not always equivalent. The two issues are whether the axis of symmetry is included in the spacetime and if so, the dimensionality of the axis. We will here follow Carter [30], who makes a distinction between an axisymmetry and a “true” U(1) symmetry. Definition 1. A non-singular spacetime (M , g)is said to have a cyclic sym- metry (also known as U(1) symmetry) if it admits an isometry with closed orbits, i.e., there is an effective action ξ : U(1) × M → M of the one- parameter cyclic group U(1) which preserves the metric, i.e., for every point φ ∈ U(1), ξ∗g = g. It is standard to use 0 ≤ φ ≤ 2pi as the parameter of the group. The tangent vector to the orbits of ξ, ξa = (∂/∂φ)a is a Killing vector of the spacetime. Definition 2. An n-dimensional spacetime with cyclic symmetry is called axisymmetric if the set of fixed points of the isometry is a submanifold F of n − 2 dimensions, properly embedded in M . F is called the axis of symmetry. Definition 2 is required by some authors but not by others (see [30–34]), and those who do require the axis to be included in the spacetime do not always place the same set of restrictions on it. Except for the discussion in section 5.4, we will always deal with axisymmetry in this thesis. An example of a three-dimensional spacetime with a cyclic symmetry that is not an axisymmetry according to our definition is a direct product of a two-dimensional flat torus T 2 with a timelike line. In this case the spacetime is not simply connected. An example of a cyclic symmetry that is not an axisymmetry on a simply connected four-dimensional spacetime with the topology S3 × R is the Hopf fibration of S3 (see e.g., [35]), which has no fixed points5. When we discuss the U(1) symmetry we assume that it is not an axisymmetry, i.e., the action of the symmetry group on the manifold is free. In other words, all the fibers of the map pi : M /U(1) → M̄ are homeomorphic to a circle. The axis of symmetry F has several important properties listed below. Property 1. The norm of the Killing vector ξa, λ = gabξ aξb vanishes on the axis F . 5S3, of course, has other cyclic symmetries that are axisymmetries. 16 2.6. Dimensional Reduction Property 2. (Elementary Flatness Condition) On the axis of symmetry ∇aλ ∇aλ 4λ → 1. (2.17) Elementary flatness is equivalent to regularity on the axis. This is a generalization of the Minkowski case, where in the cylindrical chart the metric (outside the axis) can be written as ds2 = −dt2 + dρ2 + ρ2dφ2 + dx21 + · · ·+ dx2n−3, (2.18) where 0 ≤ φ ≤ 2pi. Here λ = ρ2. For the proof of this condition, see e.g., [32]. Property 3. F is an autoparallel ( totally geodesic) submanifold of M . That is, for any two vector fields X and Y on F Xa∇aY is tangent to F . The proof can be found in many differential geometry texts. For a succinct version, see e.g., [36]. The consequence of this is Property 4. F has identically vanishing extrinsic curvature. This property is what ensures the existence of a non-singular extension (a doubling) of the reduced spacetime, as described in section 3.8. Property 5. F is timelike, i.e., there is a timelike curve through every point. See e.g., [32, Theorem 1] for the proof. An interesting special case of axisymmetric spacetimes are those admit- ting another Killing vector ζ, tangent to the axis of symmetry. In this case the two Killing vectors commute, [ξ, ζ] = 0. Carter [30] proved that such a Killing vector exists for any stationary spacetime (in which case ζ is ev- erywhere timelike). Kerr blackhole metric is the best known example of an asymptotically flat stationary spacetime. If ζ is spacelike, we have a cylin- drical symmetry [31]. The van Stockum cylinder, discussed in section 6.2, is one example of such a spacetime. 2.6 Dimensional Reduction The reduction procedure we employ was pioneered in four spacetime di- mensions by Geroch in [37], where it was used in his solution-generating technique. The Hamiltonian approach to this reduction is discussed in e.g., 17 2.6. Dimensional Reduction [38]. We first present it for a four-dimensional spacetime, and then generalize to any number of dimensions. The formalism here applies to any spacetime with a Killing vector, the axisymmetric case is discussed at the end of the section. A related type of reduction, the conformal one, is discussed in sec- tion 3.9. The advantage of the reduction employed here is that it preserves the fall-off of the metric, which is essential for keeping the reduced spacetime asymptotically flat. Consider a four-dimensional spacetime M with metric gab. We want to identify points along the orbits of an axisymmetric spacelike Killing field ξ, which has the norm λ = ξaξa. The metric of the reduced three-dimensional spacetime is ḡab = gab − ξaξb λ . (2.19) It follows from the Gauss-Codazzi relations [2] that R̄ab = 1 2λ2 (ωaωb − ḡabωcωc) + 1 2λ ∇̄a∇̄bλ− 1 4λ2 ∇̄aλ∇̄bλ+ ḡcaḡdbRcd, ∇̄2λ = 1 2λ ∇̄cλ∇̄cλ− 1 λ ωcωc − 2Rcdξcξd, ∇̄aωa = 3 2λ ωc∇̄cλ, ∇̄[aωb] =− abcdξcRdeξe. (2.20) Here Rab = Tab− 12Tgab is the full spacetime Ricci tensor, R̄ab its counterpart in the reduced spacetime, ωa = abcdξ b∇cξd is the twist of ξ and ∇̄ is the covariant derivative on the reduced spacetime. The reduction from five to four dimensions is described in [39]. The main difference from the four-dimensional case is that the twist ωab = abcdeξ b∇dξe is conveniently described by a two-form instead of a one form. In higher dimensions calculating twist as a dual to ξa∇bξc does not reduce the number of indices, so it makes sense to use the projection on the reduced spacetime ωab = ga cgb d∇cξd instead of any other form of twist. The following describes the reduction procedure in D spacetime dimen- sions. The capital indices refer to the full spacetime and the lower case ones to the reduced spacetime. Let (M , gAB) be a D-dimensional spacetime admitting a Killing vector ξA with ∇(AξB) = 0. In the following we use ξ as a fiber of a bundle, as a vector field, and as a parameter of a curve describing orbits of ξA. The meaning, when not explicitly given, should be unambiguous from the 18 2.6. Dimensional Reduction context. One can consider M a lift of a (D − 1)-dimensional manifold M̄ with ξ as the fiber of a fiber bundle: ξ : M → M̄ . This is possible as long as all the fibers have the same topology. Since orbits of ξ are one-dimensional, there are only two choices. If the orbits are open, then the fiber bundle is a principal R-bundle, and if they are closed, then it is a principal U(1)-bundle (the rotation axis, if any, has to be treated separately). The base space M̄ is then a quotient space M̄ = M /ξ. The map from a tensor field TC..DA..B on M to a tensor field T̄ c..da..b on M̄ is an isomorphism if the tensor in the full space is horizontal and does not explicitly depend on the Lie group parameter ξ: ξ · TC..DA..B = 0 (2.21) LξT C..D A..B = 0, (2.22) where the top equation means vanishing of all possible dot products. In particular, the projection ḡab ∼= ḡAB = gAB − ξAξB λ , where λ = ξCξC is isomorphic to a tensor ḡab in the base space, and the symbol ∼= denotes the isomorphism between the tensors in base and total spaces. Similarly, ḡba ∼= ḡBA = gBA − ξAξ B λ , and ḡab ∼= ḡAB = gAB − ξ AξB λ . In the following we will use ‘=’ instead of ‘∼=’, as there is no reason to make a distinction between the two quantities. For an arbitrary tensor TC..DA..B the map is given explicitly as T̄ c..da..b = ḡ A a · · · ḡBb ḡcC · · · ḡdDTC..DA..B . (2.23) This also applies to the derivative of any ξ-independent horizontal tensor: ∇̄eT̄ c..da..b = ḡEe ḡAa · · · ḡBb ḡcC · · · ḡdD∇ETC..DA..B . (2.24) The next step is to project the Einstein equation GAB = TAB on M̄ . A symmetric tensor UAB can written in terms of the horizontal and vertical tensors as Uab = ḡ A a ḡ B b UAB + 2 λ ξ(aḡ A b)UABξ B + 1 λ2 ξAξBUABξaξb. (2.25) 19 2.6. Dimensional Reduction Thus, just like in the case of reduction from four to three dimensions, the Einstein equation in the full space is equivalent to three equations in the reduced spacetime, a tensor, a vector and a scalar one. The tensor equation involves the curvature of the reduced spacetime and so can be treated as the Einstein equation in the reduced spacetime with extra matter terms. The other two are in essence the equations of state for the scalar and vector induced fields. There are different ways to project the Einstein equation [1, 37]. For example, one can use the difference of two connections, one for the original and one for the reduced metric [14]: ∇̄a∇̄bf = ∇a∇bf − Ccab∇̄cf, (2.26) where f is a scalar in both the full and the reduced spacetimes and Ccab = 1 2 ḡcd (∇aḡbd +∇bḡad −∇dḡab) . (2.27) Alternatively, one can express the Riemann tensor of the reduced space- time directly using that for a vector field ka the identity ∇[a∇b]kc = Rdabckd holds in both the original and the reduced spacetime. We then get, after some algebra, R̄abcd = g A [ag B b] g C [cg D d] ( RABCD + 2 λ ∇AξB∇CξD + 2 λ ∇AξC∇BξD ) . (2.28) It is convenient to express the quantities in the reduced spacetime using the Killing vector ξa itself and its two lower-dimensional characteristics, the norm λ and the twist 2-form6: ωab = ḡ A a ḡ B b ∇AξB = ∇aξb + 1 λ ξ[a∇b]λ, (2.29) and (2.28) becomes R̄abcd = g A [ag B b] g C [cg D d]RABCD + 2 λ (ωabωcd + ωacωbd) . (2.30) From this expression it is clear that in the hypersurface-orthogonal case, where the twist vanishes, the Riemann tensor in the reduced spacetime is simply the image of the Riemann tensor in the full spacetime. The connection coefficients are: Ccab = − 2 λ ξ(aωb) c + 1 2λ2 ξaξb∇̄cλ, (2.31) 6Also known as the vorticity tensor if ξa is an arbitrary timelike field, see e.g., [2, §4.12]. 20 2.6. Dimensional Reduction and the Ricci tensor in the reduced spacetime R̄ab is: R̄ab = ḡ A a ḡ B b RAB + 2 λ ωacωb c + 1 2λ ∇̄a∇̄bλ− 1 4λ2 ∇̄aλ∇̄bλ. (2.32) The induced curvature tensors remain finite in the dimensional reduction with respect to a Killing vector described in this section as long as the norm of the Killing vector is non-zero. However, the latter is precisely the case on the axis of symmetry, so the axis requires a special consideration. In particular, the expression (2.28) must remain bounded in the neighborhood of the axis. This follows from the fact that for a regular spacetime the projector ḡba is bounded and the twist ωab vanishes on the axis. See also section 3.8. 21 Chapter 3 Examples of Dimensional Reduction In this chapter we dimensionally reduce a few well-known spacetimes to observe the arising pathologies in a very concrete way. While the exam- ples presented here are quite elementary, we have not been able to find a discussion of energy conditions in dimensionally reduced spacetimes in any previously published work. We begin by applying dimensional reduction to four-dimensional asymp- totically flat spacetimes. We find that, although the dimensional reduction produces a three-dimensional spacetime geometry that is asymptotically flat, it now violates all energy conditions. In addition, the resulting spacetime now exhibits a boundary at the axis of symmetry, though it can be removed by doubling. We then consider dimensional reduction of higher dimensional asymp- totically flat spacetimes and of spacetimes with cosmological constant such as (A)dS spacetime. Next we discuss removal of the boundary present in certain of these reductions by doubling. Finally we discuss an alternate reduction, the conformal reduction. This reduction satisfies the DEC but exhibits two pathologies: 1. it induces a naked timeline singularity in axisymmetric reductions and 2. the reduced spacetime is no longer asymptotically flat. These pathologies limit the usefulness of this dimensional reduction for the study of axisymmetric spacetimes. 3.1 Schwarzschild Blackhole A simple example of dimensional reduction of a non-flat axisymmetric space- time with respect to its rotational Killing vector is that of a four-dimensional Schwarzschild blackhole. Despite being probably the simplest non-trivial asymptotically flat case, this example shows all the essential properties of 22 3.1. Schwarzschild Blackhole an axisymmetric dimensional reduction preserving asymptotic flatness (the definitions of asymptotic flatness are discussed in section 2.3). Specifically, the pathologies we encounter in this case are the same ones seen in general. The Schwarzschild spacetime is a vacuum spacetime, so all energy con- ditions (section 2.2) are trivially satisfied. The Schwarzschild singularity is shrouded by an event horizon. However, the dimensional reduction un- der consideration leads to a rather pathological spacetime. Specifically, the stress-energy tensor induced in the reduced spacetime by the reduction is seen as having negative energy density by all timelike and null observers everywhere. Additionally, the momentum flow seen by timelike observers is tachyonic for timelike observers with a large enough angular velocity. Thus the null, the weak and both parts of the dominant energy conditions are violated. The reduced spacetime has a boundary, but the doubling proce- dure (section 3.8) restores the asymptotic flatness of the spacetime, so the global quantities such as the ADM mass can be calculated. The details are described in this section. The four-dimensional Schwarzschild metric in the Schwarzschild coordi- nates (t, r, θ, φ) is ds2 = − ( 1− 2M r ) dt2 + dr2 1− 2Mr + r2dθ2 + r2 sin2 θdφ2. (3.1) This coordinate chart breaks down on the event horizon, r = 2M , and there are other charts, such as the Kruskal-Szekeres one ([2], section 5.5), which cover all of the spacetime, all the way to the singularity at r = 0, but, given that we are only interested in the asymptotic properties of the reduction, the Schwarzschild chart is adequate. It covers the DOC, the part of the spacetime accessible to a remote observer. A spatial slice of the horizon is topologically a two-sphere. This manifestly axisymmetric spacetime can be reduced with respect to the Killing vector φ. The metric of the reduced spacetime ḡab = gab−r−2φaφb is ds̄2 = − ( 1− 2M r ) dt2 + dr2 1− 2Mr + r2dθ2, (3.2) where 0 ≤ θ ≤ pi. While this is simply the metric (3.1) with the last term “chopped off” (because the Killing vector φ is hypersurface-orthogonal and the coordinate system is adapted to it), if we treat it as a metric of a three- dimensional spacetime, it will no longer correspond to a vacuum solution. The induced stress-energy tensor is Ḡab = M r3 ( − tatb (tctc)2 + rarb (rcrc)2 − 2 θaθb (θcθc)2 ) , (3.3) 23 3.1. Schwarzschild Blackhole where ta, ra, θa are the coordinate vectors. In an orthonormal basis locally coinciding with the coordinate one this expression can be written as Ḡâb̂ = M r3  −1 0 00 1 0 0 0 −2   . (3.4) This stress-energy tensor can also be modeled as an anisotropic fluid with negative energy density ρ = −M/r3, equal and opposite positive radial pressure p‖ = −ρ, and a large negative tangential pressure p⊥ = 2ρ. From the form of the reduced stress-energy tensor (ρ < 0) one can already see that the WEC, NEC and DEC are all violated. To see the violations in detail, we consider a general timelike observer with the velocity ua. In an orthonormal frame it can be written as ua = γ ( t̂a + r̂av cosα+ θ̂av sinα ) , where v is the observer’s instantaneous spatial velocity relative to a static observer ta, α defines its direction of motion and γ = 1/ √ 1− v2. The energy density seen by this observer is ρ = γ M r3 ( −1 + v2 cos2 α− 2v2 sin2 α ) = −M r3 ( 1 + 3γ2v2 sin2 α ) . (3.5) This expression is negative everywhere in the reduced spacetime for a pos- itive mass Schwarzschild solution. Thus, the weak energy condition, ρ ≥ 0, is maximally violated in this case. The null case can be obtained by taking a limit v → 1 and rescaling, resulting in ρ = −M r3 sin2 α, (3.6) so the NEC is also violated. Another issue with this spacetime is the range of θ and the topology of the horizon r = 2M after reduction. Since the Schwarzschild coordinates break down at the horizon, we use the Kruskal-Szekeres chart, which remains regular on the horizon. The reduced metric in this chart is ds̄2 = 16M2 r e− r 2M ( −dt′2 + dx′2 ) + r2dθ2, (3.7) where t′2 − x′2 = −(r − 2M)e r2M . (3.8) 24 3.1. Schwarzschild Blackhole Figure 3.1: A schematic representation of axisymmetric dimensional reduc- tion and doubling of the Schwarzschild blackhole. One spatial slice is shown. Arrows show the orbits of the Killing vector φ. Spherical horizon r = 2M in three dimensions becomes a half-circle after reduction, connected to the boundary at sin θ = 0. After doubling, the spacetime has no boundary and there are non-contractible causal curves starting and ending on I in this spacetime. One such curve is shown. The horizon r = 2M , 0 ≤ θ ≤ pi is manifestly regular and corresponds to a null half-cone t′ = x′, no longer a closed surface. The set of points for which sin θ = 0 can be attached as a boundary to the reduced spacetime, thus closing the horizon. The DOC r > 2M is simply connected. One can remove the boundary by identifying θ = 0 ∼= θ = pi, but this breaks the ele- mentary flatness condition (2.17) and creates a conical singularity. A better way to remove the boundary is by “doubling” this spacetime. This corre- sponds to changing the range of θ to [0, 2pi] and identifying the end points. Then the only remaining singularity in this spacetime is the original one at r = 0. The horizon in the doubled spacetime is now S1 × R, and the DOC is no longer simply connected. The consequences of this fact are discussed in chapter 5. The process of reduction and doubling is shown on Fig. 3.1. Interestingly enough, for an (unphysical) negative mass Schwarzschild so- lution (M < 0), the weak energy condition (though not DEC) is actually satisfied in the reduced spacetime, since the energy density is now positive. Of course, the latter solution has other pathologies in both four and three dimensions: it describes a naked singularity rather than a blackhole7, and it is perturbatively unstable in four dimensions (see e.g., [40] and references therein). This example is symptomatic of how dimensional reduction can 7Though this can be ameliorated by matching to a “negative density star”. 25 3.1. Schwarzschild Blackhole create, cure, and trade spacetime pathologies. We now consider the second part of the dominant energy condition — subluminal matter flows seen by all timelike observers. We already know it is violated, since the tangential pressure exceeds the energy density in (3.4). To see which observers see superluminal matter flows we calculate the norm of observed momentum: p2 = −M 2 ( 1− 3γ2v2 sin2 α) r6 . (3.9) Thus fast-rotating observers (with the linear rotational velocity above c/ √ 3 in the static frame) see superluminal matter flow in this spacetime, this time regardless of the sign of the blackhole mass M . Superluminal matter flows present a challenge from an initial value point of view. In particular, as shown in [2, §4.3], DEC ensures that a subset of Cauchy surface has a future Cauchy development. See also [41, Appendix IV]. The light cone is necessary in order to bound from above any changes in the spatial metric during evolution and consequently to prove that the Cauchy problem is well posed, i.e., that the solution exists, is unique and continuously depends on the initial data. When the DEC is violated, the resulting apparent superluminal matter flows are often described as “non- local” when discussed in the context of solving a wave map (a non-linear generalization of the wave equation). A lack of bound on the propagation speed means that the standard techniques and results for hyperbolic systems do not apply and well-posedness must be examined on a case-by-case basis. For example, DeTurck [42] shows existence and uniqueness of solutions to the Einstein equation with invertible stress energy tensor (though not necessarily their continuous dependence on initial conditions). To consider the ANEC (2.6), we first note that any geodesic without φ-dependence is mapped into a geodesic by the reduction (and when the Killing vector is hypersurface orthogonal any geodesic initially orthogonal to the Killing vector remains so). In fact, in a general case an image of a null geodesic commuting with the Killing vector used for the reduction is still a null geodesic in the reduced spacetime. Indeed, given any ka such that kaka = 0, k aφa = 0, [k, φ] a = 0, (3.10) (that is, ka is null and unaffected by the reduction), we have for its image, 26 3.1. Schwarzschild Blackhole k̄a, 8 k̄a∇̄ak̄b = ka∇akb = 0. (3.11) Thus any null geodesic lying in a cross-section of the spacetime by φ = const remains a null geodesic after the dimensional reduction. Any such geodesic must, of course intersect the axis of symmetry, which does not present an issue, as long as the reduced spacetime is doubled. If we integrate the everywhere negative quantity Tabk akb along this geodesic we get a negative number. Thus ANEC is also violated in this example. It is also interesting to consider whether an observer in the reduced Schwarzschild spacetime would see gravity as an attractive force. In the Newtonian approximation, positive energy density is associated with attrac- tion and negative energy density is associated with repulsion. In the reduced Schwarzschild spacetime energy density is negative everywhere outside the central object. Since the stress-energy tensor induced by an axisymmetric reduction of the Schwarzschild metric is traceless, the strong energy con- dition is equivalent to the WEC (we always assume the Einstein equation to hold in the reduced spacetime). Since the WEC is violated, so is the SEC. On the other hand, since geodesics commuting with the Killing vector in the full spacetime are mapped into geodesics in the reduced spacetime, bound orbits are mapped into bound orbits, indicating attraction, rather than repulsion. This can be confirmed by calculating the geodesic expan- sion ∇aξa for a given geodesic ξa and its rate of change along the geodesic, which remains negative everywhere. A convenient way to think about this is that the negative energy density manifests itself as a sort of a “vacuum polarization”, screening the central object from distant observers, rather than being an independent source of repulsive gravity, in a way, say, the cosmological constant is. In other words, the central object still attracts, but its attractive force falls off faster than it would in a vacuum spacetime. This model, while valid for reductions to four or more dimensions, does not quite apply to reductions to three dimensions, because any three-dimensional vacuum spacetime is locally flat, the once parallel geodesics remain parallel, and locally there is neither attraction nor repulsion. Any interesting effects in three-dimensional gravity result from matter sources (including the cosmological constant) and/or from global identifications. Still, one can think of the 1/r fall-off of the “attractive 8The image of ka, written k̄a, is defined by k̄ a − 1 λ k c ξcξ a = ka. 27 3.2. Charged Blackhole force” induced by reduction from four to three dimensions as faster than the “no fall-off at all” for a vacuum spacetime. Next we calculate the mass of the reduced Schwarzschild spacetime using (2.15). For the four-dimensional Schwarzschild metric in the Schwarzschild coordinates we get E =M , as expected. Note that the integrand is −M/r2 and the integration measure is r2dΩ2. This calculation can also be applied to the reduced (and doubled) Schwarzschild spacetime. The initial data set on the reduced spacetime is induced by the reduction. The integrand is unchanged, however the boundary of the resulting two-dimensional initial surface is S1, not S2. As a result, the integral over the one-sphere gives −4piM/r, and the limit E = − 1 8pi lim r→∞ ( −4piM r ) vanishes due to the fall-off of the integrand. Thus the total energy of the reduced Schwarzschild spacetime is zero. If the positive energy theorem (see section 4.2) can be applied to this (non-flat) spacetime, we can conclude that the dominant energy condition must be violated, which is confirmed by the calculations such as (3.5) and (3.9). The main issues with the axisymmetric reduction of the Schwarzschild metric persist for other highly symmetric AF spacetimes, as well, as dis- cussed below. 3.2 Charged Blackhole In the previous section we considered the axisymmetric reduction of the Schwarzschild spacetime. This spacetime is asymptotically a good model of an isolated gravitational source of finite size, without any charge, spin or contribution from other non-gravitational interactions. The Schwarzschild spacetime is vacuum, so it is reasonable to inquire whether adding matter (with positive energy density) to the mix would prevent or alleviate the pathologies of the stress energy tensor induced by the axisymmetric reduc- tion preserving asymptotic flatness. One of the simplest cases to analyze is the charged (Reissner-Nordstrom) blackhole, where the matter content is purely due to the static spherically symmetric electric field [14]. In the Schwarzschild coordinates the metric is ds2 = − ( 1− 2M r + Q2 r2 ) dt2+ dr2 1− 2Mr + Q 2 r2 +r2dθ2+r2 sin2 θdφ2. (3.12) 28 3.3. Spinning Blackhole Here, Q is the electric charge (as seen from infinity, using the Gauss law). The reduced metric is simply ds2 = − ( 1− 2M r + Q2 r2 ) dt2 + dr2 1− 2Mr + Q 2 r2 + r2dθ2. (3.13) The stress-energy tensor of the reduced spacetime in an orthonormal frame locally coinciding with the Schwarzschild coordinates is Gâb̂ =   − ( M r3 − Q 2 r4 ) 0 0 0 ( M r3 − Q 2 r4 ) 0 0 0 − ( 2M r3 − 3Q 2 r4 )   , (3.14) or, in terms of the coordinate vectors Ḡab = M r3 ( − tatb (tctc)2 + rarb (rcrc)2 − 2 θaθb (θcθc)2 ) + Q r4 ( tatb (tctc)2 − rarb (rcrc)2 + 3 θaθb (θcθc)2 ) . (3.15) We see that the electric field adds a positive term to the overall energy density, however its contribution falls off as r−4, faster than that of the pure reduced Schwarzschild, so adding electric charge to an isolated body does not cure the pathological stress energy tensor, at least not asymptotically. Even close to the horizon the situation is not much better. Indeed, the WEC is violated when T00 < 0, or r > Q 2/M . In the non-extremal case the metric has two horizons: r− = Q2 M + √ M2 −Q2 , r+ = Q2 M −√M2 −Q2 (3.16) and the positive energy density region lies wholly inside the outer horizon r+. Thus the energy density in this spacetime is negative everywhere in the DOC (r > r+). The second part of the dominant energy condition (subluminal matter flows) is violated for r > 2Q2/M , which is barely outside the outer horizon even for near-extremal blackholes. 3.3 Spinning Blackhole Another way an axisymmetric reduction could potentially induce a non- pathological stress-energy tensor is when the Killing vector field used for 29 3.3. Spinning Blackhole reduction has a non-zero twist (2.29). A standard example of such an axi- symmetric spacetime is the rotating (Kerr) blackhole, and its charged ver- sion, the Kerr-Newman blackhole. The Kerr metric can be written as ds2 = −∆ r20 ( dt− a sin2 θdφ )2 + sin2 θ r20 (( ρ2 + a2 ) dφ− adt )2 + r20 ∆ dρ2+r20dθ 2, (3.17) where ∆ = ρ2 − 2Mρ + Q2 + a2 and r20 = ρ2 + a2 cos2 θ and M , Q, and a are respectively mass, electric charge and angular momentum. (∂/∂φ)a is a Killing vector with a non-zero norm outside the rotation axis sin θ = 0, so the metric can be dimensionally reduced with respect to it. The relevant calculations are straightforward but long, so they were generally performed with assistance of a computer algebra software. The expressions for the energy density and the norm of the momentum density seen by a timelike observer on a worldline with the unit tangent vector (ut, ur, uθ) can be expanded in the powers of O(1/r), with the three leading terms being ρ = −M r3 ( 1 + 3uθ 2 ) + Q2 r4 ( 1 + 4uθ 2 ) − 3Ma 2 r5 (( ur 2 − 7uθ2 − 2 ) cos2 θ + 6uruθ sin θ cos θ − ur2 + uθ2 ) +O ( 1 r6 ) , (3.18) and p̄ap̄a = −M 2 r6 ( 1− 3uθ2 ) + 2Mq2 r7 ( −1 + 5uθ2 ) + Q4 r8 ( 1− 8uθ2 ) + 6Ma2 r8 (( ur 2 + 8uθ 2 − 2 ) cos2 θ − 3uruθ sin θ cos θ − 6ur2 − 2uθ2 ) +O ( 1 r9 ) . (3.19) The expansion shows that these quantities display the following property: the highest order contribution comes solely from the mass term, the next order comes solely from the charge parameter, and only the one after that depends on the angular momentum. Thus, for large enough values of r, the leading order contribution coincides with that of the reduced Schwarzschild 30 3.4. Axisymmetric Gravitational Radiation metric, and so the energy conditions are still violated, at least asymptoti- cally. 3.4 Axisymmetric Gravitational Radiation All the spacetimes considered thus far in this section were either station- ary or static. It is natural to ask whether non-stationary spacetimes also violate the energy conditions, at least asymptotically. An example of a non- stationary axisymmetric spacetime is axisymmetric gravitational radiation on a stationary background. While not an exact solution, it is an asymptotic approximation of radiation from axisymmetric gravitational wave emitters. In the original spacetime gravitational radiation does not contribute to the stress energy tensor, but, just like in the Schwarzschild case, it shows up as an induced stress-energy tensor in the reduced spacetime. Given that radi- ation falls off slower than a static field, one might conceivably expect that it becomes a dominant contribution to the stress-energy tensor asymptotically and could thus “cure” the pathology induced by the axisymmetric reduction. In this section we investigate the properties of the induced stress-energy tensor in the simplest case of gravitational radiation on a flat (Minkowski) background. The radiation in the far zone and in the slow motion approximation from the source with the quadrupole moment Qab in the full four-dimensional spacetime can be described by the (perturbation) metric and is, up to a constant factor, hab = 1 r d2Qab dt2 , (3.20) where r is the distance from the source (see e.g., [14, Eq. 4.4.49]). The radiation will be axisymmetric when d2Qab/dt 2 is axisymmetric. Since we are working in the linear approximation, we can expand this term into Fourier modes characterized by their frequency ω 6= 0. For a given fre- quency ω we can write the gravitational field hab in the transverse-traceless gauge as the real part of hab(ω) = f(θ, ω) eiω(r−t) r ( eθae θ b − eφaeφb ) , (3.21) where eθa and e φ a are two of the unit vectors of the orthonormal basis locally corresponding to the spherical coordinates (t, r, θ, φ), ω is the oscillation fre- quency and f(θ, ω) depends on the details of the source of the gravitational wave. Because of the axisymmetry, only one polarization mode is present 31 3.4. Axisymmetric Gravitational Radiation (see e.g., [43]). We note that the expression (3.21) is only valid in a local coordinate chart, which is sufficient for our purposes. Because this metric is linearized, its Ricci tensor does not vanish identically. It is, however, trace- less, as radiation ought to be. Next we perform the axisymmetric reduction with respect to the Killing vector φ. It is twist-free, so the reduced metric is simply a hypersurface defined by φ = const: h̄ab = f(θ) eiω(r−t) r eθaeθb. (3.22) This reduced metric in three dimensions is no longer traceless, as the ra- diation now corresponds to the induced “matter” term, which matches the fact that there is no “true” gravitational radiation in three dimensions. The linearized Einstein equation in the reduced spacetime reads ∇̄c∇̄ch̄ab = T̄ab. (3.23) The leading order of the energy density associated with this stress-energy tensor for a timelike vector ua in the reduced spacetime is ρ̄ = −f(θ)ω(u θ)2 r2 sin (ω(t− r)) +O ( 1 r3 ) . (3.24) This induced energy density falls off as r−2, which is slower than the r−3 fall-off from the Schwarzschild reduction, as expected. Importantly, it also vanishes for static observers (uθ = 0) and so does not affect the energy density they measure. In this linearized approximation a general gravitational wave can be ex- panded in the basis of the waves of the type (3.22). Since ω 6= 0, the complete expansion for the energy density has zero mean, and so is negative “half the time”, thus violating the weak energy condition. Moreover, for static observers the reduced radiation does not contribute to the induced stress-energy tensor, so any negative energy present in the background re- duced spacetime is also present when the contribution from the reduced gravitational radiation is accounted for. Thus gravitational radiation, while significantly affecting the asymptotics of an axisymmetrically reduced space- time, does not cure the weak energy condition violations induced by the reduction, and might even make them worse. 32 3.5. Five-dimensional Schwarzschild-like Spacetime 3.5 Five-dimensional Schwarzschild-like Spacetime All the examples of axisymmetric dimensional reduction so far considered created pathologies, rather than cured them. Here we mention a rather artificial case where a somewhat pathological five-dimensional spacetime is turned into a well-behaving one by dimensional reduction. We start with a spherically symmetric spacetime with the metric ds2 = − ( 1− 2M r ) dt2 + dr2 1− 2Mr + r2dΩ23, (3.25) where dΩ23 is the metric on a unit three-sphere. This is not a Schwarzschild metric in five dimensions, since the radial fall-off is r−1 and not r−2. Its stress-energy tensor is Gab = 3M r3 ( tatb (tctc)2 + rarb (rcrc)2 ) . (3.26) While the energy conditions manifestly hold for this spacetime, the ADM mass of a t = const slice diverges. Yet an axisymmetric reduction with re- spect to one of the Killing vectors of the three-sphere produces the standard Schwarzschild metric in four spacetime dimensions. If the chosen Killing vec- tor corresponds to an S1 fiber of the Hopf fibration of S3, then the reduction is U(1) (not an axisymmetric one) and the reduced spacetime is already regu- lar, without the need for doubling. Alternatively, we can use the “standard” spherical coordinates with the metric ds2 = dψ2 + sin2 ψ ( dθ2 + sin2 θdφ2 ) , then reduce with respect to the Killing vector φ and then double the space- time to remove the axis. 3.6 Higher-dimensional Schwarzschild Spacetime In section 3.1 we analyzed the effects of an axisymmetric Killing reduction on the four-dimensional Schwarzschild blackhole. The situation is similar in higher dimensions, with only the radial fall-off of the (pathological) stress- energy tensor dependent on the dimension. Specifically, for a static spher- ically symmetric D-dimensional vacuum blackhole defined by its horizon radius rh, the metric in the Schwarzschild coordinates is ds2 = − ( 1− ( rh r )D−3) dt2 + dr2 1− ( rhr )D−3 + r 2dΩ2D−2, (3.27) 33 3.7. Effects of the Cosmological Constant where dΩ2D−2 is the metric on a (D − 2)-dimensional unit sphere. Like in four dimensions, this chart is singular on the horizon, however, it is always possible to construct a regular one, similar to the Kruskal-Szekeres chart discussed in section 3.1. If we now pick an axisymmetric Killing vector in this spacetime and perform the dimensional reduction, given that the Killing vector in question is manifestly hypersurface-orthogonal, the reduced spacetime is a subspace of the original space, with the metric ds2 = − ( 1− ( rh r )D−3) dt2 + dr2 1− ( rhr )D−3 + r 2dΩ2D−3. (3.28) After doubling the reduced spacetime to get rid of the boundary created by the axis, the term dΩ2D−3 describes a (D− 3)-dimensional sphere. The Ein- stein tensor (i.e., the induced stress-energy tensor) of the reduced spacetime is: Gab = ( rh r )D−3 [D − 2 2 ( tatb −(tctc)2 + rarb (rcrc)2 ) − θ3aθ3b (θ3cθ3 c)2 − · · · − θD−1aθD−1b (θD−1cθD−1 c)2 ] , (3.29) where (θ3 · · · θD−1) are the spherical coordinates on the (D−3)-dimensional unit sphere. We see that the WEC is always violated, but the tangential pressures do not exceed the energy density for D > 4, so in higher di- mensions no observers in the reduced spacetime see superluminal matter flows. Consequently the only pathology of the reduction of a negative mass higher-dimensional Schwarzschild solution, unlike that of the one in four dimensions, is the naked singularity at r = 0. 3.7 Effects of the Cosmological Constant So far all the examples considered described a dimensional reduction that preserves asymptotic flatness. However, the observed universe is known to consist mostly of the “dark energy” [44], which is currently best modeled using a positive cosmological constant. Another reason to consider non- AF spacetimes is the famous AdS/CFT duality conjecture [45], where the negative cosmological constant is an essential part of the setup. Thus it may be interesting to see how adding the cosmological constant affects the axisymmetric reduction. 34 3.7. Effects of the Cosmological Constant 3.7.1 De Sitter and Anti-de Sitter Spacetimes For the maximally symmetric case, the Minkowski, de Sitter and anti-de Sitter spacetimes, the metric in D spacetime dimensions can be written in the static chart as ds2 = − ( 1− Λr 2 (D − 1)(D − 2) ) dt2 + dr2 1− Λr2(D−1)(D−2) + r2dΩ2D−2, (3.30) where dΩ2D−2 is a metric on the (D− 2)-dimensional unit sphere and Λ = 0 for the Minkowski spacetime. As before, the reduction with respect to any of the sphere’s axisymmetries results in the metric of the same form, only with D reduced by one. The reduced spacetime is again (A)dS, only with a re-normalized cosmological constant, ΛD−1 = ΛD(D − 3)/(D − 1). The reduced Minkowski space (ΛD = 0) remains flat. Thus, the reduction has no effect at all on the energy conditions. 3.7.2 Schwarzschild-(Anti-)de Sitter Combining a Schwarzschild blackhole with the cosmological constant results in the (A)dS stress energy tensor being added to the Schwarzschild one in both the full and dimensionally reduced spacetimes. Indeed the metric is ds2 =− ( 1− ( rs r )D−3 − Λr 2 (D − 1)(D − 2) ) dt2 + dr2 1− ( rsr )D−3 − Λr2(D−1)(D−2) + r 2dΩ2D−2, (3.31) and the stress energy tensor is (including the contribution from the cosmo- logical constant) Tab = T S ab − Λgab, (3.32) where T Sab is the stress energy tensor of the Schwarzschild blackhole alone, and Λ is the original (for the full spacetime) or the re-normalized (for the reduced spacetime) cosmological constant. The WEC (and so the DEC) is violated maximally for the anti-de Sitter case, since both terms yield negative energy, and the near-horizon energy density can become negative in the de Sitter case if the cosmological constant is not very large. However, the DEC is always violated for the de Sitter case, as the positive tangential pressure induced by the reduction adds to the Λ-term, making it always exceed the energy density. Thus adding the cosmological constant to the mix does not cure the pathologies in this example, though it may alleviate some of them. 35 3.7. Effects of the Cosmological Constant 3.7.3 Four-dimensional AdS Soliton In sections 3.1 and 3.7 we saw that even though the axisymmetric reduction of the Minkowski spacetime does not locally change the stress-energy tensor, such a reduction of another vacuum spacetime, the Schwarzschild one, leads to a pathological stress-energy tensor. Given that the dimensional reduc- tion of an (A)dS spacetime results in only a rescaling of the cosmological constant, it is interesting to see what happens if we axisymmetrically reduce a non-AdS spacetime with the stress energy tensor identical to that of the AdS. One of the best known examples of such a spacetime is the AdS soliton. The AdS soliton [46] in four dimensions is described by the metric ds2 = −r 2 l2 dt2 + dr2 V (r) + r2 l2 dθ2 + V (r)dφ2, (3.33) where V (r) = r2 l2 ( 1− r 3 0 r3 ) is obtained from a four-dimensional p-brane metric by a double analytic continuation. The stress-energy tensor is that of the AdS spacetime: Tab = − 3 l2 gab, (3.34) (and so it satisfies all the energy conditions) but the spacetime is not AdS, because its curvature is not constant, as can be seen from the square of the Riemann tensor, known as the Kretschmann scalar: RabcdR abcd = 24 l4 + 12r60 l4r6 . However, it is asymptotically locally AdS, with the cosmological con- stant Λ = −3/l2. Just like in the Schwarzschild case, the spacetime has a singularity at r = 0, but its global structure is different. First, we consider the necessary identifications. Since ∂/∂φ is a Killing vector with the norm vanishing at r = r0, to avoid a conical singularity at r = r0, the elementary flatness condition (2.17) must hold. This results in the coordinate φ being periodic, with the period 43pil 2/r0. Additionally, this restricts the range of r to r ≥ r0. No other coordinates have to be periodic, although imposing periodicity on θ results in the finite total energy of this spacetime [46]. The conformal infinity of the spacetime is homotopy-equivalent to a torus T 2. 36 3.7. Effects of the Cosmological Constant Identified or not, the AdS soliton is an axisymmetric spacetime with a reg- ular axis of symmetry r = r0, where φ is a periodic coordinate suitable for axisymmetric dimensional reduction. The standard reduction ḡab = gab − 1 V (r) φaφb leads to the following reduced metric: ds̄2 = −r 2 l2 dt2 + dr2 V (r) + r2 l2 dθ2. (3.35) Since this is a U(1) reduction, and not an axisymmetric one, the reduced spacetime is regular and without boundary. The stress-energy tensor of this three-dimensional spacetime in the or- thonormal frame (t, r, θ) is Ḡâb̂ =   1 l2 ( 1− r30r3 ) 0 0 0 − 1l2 ( 1− r302r3 ) 0 0 0 − 1l2 ( 1− r302r3 )   , (3.36) and the observed energy density for the timelike observers is ρ̄ = r30 l2r3 γ2 ( 1− v 2 2 ) , (3.37) while for the null observers it is ρ̄ = r30 2l2r3 . (3.38) Both expressions are positive everywhere, so both the WEC and NEC (and ANEC) are satisfied. The DEC, is, however, violated: p2 = 1 l2 γ2  − ( 1− r 3 0 r3 )2 + v2 ( 1− r 3 0 2r3 )2 , (3.39) which is positive when v > 1− 1 2 r 3 r3 0 − 1 . This condition can be satisfied for some observers at any point in this space- time, even for asymptotically static observers. In this case we have traded one pathology for another: negative energy for superluminal matter flows. 37 3.8. Doubling Reduced Spacetime 3.8 Doubling Reduced Spacetime As we have seen, the common problem of any spacetime obtained by an axisymmetric reduction is the remnant of the axis, manifesting itself as a timelike singularity. The singularity extends “all the way to infinity”, which makes computing Komar integrals, such as the ADM mass, impossible. In this section we discuss a way to remove such a singularity by “doubling” the reduced spacetime. This procedure has already been applied in the Schwarz- schild example in section 3.1. In this section we first review it for the flat four-dimensional spacetime and then generalize to arbitrary axisymmetric spacetimes. 3.8.1 Doubling Reduced Minkowski Spacetime For the simplest possible case of the four-dimensional Minkowski spacetime with the metric in polar coordinates ds2 = −dt2 + dr2 + r2 ( dθ2 + sin2 θdφ2 ) , (3.40) where r ≥ 0, 0 ≤ θ ≤ pi and 0 ≤ φ ≤ 2pi, the spacetime obtained by the axisymmetric reduction with respect to the Killing vector θa is just ds2 = −dt2 + dr2 + r2dθ2, (3.41) where r ≥ 0 and 0 ≤ θ ≤ pi. This chart describes a half-plane for each t = const, so the spacetime is “one-half” of the full three-dimensional Minkowski spacetime, with the boundary described by θ = 0 and θ = pi, or, after transforming to the Cartesian coordinates, by x = 0, where x = r cos θ and y = r sin θ. It is clear in this example that this spacetime can be extended through the boundary by gluing another copy of it, obtained by reflecting the original across the boundary (x 7→ −x). Explicitly, the doubled spacetime is ds2 = −dt2 + dx2 + dy2, (3.42) where x, y ∈ R. This doubled spacetime is no longer singular, and we can calculate asymptotic quantities, such as the ADM mass (which is, of course, zero in this case). This argument can be generalized to an axisymmetric reduction of any axisymmetric spacetime in any number of dimensions. The four-dimensional case is described in detail in [34, §2.2.3, Proposition 2.2.1]. The main points are: 38 3.8. Doubling Reduced Spacetime • The spacetime obtained by axisymmetric reduction has a boundary where the axis of rotation used to be. • The boundary is a totally geodesic submanifold of the total (and re- duced) spacetimes. As noted before, an axisymmetric reduction maps the symmetry axis into a timelike singularity in the reduced spacetime. Our aim is to construct an extension of the reduced spacetime that removes this singularity. To prove the first point, we note that the symmetry axis is a boundary of the reduced spacetime. First, we recall from section 2.5 that F is a totally geodesic submanifold of M with vanishing extrinsic curvature. We can find an axisymmetric neighborhood of F U of F , where we can introduce a cylindrical chart (t, ρ, φ, x3, . . . , xD), ρ > 0, 0 ≤ φ ≤ 2pi mapping U → R n\F . The (D−1)-dimensional manifold defined by φ = 0 is a cross-section for the action of U(1) on U , with the coordinates (t, ρ, x3, . . . , xD), ρ > 0. We can then add the axis F (corresponding to ρ = 0) to the reduced manifold Mr = (M \F )/U(1) and obtain a manifold with boundary M̄ = Mr ∩F . The second point follows from the properties of the axis, described in section 2.5. Indeed, by the above construction one can see that the properties of F ∈ M are retained by F̄ ∈ M̄ . Specifically, F̄ is totally geodesic with vanishing extrinsic curvature in M̄ . Next, we consider a copies of M̄ , M̄ ′, covered by the charts (t′, ρ′, x′3, . . . , x′n) near the boundary. We now join the two spacetimes along the common boundary F = F ′, ρ = ρ′ = 0, such that at the boundary all the coordi- nates match: t = t′, x3 = x′3, etc. We note that the junction conditions [47] do not contribute to the stress-energy tensor at F . Indeed, given that F is a totally geodesic submanifold, its extrinsic curvature vanishes. Moreover, by construction the metric is continuous across F . Thus the surface stress- energy tensor vanishes and the symmetry axis essentially “disappears” after doubling. Once the spacelike boundary has been eliminated, we can construct “spheres near infinity” and calculate the ADM mass of the doubled reduced spacetime. If the original spacetime has an event horizon crossing the symmetry axis, as it does in the Schwarzschild case, considered in section 3.1, the horizon in the reduced space gains a boundary, which is removed by the doubling procedure. If the horizon topology before reduction is SD−2 × R, then after the reduction and doubling it is SD−3 ×R. The resulting change in the topological properties of the DOC can sometimes be used to constrain its physical properties, as discussed in chapter 5. 39 3.9. Axisymmetric Conformal Reduction 3.9 Axisymmetric Conformal Reduction Of course, not every reduction has the same pathologies. Here we consider an axisymmetric reduction that results in a non-asymptotically flat spacetime, the conformal reduction. We show that, while the induced stress energy tensor satisfies the energy conditions, there is a naked timelike curvature singularity in the reduced spacetime. Moreover, one cannot sensibly define any asymptotic quantities, such as the total mass and charge, in the reduced spacetime. The properties of this reduction are investigated in this section. We rescale the metric obtained by the usual Killing reduction, ḡab = gab− ξaξb/λ by the conformal factor equal to the norm of the Killing vector: g̃ab = λgab − ξaξb. (3.43) The Einstein-Hilbert action S = ∫ dDx √−gR (3.44) can be written as a sum of the Einstein-Hilbert action for the conformally reduced9 metric g̃ab and two fields, a scalar λ and a twist tensor ωab = 1 λ g̃cag̃ d b∇cξd, giving S = ∫ dD−1x √−g̃ (R̃− 1 2λ2 ( ∇̃aλ∇̃aλ+ ∇̃aωbc∇̃aωbc )) . (3.45) Because the action is quadratic in the induced fields, the induced stress energy tensor obeys all energy conditions. However, this nice property comes at a price. In particular, the following issues arise: • The reduced spacetime is no longer asymptotically flat, making its total mass and angular momentum undefined. • The reduced spacetime has a naked timelike curvature singularity on the symmetry axis, at least for the reduction from four to three di- mensions. 9See [48, Eq. 9] and [49, Eq. 35] for the conformal reduction from four to three dimen- sions, and [39, Eq. 46] for the conformal reduction from five to four dimensions. 40 3.9. Axisymmetric Conformal Reduction We now look at these pathologies in more detail. In fact, they are already apparent for the conformal reduction of the Minkowski space, so we consider this simpler example instead. The metric of the conformally reduced four- dimensional Minkowski spacetime in spherical coordinates is ds2 = r2 sin2 θ ( −dt2 + dr2 + r2dθ2 ) . (3.46) After rescaling r′ = 12r 2, θ′ = 2θ we get ds2 = sin2 θ′ 2 ( −r′dt2 + dr′2 + r′2dθ′2 ) . (3.47) In this form the conformally rescaled metric has a manifest r′-dependent time-dilation term, and no asymptotic time translation symmetry, since the norm of the time translation Killing vector diverges at the “spatial infinity”. Moreover, the Ricci scalar for this spacetime is R = 1 2r′2 sin4 θ′2 , (3.48) which diverges on the symmetry axis, where r sin θ = 0, resulting in a naked curvature singularity. The curvature term responsible for the divergence is present in any conformal reduction and it dominates any other contribution to the stress energy tensor from a regular axisymmetric spacetime. The curvature singularity in question corresponds to the original rotation axis, and so it cannot be completely shrouded by an event horizon. Indeed, if it were, the original four-dimensional spacetime would have included a black string, which is not asymptotically flat in four dimensions. In addition, the conformal term makes this spacetime non-asymptotically flat, even though the Ricci scalar vanishes at infinity. This becomes obvious when one considers timelike and null geodesics in this spacetime. The energy conservation for a timelike radial geodesic leads to the following first order equation of motion: (ṙ′)2 = k2 (r′)4 − 1, (3.49) where k = const and corresponds to the energy of the particle. The second term tends to zero at large r, constraining the allowable values of r′. Indeed, the turning points are at r′ = √|k′|. Thus a free test particle of finite energy cannot escape to infinity, no matter how far from r = 0 it is or how much energy it has. For the null geodesics the radial equation of motion is (ṙ′)2 = k2 (r′)4 . (3.50) 41 3.9. Axisymmetric Conformal Reduction Thus light can “almost” escape to infinity, but it is infinitely red-shifted in the process. These conclusions also hold for non-radial geodesics. In a non-Minkowski spacetime the term (3.48) is added to the induced curvature, which falls off as 1/r3, and so asymptotically the energy density is determined by the conformal factor, and the conclusions made for the Minkowski space remain valid. 42 Chapter 4 Violations of Energy Conditions in Dimensionally Reduced Spacetimes We have seen that the pathologies arising from the dimensional reduction of the Schwarzschild spacetime with respect to one of its rotational Killing vectors to an asymptotically flat spacetime manifest themselves in other re- ductions that preserve asymptotic flatness. Many such spacetimes can be thought of as asymptotic perturbations on the Schwarzschild one. In this chapter we clarify the meaning of asymptotic perturbation on Schwarzschild and construct a perturbative argument to show that the weak energy condi- tion must be asymptotically violated in an axisymmetric reduction that pre- serves asymptotic flatness. This argument holds for the reduced spacetimes in three or more spacetime dimensions. We then present a non-perturbative argument, based on the positive energy theorem, that the dominant energy condition is also violated. 4.1 Perturbative Calculation The definition of a perturbation in our case will be somewhat different from the standard one in [14]. Usually one can define a metric perturbation on a given manifold M using a smooth one-parameter map representing a family of metrics gab(s), such that gab(0) = g 0 ab, where g 0 ab is the unperturbedmetric. The parameter s is the same for all points on the manifold. A power series expansion near s = 0 gives gab = g 0 ab + sγab +O ( s2 ) . (4.1) The tensor γab does not have to be “small”, only sγab does. Additionally, the derivatives of γab are presumed to exist and be bounded on the domain of interest. Since the metric is non-degenerate everywhere, the inverse metric 43 4.1. Perturbative Calculation can be similarly perturbatively expanded: gab = g0 ab − sγab +O ( s2 ) . (4.2) The perturbation of the inverse metric can be expressed using (4.1) and (4.2) as γab = g0 acg0 bdγcd, (4.3) where the terms of order O ( s2 ) are omitted. For the connection difference between the perturbed and the original metric we have [14, §7.5.12]: ∇aub = ∇0aub + sC0 cab uc = ∇0aub + s 1 2 g0 cd ( ∇0aγbd +∇0bγad −∇0dγab ) uc +O ( s2 ) (4.4) and for the Ricci tensor we have [14, §7.5.15]: Rab = R 0 ab+ s ( −1 2 ∇0a∇0bγ − 1 2 ∇0 c∇0cγab + 1 2 ∇0 c∇0bγac + 1 2 ∇0 c∇0aγcb ) , (4.5) where γ = gabγab. This approach cannot be repeated verbatim in the case under considera- tion, for an asymptotically flat perturbation on an asymptotically flat space- time. We recall that any AF spacetime (M , gab) admits a scalar field Ω van- ishing on the boundary ∂M of the compactified manifold M̃ , M = M̃ \∂M , which is at least twice differentiable and has a non-vanishing gradient on the boundary, ∇aΩ on ∂M . In the case of the Schwarzschild spacetime the field Ω can be identified with the reciprocal radial coordinate, r = 1/Ω. Both the original and the perturbed metric tend to the flat one at large r, so there is no r-independent parametrization of the form (4.1) for which γab is not required to be small. Here r is the radial coordinate in the Schwarzschild chart, which is our unperturbed metric. Hence, we need a new definition of what constitutes a perturbation, one that allows for an expansion parameter that is not independent of the metric. Without loss of generality we can assume that the perturbed spacetime has the same ADM mass as the background Schwarzschild spacetime (if it does not, we can always choose a different background Schwarzschild space- time, with the Schwarzschild mass parameter equal to that of the perturbed spacetime, and the recalculated perturbation will not contribute to the ADM mass). 44 4.1. Perturbative Calculation By an asymptotically flat perturbation on Schwarzschild we mean a faster radial fall-off in the components of γab compared to the components of the Schwarzschild metric gSab in the same orthonormal basis. More precisely, given a manifold RD minus a timelike cylinder BD−1 ×R, parametrized in the Schwarzschild-like coordinates (t, r, θ, φ, . . .), r > r0, we can put two metrics on this manifold, gab and g S ab, such that in an orthonormal basis (t̂, r̂, θ̂, φ̂, . . .) gαβ − gSαβ = O ( 1 rD−2 ) (4.6) for all α, β. The excised timelike cylinder is selected in such a way as to hide all the features peculiar to the perturbed spacetime, such as any unusual topology, event horizons and singularities. Since the spacetime is asymptotically flat, it is always possible to find a large enough r0 such that these features lie inside the excised cylinder. If a spacetime is axisymmetric, this cylindrical boundary can always be chosen to coincide with the orbits of the axisym- metric Killing vector. In the remaining part of the spacetime the metric can be expressed as gab = g S ab + 1 rD−2 γab +O ( 1 rD−1 ) , (4.7) where γab does not depend on r (all remaining r-dependence is of the order O(1/rD−1) or higher and so is confined to the neglected terms). This defi- nition lets us keep the tensor part of the perturbation, γab, non-small, and confine the “small” part to the scalar 1/rD−2 and the neglectable part of the perturbation to the terms of the order 1/rD−1 and higher. Additionally, the r-independence of γab allows for some simplifications later on. We next consider what happens to the axisymmetric Killing vector after the perturbation. The perturbed metric is assumed to remain axisymmetric in order for the reduction to make sense. However, the Killing vector of the perturbed metric ξa is not necessarily the same as that of the unperturbed metric ξ0a. We can safely assume however, that the original and perturbed spacetime are homotopic. Indeed, the two relevant points are: • the topology of the spacetime (with a suitable timelike cylinder ex- cised) is unaffected by the perturbation (otherwise it would not be a perturbation to begin with), so there is a continuous smooth map between an unperturbed and a perturbed spacetimes, in both the full and the reduced spacetimes, 45 4.1. Perturbative Calculation • the curves and tangent vectors to them do not depend on the metric (although their norms and other inner products do). The first point guarantees that the same atlas can cover both the unper- turbed and perturbed spacetimes, and the second point implies that the tangent vectors remain unchanged when mapped from an unperturbed met- ric to a perturbed one (a given atlas, which is valid in both spaces, fixes the map). If desired, we can choose an atlas where the Killing vector, per- turbed or unperturbed, is tangent to one of the coordinates (ξ = ∂/∂φ). The perturbation of the metric may affects both the norm and the twist of the perturbed Killing vector. While asymptotically the perturbed norm is that of the original Killing vector, and the twist vanishes, we need to bound the two in order to calculate the effects of the perturbation on the stress-energy tensor of the reduced spacetime. After the axisymmetric reduction ḡab = gab − ξaξb ξcξc the reduced perturbed metric is ḡab = ḡ S ab + ( 1 r )D−2 γ̄ab +O ( 1 rD−1 ) . (4.8) For the hypersurface-orthogonal case the reduced spacetime M̄ is simply a hyper-surface in M (with zero extrinsic curvature, since Lξgab = 0), so the fall-off of the reduced metric is the same as the original one. Another way to see that is that in the coordinates adapted to the Killing vector ξ, with the norm λ and parametrized by 0 ≤ φ ≤ 2pi the metric can be written as ds2 = ds̄2 + λdφ2 (4.9) The effects of the twist in the non-hypersurface orthogonal case can be bounded by the contribution to the angular momentum of the original spacetime. Indeed, for the limit of the Komar integral (2.16) J = lim S→I0 ∫ S ∗∇aξb (4.10) to be finite, the fall-off of the cross term gxφ in the Killing vector-adapted coordinates must be 1/rD−3 or faster, resulting in the twist contribution to the reduced metric, −ξaξb/λ, having the fall-off at least as fast as 1/r2D−4. 46 4.1. Perturbative Calculation For example in the case of the four-dimensional Kerr metric the cross term falls off as 1/r, and the contribution to the reduced metric falls off as 1/r4. Given this, we can safely assume that in (4.8), ḡSab is reduced with respect to its Killing vector ξSa , while gab and γab are reduced with respect to their own Killing vector ξa. Defining the perturbation using the asymptotic fall-off conditions of the form r−n makes it possible to say that the Schwarzschild metric is an asymp- totically flat perturbation on Minkowski (n = D−3), the charged (Reissner- Nordstrom) metric is an asymptotically flat perturbation on Schwarzschild, and the charged rotating (Kerr-Newman) metric is an asymptotically flat perturbation on Reissner-Nordstrom. In any given approximation the re- duced perturbation γ̄ab does not depend on r, as the r-dependence has been made explicit in all relevant orders by the expansion (4.8). The only other restriction on γ̄ab is that it is twice differentiable. Comparing with (4.1), we see that r−n plays the role of the expansion pa- rameter s. However, since it is no longer a constant for the whole spacetime, the expressions for the curvature perturbations have to be re-evaluated. As a starting point, we can use the difference between the two connections, those for the perturbed and the unperturbed reduced metrics, ∇̄P and ∇̄S . ∇̄Pa ub − ∇̄Saub = Ccabuc = 1 2 gcd ( ∇̄Sa gbd + ∇̄Sb gad − ∇̄Sd gab ) uc. (4.11) In the remainder of this section we only use the unperturbed connection and only the reduced metric, so we can safely omit the overbar and the superscripts P and S for brevity. The difference between the Ricci tensors of the two spacetimes at the same point [14, §7.5.9]: Rab = R S ab −∇aCcbc +∇cCcab + CdabCccd − CdbcCcad, (4.12) where RSab is the stress-energy tensor of the reduced Schwarzschild metric, and the connection coefficients are Ccab = 1 2 ( − (D − 2) r−(D−1) (γcb∇ar + γca∇br − γab∇cr) +r−(D−2) (∇aγcb +∇bγca −∇cγab) ) +O ( 1 r2D−4 ) . (4.13) It is tempting to simply discard all terms of the order O(1/rD−1). How- ever, this is the order of R̄Sab (3.28), so for now we have to keep this order and only discard the higher order terms O(1/rD). In fact, we ought to keep 47 4.1. Perturbative Calculation one more term in the expansion (4.8), but it turns out that this term only contributes in the orders we already discard. Moreover, Ccab = O(1/r D−2) or smaller, so the terms quadratic in Ccab are at most O(1/r2D−4), which is smaller than O(1/rD−1) for D > 3, which covers all the interesting cases. These terms can, therefore, be discarded from (4.12). The contribution to the reduced Ricci tensor is (after discarding the terms containing second derivatives of r, which only contribute in the order O(1/rD)): Rab =R S ab + D − 2 2 r−(D−1) (∇aγ∇br +∇bγ∇ar −∇cγcb∇ar −∇cγca∇br +(−∇aγcb −∇bγca + 2∇cγab)∇cr) + 1 2 r−(D−2) ( ∇c∇aγcb +∇c∇bγca −∇2γab −∇a∇bγ ) +O ( 1 r2D−4 ) , (4.14) where γ = γaa denotes the trace of the perturbation. The term without any derivatives of r matches that in [14, §7.5.15], as expected. Next simplification comes from recalling that γab does not depend on r, so all r-derivatives of it10 vanish: Rab =R S ab + D − 2 rD−1 ( ∇(aγ∇b)r −∇cγc(b∇a)r ) + 1 2rD−2 ( 2∇c∇(aγcb) −∇2γab −∇a∇bγ ) +O ( 1 r2D−4 ) (4.15) The Ricci scalar in the reduced space is R = D − 2 2 r−(D−1) (2∇aγ∇ar −∇cγca∇ar −∇cγca∇ar) + 1 2 r−(D−2) ( 2∇c∇aγca − 2∇2γ ) +O ( 1 r2D−4 ) , (4.16) where we used the fact that the Ricci tensor of the reduced Schwarzschild metric is traceless. As before, using the r-independence of γab, we discard 10Represented in the above expression by the term (−∇aγ c b −∇bγ c a + 2∇ c γab)∇cr. 48 4.1. Perturbative Calculation the first term and get R = 1 2 r−(D−2) ( 2∇c∇aγca − 2∇2γ ) +O ( 1 r2D−4 ) (4.17) Before calculating energy density seen by observers in the reduced per- turbed spacetime, we have to address the issue of what kind of observers we are interested in. We note that it is sufficient to only consider ob- servers that are static in the unperturbed Schwarzschild spacetime, i.e., those moving along the trajectories of the Schwarzschild coordinate t in the unperturbed reduced spacetime. Since the metric and its perturbation are smooth, if these observers see negative energy density, so do those on sufficiently close trajectories. These “Schwarzschild-static” observers are not necessarily static in the perturbed spacetime. Indeed, the notion of a static observer might not even make sense for a general perturbation. They are only static in the background Schwarzschild metric, which is sufficient for our purposes since all the calculations are done using the background metric and connection. For these observers the calculation of the energy density can be simplified considerably. Indeed, the terms containing ∇ar do not contribute to the energy density and we get Gabu aub =RSabu aub + 1 2 r−(D−2) ( ( ∇c∇aγcb +∇c∇bγca −∇2γab −∇a∇bγ ) uaub + ( ∇c∇aγca −∇2γ ) ) +O ( 1 r2D−4 ) . (4.18) The first term is RSabu aub = −(D − 3)rD−3S /rD−1, where rS is the Schwarz- schild radius in D dimensions, which is determined by the blackhole’s ADM mass. To estimate the remaining terms it is convenient to introduce a pro- jection operator on the D − 2 spacelike hypersurface orthogonal to ua (i.e., the one given by setting t = const in the reduced Schwarzschild chart): pab = ḡab + uaub. (4.19) For Schwarzschild-static observers this operator is time-independent. Ex- 49 4.1. Perturbative Calculation plicitly, in the Schwarzschild spherical coordinates it is pba =   0 1 1 . . . 1   . (4.20) The observed energy density can now be expressed as Gabu aub =− (D − 3) r D−3 S rD−1 + 1 2rD−2 ( −pabpcd + pacpbd ) ∇c∇dγab +O ( 1 r2D−4 ) . (4.21) This shows that any time-dependence in the perturbation does not con- tribute to the energy density seen by static observers. We have already noted that in section 3.4 in the special case of linearized axisymmetric radiation. Any other dependence is guaranteed to drop at least two orders of r when differentiated twice, bringing the fall-off to the slower of O(1/r2D−4) and O(1/rD). This implies that the contribution of the perturbation to the en- ergy density seen by static observers falls off faster than that of the energy density seen by these observers in an unperturbed reduced Schwarzschild spacetime, and ensures that an asymptotically flat axisymmetric reduction of an asymptotically flat spacetime with positive energy density (and hence positive ADM mass) will violate the weak energy condition. This conclusion has been checked in a special case, a conformal pertur- bation, where the expression (4.21) can be computed by hand. For example, in the conformal case, gPab = gab + f(r, . . .)gab, (4.22) where f (r, . . .) = O(1/r) we get Gabu aub = −(D − 3)r D−3 S rD−1 − D − 2 2rD−2 pab∇a∇bf +O ( 1 r2D−4 ) . (4.23) The second term is of the order O(1/rD), so it does not asymptotically contribute to the observed energy density. A similar argument can be advanced to show that the NEC is also vio- lated, however in this case we cannot use static observers, as null geodesics 50 4.2. Violation of the Dominant Energy Condition in Axisymmetric Reduction are never static. The relevant calculations are beyond the scope of this the- sis. From analyzing the axisymmetric reduction of the Schwarzschild metric in higher dimensions (section 3.6), we know that in general the reduction does not induce superluminal matter flows, so the second part of the DEC may well survive the reduction. 4.2 Violation of the Dominant Energy Condition in Axisymmetric Reduction Here we offer an argument that shows that the dominant energy condition is violated by any axisymmetric reduction that preserves asymptotic flatness. This argument is based on the positive energy theorems (PET), see [50–53]. The combined statement of the PET is (see e.g., [54] for review) as follows: Theorem 1 (Positive Energy Theorem). Let (M , g) be a a D-dimensional spacetime admitting an asymptotically flat spacelike hypersurface (M̄ , ḡ) sat- isfying the dominant energy condition. Then the total ADM mass m satisfies m ≥ 0, with equality if and only if (M , g) is flat (Minkowski). We next perform the dimensional reduction with respect to the axisym- metric Killing vector. In general a spacetime obtained by an axisymmetric reduction is not geodesically complete, due to a timelike quasi-regular sin- gularity [55] on the axis. We remove the singularity by doubling the reduced spacetime (see section 3.8). This enables us to calculate the surface inte- grals at infinity and also to apply the PET. The doubled reduced spacetime is again asymptotically flat by assumption, and we can calculate its ADM mass. We show that it necessarily vanishes. The PET applied to the doubled reduced spacetime implies that the dominant energy condition is violated or that the spacetime is flat. The DEC can be violated one of two ways: the WEC is violated, i.e., there is negative energy density somewhere in the spacetime, or the spacetime contains regions of superluminal matter flows. The examples considered in chapter 3 show that either or both can happen. Given that the doubled spacetime consists of two identical copies of the re- duced spacetime, the same applies to the non-doubled reduced spacetime, as well. This result can be stated as follows: Theorem 2. Let (M , g) be an asymptotically flat D-dimensional axisym- metric spacetime (D ≥ 4) obeying the geometric conditions (but not nec- essarily the DEC) required for the proof of one of the positive energy the- orems. Let (M̄ , ḡ) be the (D − 1)-dimensional spacetime obtained by an 51 4.2. Violation of the Dominant Energy Condition in Axisymmetric Reduction axisymmetric reduction with respect to the Killing vector ξ generating the axisymmetry in M . Then the stress-energy tensor of the reduced spacetime violates the dominant energy condition. Note that we do not actually require that the DEC be valid in the full spacetime, only that any other conditions required for the proof of the PET are satisfied. For example, the negative mass Schwarzschild solution violates the WEC and consequently the DEC. The axisymmetrically reduced nega- tive mass Schwarzschild actually does obey the WEC, but not the DEC, as some timelike observers can see spacelike matter flows (see section 3.1). We prove the following lemma first: Lemma 1. The ADM mass of an asymptotically flat spacetime obtained by an axisymmetric reduction of a regular asymptotically flat spacetime in four or more dimensions vanishes. Proof. We note that in the expression for the ADM mass (2.15) the integra- tion measure is proportional to rD−2, as it describes a (D − 2)-dimensional sphere of radius r. Given that the limit (2.15) is finite, the integrand must have a radial fall off 1/rD−2 or faster. After the reduction and doubling the new integration measure is proportional to rD−3. As shown in section 4.1, the reduction does not affect the fall-off on the metric and its derivatives, so the integral in (2.15) for the reduced spacetime falls off at least as 1/r and the corresponding limit is zero. Theorem 2 immediately follows once we assume that the conditions of the PET are satisfied. We illustrate this result by a direct calculation of the ADM mass of the reduced four-dimensional Reissner-Nordstrom metric. The spatial part of the metric is: ds2 = dr2 1− 2Mr + Q 2 r2 + r2dθ2 + r2 sin2 θdφ2. (4.24) The reduction with respect to the azimuthal Killing vector φ gives ds̄2 = dr2 1− 2Mr + Q 2 r2 + r2dθ2 = ( 1 + 2M r )( dx2 + dy2 ) +O ( 1 r2 ) . (4.25) Since θ is a Killing vector of the reduced metric, the metric is regular everywhere except r = 0, including the boundary resulting from the reduc- tion, θ = 0 and θ = pi, and so we can extend this spacetime by extending 52 4.2. Violation of the Dominant Energy Condition in Axisymmetric Reduction the range of the coordinate θ to 0 ≤ θ ≤ 2pi from a half-plane to a full plane. The ADM mass of this spacetime is M̄ADM = − lim r→∞ 2pir d dr ( 1 + 2M r ) = 0. (4.26) This theorem does not specify which part of the DEC is violated, the positive energy or the non-spacelike matter flows, or both. From the exam- ples in section 3 we see that both parts are violated at least asymptotically for the reduction to three dimensions, but in higher dimensions it is the WEC that is likely to break, as there no induced superluminal matter flows in the higher-dimensional Schwarzschild reduction (see 3.6). 53 Chapter 5 Topological Restrictions on the Properties of Dimensionally Reduced Spacetimes 5.1 Introduction In chapter 3 we gave several examples of axisymmetric reduction preserving asymptotic flatness. These examples show what goes wrong with the energy conditions in the reduced spacetime. In chapter 4 we gave an argument for why this happens, based on the fall-off conditions. Specifically, The fall-off conditions on the metric tensor and the Killing vector used for the reduction force the violation of the weak and null energy conditions in the reduced spacetime, at least asymptotically. In section 4.2 we invoked the positive energy theorem to show that the DEC is also asymptotically violated in such reductions. Given the pervasiveness of these pathologies, it is natural to ask whether there is a more general reason for the reductions to be pathological. In this chapter we provide two such arguments, based on the principle of topological censorship (PTC). First, we discuss the nicer case of the U(1) reduction, rather than the axisymmetric reduction, and only from four to three spacetime dimensions. We show that there is no U(1) reduction that preserves global hyperbolicity, asymptotic flatness and positive energy at the same time. This is a conse- quence of the topological rigidity theorem [5], a proof of which is given for the U(1) reduction and outlined for a reduction with respect to a general Lie group. This theorem is an application of the PTC, which restricts the topol- ogy of the interior of the spacetime, given the topology of its (conformal) boundary. Next we use the PTC to show that there is no axisymmetric reduction of a spacetime with two or more spherical horizons resulting in a three- 54 5.2. Topological Censorship dimensional AF or ALADS globally hyperbolic spacetime with everywhere positive energy. We then discuss the implications of this to the 2 + 1 + 1 formalism in numerical relativity. These topological arguments still do no imply that any axisymmetric reduction is necessarily pathological. Indeed, the Minkowski spacetime can be reduced from four to three dimensions without any ill effects, other than the presence of a timelike boundary where the axis of rotation used to be. However, the arguments in the preceding chapters show that this is the only such case, as any spacetime not devoid of matter is either already pathological or results in a pathological spacetime after the reduction. This chapter is organized as follows: in section 5.2 we review the princi- ple of topological censorship, which is central to the arguments that follow. In section 5.3 we review the PTC corollary restricting the topology of any Cauchy surface in a “nice” three-dimensional spacetime. In section 5.4 we prove the topological rigidity of a U(1) reduction from four to three dimen- sions. In section 5.5 we use the topological rigidity and the PTC to show that there is no U(1) reduction from a four-dimensional AF globally hyper- bolic spacetime to a three-dimensional AF or ALADS spacetime satisfying NEC. In section 5.6 we review the examples of spacetimes that are and are not a subject to these restrictions. In section 5.7 we return to the axisym- metric reduction and show that there can be at most one blackhole in any “nice” reduced spacetime. In section 5.8 we consider the implications of the arguments presented in this chapter for axisymmetric numerical schemes. 5.2 Topological Censorship The (weak) cosmic censorship conjecture, advanced by Roger Penrose in [56] states that any singularities forming as a result of a Cauchy development of regular initial data are hidden by an event horizon. A related topologi- cal censorship conjecture, asserting that not only singularities, but also any non-trivial topology cannot be observed by an asymptotic non-spacelike ob- server, and so must be hidden behind an event horizon, as well, was first advanced in [57]. It was proven in [20] for globally hyperbolic asymptotically flat spacetimes obeying the average null energy condition. Multiple gener- alizations and related results for the topology of event horizons followed: [3, 58–64]. The result essential to the topological rigidity was first proven in [3, Proposition 3.1], and it can be stated as follows: Let D be the DOC of a boundary at infinity I of a conformally com- pactified spacetime M . If the PTC holds for this spacetime, i.e., that if 55 5.3. PTC Corollary for Three-dimensional Spacetimes every causal curve in D with endpoints on I can be deformed to I , then the map of the fundamental group of I onto the fundamental group of D , induced by inclusion, is surjective. 5.3 PTC Corollary for Three-dimensional Spacetimes In a general case the topological restriction on the DOC by the topology of its conformal boundary is rather mild, but for the case of three dimensions the topology of the DOC is restricted quite severely. This has been formulated in[60, Theorem 6]. We will use an equivalent statement given in [5]: Corollary (Topological Censorship). Let D2+1 be the DOC of a globally hyperbolic AF or ALADS spacetime-with-boundary satisfying NEC, and let V be a (two-dimensional) Cauchy surface in D2+1. Then V is either a two-ball B2 or an annulus, I × S1. We review the necessary definitions first. Let M be an AF or an AL- ADS spacetime, as described in section 2.3 with I as its boundary. Let M ′ = M ∩ I be the conformally compactified spacetime with boundary. In four spacetime dimensions each connected component of I is homeo- morphic to S2 × R. If the spacetime M ′ is globally hyperbolic, it admits a Cauchy surface V ′. V ′ is a two-dimensional manifold with boundary, and its boundary Σ∞ is a subspace of I . A manifold without boundary V = V ′ \ Σ∞ is a Cauchy surface in the original spacetime M . The DOC is the portion of the spacetime exterior to the event horizons, the intersection of the causal past and the causal future of I . A spacetime with boundary D′ = D ∩I is the DOC of M ′. Proof. First we summarize the proof strategy. 1. The conformal boundary Σ∞ of V is S1. 2. The compactified Cauchy surface is a subspace of the D′. 3. The inclusion map i : Σ∞ → V ′ implies that the homomorphism of fundamental groups i∗ : pi1(Σ∞)→ pi1(V ′) is surjective. 4. pi1(V ′) is a subgroup of pi1(Σ∞) 5. Any subgroup of integers is isomorphic to a cyclic group Zm. 56 5.3. PTC Corollary for Three-dimensional Spacetimes 6. the only choices for m for two-dimensional manifolds are 0 and 1, corresponding to the annulus and the disk respectively. Now, we prove each step. Step 1 V is a two dimensional manifold, so its conformal compactification boundary Σ∞ is a one-dimensional manifold. The only closed con- nected one-dimensional manifold is S1, and the only connected non- compact manifold is R, so these are the only boundary choices. For the AF and ALADS spacetimes the boundary is compact, which leaves S1 as the only choice. Step 2 Σ∞ ⊂ I ∈ D′ and D ⊂ D′, so V ′ ⊂ D′. Step 3 First, we recall the definition of a map induced by inclusion (see [65, §52]). The inclusion map i : Σ∞ → V ′ induces a map between fun- damental groups of the two spaces. Specifically, for a given basepoint x0 ∈ Σ∞ and the map i, the map i∗ : pi1(Σ∞) → pi1(V ′) is defined by the equation i∗([f ]) = [i ◦ f ], where f : I → Σ∞ is a curve in V and [f ] is the set of equivalence classes of curves under homotopy. For each path connected piece the fundamental groups with different base points are isomorphic, so we can safely omit mention of the base point. Similarly, the inclusion map V ′ → D′ also induces a homomorphism of fundamental groups between these spaces. The PTC states that every causal curve in D′ with endpoints on I can be deformed to a curve on I , and it is shown in [3, Proposition 3.1] that the map pi1(I ) → pi1(D′) is surjective. Moreover, the map of their respective Cauchy surfaces, i∗, is also surjective (see [60, Theorem 4]). Step 4 Surjectivity of homomorphisms implies that pi1(V ′) is a subgroup of pi1(Σ∞). Since Σ∞ is S1, its fundamental group is the group of integers Z. Step 5 Z is the only infinite cyclic group, up to an isomorphism (see e.g., [66, Chapter 6, Example 2]), and every subgroup of a cyclic group is cyclic11. A finite cyclic group of order m is isomorphic to the group of integers under addition modulo m, Zm = Z/mZ. Step 6 Any compact real two-dimensional manifold without boundary is a connected sum of spheres S2, tori T 2 and real projective spaces RP 2 11This is sometimes referred to as the fundamental theorem of cyclic groups. 57 5.4. Topological Rigidity of U(1) Reduction to Three Dimensions [65, Chapter 12]. A manifold with boundary can be obtained from the one without boundary by removing a number of disks B2. For a two- dimensional manifold with boundary, one also gets the contribution from the removed disks. It follows from the Van Kampen theorem that the fundamental group of a two-dimensional manifold is the free group on the generators of the fundamental group of each of the spaces used in its construction. This group is non-Abelian except when the compact manifold is S2, T 2 or RP 2. Furthermore, an orientable two- dimensional manifold with a non-empty boundary with an Abelian fundamental group is either a sphere with one disk removed, or a sphere with two disks removed (since a torus with a disk removed has a non-Abelian fundamental group and so does a sphere with three or more disks removed). In the first case the fundamental group is trivial, m = 1, and in the second case it is Z, corresponding to m = 0. In the spacetimes under consideration (AF or ALADS) the I is ori- entable, and by [3, Corollary 3.3] the DOC D′ is orientable, too. This means that the Cauchy surface V ′ of D′ is also orientable. Since V ′ is an orientable two-dimensional manifold with boundary, it satisfies the above conditions. The Cauchy surface V of D is V ′ with the boundary removed, and so it is either a disk B2 or an annulus, I × S1. 5.4 Topological Rigidity of U(1) Reduction to Three Dimensions The topological censorship corollary for three-dimensional AF or ALADS spacetimes restricts the topology of the Cauchy surface to being either a disk or an annulus. This topological restriction makes it possible to con- strain the topology of a total spacetime dimensionally reduced to a “nice” three-dimensional spacetime, subject to some reasonable conditions. Specif- ically, the total space topologically is necessarily a product of the reduced space and the group of spacelike symmetries generated by the Killing vec- tors employed in the reduction. In this section we prove this for a spacelike U(1) reduction. As discussed in the section 2, this applies to any cylindrical reduction without fixed points (there is no rotation axis in the spacetime). For example, a toroidal wormhole spacetime admits such a reduction. The metric of one such spacetime metric can be written as ds2 = −dt2 + dr2 + (r2 + a2)(dθ2 + dφ2). (5.1) 58 5.4. Topological Rigidity of U(1) Reduction to Three Dimensions Since −∞ < r <∞, the spatial part of this metric is topologically R× T 2, and its spatial Killing vectors with closed orbits, such as ∂θ, have no fixed points and so can serve as principal U(1) fibers. Thus a reduction with respect to one of those vectors is a U(1) reduction. The resulting three- dimensional spacetime is regular, with the topology R× S1 and the metric ds̄2 = −dt2 + dr2 + (r2 + a2)dθ2. (5.2) We note that neither the original nor the reduced spacetime obey the NEC or are asymptotically flat, and the topology of the full spacetime is a product topology, so this example, while simple, is not very illuminating for the purposes of this discussion. An example of a non-trivial bundle is the U(1) reduction of a four-dimensional spacetime with a spherical spatial part, S3× R, such as the closed FLRW model (see e.g., [14]): ds2 = −dt2 + a2(t)ds23, (5.3) where ds23 is a metric on S 3. Since the Hopf fiber is an isometry of S3, we can use it as a Killing vector field for dimensional reduction. Applying the Hopf fibration U(1) : S3 → S2, we can reduce the spacetime to be topologically R× S2, with the metric ds̄2 = −dt2 + a2(t)ds22, (5.4) where ds22 is a metric on S 2. An explicit construction for electrovacuum spacetimes is given in [67]. Both the total and the reduced spacetimes are globally hyperbolic and satisfy the energy conditions, but are not AF (or ALADS). Theorem 3 (Topological Rigidity of U(1) Reduction). Let D2+1+1 be the DOC of a globally hyperbolic spacetime with boundary with a spacelike U(1) symmetry group. If D2+1+1 admits a globally hyperbolic bundle reduction D2+1 = D2+1+1/U(1) that is both AF or ALADS and satisfies NEC, then either D2+1+1 = R3 × U(1) or D2+1+1 = R2 × S1 × U(1). Proof. The topological censorship corollary applied to D2+1+1/U(1) directly implies thatD2+1+1/U(1) = R3 orD2+1+1/U(1) = R2×S1. If the base space is contractible, any bundle over it is trivial (a product bundle), since the local trivialization can be extended to be global. This proves the theorem for the case of the contractible base space D2+1 = R3. For the non-contractible case D2+1 = R2 × S1 we need to consider the U(1) bundles over S1. There are two possible bundles of S1, the topological space of U(1), over S1: the 59 5.5. No AF U(1) reduction from Four to Three Dimensions trivial bundle, resulting in the T 2 total space, and the twisted bundle, known as the Klein bottle K. The latter is non-orientable, making the potential Cauchy surfaceK×R non-orientable, as well, which contradicts [3, Corollary 3.3]. Alternatively, one can show that there is no non-trivial principal U(1) bundle over S1, without invoking the orientability argument. Indeed, the classifying space BU(1) is CP∞, which is the model space of Eilenberg- MacLane K(Z, 2) and so is simply connected (see [35, Chapter 4]). Theorem 3 can be extended to cover an arbitrary Lie group as a principal bundle. This general theorem and its proof were first given in [5] and also generalized for a special case of a non-bundle reduction, specifically, for Seifert fiber manifolds. 5.5 No AF U(1) reduction from Four to Three Dimensions The topological rigidity theorem requires the reduced three-dimensional spacetime to obey the PTC, but it imposes no such restriction on the total (four-dimensional) spacetime. If one further requires the four-dimensional spacetime to be asymptotically flat, the already short list of possible reduc- tions disappears completely. This is a straightforward corollary from the PTC in combination with the topological rigidity theorem 3: Corollary (No U(1) reduction). There is no U(1) bundle reduction of a four-dimensional asymptotically flat spacetime to a three-dimensional asymp- totically flat or ALADS spacetime that satisfies the NEC. Proof. If the PTC holds in the full four-dimensional spacetime, its DOC D2+1+1 is simply connected if I is. Consequently, the DOC of a four- dimensional asymptotically flat globally hyperbolic spacetime obeying NEC is simply connected. Theorem 3 implies that its topology is a product topol- ogy, where one of the terms is S1. But a product space where one of the terms is non-simply connected is again non-simply connected12 (see e.g., [65, Theorem 60.1]). Thus no such reduction exists. The proof relies on the four-dimensional spacetime having a simply con- nected DOC, hence the restriction to the AF spacetimes. Other spacetimes obeying the PTC may still be reduced to a PTC-obeying three dimensional spacetime. Such a spacetime may, however, still turn out to be pathological for other reasons, as we will see in the following section. 12Specifically, D2+1+1 = R2 × S1 × R or D2+1+1 = S1 × R× S1 × R. 60 5.6. Examples of Reduction 5.6 Examples of Reduction A simple example of a U(1) reduction from four to three dimensions where both the total and the base space obey the PTC is the U(1) reduction of the four-dimensional black string13 solution. We can construct the latter from a three-dimensional BTZ blackhole [68] by taking a direct product with another spatial dimension. For an explicit construction, see e.g., [69]. The resulting metric is ds2 = − ( α2r2 −m+ J2/r2 ) dt2+ dr2 α2r2 −m+r 2dφ2−2Jdtdφ+dz2, (5.5) where α2 = −Λ, m is the blackhole mass (in three dimensions) and J is its angular momentum. The important point is that the (quasi-regular) singularity at r = 0 (hidden from external observers by event horizons when m > 2αJ), is not a part of the spacetime, so the orbits of the Killing vector ∂φ are circles everywhere. This is different from, say, the Schwarzschild case, where the axis of rotation corresponding to the fixed points of the Killing vector ∂φ is a part of the spacetime. Thus a Killing reduction with respect to φ is a U(1) reduction. We now discuss the properties of the original and the reduced spacetimes. The stress energy tensor is that of an AdS space, Gab = −α2gab, so the WEC and NEC hold. The spacetime is also globally hyperbolic and ALADS. The metric of the reduced spacetime after the U(1) reduction with respect to φ is ds̄2 = − ( α2r2 −m+ J2/r2 ) dt2 + dr2 α2r2 −m+ J2/r2 + dz 2. (5.6) The stress-energy tensor of the reduced spacetime is Gab = ( α2 + 3J2 r4 ) zazb, where za is the unit tangent vector along z (note that the Weyl tensor vanishes and so the spacetime is conformally flat). Thus NEC (and WEC) is preserved: for a null (or timelike) unit vector ka the observed energy density is kakbTab = ( α2 + 3J2 r4 ) (kz)2 ≥ 0, 13Here a black string is a cylindrical AdS blackhole. 61 5.7. No Blackhole Horizons in Reduction to Three Dimensions where kz is the z-component of ka in an orthonormal frame. Note, however that the dominant energy condition does not hold, as the pressure in the z direction exceeds the (vanishing) energy density. While the reduced spacetime is neither AF nor ALADS, the PTC still ap- plies and the equivalent of theorem 3 holds: the DOC of the four-dimensional spacetime is topologically R3×U(1). This is an example of a U(1) reduction from a four-dimensional globally hyperbolic ALADS spacetime satisfying NEC to a three-dimensional globally hyperbolic simply connected spacetime satisfying NEC. It is not a counterexample to the corollary in section 5.5, since the four-dimensional spacetime is not AF. Another example of a U(1) reduction that is not a counterexample to the corollary is the four-dimensional AdS solution reviewed in section 3.7.3. The total space is ALADS, not AF, and its DOC is not simply connected, so a U(1) reduction preserving NEC is allowed. As mentioned in section 3.7.3, the solution is still rather pathological because some observers see superlu- minal matter flows. 5.7 No Blackhole Horizons in Reduction to Three Dimensions In four dimensions the PTC does not restrict the number of spherical black- holes allowed in an asymptotically flat spacetime, since a spatial cut of each horizon is a two-sphere and so is simply connected. However, in three dimen- sions a spatial cut of the blackhole horizon is S1, the only closed connected one-manifold. Similarly, the spatial boundary at infinity Σ∞ is also S1. So far there is no contradiction to the PTC. But if we consider a three- dimensional spacetime with more than one horizon, the topology of the DOC becomes more complicated than that of Σ∞, and no surjective homo- morphism from pi1(Σ∞) to pi1(DOC) exist. Concretely, for a two-blackhole spacetime the fundamental group of the DOC is the free group on two gen- erators, which is non-Abelian, and so not a subgroup of pi1(Σ∞) = Z. Since any globally hyperbolic AF or ALADS spacetime satisfying ANEC is a sub- ject to the PTC, any globally hyperbolic three-dimensional AF or ALADS spacetime with two or more horizons must violate ANEC. It would be premature, however, to conclude that an AF dimensional re- duction from four to three dimensions of a spacetime with multiple horizons contradicts the PTC. To see the problem we first return to the axisymmetric reduction of a Schwarzschild spacetime and its topology. The reduced space- time is described in section 3.1. Due to the timelike boundary remaining 62 5.8. Considerations for Axisymmetric Numerical Schemes from the rotation axis (whether included into the spacetime or not), the DOC of the reduced spacetime is simply connected and so its fundamen- tal group is trivial. Thus, the PTC is trivially satisfied (though the reduced spacetime itself does, of course, maximally violate all the energy conditions). To make the fundamental group non-trivial without changing the local properties of the spacetime we can extend the reduced spacetime through its timelike boundary by gluing a second copy to it, as described in section 3.8. This procedure removes the boundary, and the DOC of the resulting space- time is no longer simply connected. The fundamental group of both the boundary at infinity and the DOC is still Z, so the PTC holds and we can- not use it to infer that the reduced spacetime is pathological. However, we can apply the results of [70, 71] to the doubled spacetime to show that the DEC is violated. In particular, the result states that any 2 + 1 ini- tial data set, which obeys the dominant energy condition with non-negative cosmological constant and which satisfies a mild asymptotic condition (a “null mean convexity condition”), must have trivial topology. Moreover, any data set obeying these conditions cannot contain horizons (or, more precisely, marginally outer trapped surfaces). Since the doubled spacetime is asymptotically flat, it satisfies the required conditions and so must be either horizon-free or violate the DEC. However, less restrictive energy con- ditions, such as ANEC, might still hold. We will now consider an axisymmetric AF spacetime with two or more blackholes. After the reduction with respect to the rotational Killing vector we get a spacetime with two horizons, but still simply connected. However, after extending the reduced spacetime through the timelike boundary we get a spacetime with the DOC whose fundamental group is not only non- trivial, but also non-Abelian, as this doubled reduced spacetime is homeo- morphic to the one discussed in the beginning of this section. Thus ANEC is violated in this spacetime, and since it consists of two copies of the re- duced spacetime, ANEC is also violated in the reduced spacetime. Thus any axisymmetric reduction of a four-dimensional two-blackhole spacetime that preserves asymptotic flatness necessarily induces a pathological stress energy tensor in the reduced spacetime. 5.8 Considerations for Axisymmetric Numerical Schemes Until the recent breakthroughs in the full 3D numerical evolution codes [72] the axisymmetric numerical schemes were often considered for suitable 63 5.8. Considerations for Axisymmetric Numerical Schemes problems, such as axisymmetric blackhole collisions, collapsing Brill waves [6], axisymmetric scalar field collapse [1]. So far the success has been in- termittent, apparently due to numerical stability issues. While the topic of numerical relativity is outside of the scope of this thesis, we would like to remark that it is possible that some of these difficulties are related to the pathological stress-energy tensor of the reduced (2 + 1)-dimensional space- times. Specifically, as shown in chapters 3 and 4, the induced stress-energy tensor in reduced asymptotically flat spacetimes generically has negative energy density, and appears to have superluminal matter flows for the ob- servers with large lateral velocity. As discussed in section 3.9, it is possible to trade a pathological stress- energy tensor for a naked timelike curvature singularity on the axis, by rescaling the reduced metric by the norm of the Killing vector. Then the numerical evolution challenge is to control the divergent curvature terms near the rotation axis, rather than to deal with the misbehaving stress energy tensor in the bulk. However, if both the total and the reduced spacetime are AF, the argument of the section 5.7 can be applied to show that a numerically evolved reduced spacetime must violate ANEC. We discuss the point in more detail in this section. We assume that a strongly hyperbolic [16] well-posed 2+1+1 numerical scheme is used to evolve a two-dimensional spatial slice, using some form of evolution equations derived from the Einstein equation reduced from four to three dimensions. The resulting three-dimensional spacetime is then neces- sarily isomorphic to what one would obtain by numerically evolving the full three-dimensional axisymmetric spatial slice and then reducing with respect to the spacelike Killing vector corresponding to the axisymmetry, though the type of the reduction may depend on the particulars of the 2 + 1 + 1 numerical scheme used. We further assume, for the purposes of this anal- ysis, that the two-dimensional spacelike slice being evolved numerically is asymptotically flat, and so is the full four-dimensional spacetime, once re- constructed. Given these assumptions, we can apply the PTC to both the reduced and the total spacetimes. But first we have to get rid of the time- like boundary, using the spacetime doubling, discussed in section 3.8. The Cauchy surface of this doubled spacetime no longer has a timelike boundary and we can apply the argument of [3, Section IV] to establish the conditions required for the PTC to hold in the reduced doubled spacetime. Since the PTC does not hold in an AF spacetime with two or more blackholes, the reduced spacetime must violate ANEC. The retraction by the timelike field argument in the remark to [3, Theorem 4.1] establishes the Lemma 4.2 and the remainder of the proof of the topological rigidity theorem. Then if an 64 5.8. Considerations for Axisymmetric Numerical Schemes axisymmetric reduction preserves asymptotic flatness (or it is a reduction from an AF spacetime to an ALADS one), it must violate the NEC. The paper [73] claims to have constructed “a strongly hyperbolic first-order evo- lution system” for an axisymmetric reduction. If such a hyperbolic scheme is constructed, or is claimed to be constructed, then the topological rigidity corollary should imply that the reduced spacetime is pathological. 65 Chapter 6 Unwrapping Closed Timelike Curves 6.1 Introduction Closed timelike curves (CTCs) are closed curves in a spacetime that a time- like test observer can trace [2]. Although spacetimes with CTCs cannot be constructed by evolution, they are still solutions of the Einstein equation, just not of its initial value formulation. CTCs are considered problematic because their presence appears to lead to causality violations. Spacetimes with CTCs are usually dealt with in one of two ways: either a spacetime with CTCs is declared not physically relevant or it is modified globally in such a way that the CTCs are absent. In many cases the CTCs are manifest isometries of the spacetime and follow coordinate curves of a periodically identified coordinate. If this periodic identification is removed, the global structure of the spacetime changes. This global modification is informally known as unwrapping. One is then tempted to declare the unwrapped space- time to be “more natural” than the original one. This is the content of the claim of Cooperstock and Tieu in [12], who declare that “the imposition of periodicity in a timelike coordinate is the actual source of CTCs, rather than the physics of general relativity”. We investigate this claim in detail. Two natural questions that arise are: Do spacetimes extended by un- wrapping contain any CTCs not explicitly removed by unwrapping? Do any other pathologies arise as a result of unwrapping CTCs? To answer these questions we consider the two spacetimes where CTCs are concluded to be artificial by Cooperstock and Tieu, the Gödel universe [10] and the Gott spacetime [13]. These spacetimes serve as nice toy models, as they are highly symmetric and essentially (2 + 1)-dimensional (each is a direct products of a (2 + 1)-dimensional spacetime with a spacelike real line). Implicit in the claim of [12] seems to be the requirement that the metric of the spacetime without the periodic identifications is locally the same as of the original spacetime with identifications. An important issue is whether, 66 6.1. Introduction if the original spacetime is regular everywhere, the spacetime where the CTCs are unwrapped should be as well. Otherwise one ends up trading CTCs for singularities. We find that in axisymmetric spacetimes, where the axis of symmetry is a regular (n− 2)-dimensional subspace, such as the Gödel spacetime, there is an obstacle to having this regularity everywhere, which manifests itself as a special kind of singularity in the unwrapped spacetime. This singularity is of the type known in literature as a quasi- regular singularity [55], where the spacetime curvature is bounded along each incomplete curve. This result also holds for the Gott spacetime. This “unwrapping” singularity is similar to the one describing infinitely thin straight cosmic strings in 3 + 1 dimensions, where strings are (1 + 1)- dimensional timelike singularities. If a (3 + 1)-dimensional spacetime is a direct product of a (2 + 1)-dimensional spacetime and a real line, the third spatial dimension can be projected out, transforming the string into a point particle. The main difference between the unwrapping singularity and a point particle singularity is topological: there are no closed curves winding around the unwrapping singularity. Observationally, the effect of such a sin- gularity is also significantly different from a regular cosmic string: instead of the usual gravitational lensing we observe a total lack of correlation be- tween the images from the opposite sides of an unwrapping singularity, as they come from completely different parts of the universe. We show that, while unwrapping the Gott spacetime results in a (singu- lar) spacetime with no CTCs, unwrapping the Gödel space does not remove all of the CTCs. In the case of the Gödel spacetime, where the CTCs still persist after unwrapping, we investigate a possibility of a “multiple unwrapping”, where multiple families of CTCs are unwrapped all at once. In this procedure mul- tiple “strings” are removed, and the resulting multiply connected spacetime is then unwrapped by constructing its universal cover in order to get rid of the CTCs winding around each removed string. This removes all of the circular CTCs winding around each such string and we conjecture that no other CTCs remain, either. Our example of extending a locally Gödel chart in a way that apparently does not give rise to CTCs results in a spacetime with pervasive quasi- regular naked singularities instead. Similarly, a CTC-free extension of the Gott spacetime, even though it is locally Minkowski almost everywhere, also results in a naked quasi-regular singularity. These examples support the view that any attempt to get rid of CTCs by alternate extensions, even if successful, is likely to result in other pathologies. This chapter is organized as follows. In section 6.2 we give a brief re- 67 6.2. Spacetimes with CTCs view of some of the spacetimes admitting CTCs and describe how the and Gott solutions fit into the picture. In section 6.3 we define what we mean by unwrapping and investigate the nature of the resulting singularities. In section 6.4 we unwrap the Gott spacetime and show that there are no CTCs in the unwrapped (singular) spacetime. In section 6.5 we unwrap the Gödel space and show that the unwrapped space is singular, and moreover, that CTCs are still present. In section 6.6 we “improve” our unwrapping proce- dure in order to remove the remaining circular CTCs and discuss the prop- erties of the resulting spacetime. The research presented in this chapter has been published in [74]. 6.2 Spacetimes with CTCs Spacetimes with CTCs can arise in a variety of ways. In some cases, such as the van Stockum cylinder, the Gödel universe and the Kerr blackhole, CTCs are produced by the “frame dragging” effect of the rotating matter. In other cases, such as the spinning string, they are due to coordinate identifications. In yet other cases the CTCs arise due to the non-trivial topology of the spacetime itself (wormholes). We give a brief overview of some of these spacetimes in this section, with the emphasis on the Gödel spacetime. The first spacetime where the CTCs are manifest, the van Stockum cylinder, was constructed by Lanczos in 1924, then rediscovered by van Stockum [75] and analyzed by Tipler [76]. This spacetime is stationary and axisymmetric (it admits two commuting Killing vectors, one timelike and one spacelike with closed orbits and a regular (1 + 1)-dimensional axis), its metric is of the Weyl-Papapetrou type [33]: ds2 = −A(r)dt2 +B(r)dtdφ+ C(r)dφ2 +H(r)(dr2 + dz2). (6.1) Here φ is 2pi-periodic, and so the CTCs appear whenever C(r) < 0. Con- ventionally, the solution consists of a spinning dust cylinder matched to an external vacuum solution. For certain values of the cylinder size and angu- lar momentum the CTCs occur in the external vacuum only. See [76] for details. One of the most famous solutions of the Einstein equation which admit CTCs was obtained in 1949 by Gödel [10]. Its geodesics were computed in [11] and its properties are discussed in [2]. Following [2, §5.7], we write the metric of the Gödel universe as ds2 = −dt2 + dx2 − 1 2 e2 √ 2ωxdy2 − 2e √ 2ωxdtdy + dz2. (6.2) 68 6.2. Spacetimes with CTCs Here ω = const and (t, x, y, z) take all real values. In the following we will call this coordinate system Cartesian. The manifold of the Gödel metric is R 4 and the spacetime is homogeneous. The matter source in this space can be written as Tab = ρuaub + 1 2 ρgab, (6.3) where ρ = 2ω2 is the energy density in the units where 8piG = 1 and c = 1. Here ua is the timelike unit vector field tangent to the coordinate curves of t in (6.2). If the second term in (6.3) is associated with a negative cosmological constant Λ = −ρ/2, then the matter content of the Gödel universe is rotating dust (pressureless perfect fluid) with density ρ, and ω is the magnitude of its vorticity flow. This spacetime is a direct product of a three-dimensional spacetime with a real line R, parametrized by the coordinate z, which does not add any interesting features and can be safely ignored. In the following we set ω = 1/ √ 2, so that ρ = 1. This is equivalent to rescaling the coordinates, up to a constant overall factor in the metric. The Gödel space is highly symmetric, admitting 4 out of a possible 6 Killing vectors in the (t, x, y) subspace. These are ∂t, ∂y, ∂x − y∂y, −2e−x∂t + y∂x + (e−2x − 12y2)∂y. The first one of these commutes with the rest, the last three form an SO(2, 1) Lie algebra. The rotating matter in the Gödel space leads to CTCs, some of which are manifest after a coordinate transformation to a cylindrical-like chart (τ, r, φ), where r > 0, τ can take any real values and φ is a 2pi-periodic angular coordinate: ex = cosh 2r + cosφ sinh 2r yex = √ 2 sin φ sinh 2r tan 1 2 ( φ+ t√ 2 − √ 2τ ) = e−2r tan 1 2 φ. (6.4) The new metric, after omitting the irrelevant z-coordinate, is ds2 = −dτ2 + dr2 + sinh2 r ( 1− sinh2 r ) dφ2 − 2 √ 2 sinh2 rdτdφ. (6.5) The validity of imposing periodicity on φ follows from the fact that the last of the equations (6.4) is 2pi-periodic in φ, and from the regularity condition on the axis. Specifically, a spacetime admitting an axial (U(1)) Killing vector ξa, parametrized by a 2pi-periodic coordinate φ is regular on the rotation axis 69 6.2. Spacetimes with CTCs (a set of fixed points of ξa) if and only if the elementary flatness condition (2.17) holds: ∇a(ξcξc)∇a(ξcξc) 4ξcξc → 1, (6.6) where the limit corresponds to the rotation axis [33]. This is further dis- cussed in section 6.3. This condition holds on the axis r = 0 of (6.5). The tangent to the coordinate curve of φ is future pointing when it is timelike, resulting in CTCs. This occurs for sinh r > 1. Given that the Gödel spacetime has CTCs passing through every point, it has generally been discarded as pathological. Other CTC-admitting solu- tions have been harder to dismiss, however. There are several known examples of non-simply connected spacetimes with matter sources satisfying the Null, Weak and Dominant energy condi- tions where CTCs are present. In such cases Carter, in his investigation of the spinning blackhole metric [77], makes a distinction between the “trivial” CTCs (those that are not homotopic to zero, i.e., non-contractible) and the “non-trivial” (contractible) ones, such as those present in the Gödel space- time. The trivial CTCs can be removed by going from a given non-simply connected spacetime to its universal cover, without changing the metric lo- cally. The non-trivial ones obviously persist even in that case. A standard example of a spacetime admitting trivial CTCs is Minkowski spacetime ds2 = −dt2 + dx2 + dy2 + dz2 (6.7) with the periodically identified timelike coordinate (t ∼ t+ T ). This space- time is homeomorphic to a cylinder S × R3, all CTCs are trivial and can be removed by going to its universal covering space, the R4 manifold with Minkowski metric. Another example, relevant to some of the unwrapping constructs below, is the spacetime of an infinite spinning cosmic string. A single straight non-spinning cosmic string along z-direction is described by the metric ds2 = −dt2 + dr2 + ( 1− m 2pi )2 r2dφ2 + dz2, (6.8) 0 ≤ φ ≤ 2pi, m 6= 0. This was first analyzed by Marder [78], who described the conical singularity at r = 0, without assigning any physical meaning to it. For the spinning string the metric is ds2 = − ( dt+ a 2pi dφ )2 + dr2 + ( 1− m 2pi )2 r2dφ2 + dz2, (6.9) 70 6.2. Spacetimes with CTCs 0 ≤ φ ≤ 2pi, where a is the angular momentum per unit length, and m is the mass per unit length (string tension), respectively. This spacetime can be obtained from that of a non-rotating string (6.8) by replacing the identification (t, r, φ, z) ∼ (t, r, φ + 2pi, z) with (t, r, φ, z) ∼ (t + a, r, φ + 2pi, z) and substituting t 7→ t + aφ/2pi (see e.g., [25]). This is an example of “topological frame dragging”, where an observer on the Killing horizon (corresponding to the zeros of the norm of ∂φ) appears to be rotating relative to an observer at infinity ([79, 80]). The manifold of (6.9) is regular and flat everywhere except at r = 0, where there is a conical singularity due to the mass term (the deficit angle is equal to m). The coordinate curves of φ are closed and become timelike sufficiently close to the string, i.e., r < a/(2pi −m). The conical singularity can be smeared out by a suitable matter distribution [81], in which case the CTCs become contractible. Instead of a single rotating string one can produce CTCs with a pair of non-rotating strings moving with respect to each other with a non-zero impact parameter, as discovered by Gott [13]. The coordinate chart in the vicinity of each string is just (6.8), and the two charts can be smoothly connected, such that there is a boost along the junction, as discussed in detail in section 6.4.1. This causal structure of this spacetime is, in a sense, an opposite of the spinning string one, as the CTCs in it have minimum size and extend all the way to infinity [82]. Thus the Gott spacetime violates the “physical acceptability conditions” proposed in [83]. The CTCs of this spacetime are discussed in detail in section 6.4.2. In all of the above examples the CTCs exist for all times, and so such spacetimes are usually considered unphysical, since they do not admit a Cauchy surface from which such a spacetime could evolve. Another well-known example of a vacuum spacetime with CTCs is the Kerr blackhole. In this metric there exist both the CTCs that wrap around the ring singularity and those that do not [77]. Both kinds are hidden from an external observer by the event horizon of the blackhole. A different class of spacetimes admitting CTCs are those with worm- holes and “exotic” matter sources (those violating the energy conditions), but without singularities. Exotic matter is required to keep the wormholes traversable. The first traversable wormhole metric was given in [84]. An example of such a metric is ds2 = −dt2 + dr2 + ( r2 +R (r)2 ) ( dθ2 + sin2 θdφ2 ) , (6.10) where R(r) has compact support, r ∈ R and R(0) 6= 0. Positive and negative values of r correspond to two different flat asymptotic regions. These can 71 6.3. Unwrapping be patched together far enough from the wormhole throats, where R(r) = 0. To do that we can identify a 3-plane (e.g., x = r sin θ cosφ = const) in one of these asymptotic regions with a corresponding 3-plane in the other. Since the spacetime is flat there, both the intrinsic and extrinsic curvatures of the planes vanish, and so a spacetime with such an identification satisfies the Einstein equation without the need for any additional matter sources. Once a wormhole exists in a given spatial slice t = const, it can be manipu- lated into producing CTCs by a variety of means, all based on changing the relative rates of time flow between the throats, as seen by an asymptotic ob- server, until there is a CTC threading through them. This can be achieved using either the special relativistic time dilation effect, as described in the original paper, or the gravitational one [85]. Since all such CTCs result from the underlying spacetime not being simply connected, going to the simply connected universal cover gets rid of the CTCs. Another evidence of the ubiquity of CTCs is the solution constructed by Ori [86], where a regular Cauchy horizon bounded by a closed null geodesic may be constructed as a limit of an evolution of a regular spatial slice and the matter sources satisfying energy conditions. There is a number of conjectures and theorems that deal with the CTCs and their appearance. Tipler [87] has shown that CTCs cannot evolve from non-singular initial data in a regular asymptotically flat spacetime. Hawking [88] has advanced the chronology protection conjecture, which states that, if the NEC holds, then the Cauchy horizon (the null boundary of the domain of validity of the Cauchy problem, see e.g., [2]) cannot be compactly generated. Moreover, even if the NEC is violated, quantum effects are likely to prevent the Cauchy horizon from appearing. 6.3 Unwrapping Since one of our goals is to show that unwrapping contractible CTCs creates singularities, we first review the definition of singularities in general relativ- ity. Next we discuss the quasi-regular singularities resulting from changing the angular coordinate identifications in Minkowski spacetime, as described in [55]. Finally, we consider one particular kind of quasi-regular singular- ities, the unwrapping singularity, one that results from unwrapping closed timelike curves in axisymmetric spacetimes with a regular symmetry axis. 72 6.3. Unwrapping 6.3.1 Singular Spacetimes Intuitively, one tends to think of a singularity in a spacetime as a place in it where the curvature diverges or something else pathological happens. This approach, while satisfactory in other field theories, where the spacetime is provided a priori, does not work in general relativity, because the spacetime is not given in advance. Instead it is determined by solving the Einstein equation, which relates the matter content to the spacetime curvature. This means that the manifold and the metric must be smooth enough to keep the Einstein tensor finite everywhere. Hence, any singular point is not a part of the spacetime manifold and cannot be described as a “place” [2], so a different definition of singularity is required. A singularity can be detected by the existence of curves of “finite length” which are inextendible in at least one direction. An inextendible curve γ is defined as a continuous map γ : [0, 1) → M for which there is no end point, i.e., there is no continuous map γ′ : [0, 1] → M , γ ⊂ γ′. However, while “length along a curve” is well-defined for manifolds with Riemannian metric, a Lorentzian metric does not give rise to a distance function for an arbitrary curve. For a geodesic curve one can use its affine length [2], but there is no unique or natural prescription for a distance between two points on a general curve. The definition of singularities through the existence of incomplete inex- tendible geodesics is successfully used in the proofs of the singularity the- orems [2]. These theorems are generally associated with the most familiar type of singularities, the curvature singularities, such as that of the Schwarz- schild metric for r → 0. This is, however, not the only type of singularities possible. A commonly accepted classification of singularities is given in [55]. In the case of curvature singularities some of the curvature invariants, such as R, RabR ab or RabcdR abcd, grow unbounded along an incomplete curve. A different case is the parallel-propagated curvature singularities, where some of the components of the Riemann tensor cannot be bounded along some incomplete curves, even though all of the curvature invariants remain finite or even vanish, such as in the case of singularities formed by gravitational waves. Yet another case, and the one most relevant to the subject of this chap- ter, is that of quasi-regular singularities. There the curvature tensors remain smooth and bounded everywhere along an incomplete curve, yet the curve is still inextendible. To avoid the case where singularities are created artifi- cially by removing a regular point from a given spacetime, only inextendible spacetimes are considered. A spacetime is inextendible if it is not isometric 73 6.3. Unwrapping to a proper subset of another regular spacetime of the same dimension. A classic example of a quasi-regular singularity is the two-dimensional cone. The differentiable structure and the metric on that space can be taken to be induced from its embedding into a three-dimensional flat space. The space is smooth (even flat) everywhere outside the apex of the cone. However, an attempt to include the apex into the space and continue the geodesics through it leads to problems. One can see that there is a curvature singularity at the apex by considering a cone with a spherical cap (as seen from its three-dimensional embedding) and taking a limit where the cap radius goes to zero (see e.g., [89]). The induced differentiable structure breaks down at the apex of the cone. In fact, this space is an example of a conifold, which is a generalization of a manifold, and it allows quasi-regular singularities of the conical type [90]. Cosmic string, described by the metric (6.8), is another example of a conical singularity. Mars and Senovilla [32] have shown that an axisymmetric spacetime is regular on the axis if and only if the regularity condition (6.6) holds. It is obviously violated in the case of a cosmic string with non-zero mass (6.8). Here we briefly review some of the definitions of a singularity, each tai- lored to a particular class of problems. For the simple examples of quasi- regular singularities considered in this paper, all the definitions agree. A detailed review of the topic can be found in [91]. One of the first definitions was Geroch’s g-boundary [92], defined by existence of incomplete geodesics, whether timelike, spacelike or null. The timelike geodesic incompleteness is the most severe case, as there would exist inertial observers whose existence comes to an end within a finite proper time. This definition does not address the case of singularities that can only be reached by non-geodesic curves. Another definition of singularity, originally due to Penrose [22], uses the concept of a conformal boundary. There a spacetime is embedded in another “unphysical” Lorentzian manifold conformally, rather than properly, in effect bringing the “infinity” to the finite values of the coordinates. This allows one to attach a conformal boundary to the spacetime. See e.g., [2]. If such a boundary is reached by a geodesic curve with a finite affine parameter value in the original spacetime, the conformal boundary is singular. This idea works well in highly symmetric cases, where the unphysical spacetime is easy to construct. A famous attempt to assign a causal boundary to a spacetime without resorting to an “external” concept, such as the unphysical conformal space- time, is that of Geroch, Kronheimer and Penrose [93]. They attach a “causal boundary” to any spacetime subject to certain causality restrictions. 74 6.3. Unwrapping Schmidt [94] generalized the geodesic affine parameter to non-geodesic curves, in order to characterize singularities that cannot be reached by a freely falling observer. The basic idea is to assign a positive-definite distance function to points on an arbitrary curve using a Riemannian metric on a frame bundle parallel-propagated along the curve. See e.g., [2]. This in turn enabled the use of Cauchy completion to assign an end point to an incomplete curve. These endpoints define a so called b-boundary. Once the b-boundary points are found, one can talk about a neighborhood of a singularity and the behavior of tensors, such as the Riemann tensor, along a curve approaching the singularity. Scott and Szekeres [95] suggested that the boundary definition need not be restricted to using only the objects intrinsic to the Lorentzian manifold. Moreover, it should be definable for any differentiable manifold with an affine connection, thus accommodating theories other than just the Einstein’s gen- eral relativity, such as the gauge theories, Einstein-Cartan and others. The abstract boundary approach is based on the idea of an “envelopment”, a way to embed a manifold in a “larger” manifold of the same dimension. The boundary points are then regular points of the topological boundary of the embedding. The abstract boundary points and sets are formed by equivalence classes of envelopments that cover each other. The set of all abstract boundary points is called the abstract boundary, or a-boundary. The singularities are classified as either removable, if they can be covered by a non-singular boundary set or essential if they cannot. For our purposes the timelike geodesic incompleteness provides an ad- equate definition of singularities. Indeed, any singularity resulting from a change in coordinate identifications appears in place of a formerly regular point. Since there are timelike geodesics passing through any regular point, these geodesics become incomplete after the change in identifications. In the case of singularities constructed by changing periodic coordinate identifications in an axisymmetric spacetime, the singular boundary coin- cides with the set of the fixed points of the periodic coordinate before the identification is changed. For example, for the two-dimensional cone with the metric ds2 = dr2+r2dφ2, where 0 ≤ φ ≤ Φ < 2pi, r = 0 is its singular bound- ary, due to the deficit angle. A conical singularity in the four-dimensional spacetime with the metric (6.8) is an example of a two-dimensional singular boundary which is a flat R2 manifold. The r = 0 singularity of the Schwarz- schild spacetime can be described as a singular b-boundary with a rather peculiar structure [96]. It is worth noting that the singular boundary is different from the bound- ary of an n-dimensional “manifold with boundary”. The latter is itself an 75 6.3. Unwrapping (n − 1)-dimensional manifold without boundary. In contrast, the singular boundary can be of any dimension, it may or may not be a manifold itself, and may or may not have a boundary (or a singular boundary). See [55] for examples and counter-examples. 6.3.2 Quasi-regular Singularities in a Flat Spacetime Here we describe the construction of a conical as well as unwrapped sin- gularities in a flat spacetime. Since changing the coordinate identifications does not change the metric at any of the regular points, the unwrapping singularities are always quasi-regular (they are not associated with a cur- vature divergence). Provided the original spacetime is asymptotically flat and has no event horizon, neither does the unwrapped one, so the resulting quasi-regular singularity is naked, i.e., there are future directed null curves originating arbitrarily close to the singularity that reach the future null in- finity. We first describe the unwrapping singularity for the four-dimensional Minkowski spacetime and then generalize the definition to apply to the spacetimes of interest. A conical singularity in a four-dimensional Minkowski spacetime can be obtained as follows [55]: 1. Remove the timelike two-plane x = y = 0 in the Cartesian chart. This is precisely where the cylindrical chart is not defined (r2 = x2+y2 = 0). This space is no longer simply connected and has the topology of S 1 × R3. 2. Unwrap the resulting space to obtain its universal covering (M̄ , ḡ), with the same flat metric in the cylindrical chart, but with the range of the new angular coordinate φ̄ extended to φ̄ ∈ R, instead of 0 ≤ φ ≤ 2pi. A local Cartesian chart (with the non-negative x-axis removed) is obtained by the standard transformation (x = r cosφ, y = r sin φ). A two-dimensional slice of this spacetime in the x− y plane (t = z = 0) is shown in Fig. 6.1 and the three-dimensional embedding of this slice into R3 is shown in Fig. 6.2. A well-known version of this space is the Riemann surface forming the domain of the complex log function. 3. Identify the points under translation through an angle Φ 6= 2pi, taking care to preserve the rotational isometry r = const, i.e., (t, r, φ, z) ∼ (t, r, φ+Φ, z). For Φ 6= 2pi we get the metric of the cosmic string (6.8) with the string tension m equal to the deficit angle m = 2pi − Φ. If Φ > 2pi, the string 76 6.3. Unwrapping identify identifyidentifyidentify y x M y x M’ Remove fixed points construct universal cover 0 ≤ φ ≤ 2pi0 ≤ φ ≤ 2pi 0 ≤ φ ≤ 2pi 0 ≤ φ ≤ 2pi 0 ≤ φ ≤ 2pi 0 ≤ φ ≤ 2pi −2pi ≤ φ̄ ≤ 0 0 ≤ φ̄ ≤ 2pi 2pi ≤ φ̄ ≤ 4pi 4pi ≤ φ̄ ≤ 6pi Figure 6.1: Unwrapping Minkowski space: a) pick a cylindrical chart, b) remove the fixed points of φ at r = 0, c) go to the universal cover by extending the range of φ to all reals. A t = z = 0 slice is shown as a collection of Cartesian charts with y = 0, x ≥ 0 removed and the surfaces φ = 2pi of one chart identified with the surface φ = 0 of the next chart. φ̄ is the global angular coordinate, φ̄ ∈ R. Figure 6.2: A visualization of a two-dimensional spatial slice of the un- wrapped Minkowski space. 77 6.3. Unwrapping has negative tension. The presence of a conical singularity is reflected in the focusing (or defocusing) of geodesics passing on the opposite sides of the singularity. This can also be seen from the violation of the regular- ity condition (6.6). This condition is obviously satisfied for the ordinary Minkowski spacetime, where ξcξc = r 2 in the cylindrical chart, but not after the identifications where the deficit angle m 6= 0, as can be seen by a coor- dinate transformation where the deficit angle is traded for a constant factor in ξcξc, such as in (6.8), breaking (6.6). If the last step is omitted, we get the unwrapped space (M̄ , ḡ), and the singularity is not conical in the standard sense, as there is no closed curve wrapping around it. We will call this type of quasi-regular singularity the unwrapping singularity. The subspace x = y = r = 0 removed in the step 1 forms the “boundary” of each of the three spaces corresponding to the three steps above. (We put the word boundary in quotes to indicate that it is neither a boundary of a manifold, nor yet a singular boundary, but rather an artificial “hole” in the spacetime.) After step 1 this boundary is regular, since it can be included back in to form the original inextendible Minkowski spacetime. After step 2 the boundary is no longer regular. Indeed, if it were regular, we would be able to include it back into the spacetime and show that any neighborhood of a point on the boundary point is homeomorphic to an open ball in R4. This would in turn imply that there are closed curves around any such point. Furthermore, any such closed curve is homotopic to a closed curve r = const lying inside the open ball. However, step 2 explicitly removes all such curves, contradicting the regularity assumption. The singularity is quasi-regular, because the Riemann tensor vanishes for all points r > 0. After the step 3 the boundary is still singular, as any arbitrarily small curve around it has the same fixed deficit angle, breaking the regularity condition (6.6). The argument that the singular boundary of the unwrapped spacetime is quasi-regular applies to a class of regular spacetimes that is larger than just the Minkowski space. For the steps 1 and 2 to be applicable for a certain spacetime, it is sufficient to have the spacetime covered by a single Cartesian chart. In particular, it is valid for the Gott and Gödel spacetimes and is generalized in section 6.3.3 to locally axisymmetric spacetimes. Moreover, for any future-directed null curve connecting the origin point O in this Cartesian chart with a certain point P in the original spacetime there is a future-directed null curve connecting the boundary in the un- wrapped spacetime with one of the infinitely many copies of the point P resulting from unwrapping. Thus, if the unwrapping point is not hidden 78 6.3. Unwrapping by an event horizon in the original asymptotically flat spacetime, the quasi- regular singularity in the unwrapped spacetime is necessarily naked. For completeness, we mention another type of quasi-regular singularity, the Misner singularity, constructed by removing a spacelike two-plane t = z = 0 from Minkowski space, then periodically identifying points under a given boost t2 − z2 = const, to obtain the four-dimensional Misner space14. The topology of the resulting space is S × R3, it contains CTCs through every point, and the surface t = z = 0 is a quasi-regular singularity. See [55] for detailed examples. If we omit the identification step, we obtain an “unwrapped” Misner space. One can construct rather complicated quasi-regular singularities by cut- ting and gluing together spacetime pieces with different properties. For example, Krasnikov [98] describes string-like singularities that are loops or spirals embedded in otherwise flat spacetime. 6.3.3 Singularities Created by Unwrapping We now generalize the unwrapping procedure to a general axisymmetric spacetime and review the properties of the resulting singularity. As discussed in section 2.5, we define an axisymmetric spacetime (M , g) as the one which is invariant under the action Φ : SO(2) ×M → M of the one parameter rotation group SO(2) ≡ U(1), such that the set of the fixed points of Φ is an (n − 2)-dimensional embedded surface F . If the latter condition is not required to hold, the spacetime is cyclic. A cyclic, but not axisymmetric, spacetime does not necessarily becomes singular after unwrapping - for example one can unwrap a two-dimensional torus in an obvious way (by going to its universal cover) and get a two-dimensional plane. Note that the axis of symmetry F is a set of regular points in M . By this definition, the cosmic string spacetime is, while cyclic, not axisymmetric, unless the infinitely thin string is thickened and smoothed into a one of finite diameter. To unwrap an axisymmetric spacetime we follow the same steps as in section 6.3.2. See also [36, 98]: • Start with an axisymmetric spacetime (M , g) • Remove the fixed point set F (the symmetry axis) to obtain (M ′ = M \F , g) 14A four-dimensional Misner space is a direct product of the two-dimensional Misner space introduced in [97] with the x− y plane 79 6.4. Unwrapped Gott Spacetime • Go to the universal covering space M̄ of M ′, Z : M̄ → M ′. The unwrapped spacetime (M̄ , g) is singular by construction (we ex- cised a set of regular points from its base space), and inextendible, provided (M , g) is inextendible. We cannot use the regularity criterion (6.6) to show inextendibility, as it only applies to axisymmetric spacetimes, and M̄ is not axisymmetric, because the rotation Φ is lifted to the translation Φ̄. In- stead we mirror the argument from [55] reproduced in section 6.3.2 for the unwrapped Minkowski spacetime. Suppose there is an extension M̂ of M̄ that includes a point of F as a regular point. Any compact neighborhood Ô of a point p ∈ F ⊂ M̂ includes a neighborhood Ō of the corresponding point on the singular boundary of the unwrapped space M̄ . However, Ō includes a lift Φ̂ of some orbits of the rotation Φ, which are non-compact (and infinitely long, as measured by the generalized affine parameter) in the unwrapped spacetime M̄ . Thus Ô is also non-compact, leading to a contra- diction. Any attempt to compactify Ô while keeping Φ̂ an isometry would make the orbits of Φ̂ closed, thus violating the regularity condition. The boundary of the unwrapped spacetime is again a quasi-regular sin- gularity15, as there is no curvature divergence anywhere along a lift of the curves from M to M̄ . The singularity is naked, provided the rotation axis is not hidden by the event horizon in M . The steps described in this section can be generalized to remove the as- sumption of axisymmetry. Indeed, since we are interested in unwrapping contractible CTCs, we only need to make them non-contractible by remov- ing a relevant subspace F (typically (D − 2)-dimensional, such as the axis of symmetry) from the original manifold M . We can then go to the uni- versal cover of the resulting non-simply connected manifold M̄ . This lift “unwraps” the CTCs in question, at the expense of creating quasi-regular singularities. In this thesis we restrict our attention to just two examples of spacetimes containing CTCs, and their unwrapping, so we do not use this more general form. 6.4 Unwrapped Gott Spacetime The Gott spacetime, discussed in detail in this section, admits CTCs. Coop- erstock and Tieu suggested that such a matching is artificial and that iden- tification “before the Lorentz boost is applied” is “more natural” [12]. Since 15A singularity resulting from unwrapping of a blackhole, such as BTZ one, [68] may or may not remain shrouded by its (unwrapped) event horizon. 80 6.4. Unwrapped Gott Spacetime their claim is based on a different spacetime16 and relies on a closed curve crossing these ribbon singularities, it is hard to evaluate. Instead, in keeping with the procedure described in section 6.3.3, we unwrap the Gott space- time, such that the Gott CTCs correspond to open curves in the unwrapped spacetime. To do that without changing the metric locally, we remove a timelike line from the Gott spacetime and construct the universal cover of the resulting multiply connected (and now singular) spacetime. We show that no other CTCs are present in the new spacetime. We also construct some alternative extensions of the Gott spacetime with a timelike line re- moved and discuss their properties. 6.4.1 Construction We first review the way the (2 + 1)-dimensional Gott spacetime is con- structed. Following Gott [13], we start with the (2+ 1)-dimensional version of the straight cosmic string spacetime (6.8), where the string is represented by a point particle. The metric of a single point-particle in 2+1 dimensions ds2 = −dt2 + dr2 + ( 1− m 2pi )2 r2dφ′2, (6.11) where 0 ≤ φ′ ≤ 2pi and the particle mass m is the deficit angle. The deficit angle can be made explicit by the substitution φ′ = φ/(1 − m/2pi), where now 0 ≤ φ ≤ 2pi −m: ds2 = −dt2 + dr2 + r2dφ2. (6.12) The coordinate identification φ ∼ φ+2pi−m corresponds to removing a time- like wedge centered at the particle and identifying the opposite faces of the wedge. The angle φ0 the wedge makes with the horizontal axis corresponds to the remaining coordinate gauge freedom and can be chosen in a way that simplifies a particular calculation. Two such strings can be joined together, and the relative orientation of the wedges is a matter of convenience. Gott has chosen the wedge angle in a way that identifies the surfaces φ0 = pi 2 − m 2 ∼ φ1 = pi 2 + m 2 , which simplifies his proof of the existence of CTCs. Cutler [99] used the identification φ0 = pi/2 − m ∼ φ1 = pi/2 for one of the strings to show 16They appear to remove two timelike ribbons from Minkowski space and boost the resulting singularities relative to each other. 81 6.4. Unwrapped Gott Spacetime                                                                           CTC p p p’p’ 12 12 Figure 6.3: Conical wedge identification choices in the Gott spacetime. Wedge fill lines indicate the identified points. The strings are shown at the moment of the closest approach. Left: Gott identification, where CTCs are manifest (one CTC is shown). Center: Cutler identification, used to prove the existence of points not lying on any CTCs. Right: Carroll identi- fication, used to visualize the existence of CTC-free spacelike hypersurfaces, as the opposite sides of the wedge are identified at equal times. the existence of a timelike cylinder enclosing both strings which no CTCs enter. Carroll et al. [100] identified φ0 = −m/2 ∼ φ1 = m/2 to visualize the existence of spacelike hypersurfaces through which no CTCs pass. These identification choices are illustrated on Fig. 6.3. As our goal is to investigate the CTCs in the Gott spacetime, we use the Gott’s choice of φ0. We write each face of the wedge (denoted by the indices 1 and 2) in the Cartesian coordinate system, expressed in terms of two parameters t and x: t1 = t, x1 = x, y1 = x cot m 2 , t2 = t, x2 = −x, y2 = x cot m 2 . (6.13) Points corresponding to the same values of (t, x) on both faces are identified. It is possible to have multiple strings in the same spacetime, as long as the total deficit angle does not exceed 2pi, otherwise the topology of the spacetime becomes S2 × R instead of R3, where the total deficit angle is equal to 4pi [25]. Next step is to boost the wedge in the positive x direction with the ve- locity v < c = 1. The coordinates of the faces are then Lorentz-transformed into the laboratory frame as tL = γ(t+ vx), xL = γ(x+ vt), yL = y: t1L = γ(t+ vx), x1L = γ(x+ vt), y1L = x cot m 2 , t2L = γ(t− vx), x2L = γ(−x+ vt), y2L = x cot m 2 . (6.14) One can see that a pair of identified points p1 = (t1L, x1L, y1L) and p2 = 82 6.4. Unwrapped Gott Spacetime (t2L, x2L, y2L) have different values of the time coordinate tL in the labora- tory (center of momentum) frame. Specifically, the time difference between the two is ∆tL = t2L − t1L = −2γx = v(x2L − x1L), (6.15) and is always negative. To describe two boosted strings of masses m moving along the x-axis in opposite directions with the velocities v and −v and the impact parameter 2b, we shift the boosted wedge (6.14) by b in the positive y direction and introduce a second wedge, v 7→ −v′ and y 7→ −y: t1L = γ(t+ vx), x1L = γ(x+ vt), y1L = b+ x cot m 2 , t2L = γ(t− vx), x2L = γ(−x+ vt), y2L = b+ x cot m 2 , t′1L = γ(t ′ − vx′), x′1L = γ(x′ − vt′), y′1L = −b− x′ cot m 2 , t′2L = γ(t ′ + vx′), x′2L = γ(−x′ − vt′), y′2L = −b− x′ cot m 2 , (6.16) where primed variables describe the second wedge. We have now con- structed the Gott spacetime in a single Cartesian chart (tL, xL, yL), cor- responding to the laboratory frame, subject to the two wedge identifications (t1L, x1L, y1L) ∼ (t2L, x2L, y2L) and (t′1L, x′1L, y′1L) ∼ (t′2L, x′2L, y′2L). If we choose t′ and x′ such that γ(t+vx) = γ(t′−vx′) and γ(x+vt) = γ(x′−vt′), then the closest approach of the strings corresponds to tL = 0. 6.4.2 CTCs of the Gott Spacetime Following Gott, we now consider the curves composed of two pieces of geodesics with xL = const, as shown on the Fig. 6.3 left. The first piece con- nects the two wedges at the points p′1 = (t′1L = −T, x′1L = a, y′1L = −V T ) and p1 = (t1L = T, x1L = a, y1L = V T ). Here V is the velocity of the observer traveling the geodesics. The travel time along the geodesic T is T = b+ aγ cot m2 V + vγ cot m2 . The second piece connects the two wedges at the points p2 = (t2L = −T, x2L = −a, y2L = V T ) and p′2 = (t′2L = T, x′2L = −a, y′2L = −V T ). For the two geodesics to form a closed curve, the initial point of one must be the final point of another: p1 ∼ p2 and p′1 ∼ p′2. In this case the co- ordinate time 2T taken to travel from one wedge to another is balanced 83 6.4. Unwrapped Gott Spacetime exactly by the backward time jump across the wedge between x1L = a and x2L = −a. Using (6.15), we get 2T +∆tL = 2T + v(x2L − x1L) = 0, (6.17) or T = av, resulting in the relation V = b/(av)+ 1γv cotm/2. The curve is a CTC for V < 1, which is possible for large enough a whenever γv > cotm/2. This also sets the lower limit on a for a given boost: amin = γb γv − cot m2 for this choice of geodesics. These xL = const CTCs are not the only ones possible. Cutler [99] has determined the (null) boundary of the region containing CTCs, and it turns out that, while there are closed null curves passing closer to the origin than amin, no CTCs pass through the origin and all CTCs go counter-clockwise around the origin. Moreover, even when CTCs are present, there always exist a neighbor- hood of each string free of CTCs. As a consequence, “smoothing out” the strings in a small enough neighborhood does not affect the CTCs, as noted by Cutler. 6.4.3 Unwrapping Gott Spacetime The existence of the minimum value for a CTC’s distance from the x = y = 0 line implies that all CTCs wrap around it in the direction opposite to the relative motion of the strings (they also wrap around both strings at some finite distance from each). The Gott spacetime is flat away from the strings, so we can apply the same unwrapping procedure as in section 6.3: we remove the subspace x = y = 0 and construct the universal covering space of the resulting non-simply connected manifold (we assume that the original Gott spacetime is simply connected, as the strings can be smoothed out). Since in the Gott’s choice of identifications neither wedge crosses the x-axis, the resulting space can be described using countably many copies of the Cartesian (t, x, y) charts, with the charts n and n + 1 joined along the non-negative x-axis of each one, as shown on Fig. 6.4a. This unwrapped Gott spacetime is simply connected, has a quasi-regular singularity at x = y = 0 and admits no CTCs, no matter how fast the strings are moving relative to each other. It also contains a countable infinity of pairs of boosted strings. The presence of the singularity removes the 84 6.4. Unwrapped Gott Spacetime                                                                                                                                                                                                                                     a) b) Remove (y=0,x>0) half−plane Identify multiple copies along (y=0,x>0) half−plane n−1 n n+1 n+2 n−1 n n+1 n+2 CTC Figure 6.4: Unwrapping Gott spacetime. a) The x = y = 0 subspace is removed and the universal cover is constructed by patching multiple Carte- sian charts together. The former CTCs (indicated by the vertical arrows) are now open curves passing from chart to chart. The identifications be- tween charts are indicated by the wavy arrows. b) An alternative way to unwrap: a single Gott chart is matched to a collection of Minkowski charts. The former CTCs wrap around just the two strings in the single Gott chart. 85 6.5. Unwrapping Gödel Spacetime restriction on the total mass (deficit angle) of all strings, which can now be arbitrarily large, though each string’s mass still cannot exceed pi. A timelike or null observer starting on the nth chart and traveling around both strings counter-clockwise ends up on the (n+1)th chart after crossing the positive x-axis. This ensures that no such curve is closed, and so no CTCs are present in the unwrapped Gott spacetime. Instead, what used to be CTCs are now open curves winding around two strings per turn. It is worth noting that the unwrapped Gott spacetime described above is not the only way to unwrap the CTCs. Since the surface where each two Cartesian charts are joined is locally flat, we do not have to have the two moving strings present on more than one chart. For example, all but one chart can be Minkowski, as shown on Fig. 6.4. We have shown that one can indeed change the identifications in the Gott spacetime such that no CTCs are present. The trade-off for the CTCs removal is introduction of a naked quasi-regular singularity. This singularity is timelike and so it is present in any possible initial data set, making any initial value formulation problematic. We next turn to another (2 + 1)-dimensional spacetime with CTCs, the Gödel universe, and demonstrate that a straightforward CTCs unwrapping does not work there. 6.5 Unwrapping Gödel Spacetime In this section we construct a spacetime which is locally Gödel at every point, but with a different global structure, such that a given set of CTCs in the original spacetime is not longer closed in the unwrapped spacetime. This is not the only way one can get rid of the Gödel CTCs. One obvious way to remove CTCs from the Gödel space is to restrict the radial coordinate in the chart (6.5), thus creating a boundary where the orbits of φ are still spacelike. Examples of this are given in [101, 102], where a preferred holographic screen is constructed at the radial distance sinh r = 1/ √ 2, where the expansion of the congruence of null geodesics emanating from a point at r = 0 is zero. A generalization of this approach to higher-dimensional Taub-NUT- AdS spacetimes is done in [103]. Another approach, considered in [104], is to match the Gödel interior to an exterior spacetime without CTCs. There the metric is explicitly changed locally in the regions where the CTCs used to exist. 86 6.5. Unwrapping Gödel Spacetime 6.5.1 Unwrapping Procedure and Circular CTCs We now consider a particular example of unwrapping the Gödel space with respect to a given family of CTCs. The metric is given in the (τ, r, φ) coordinate chart by the expression (6.5). This chart is singular at r = 0, but this is just a coordinate singularity, as the transformation (6.4) shows. Since the spacetime is homogeneous, this chart can be constructed using any point in the spacetime as its origin. The CTCs manifest in this chart are the coordinate curves of φ when sinh r > 1. They are not geodesics, but they are isometries of the spacetime, since the metric in this chart does not depend on φ. These CTCs have been extensively studied (see e.g., [2]). Cooperstock and Tieu, in [12], “question the continuation of identifying the φ values of 0 and 2pi when φ becomes a timelike coordinate”. Since it is impossible to abruptly change the coordinate identification of φ only at sinh r ≥ 1 without breaking the regularity of the spacetime at sinh r = 1, we will instead remove this identification everywhere. The Gödel spacetime is axisymmetric, and so the unwrapping procedure is rather straightforward. We start with the chart (6.2) of the Gödel space- time M . In this chart the CTCs that are coordinate curves of φ cross the positive x-axis in the counter-clockwise direction, like the CTCs of the Gott spacetime. We remove the x = y = 0 subspace (corresponding to a single fiber of the Killing vector ∂t, which also coincides with ∂τ at r = 0) and construct a universal covering space of the resulting non-simply connected manifold M ′. The resulting spacetime can be described by either a single global chart (6.5) with φ ∈ R or by a collection of countably infinitely many Cartesian charts with the charts n and n + 1 joined along the positive x- axis of each one. The description using a single cylindrical chart is possible because the subspace r = 0, where (6.5) is not defined, has been removed, and so this chart is valid everywhere in the unwrapped spacetime. As expected, the removed subspace makes the unwrapped Gödel space singular, and inextensible. Any curve that passed through the subspace r = 0 in M is incomplete in M ′ and hence in the unwrapped space, as well. All circular CTCs cross the positive x-axis in M ′ at least once, so their unwrapped-space analogs in the nth Cartesian chart end up on the n + 1 chart after crossing the axis and so cannot be closed. This corresponds to the orbits of φ in the chart (6.5) being open in the unwrapped space. 87 6.5. Unwrapping Gödel Spacetime 6.5.2 Remaining Circular CTCs While there are no r = const > sinh−1 1 CTCs in the unwrapped Gödel spacetime, this is not the only kind of circular CTC present in the Gödel metric. Since the Gödel spacetime is homogeneous, we can construct a cylindrical chart (6.5) around any point and obtain CTCs winding around that point. If this new, shifted origin of the cylindrical chart is “far enough” from the old, unshifted one, then the CTCs around it will lie wholly on a single sheet of the space unwrapped around the unshifted origin, and so will remain CTCs even in the unwrapped space. To demonstrate this, it is convenient to use the quotient space M̃ of the (2 + 1)-dimensional Gödel space M with the fiber defined by the orbits of the timelike Killing vector y : M → M̃ . The metric hab of the quotient space is calculated as hab = gab − yayb ycyc (see e.g., [37]). Since ya is timelike everywhere, with a non-vanishing norm, hab is nowhere singular, the reduced space is Riemannian, and its line el- ement in the original Cartesian coordinates is just a flat two-dimensional space ds2 = dx2 + dt2. (6.18) An arbitrary circular CTC in the full space, parametrized by 0 ≤ φ ≤ 2pi, is completely defined by its center (t0, x0, y0) and radius R. It can be written using the equivalent of (6.4) as ex−x0 = cosh 2R+ cosφ sinh 2R (y − y0)ex = √ 2 sin φ sinh 2R tan 1 2 ( φ+ t− t0√ 2 ) = e−2R tan 1 2 φ. (6.19) An image of this CTC in the flat quotient space M̃ is obtained by omitting the coordinate y from (6.19): ex−x0 = cosh 2R + cosφ sinh 2R tan 1 2 ( φ+ t− t0√ 2 ) = e−2R tan φ 2 , (6.20) which can be rewritten in an explicit form as x = x0 + ln cosh 2R+ cosφ sinh 2R t = t0 + 2 √ 2 tan−1 (e−2R − 1) tan φ2 1 + e−2R tan2 φ2 . (6.21) 88 6.6. Multiple Snwrapping of the Gödel Spacetime All CTCs with the same values of x0 and t0, but with a different y0 are mapped into the same closed curve. The image of the singular boundary of the unwrapped space is the point x = t = 0 of the quotient space. We note the following properties of a curve described by (6.21): 1. It is inscribed in a rectangle centered at the point (x0, t0) and with sides 2R and 2 tan−1 sinhR, and 2. It circumscribes an ellipse inscribed into this rectangle, as described by equation (6.22). ( x− x0 2R )2 + ( t− t0 2 √ 2 tan−1 sinhR )2 = 1. (6.22) The property 1 follows from the ranges of x and t in (6.21) for a given R, while the property 2 can be shown by substituting (6.21) into the left-hand side of (6.22), finding the four local minima of the resulting function of φ and showing that they give exactly the equality (6.22). At small R the curve (6.21) tends closer to (6.22), while at large R it asymptotically approaches the rectangle, as shown on Fig. 6.5. We can now show that there exist CTCs that are not unwrapped by the singular boundary resulting from unwrapping around the origin in the chart (6.5). Since the image of any CTC winding around the boundary has to wrap around the image of the boundary in the reduced space, constructing the image of a CTC that does not wrap the image of the singularity is enough to show the existence of CTCs that are not unwrapped. Since the image of any CTC of radius R with, say, |x0| > 2R or |t0| > 2 √ 2 tan−1 sinhR does not wrap around x = t = 0, all such curves remain closed after unwrapping. Thus the näıve coordinate identification change of [12] fails to unwrap at least some of the circular CTCs. 6.6 Multiple Snwrapping of the Gödel Spacetime By unwrapping the Gödel spacetime we have introduced a quasi-regular sin- gularity into it, yet we did not accomplish the goal of removing all CTCs from it. If one remains intent on also removing circular CTC, one may consider a multiple unwrapping instead, as described at the end of the sec- tion 6.3.3. 89 6.6. Multiple Snwrapping of the Gödel Spacetime -1 -0.5  0  0.5  1 -1 -0.5  0  0.5  1 x 2R t 2 √ 2 t a n − 1 s in h R R = Rmin R = 5 R = 20 Figure 6.5: Normalized concentric CTCs of radius R illustrated in the co- ordinates ( x 2R , t 2 √ 2 tan−1 sinhR ) . The CTCs lie inside a square with the side equal to two, but outside of a circle inscribed into it. 90 6.6. Multiple Snwrapping of the Gödel Spacetime 6.6.1 Double Unwrapping We first consider what happens when we unwrap the CTCs in just two charts. We show that even in this simple case the unwrapped space contains a countably infinite number of quasi-regular singularities. This “double unwrapping” procedure would look as follows. We consider two different families of circular CTCs, one centered at x = x1, y = y1 and x = x2, t = t2 respectively. The subspaces (x = x1, t = t1) and (x = x2, t = t2) are the fixed points of the corresponding U(1) isometries represented by the CTCs. As before, we first remove these fixed points from the spacetime M , only this time we have to remove both sets, resulting in the singular space M ′′. Next we construct the universal covering space M̄ of M ′′ and lift the metric tensor to M̄ . Since the original (2 + 1)-Gödel manifold is R 3, we do not need to worry about accidentally removing any topological features unrelated to unwrapping. In this case M ′′ is homotopy-equivalent to the wedge sum of two circles S1 ∨ S1, known as the “figure 8” (see e.g., [35, Chapter 1]). This can be shown by explicitly constructing a deformation retraction, an operation that preserves the fundamental group of a manifold. To do that, we first note that, since the orbits of y are open lines, they can be retracted into points by the continuous map fs : (t, x, y) 7→ (t, x, (1−s)y). f0 is the identity map, and f1 maps M ′′ into a two-dimensional plane with two points, (x1, t1) and (x2, t2), removed. Following [35], we next retract the plane first onto two circles (one around each removed point) connected by a line segment, then contracting the connecting segment into a point. The resulting space is the “figure 8”. As a result of the retraction, the singularities now “fill the inside of the circles”. The fundamental group of M ′′ is the fundamental group of the “figure 8”, which is just the free product of two copies of Z, pi1 = Z ∗ Z. Each element of the group corresponds to winding around one of the two singularities in M ′′. The universal cover of the “figure 8” is well known, it is a tree with countably infinitely many edges and each node connecting four edges (see e.g., [35] for construction). The process of constructing the universal cover of the twice punctured plane is shown schematically on Fig. 6.6. Each edge of this graph corresponds to a CTC winding around one of the removed subspaces of fixed points. In the unwrapped space this CTC becomes open and corresponds to a given path along the graph. Traversing one edge corresponds to “going around” one of the singularities, so there is a one-to- one correspondence between the singularities in the twice-unwrapped Gödel and the edges in the graph. We can now conclude that unwrapping around two axes at once in a 91 6.6. Multiple Snwrapping of the Gödel Spacetime Figure 6.6: Constructing the homotopy equivalence of the Gödel space after double unwrapping. We start with the (t, x, y = 0) subspace with two points removed (left), then retract the space onto first two circles connected by a line segment, then the “figure 8”, and finally construct the universal cover of the “figure 8” (only the first four levels of nodes are shown). Each edge corresponds to a circle wrapping around one of the two singularities in the original space, so the unwrapped space contains infinitely many singularities. simply connected spacetime results in a spacetime with a countable infinity of quasi-regular singularities of the type discussed in section 6.3.2. 6.6.2 Multiple Unwrapping As discussed in section 6.5.2, there are infinitely many families of concentric CTCs parametrized by (t0, x0, y0) for which at least some CTCs persist after unwrapping around (x = 0, t = 0). According to the property 2, each CTC lies outside an ellipse (6.22). All such ellipses are larger than the image of a closed null curve corresponding to sinhR = 1. If we modify the spacetime in a way that transforms any such ellipse into an open curve, the resulting spacetime will not have any circular CTCs. This can be accomplished by first tessellating the quotient space (x, t) with a dense enough triangular lattice, such that there is a vertex inside any ellipse with sinhR = 1, then lifting it into the full three-dimensional Gödel spacetime using y as the fiber and finally by going to the universal covering space, in a procedure analogous to the one described in section 6.6.1. The tessellation is shown schematically on Fig. 6.7. The fundamental group of the tessellated space is the free group on Z generators, one for each removed point. A small patch of the unwrapped quotient space is illustrated on the Fig. 6.8. Each helix corresponds to a family of unwrapped concentric circular CTCs. The price to pay for removing all circular CTCs is the introduction of a naked singular boundary consisting of a countable 92 6.6. Multiple Snwrapping of the Gödel Spacetime x 2R t 2 √ 2 t a n − 1 s in h R Figure 6.7: A tessellation of the two-dimensional quotient space of the Gödel spacetime. Each vertex corresponds to a timelike line in the full spacetime and each circle is an image of a closed null curve R = Rmin. The tessellation is dense enough to make any circular CTCs wrap around at least one such line, as shown. Once the lines are removed and the resulting non-simply connected spacetime is lifted into the full spacetime and then into its uni- versal cover, no circular CTCs are present in this “multiply unwrapped” spacetime. Figure 6.8: A visualization of the two-dimensional quotient space of the Gödel spacetime unwrapped at three points at once. Only a small patch of this space is shown, as unwrapping around two or more points results in a countable infinity of singular boundaries. 93 6.6. Multiple Snwrapping of the Gödel Spacetime infinity of disjoint pieces. 6.6.3 Sector-like CTCs in the Gödel Spacetime One can ask whether any other types of CTCs are present in the multiply unwrapped Gödel spacetime. For example, is it possible to weave one’s way in between the vertexes and come back to the starting point along a CTC? This seems unlikely and we conjecture that no such CTCs exist. To support this conjecture we describe a different kind of CTCs, we call sector-like CTCs, and show that they, too, are transformed into open curves by the multiple unwrapping procedure of the section 6.6.2. The idea of constructing a CTC surviving the tessellation of the sec- tion 6.6.2 is to exploit the property of the Gödel spacetime where an arc with a larger radius but a smaller angular distance can get us just as far back in time in the chart (6.5). The hope is then to go far along a timelike curve in a radial direction, then along an arc, then back to the starting point, thus covering a sector instead of a full circle in this chart. If the resulting sector is thin enough, then we can try to fit it in the tessellated space in such a way that no vertexes are inside the sector. To check if this can be done, we calculate the angular and linear distance along the arc required to overcome the time lost traveling forward and back along the two radial directions. Since a closed null sector would be “thinner” than the corresponding timelike sector, we analyze the null sector first. The (negative) time change in the coordinate τ along a null arc of radius R and angle ∆φ in the chart (6.5) can be calculated as ∆τ = (sinh2R( √ 2− cothR))∆φ, (6.23) and it has to compensate for the positive time change of 2R along the two radii of the sector, resulting in the total angular change of ∆φ = 2R sinh2R( √ 2− cothR) . (6.24) To see if the sector is thin enough to fit between the vertexes of the lattice for large enough R, we project it into the flat quotient space (6.18). The three legs of the path written in the (τ, r, φ) coordinates and parametrized 94 6.6. Multiple Snwrapping of the Gödel Spacetime -8 -6 -4 -2  0  2  4  6 -5  0  5  10  15 t x 1 2 3 4 5 6 1 Circular Closed Null Curve (CNC) 2 Sector-like CNC, R=Rmin+0.2 3 Sector-like CNC, R=Rmin+0.5 4 Sector-like CNC, R=Rmin+1 5 Sector-like CNC, R=Rmin+2 6 Sector-like CNC, R>>Rmin Figure 6.9: Images of sector-like closed null curves in the (x, t) coordinates for a range of values R. Arrows indicate the future null directions. For large R the curve tends to a triangle with a fixed minimum angle. The angle increases as R goes down, and for R close to the minimum value of sinhRmin = 1 the null curve has to wrap around the origin several times to compensate for the time lost along the radial paths. by λ are  (λ, λ, 0) if 0 ≤ λ ≤ R,( 2R − λ,R, λ−R sinh2R( √ 2− cothR) ) if R ≤ λ ≤ 3R,( λ− 4R, 4R − λ, 2R sinh2R( √ 2− cothR) ) if 3R ≤ λ ≤ 4R. (6.25) The same curve in the (x, t) chart of (6.18) can be described using the ex- plicit coordinate transformation (6.21). The resulting curves are plotted for several values of R on Fig. 6.9. For large R the three turning points of the path in the (x, t) chart asymptotically approach (0, 0), (2R, 2 √ 2R), (2R,−2√2R) respectively. Thus, no matter how large R is, the tessellation 95 6.7. Conclusion dense enough to unwrap circular CTCs also unwraps the sector-like closed null (and therefore timelike) curves. In this sense, the original CTCs de- scribed by Gödel appear to have the smallest “footprint” in the flat quotient space. Whether or not there are other CTCs that persist in the multiply un- wrapped spacetime, it is quite clear that removing CTCs from the Gödel spacetime solely by changing the coordinate identification results in a rather contrived space, given its countable infinity of naked quasi-regular singular- ities. 6.7 Conclusion In this chapter we have defined and investigated “unwrapping” CTCs in two (2 + 1)-dimensional toy models, the Gödel spacetime and the Gott space- time, as a concrete implementation of the claim by Cooperstock and Tieu [12] that the periodic identification of a timelike coordinate is “purely arti- ficial”. The procedure requires removing a timelike line from the spacetime and constructing a universal cover of the resulting non-simply connected spacetime. We have demonstrated that such an unwrapping creates a naked quasi-regular singularity, corresponding to the removed timelike line in the original space. The same argument was extended to any locally axisym- metric spacetime where CTC wrap around the axis, as is the case in the Gott spacetime. While the unwrapped Gott spacetime is devoid of CTCs, the unwrapped Gödel spacetime still contains them. We have defined a “multiple unwrap- ping” of the Gödel spacetime in order to remove the remaining circular CTCs. As a result, this multiply unwrapped spacetime contains a countably infinite number of singularities. We conjecture that this multiple unwrap- ping removes all other CTCs as well, and support it by giving an explicit example of a sector-like CTC, which is also removed by the multiple un- wrapping. Our investigation into the ways of removing the CTCs by means of chang- ing coordinate identifications resulting in unwrapping suggests that CTCs appearing in the solutions of the Einstein equation are not a mathematical artifact of arbitrary coordinate identifications, but rather are an unavoid- able, if an undesirable, consequence of general relativity. Different ways to extend the same local coordinate patch of a pathological spacetime may lead to different pathologies, such as CTCs or naked quasi-regular singularities, but are unlikely to result in a physically acceptable regular spacetime. 96 Chapter 7 Conclusion This thesis describes the research in the area of axisymmetric dimensional reduction in general relativity, with an emphasis on the pathologies created by the reductions preserving asymptotic flatness. Starting with some simple and highly symmetric four-dimensional examples and proceeding to rather complicated ones, we have shown that treating an axisymmetric spacetime as the one with one lower dimension without this symmetry, while an attractive idea, results in a number of issues which are likely to hamper an investigator unless the pathologies in question are explicitly dealt with. Specifically, the stress-energy tensor of a reduction preserving asymptotic flatness is shown to induce negative energy density into the reduced spacetime. In the reduction from four to three dimensions the pressures exceed energy density leading to superluminal matter flows being seen by some observers. We have shown this by treating a general asymptotically flat spacetime as a perturbation on the Schwarzschild one. For the special, but still interesting case of a U(1) reduction from four to three dimensions, we applied the topological censorship as well as the topo- logical rigidity theorem (proved in this thesis for the reduction in question) to show that no such U(1) reduction from an AF spacetime to either an AF or ALADS one can satisfy the null energy condition. A next step in this research would be an investigation of the potential adverse effects the pathological stress-energy tensor may have on the cur- rently popular numerical evolution schemes that take advantage of space- time symmetries, and the ways to address them. Another potential topic of further research would be an application of the topological rigidity to an axisymmetric reduction, not just a U(1) reduction. Another application of the dimensional reduction techniques was the in- vestigation of the nature of the closed timelike curves in the Gott and Gödel spacetimes. 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Grav. 15 (1998) 357–366, gr-qc/9711011. 107 Index U(1) reduction, 3, 33, 37, 54, 58, 60, 62, 97 ADM mass, 3, 15, 23, 33, 39, 44, 49, 52 angular momentum, 2, 15, 30, 40, 46, 61, 68, 71 anti-de Sitter, 3, 4, 7, 35 asymptotically flat, 2, 4, 12–14, 17, 22, 40, 44, 50, 55, 59, 62, 72, 79, 97 average null energy condition, 10, 27, 37, 62, 64 axis of symmetry, 4, 16, 21, 37, 79 axisymmetry, 2, 16, 22, 29, 31, 33, 35, 37, 38, 40, 45, 51, 55, 63, 67, 72, 75, 79, 87, 96 black string, 61 boundary, 3, 5, 7, 11, 22, 25, 28, 34, 37, 44, 52, 58, 63, 74, 78, 84, 89, 92 Cauchy problem, 1, 11, 26, 72 Cauchy surface, 2, 7, 26, 55, 56, 64, 71 closed timelike curve, 4, 9, 66, 67, 71, 79 domain of outer communications, 3, 8, 23, 25, 29, 39, 55, 59 dominant energy condition, 1, 11, 22, 35, 51, 53, 63 doubling, 17, 22, 33, 38, 51, 63 elementary flatness condition, 17, 25, 36, 70 globally hyperbolic, 2, 3, 7, 9, 55, 59, 61, 62 initial surface, 1, 8, 11, 13, 28 Komar integral, 3, 15, 46 naked singularity, 1, 4, 5, 25, 34, 40, 64, 67, 76, 86 null energy condition, 9, 11, 24, 50, 55, 59, 72 positive energy theorem, 15, 43, 51, 54 quasi-regular singularity, 5, 51, 61, 67, 72, 73, 76, 78, 84, 89, 96 spacetime-with-boundary, 7, 8, 13, 56, 59 topological rigidity, 3, 12, 54, 55, 59, 64, 97 totally geodesic, 17, 39 weak energy condition, 3, 9–11, 24, 25, 27, 29, 32, 34, 37, 43, 50, 51, 53, 61 well-posed, 1, 26, 64 108


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