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Gravitational waves from a string cusp in Einstein-aether theory Lalancette, Marc 2008

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Gravitational waves from a string cusp in Einstein-aether theory by Marc Lalancette B.Sc., Université de Montréal, 2003 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in The Faculty of Graduate Studies (Physics) The University Of British Columbia (Vancouver) April, 2008 c
 Marc Lalancette 2008 Abstract The motivation of this thesis is to look for a signature of Lorentz violation, hopefully ob- servable, in the gravitational waves emitted by cosmic strings. Aspects of cosmic strings are reviewed, in particular how focused bursts of gravitational radiation are emitted when a cusp forms on the string. The same phenomenon is then studied in an e¤ective …eld theory with Lorentz violation called Einstein-aether theory. This is a simple theory with a dynamic preferred frame, but it retains rotational and di¤eomorphism invariance. The linearized ver- sion of the theory produces …ve wave modes. We study the usual transverse traceless modes which now have a wave speed that can be lower or greater than the speed of light. This al- tered speed produces distinctive features in the waves. They depend on two free parameters: roughly the wave speed and the acceleration of the string cusp. The pro…le of the wave is analysed in detail for di¤erent values of the parameters and explained by close comparison with the string motion. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Cosmic strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1 Topological defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 E¤ective potential and phase transition . . . . . . . . . . . . . . . . . . . . 7 2.3 Formation and evolution of a string network . . . . . . . . . . . . . . . . . . 9 2.4 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.5 String theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.6 String dynamics and cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3 Gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4 Waves from a string cusp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.1 Reaching the TT-gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.2 Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.3 Wave pro…le . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 iii 5 Lorentz violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.1 E¤ective …eld theory and the standard model extension . . . . . . . . . . . 36 6 Einstein-aether theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6.1 Field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6.2 Wave modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.3 Constraints on the aether Lagrangian parameters . . . . . . . . . . . . . . . 47 6.4 Other developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 7 Waves from a cusp in Einstein-aether theory . . . . . . . . . . . . . . . . . 50 7.1 Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 7.2 Wave pro…le . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Appendices A Coordinate transformations and di¤eomorphisms . . . . . . . . . . . . . . 72 B Calculation of the wave shape . . . . . . . . . . . . . . . . . . . . . . . . . . 78 iv List of Figures 2.1 Potential for a complex scalar …eld in the Goldstone model . . . . . . . . . . 4 2.2 Pro…le of the e¤ective potential of a complex scalar …eld . . . . . . . . . . . 7 2.3 String worldsheet around a cusp . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.1 Graph of the integral (4.18) . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2 Gravitational wave spectrum from an idealised string cusp . . . . . . . . . . 31 4.3 Gravitational wave pro…le from an idealised string cusp . . . . . . . . . . . . 32 4.4 Comparison of the wave pro…le and its source. . . . . . . . . . . . . . . . . . 34 7.1 Gravitational wave spectra from a string cusp in Einstein-aether theory . . . 55 7.2 Pro…le of superluminal gravitational waves emitted by a string cusp . . . . . 56 7.3 Pro…le of subluminal gravitational waves emitted by a string cusp . . . . . . 57 7.4 Comparison of the wave pro…le and its source, with a = 1:8; ca = 1 . . . . 59 7.5 Comparison of the wave pro…le and its source, with a = 1:8; ca = 1 . . . . . 60 7.6 Comparison of the wave pro…le and its source, with a = c = 1 . . . . . . . . 62 v Acknowledgements I would like to thank my supervisor, Prof. Moshe Rozali, for helpful discussions and for his patience and understanding. I am also grateful to the department of physics for allowing me the time to complete my thesis. Finally, I want to thank my friends and family for their support and encouragement. vi 1. Introduction Lorentz symmetry is a central component of our most successful physical theories. How- ever, the possibility of departure from it at or near the Plank scale has received increased attention lately. One might question exact scale independence since it can never be tested (to arbitrarily high energies) even in principle. But more importantly, hints of Lorentz violation have appeared in our search for a quantum theory of gravity, the need to cut o¤ high energy divergences of quantum …eld theory and perhaps even from experiment, in the possibly missing GZK cuto¤ of ultra high energy cosmic rays [1, and references]. The motivation of this thesis is to look for a signature of Lorentz violation, hopefully observable, in the gravitational waves emitted by cosmic strings. Cosmic strings are a particular case of a kind of …eld con…guration known as topological defects. These highly energetic linear objects are predicted to form by a wide variety of elementary particle models at symmetry breaking phase transitions in the early Universe. For recent reviews see [2, 3, 4], see also [5, 6, 7]. Similar con…gurations exist in condensed matter systems, like vortex lines in liquid helium, ‡ux tubes in type-II superconductors or disclination lines in liquid crystals. The existence of a network of such strings has important cosmological implications. At …rst, they were considered a candidate to explain structure formation, providing density inhomogeneities, the seeds from which galaxies and clusters can grow. But studies of the cosmic microwave background have shown that cosmic strings (or other topological defects) could not explain most of the density perturbations. However, interest has been renewed by theoretical work on hybrid in‡ation and supersymmetric grand uni…ed theories, by the discovery in string theory that fundamental strings or D-strings can in some scenarios play 1 the role of cosmic strings, and because we might soon be able to detect them. It has been shown that cusps form generically when a string loop oscillates and that these cusps emit strong focused bursts of gravitational waves. It is this phenomenon that we will study in the presence of Lorentz violation. To do this, we will use an e¤ective …eld theory that introduces a dynamic preferred frame by way of a unit timelike vector …eld dubbed the aether. The interaction of this aether …eld with the metric modi…es the usual gravitational wave modes and the emitted spectrum by the string cusp. This thesis is structured as follows. In chapter 1, we review some aspects of cosmic strings, including their nature, how they form at phase transitions and observable features of a cosmological network of such strings. We then touch on the superstring perspective before going into the dynamics of strings in more details. We brie‡y review how to calculate the gravitational waves generated by a source in chapter 2 and apply this to a string cusp in chapter 3. We obtain the spectrum and pro…le of the waves emitted by the cusp in the direction of its motion. In chapter 4, we consider Lorentz violation more closely and justify our choice of theory. Chapter 5 is a review of Einstein-aether theory, looking in details at the linearized wave modes. Finally we put the pieces together in chapter 6 where we compute the gravitational waves emitted by a cusp in Einstein-aether theory. We …nd distinctive features that may be detectable by planned gravitational wave detectors. 2 2. Cosmic strings 2.1 Topological defects To understand the nature of the string, let us consider a simple model (Goldstone model) of a complex scalar …eld with a "mexican hat" potential (…gure 2.1), described by the Lagrangian density L = @@ V () (2.1) V () = 1 2   jj2  1 2 2 2 where  and  are positive constants. This model has a global symmetry: it is invariant under the U (1) group of phase transformations  (x)! ei (x) (2.2) with  constant. The ground state (or vacuum) has an expectation value on the circle jj =  at the bottom of the potential, with an arbitrary phase  h0 jj 0i = ei (2.3) We see that the vacuum state is not invariant under the transformation, it changes the phase  !  + . The symmetry is spontaneously broken. Now let’s assume that the …eld is in a vacuum state. Generically, the phase is free to vary continuously from point to point. If we …nd a closed curve in space where the phase 3 Figure 2.1: Potential for a complex scalar …eld in the Goldstone model. The minimum of the potential is degenerate as highlighted by the red circle (left). The yellow dot marks the local maximum at the origin. The drawing on the right shows a closed curve in space (red) where the …eld is at the minimum of the potential, winding once around the circle. Two surfaces bounded by this curve are drawn which are intersected by a curve (yellow) where  = 0. 4 winds once around the circle, then continuity demands that the …eld go through zero at a point (at least) on any surface bounded by that curve (see drawing in …gure 2.1). Thus we …nd a linear region with non-zero energy density: a cosmic string. The gradient of the …eld also contributes to the energy and the string will have a certain width. When the broken symmetry is global as in this case, the solution is called a global string or vortex. Note that in this model, a static cylindrically symmetric string solution has an energy density that decays as 2 at large radius , so the energy per unit length of the string is in…nite. Strings can also form when the broken symmetry is local. A simple example is the Abelian Higgs model (or scalar electrodynamics) where we now introduce a vector …eld A. L = DD 14FF   V () (2.4) F = @A  @A D = @ + ieA The symmetry transformation is now  (x) ! ei(x) (x) (2.5) A ! A  1 e @ (x) We again have a symmetry breaking vacuum state and string solutions. However, the energy density is now much more localized near the string center and the energy per unit length is …nite. The string also caries magnetic ‡ux and this will generate a repulsive force between strings since lines of ‡ux repel each other. There is also an attractive force due to the scalar …eld because it is energically favorable to minimize the region of non-zero potential energy density. One of these forces will dominate depending on the masses of the vector and scalar modes about the vacuum. In general, to see if string solutions or other defects exist for a particular model, we need to look at the topology of the vacuum manifold, the set of minima of the potential. Given 5 a gauge group G, we can …nd an unbroken subgroup H of transformations under which a vacuum solution is invariant (the isotropy group or little group). The vacuum manifold can then be identi…ed asM = G=H. A necessary condition for the existence of stable strings is that M be not simply connected, i.e. there are non-contractible loops in M. Yet this condition is not su¢ cient in the sense that extra symmetry might extend the true vacuum manifold. For example, a semilocal model has both a global and a local symmetry and the vacuum manifold does not separate as the direct product of a local and a global part. The full vacuum manifold might be a sphere and the gauge orbit a circle on that sphere. A string con…guration could then have zero potential energy density and have a tendency to spread out. Similarly, if the vacuum manifold is not connected then there can be 2-dimensional do- main walls separating regions in space where the …eld takes a vacuum value in di¤erent disconnected parts ofM. IfM has the topology of a 2-sphere (it contains non-contractible surfaces), then there can be point-like monopoles. Finally, the topology of a 3-sphere can allow defects called textures. These are qualitatively di¤erent from other defects since the …eld is nowhere constrained to leave the minimum of the potential. Instead, they consist of non-trivial windings of the …eld when a uniform boundary condition is imposed at in…nity. E¤ectively the boundary condition compacti…es physical space. Note that stable defects cannot have free boundaries. For strings this means they will either form loops or extend to in…nity, or in special scenarios they can end on monopoles. From this point we will only be interested in strings, mainly because in models where monopoles or domain walls are formed, they soon dominate the energy density of the universe and close it, unless they are formed before in‡ation, in which case they are too rare to be of interest. 6 2.2 E¤ective potential and phase transition In the previous discussion, we treated the …elds classically for simplicity, but for quantum …elds we must take into account radiative corrections to the potential and look instead at the e¤ective potential. Depending on the model, the corrections can be negligible or can drastically alter the potential. For example, the Abelian Higgs model with quadratic potential V () = 20 jj2 (2.6) appears to have unbroken symmetry, but its e¤ective potential is Ve¤ () =  2 0 jj2 " 1 + 1 jj2 2 log jj2 2 # (2.7) where  is the renormalization scale. When the dimensionless quantity  = 16 220 3e42 becomes less than 0:37, then the absolute minimum will be away from  = 0 (…gure 2.2 a).  Φ¤ L Veff L aL  Φ¤ L Veff L + 2ΑT4 bL  Φ¤ Veff + ΑT4 cL Figure 2.2: Pro…le of the e¤ective potential of a complex scalar …eld in the Abelian-Higgs model (a, b) and Goldstone model (c). In a), we show the potential for decreasing (green to blue) values of the parameter . In b) and c) it is the temperature that decreases and we observe a phase transition of …rst and second order respectively. 7 To see how defects arise from phase transitions, we must consider the behavior of the …elds at higher temperature (away from the ground state). We can do this by looking at the "…nite-temperature e¤ective potential", which is de…ned as the free energy density Ve¤ (; T ) = (E  TS) =V (2.8) (The name comes from the fact that the diagrammatic expansion is the same as for the e¤ective potential, but with …nite temperature Green’s functions.) To …rst order in the coupling constants and for temperatures much higher than the mass thresholds, we have Ve¤ (; T )  Ve¤ () + 1 24 M2T 2   2 90 NT 4 (2.9) where M2 is a sum of particle masses and N is a sum of number of spin states. For our Higgs model we …nd Ve¤ (; T )   20 + 1 4 e2T 2  jj2 +  2 0 jj4 2 log jj2 2  2 2 45 T 4 (2.10) The pro…le of this potential for  < 0:37 at various temperatures is shown in …gure 2.2 b. At high temperatures, the T 2 jj2 term dominates and there is a single minimum at  = 0. As the temperature is lowered, a second minimum appears and below some critical temperature Tc it becomes lower than the minimum at  = 0. Such a discontinuous change in the value of jj at the minimum is a …rst-order phase transition. Note that the symmetric phase remains metastable all the way down to T = 0 for 20 > 0 (although our expression is only valid for high temperatures). As another example, the e¤ective potential for the Goldstone model is Ve¤ (; T )   2  1 6 T 2  2  jj2 +  4 jj4   2 45 T 4 (2.11) In this case, as the temperature decreases below Tc = p 6, the symmetric state becomes unstable and the value of jj moves away from 0 continuously (…gure 2.2 c). This is a second-order phase transition. 8 2.3 Formation and evolution of a string network We can now see how strings can form in the early universe. Keeping with our simple models, as the universe expands after the Big Bang, the temperature decreases and the e¤ective potential changes. Once the temperature reaches Tc,  will start moving away from the symmetric state. However, the complex phase  of  will depend on random ‡uctuations. As the new state appears at di¤erent points in space, regions with di¤erent ’s will start expanding. When these regions meet, the …eld will adjust to remain continuous, but as we saw before, if the phase ends up winding on a closed curve in space, then the …eld will have to remain at 0 along a linear region, resulting in a cosmic string. Once the phase transition is over, we end up with a tangled string network. Now in a realistic scenario, the …nal low-temperature symmetry group of the …elds must be that of the Standard Model: SU(3)U(1). We know that the weak and electromagnetic forces become united at a scale of about 100GeV, and the gauge symmetry is enlarged from the electromagnetic U (1) to an overall electroweak SU (2)  U (1). Similarly, the energy dependence of the strong and electroweak coupling strengths suggests that these interactions would also unite at an energy scale of 1015 or 1016GeV into a grand uni…ed theory. Such theories have been proposed based on various larger symmetry groups and symmetry breaking can occur in one or more steps from these groups down to SU(3)  SU (2) U(1). A detailed study of these theories shows that cosmic strings are generically formed [8]. The strength of the gravitational interaction of strings is characterized by the dimension- less parameter G, where G is Newton’s constant and  is the energy per unit length, also equal to the string tension. For strings produced at a phase transition with temperature Tc, this is expected to be roughly T 2c =M 2 p where Mp is the Plank mass. At the grand uni…ca- tion scale, this corresponds to G  106 or   1021 kg=m while at the electroweak scale, 9 G  1034 or   107 kg=m. Note that stable strings are not formed in the electroweak model, but there are extensions of the Standard Model in which they are. Once the universe is …lled with a network of wiggly cosmic strings, they quickly develop relativistic speeds because of their tension. When two strings cross, they intercommute, i.e. the ends of the strings exchange partners. This happens with probability of order 1 for both global and local cosmic strings. This also leaves the strings with kinks at the crossing point. When a string intersects itself, a loop is cut o¤. In an expanding universe, we could estimate that the total energy of in…nite strings grows as the scaling factor a (the ratio of physical distance to distance in comoving coordinates) and so their energy density would scale as a2. This would lead to strings dominating over matter and radiation which scale as a3 and a4 respectively. Energy loss mechanisms however lead to a regime where strings provide a constant fraction of the energy density in an expanding universe. First, long strings will tend to straighten, reducing the total length, but this is o¤set by the formation of new kinks by string interaction. Second and most importantly, small loops (on scales shorter than the horizon) are not stretched and behave as matter. Finally, both kinks and loop oscillations will emit gravitational radiation and reduce the energy of the network, the loops eventually decaying completely. This scaling solution is an attractor, i.e. regardless of the initial distribution, the network will tend to this regime as long as there are some in…nite strings. Simulations show that the ratio of string to matter energy density tends to 60G (in the matter era), which is much too small to be a candidate for dark matter. 10 2.4 Observations The most important observable e¤ects of cosmic strings are believed to be gravitational. One place to look for their signature is in the cosmic microwave background (CMB) where statis- tical studies, pattern search [9, 10] and model likelihood analyses [11] have been performed. Originally, the possibility that strings might be an alternative to in‡ation for producing the primordial density perturbations generated a lot of interest. However, observations of the density perturbations and more recently the polarization of the CMB gave results in agree- ment with the in‡ation model, but di¢ cult to explain by cosmic strings or other defects. On small scales, strings are expected to generate non-Gaussian perturbations. By determin- ing how much the CMB temperature ‡uctuations deviate from Gaussian, we can …nd what fraction of the ‡uctuations might be due to strings. Many statistical studies agree that the contribution from cosmic strings is at most 10% and that G has an upper bound of order 106. On the other hand, spacetime around a string is conical, i.e. the total angle around the string is less than 2, the de…cit angle given by 8G. Because of this, photons passing near a moving string will be red shifted if they pass in front and blue shifted behind. Hence strings would produce temperature step patterns that can be searched for directly in the CMB [12]. For a string with velocity  (and corresponding Lorentz factor 
), the height of the step would be T T  8G
. (The inequality represents the dependence on the orientations of the string and observer.) One can also search for gravitational lensing e¤ects due to strings. As of now, none have been found though there has been two candidates, one disproved [13], the other debatable [14]. Still, the search for lensed objects, in particular compact radio sources [15], could greatly improve constraints on cosmic string networks. 11 Other observable e¤ects are due to the gravitational waves emitted by strings. First, they could be detected directly by planned experiments. It might appear at …rst that gravitational e¤ects of strings with G  107 would be too weak to be observed, but oscillating string loops will generically develop cusps which emit strong gravitational wave bursts (see section 3). Such bursts should be detectable by the Advanced LIGO and LISA detectors for G as low as 1013. The gravitational waves emitted since the formation of the network also add up to form a background with frequencies spanning several orders of magnitude. This background introduces noise in pulsar timing experiments. Current pulsar data yields G  1:5  108, a bound that could be lowered to 5  1012 once the Parkes Pulsar Timing Array project is completed [16]. Such a bound would rule out many current models. Finally, some cosmic strings can be superconducting and produce a variety of astrophys- ical e¤ects even if they are light. They would interact strongly with magnetic …elds and could produce gamma-ray bursts and high-energy cosmic rays [17]. 2.5 String theory The cosmic strings we have been considering so far are solitons of classical and quantum …elds. We now address the possibility that superstrings might also form on large scales and act as cosmic strings [18]. For such strings to exist, there has to be a formation mechanism at the end of in‡ation, producing strings that are stable on cosmological time scale and with tensions not already ruled out by observations. Some models satisfy all these conditions and might have some distinctive characteristics. In fact, cosmic strings could provide the best observational window into string theory. In the context of perturbative string theory, cosmic strings are ruled out since funda- 12 mental strings have tensions close to the Planck scale. However, the geometry of compact dimensions was found to have more general possibilities than previously thought, and ten- sions can be much lower, all the way down to the weak scale. For example, if the gauge …elds are con…ned on a brane while gravity propagates in the bulk with extra dimensions of size R, then the tension is suppressed by some power of R Lp where Lp the Planck length. Even without large extra dimensions, the e¤ective tension can be reduced by bulk gravitational potentials. Also, the role of cosmic strings can be played by D-strings or higher dimensional D-branes with D  1 compact dimensions, which look like strings on a macroscopic scale. In models of brane in‡ation, colliding branes generate a network of cosmic strings in many cases. In fact, in what seems to be the most natural model (KKLMMT) strings are produced but not the problematic monopoles or domain walls. In these models, the value of G can be deduced from the CMB ‡uctuations. It is expected to be in the range [1011; 106]. Some cosmic strings can be subject to instabilities, even in …eld theory [19]. There are general arguments in string theory that there are no exact global symmetries. Also the no-hair theorem implies that black holes can destroy global charges and they are not exactly conserved. If the symmetry is not exact, the vacuum is not exactly degenerate. This leads to the formation of domain walls bounded by the strings and their tension forces the network to collapse. For gauge symmetries, it is generally expected in uni…ed theories that there are sources for every ‡ux. Then strings can break into short segments by creation of monopole-antimonopole pairs. In string theory, coupling of strings to form …elds can also lead to breakage. However, there are still stable candidates. In particular, in in‡ation models, features such as large dimensions or warping can stabilize strings. One of the main distinctive features of cosmic superstrings is that the probability of intercommuting can be much less than 1. For fundamental strings, it is a quantum process 13 with a probability of order g2s . Also in many cases, strings can move in the compact dimensions and so can miss each other. However in realistic compacti…cations they will be con…ned by a potential in these dimensions. For di¤erent models, the reconnection probability for fundamental strings ranges from 103 to 1 while for D-strings it goes from 101 to 1. For collisions involving one of each, the probability can vary from 0 to 1. With a probability much smaller than 1, the number of strings should be larger. This feature may help distinguish a string theory network from a …eld theory one. 2.6 String dynamics and cusps We now wish to …nd a simpli…ed general model to study the motion of a string at low temperature and small string curvature. At low energies, we do not expect signi…cant excitation of the massive modes, like oscillations of the string thickness. If we integrate out these modes, we …nd a low energy e¤ective action proportional to the area of the worldsheet, the two-dimensional surface swept out by the motion of the string (neglecting its width). We can also justify this action by general arguments. First, if the radius of curvature R is much larger than the string thickness , we can consider the string as a one-dimensional object. For gauge strings, where there is no long range interaction between string segments, we should be able to write a local action of the form S = Z d2 p
L (2.12) where 0; 1 are worldsheet coordinates and 
 is the determinant of the induced metric on the worldsheet 
ab = X  ;aX  ;bg , X  (0; 1) being the spacetime coordinates of the string. The Lagrangian density must be invariant under spacetime and worldsheet coordi- nate transformations so it can only depend on the string tension and geometric quantities such as curvature. But typically, the string thickness is & 1=2 while curvature is of order 14 R2. Thus we can neglect curvature terms and we are left with the Nambu-Goto action S =  Z d2 p
 (2.13) We next …nd the equations of motion in ‡at spacetime  = diag (;+;+;+). De…ning _X  @0X (2.14) X 0  @1X we choose the following gauge conditions on the worldsheet: X0 = 0 (2.15) _X X 0 = 0 _X2 +X 02 = 0 The …rst condition is simply to take 0 as being our time coordinate. The second condition ensures that the motion of a point on the string with …xed 1 will be perpendicular to the shape of the string at any given time. Then _X is the transverse velocity of the string. The third condition then implies that the energy is distributed uniformly in terms of the 1 parameterization. To see this, use the …rst condition to write the others as _X X0 = 0 (2.16) _X2 +X02 = 1 and then, looking back at the Lagrangian, we …nd the Hamiltonian density: H = @L @ _X i _X i  L (2.17) =  X02r _X X0 2 +  1 _X2  X02 !  15 So we see indeed that in this gauge, the energy density is simply the constant , and the total energy, E =  R d1. In terms of proper length then the energy of a segment ds = jX0j d1 is just the relativistic expression for the energy of a particle of mass  ds: dsp 1 _X2 . We also de…ne the "invariant length" of the string to be l  E  , and we can then choose 1 2 [0; l]. The last two conditions imply that the induced metric is conformally ‡at: 
ab = p
ab = 1p
  ab (2.18) and so this choice is called a "conformal gauge". Varying the action with respect to X and then imposing our gauge gives S =  Z d2 h @0  X _X    @1 (XX 0) + X  X 00  X i (2.19) The …rst term vanishes for variations with …xed initial and …nal states. The second term gives the boundary conditions for open stringsZ d0 [XX 0]l0 = 0 (2.20) implying either …xed ends (X = 0) or free ends that move at the speed of light (X 02 = 1  _X2 = 0). The last term in the variation gives us the equation of motion, a wave equation: XX00 = 0 (2.21) Our work will be simpli…ed with the use of null worldsheet coordinates  = 0  1. The general solution is then expressed as the sum of two functions (the so-called left and right movers) X  0; 1  = 1 2 [X+ (+) +X ()] (2.22) and our gauge constraints (2.15) become _X2+ = _X 2  = 1 (2.23) 16 Di¤erentiating these yields the useful relations _X  X = 0 (2.24) _X  ... X + X2 = 0 It is easy to see that the motion of a closed string will also be periodic in time (in the center-of-mass frame), but with period l 2 since X  0 + l 2 ; 1 + l 2  = X (0; 1). Periodicity in both space and time requiresZ l 0 d+ _X+ = Z l 0 d _X = 0 (2.25) With the unit constraint (2.23), this means that _X trace out a pair of curves on the unit sphere, each with its centroid at the center of the sphere. So in particular, a curve cannot lie in one hemisphere and they will be expected to cross in generic cases. When this happens, we …nd that at that point (which we’ll call X0)1, the speed on the string will be the speed of light: _X20 = 1 4 h _X+0 + _X0 i2 = _X20 = 1 (2.26) If the string has kinks, there are discontinuities in _X, meaning gaps in the curves on the sphere and it is much easier for them not to intersect and avoid these luminal points. To visualize the shape of the string around the luminal point, we use a Taylor expansion in , with X0 at the origin. X ' _X0 + 1 2 X0 2  + 1 6 ... X  0 3  (2.27) X (0; ) ' 1 2 X0 2 + 1 12 ... X+0  ... X0  3 We see that the string will develop a cusp at that moment and it is the tip of the cusp that is luminal (…gure 2.3). The relation (2.24) shows that the tip moves transversely as expected: _X0  X0 = 0 (2.28) 1The notation here shouldn’t cause any confusion since in our gauge the component X0 has been …xed and we don’t use it henceforth. 17 We now return to the action and vary it with respect to the spacetime metric. Recalling that the variation of a determinant is  detA = detA tr  A1A  (2.29) we …nd S =  Z d2 1 2 

 (2.30) =  2 Z d2 p

abX;aX;bg =  2 Z d4x Z d2 p

abx;ax;b(4) (X  x) g So the energy momentum tensor is T  (x) =  2pg L g (2.31) =  pg Z d2 p

abx;ax;b(4) (X  x) In ‡at spacetime and in our gauge, this becomes T  (x) =  Z d2 ( _x _x  x0x0) (4) (X  x) (2.32) Later, we will use the Fourier transform T  (x) = 1 (2)4 Z 1 1 d! Z d3k eikxT  (k) (2.33) T  (k) = Z 1 1 dt Z d3x eikxT  (x) In terms of our null coordinate solution, we obtain T  (k) =  Z d2  _X _X X 0X 0  eikX (2.34)   2 I ( + I )  I = Z l 2  l 2 d _X   e  i 2 kX 18 Figure 2.3: String worldsheet around a cusp. The lines on the worldsheet show the string at given times, in particular at the moment of the cusp formation (in red). Also shown are the speed vector (in yellow) and in a plane perpendicular to it, the acceleration vectors for the + and - components of the string solution (in green and blue). 19 3. Gravitational waves We now brie‡y review how the linearized Einstein equations lead to gravitational waves and we …nd an expression for the metric perturbation far from a localized source. First, we assume that the metric is nearly ‡at and write it as g =  + h (3.1) jh j  1 Indices are raised and lowered with  so to …rst order in h, the inverse metric is g  =   h . We will also use the trace-reversed metric perturbation: h = h  1 2 h   (3.2) The allowed transformations that preserve the smallness of the perturbation are global Lorentz transformations h ! h (3.3) and small coordinate transformations x ! y = x + " (3.4) J  @y @x =  + "  ; Under an in…nitesimal coordinate transformation, tensors pick up a Lie derivative [see ap- pendix A], i.e. with indices suppressed: T 0 (x) = T (x) L"T (x) (3.5) 20 where on the left, x is taken as a value of the new coordinates. On the metric this gives g0  g  ";g  ";g  "g; (3.6) h0  h  ";  ";  "h; = h  2"(;)  "h; h0 = h  2"(;) + ";  "h; We see that in order for h0 to remain small, we need ";  1. The last term is usually thrown away, taking only the terms …rst order in both h and ". This presupposes that both " and the derivatives of h are also  1. However the requirement on " is only that the product "h; be small and so in general, the term should be kept. Of course, if we choose an appropriately small " (depending on how small the derivatives are) then we can neglect the term. To …rst order in the perturbation h, the Ricci and Einstein tensors are R(1) = 2hxp;yq   1 2  (h;  h; + h;  h;) (3.7) G(1) = 2hxp;yq + hx;y To solve the linearized …eld equation G(1) = 8GT , we …rst choose a gauge. We apply the harmonic gauge condition: x  gx; = 0 (3.8) ) g = 0 ) pgg ; = 0 To …rst order in h, this corresponds to the condition h; = 0. This partial gauge choice still allows global Lorentz transformations and coordinate transformations (3.4) with " = 0. This gauge also simpli…es the Einstein tensor, and the …eld equation is now G(1) =  1 2 h; = 8GT (3.9) 21 In vacuum, the general solution is a sum of plane waves of the form h = Re  Ae ikx	 (3.10) kk  = 0 Our harmonic gauge choice further imposes Ak = 0. Next we …x (most of) the remaining freedom by imposing the transverse traceless (TT) gauge conditions [20, chapter 35.4]: A0 = A   = 0 (3.11) Equivalently, we can write the TT gauge as 8 conditions (including the harmonic gauge conditions) on h: h0 = hii = hij;j = 0 (3.12) This leaves 2 degrees of freedom (out of the 10 components of h). We are still allowed spatial rotations, so we can …x k = (!; 0; 0; !), leaving only rotations in the xy plane as remaining freedom. The two linear polarizations are called + and  and are de…ned as   26666664 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 37777775 ;      26666664 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 37777775 (3.13) Any pure wave in linearized theory can be reduced to this gauge. We also want to solve (3.9) far from a localized source, more precisely in the "wave zone" where r  jxj is much larger than the source but still much smaller than the Hubble radius. We solve this by using a Green’s function, i.e. we …nd a solution H (x) to  1 2 H; (x) =  (4) (x) (3.14) and the general solution to is then h (x) = 8G Z d4y H (x y)T (y) (3.15) 22 To solve (3.14), we go to Fourier space  1 2  Z d4k (2)4 @@e ikxH (k) = Z d4k (2)4 eikx (3.16) H (k) = 2 k2 and so H (x) = 2 Z d4k (2)4 eikx k2 (3.17) = 2 Z d3k (2)4 eikx Z 1 1 d! ei!t k2  !2 There are 2 poles on the real axis of this last integral. We can evaluate it by …rst closing a contour with a semicircle of in…nite radius in the upper or lower half plane for t < 0 or > 0 respectively (! = ei), on which the integral vanishes: lim  !1 Z  0 ei (cos i sin )t k2  2e2i ie i d   lim !1 Z  0  e jtj sin k2  2e2i d (3.18)  lim  !1    2 + k2  Z  0 d = 0 We want the retarded Green’s function, i.e. H (t < 0) = 0, so we add an in…nitesimal imaginary part to ! Z 1 1 d! ei!t  (!  jkj+ i) (! + jkj+ i) (3.19) and use Cauchy’s integral formula:I dz f (z) z  z0 = 2if (z0) (3.20) For t > 0, there is an extra minus sign from moving in the clockwise direction along the contour and we pick both poles. The residues are ! = jkj  i : e ijkjtt 2 jkj ! eijkjt 2 jkj (3.21) ! =  jkj  i : e ijkjtt 2 jkj ! eijkjt 2 jkj 23 So our integral is now H (x) =  i 2 2 Z d3k (2)3 eikx jkj  eijkjt  eijkjt  (t) (3.22) = i (t) Z 1 0 d jkj k2 (2)2 eijkjt  eijkjt jkj Z 1 1 d (cos ) eijkjjxj cos  = i (t) Z 1 0 d jkj (2)2 jkj eijkjt  eijkjt eijkjjxj  eijkjjxj i jkj jxj =  (t) 2 jxj Z 1 1 d jkj 2  eijkj(tjxj)  eijkj(t+jxj) =  (t) 2 jxj (t jxj) In the last step, the second -function vanishes because of the step function. So (3.15) becomes h (x) = 4G Z d4y  (x0  y0) jx yj  (x0  y0  jx yj)T (y) (3.23) = 4G Z d3y T (x0  jx yj ;y) jx yj = 4G Z d! 2 Z d3y ei!(x0jxyj) T (!;y) jx yj In the wave zone, r  jxj  jyj so we can approximate jy  xj = p r2  2x  y + y2 (3.24)  r  1 x  y r2 + y2 2r2  h (x)  4G Z d! 2 Z d3y ei!(x0r+ x r y)T (!;y) r (3.25) For the approximation in the phase factor to be valid, we require y 2 2r2  1j!j , so we need a large r to investigate high frequencies. De…ne the retarded time tR = t r and the wave-vector 24 in the direction of the observer kx = !  1; x r  and we get h (tR;x) = 2G  Z d! ei!tR r Z d3y eikxyT (!;y) (3.26) = 2G r Z d! ei!tRT (kx) h (!;x) = 4G r T (kx) (3.27) 25 4. Waves from a string cusp Following [21], we now …nd the gravitational wave spectrum emitted by a string cusp in the direction of its motion, k = ! _X0, far from the string. Inserting (2.34) into (3.27) we get h (!;x) = 2G r I ( + I )  (4.1) Using the expansion (2.27) about the cusp, the leading terms in our integrals are now I = Z l 2  l 2 d _X   e  i 2 kX (4.2)  Z l 2  l 2 d  _X0 + X  0  e i 2 ! _X0X To simplify the phase, we use _X0 = _X0 (2.26), and the gauge constraints (2.23) and (2.24). This gives _X0 X = _X0   _X0 + 1 2 X02 + 1 6 ... X03  (4.3) = 1 6 X2 3  and the integrals become I = Z l 2  l 2 d  _X0 + X  0  e i 12 ! X2 3  (4.4) = _X0 l 2L Z L L du eiu 3 + X0  l 2L 2 Z L L du ueiu 3 where we changed variables: u =  X20! 12 ! 1 3  (4.5) L =  X20! 12 ! 1 3 l 2 26 4.1 Reaching the TT-gauge Before going further, let’s …x the remaining gauge freedom. Writing (4.4) as I  _X0B + X0C (4.6) we get I ( + I )  = _X ( 0  ) + X ( +0 X ) 0C+C (4.7)   _X0B+B + X0B+C + X+0BC+ The following coordinate transformation will get rid of the  term: " (x) = i G 2r Z d! eikx  (!) ! (4.8) "; = G 2r Z d! ei!(tr) k ! It is easy to see that this transformation is allowed, i.e. " = 0 since k2 = 0. Also, our choice of k, _X 20 = 0 (2.26) and (2.24) imply "  ; = 0. Then from (3.6) we get h0 (x) = 2G r Z d! ei!tR  2 I ( + I )   2  G 2r Z d! ei!tR _X(0  )  (4.9) h (!;x) = 2G r X ( +0 X ) 0C+C Note that the last term in (3.6) in this case is proportional to r2 so we dropped it. Next, we rotate so that k = !ẑ. Both X+0 and X0 are now in the xy plane by (2.28) and the metric perturbation takes the form X (I +0 X J) 0 =  X+0  X0 24 cos + cos  cos (+ sin ) cos (+ sin ) sin + sin  35 (4.10) = 1 2  X+0  X0 24cos (+  ) + cos (+ + ) sin (+ + ) sin (+ + ) cos (+  ) cos (+ + ) 35 27 where I and J = 1; 2. Rotating again so that the x axis bisects the angle between X+0 and X0 gives X (I +0 X J) 0 = 1 2  X+0  X0 241 0 0 1 35+ 1 2 X+0  X0 241 0 0 1 35 (4.11) Note that we could also rotate by  4 more and it would be the other linear polarization (  ) that would survive. The important point is that there is only one polarization present. The …nal step is then to remove the trace by another transformation (3.4) with " (x) = i G 2r Z d! eikx (!) (4.12)   C+C 4! X+0  X0 (1; 0; 0;1) With k = ! (1; 0; 0; 1), we …nd k = 12C+C X+0  X0 (4.13) 0k0 = 3k3 = 1 4 C+C X+0  X0 (0k3) = 0 Most of the terms in the transformed metric (3.6) cancel and we get rid of the trace term in (4.11). We now have the gravitational wave spectrum in the TT-gauge: h (!;x) = 2G r 1 2   X+0  X0C+C (4.14) =  G r  X+0  X0  l 2 4 L2+ Z L+ L+ du ueiu 3 L2 Z L L du ueiu 3 where  is the polarization tensor (3.13). The presence of  l 2 4 suggests that we should express all lengths in units of l 2 and we shall do it from this point on. Then, r1  X+0  X0 absorbs 3 powers and the remaining one goes into h (!;x) (h (x) is dimensionless, but the Fourier transform contains the measure 28 dt). Similarly, there is one factor of l 2 in L (4.5) and it is absorbed by X 2 3 0 and j!j 1 3 . To simplify things further, we de…ne a normalized metric and frequency ~h (~!;x)  r G  X+0  X0h11 (!;x) (4.15) ~!   X+0  X0 12 ! The only remaining parameter is then the ratio of the + and - acceleration components at the cusp a   X+0 X0 (4.16) The spectrum 4.14 is then ~h (~!;x) = ~! 4 3J (a~!) J  a1~!  (4.17) J (w)  Z w 13 w 13 du ueiu 3 4.2 Spectrum We now solve the integral J (w). Note that the symmetrical limits and antisymmetric u will only pick the antisymmetric part of the exponential: i sin (u3). This integral can be expressed in terms of hypergeometric functions pFq (   ;    ; z) [22], the incomplete gamma 29 function  (a; z) [23] or the exponential integral function E (z) [24]: J (w) = 2 Z w 13 0 du u sinu3 (4.18) = 1 2 iw 2 3  1F1  3 2 ; 5 3 ;iw   1F1  3 2 ; 5 3 ; iw  = 1 3 i  1 2 + p 3 2 i    2 3 ;iw    1 2  p 3 2 i    2 3 ; iw  + 1p 3   2 3  = 1 3 iw 2 3 h E 1 3 (iw) E 1 3 (iw) i + 1p 3   2 3  Figure 4.1 shows that the integral tends to a constant at high frequencies and that low frequencies are suppressed. Two examples of the spectrum are shown in …gure 4.2, for a = 1 and 1:8. The dotted line is the approximate solution, when taking only then constant piece in the integrals (J (1) = 1p 3   2 3  ). To use this approximation, we need to introduce a cut-o¤ at low frequencies since the spectrum amplitude diverges. In …gure 4.1 we see that the amplitude drops at w  1 so we should impose j!j  12X20 . 4.3 Wave pro…le If we want to study the shape of the wave, we have to take the inverse Fourier transform of the spectrum. To be consistent with our variables (4.15), we also de…ne a normalized retarded time such that ~!~tR = !tR. The normalized metric is then ~h  ~tR   Z d~! 2 ei~!~tR~h (~!;x) (4.19) = r G  X+0  X0 Z d~! 2 ei~!~tRh (!;x) = r 12G Z d! 2 ei!tRh (!;x) = r 12G h (tR;x) 30 1 2 3 w 1 3 GI 23 M 3 J HwL Figure 4.1: Graph of the integral (4.18): I = 2 R L 0 du sinu3, a factor in the expression of the gravitational wave spectrum far from an idealised string cusp. The bound L is proportional to the string length and depends on the wave frequency as ! 1 3 . 5 Ω Ž 0.05 h Ž Figure 4.2: Gravitational wave spectrum from an idealised string cusp in the TT-gauge, for X+0 :  X0 = 1 (red) or 1:8 (black). The dotted line shows the ! 43 approximation, which needs to be cut o¤ at low frequencies. 31 Unfortunately, that integral is too complex to be evaluated in terms of known functions. It is however possible to calculate it explicitly from equation (3.23), without going to Fourier space. This is done in appendix B, giving a piecewise function containing hypergeometric functions. The other option is to compute the wave pro…le numerically either by numerical integration or much more e¢ ciently by discrete Fourier transform. We used Mathematica to compute and plot some examples for di¤erent values of a. These are shown in …gure 4.3. 1 tR Ž 0.2 h Ž a = 1 1 tR Ž0.1 h Ž a = 1.8 1 tR Ž 0.1 h Ž a = 3.2 Figure 4.3: Gravitational wave spectrum from an idealised string cusp in the TT-gauge, for di¤erent values of a   X+0 =  X0. To better understand this wave shape, we can look at which points of the worldsheet participate in producing the wave at a speci…c retarded time. From our approximation (3.24), we …nd that the points that participate are those that satisfy tR = X 0 X3 (4.20) In …gure 4.4, we show the patch of the worldsheet that is used when computing the wave pro…le. Recall that we used a square region in + coordinates. On the worldsheet, lines that satisfy (4.20) for di¤erent retarded times are displayed. The color of each of these source lines allows us to …nd the corresponding point on the wave pro…le, also shown in the 32 …gure. This particular case is for a = 1:8. Note also that the color (gray scale) of the 3d worldsheet indicates the relative speed of the string at that point, a lighter color indicating faster motion. The most obvious feature of the wave pro…le is the many singular (non smooth) points. However, studying the matching source lines shows that only the main peak at tR = 0 is a physical e¤ect, the others being due to the edges of our patch. We see that we …rst detect the perturbation produced by a corner and then extending along one edge until we reach another corner (~tR  1:2). The other four singular points are harder to explain. Careful examination of the worldsheet in 3d shows that they correspond to moments when an edge of the source line starts having a speed in an opposite direction as before (in the plane perpendicular to _X0), apparently changing abruptly the total quadrupole moment of the source line. In any case, none of these would be present in a realistic scenario. It will be useful to compare this …gure with the more complicated cases later. 33 tΣ 1 tR Ž 0.1 h Ž Figure 4.4: Comparison of the wave pro…le and its source. Top right: Shape of the worldsheet patch that was used as the source. The darker areas correspond to slower string speed. Left: Same patch in worldsheet coordinates. Lower right: Corresponding wave pro…le. Each color corresponds to a speci…c retarded time so we can identify the (linear) regions on the worldsheet that produce a speci…c point of the wave. 34 5. Lorentz violation Let us brie‡y consider the fate of Lorentz symmetry in quantum gravity research. The di¤erent approaches to the quantum gravity problem can be associated with three areas of theoretical physics [25]: particle physics, general relativity and more recently condensed matter physics, each adopting a di¤erent view on Lorentz symmetry. From the particle physics perspective, the exact symmetry is a feature of the classical spacetime background. Still, there can be spontaneous symmetry breaking and this has been studied for example in string theory [26]. The general relativity perspective, which includes loop quantum gravity and noncommutative spacetimes, instead rejects the idea of a background spacetime and attempts to describe it in a way that incorporates the fundamental limitations on mea- surements. In loop quantum gravity, the fate of Lorentz symmetry is still uncertain, but recently the attention has shifted towards the possibility of a broken or deformed symmetry. In noncommutative spacetimes, the symmetries are in general described using the structure of Hopf algebras instead of Lie algebras and one expects Lorentz symmetry to be broken (or perhaps deformed in special cases). Finally, from the condensed matter perspective, some properties of spacetime including Lorentz symmetry are viewed as being only approximate or "emergent", in the same sense that some collective degrees of freedom of a system can be relevant only near a critical point [27]. To clarify what is meant by a deformed rather than broken symmetry, consider the so-called doubly special relativity theory [28, 29], where we introduce a new observer inde- pendent high energy scale, usually taken to be the Planck scale. We then have a di¤erent (exact) symmetry that resembles Lorentz symmetry at low energies. 35 5.1 E¤ective …eld theory and the standard model extension E¤ective …eld theory is a good starting point to study Lorentz violation. It o¤ers a framework that is very ‡exible, seemingly able to describe any theory that is local and invariant under local spacetime translation above some length scale. It also allows for clear predictions and it can be constrained by experiment. The standard model and general relativity can be considered e¤ective …eld theories, as well as condensed matter systems at appropriate scales and even string theory. On the other hand, there are some e¤ects that are not describable in this framework [30], such as stochastic ‡uctuations or a "foamy" spacetime structure at very small scales. Non-commutative geometry can also lead to problematic UV/IR mixing. Still, any realistic theory, regardless of high energy e¤ects, must allow an e¤ective …eld theory description at low energies, given the success of current theories. To study Lorentz violation in the context of e¤ective …eld theory, we must introduce new tensors that break the Lorentz symmetry. If they are kept constant (explicit violation), we …nd that the energy-momentum tensor is not conserved and the Einstein equations are in- consistent, except for very speci…c solutions. Still, in the ‡at space limit, where gravitational e¤ects are negligible, or if we chose to investigate those speci…c solutions, this approach can be useful. Such a theory in ‡at space has been derived that includes all possible renormal- izable Lorentz violating terms that can be added to the standard model without changing the …eld content or violating the gauge symmetry, the so-called (minimal) Standard Model Extension (SME) [31]. This framework has been further developed, incorporating a gravity sector with Riemann-Cartan geometry and gravitational couplings in the matter and gauge sectors [32]. Before looking in more detail at the pure gravity sector, we would like to mention an 36 important issue with Lorentz violating e¤ective …eld theories: why do they have such a good approximate Lorentz symmetry at low energies? [33, 34] The Lorentz violating renormal- izable operators are already severely constrained. We might be tempted to consider higher dimension operators that would be naturally suppressed by inverse powers of the Plank mass, but without a symmetry that would forbid lower dimension operators, they would be gen- erated by radiative corrections and imply strong Lorentz violation at low energies. It has been suggested that broken supersymmetry with CPT symmetry or a braneworld scenario might resolve the issue. Doubly special relativity also evades many constraints since it has no preferred frame and thus might be more appealing phenomenologically. In any case, this …ne-tuning issue requires further attention. The pure gravity sector of the SME is de…ned using the tetrad (or vierbein) formalism, the natural language for coupling gravity to fermions. The fundamental degrees of freedom are contained in the tetrad, a (pseudo) orthonormal basis of vectors ea = ea@ with dual basis a = e a dx  such that eae a  =    ; e  ae b  =  b a (5.1) g = e a e b ab (5.2) In Riemann-Cartan geometry, the connection one-form !ba =  b ca c is independent of the tetrad and allows for non-zero torsion; in Einstein-Cartan theory (general relativity with torsion), this is the case only in the presence of an intrinsic spin density of matter. In realistic situations, torsion e¤ects are typically heavily suppressed compared to curvature e¤ects and we will only consider the Riemannian limit, with zero torsion. The low energy action in this case is S = 1 16G Z d4x e  (1 q)R 2 + sR + tR  (5.3) where e is the determinant of the tetrad,  is the cosmological constant and the coe¢ cients s and t have the symmetries of the Ricci and Riemann tensors respectively. They are 37 also de…ned as traceless (otherwise the traces can be absorbed in q and s). There are 19 independent Lorentz violating degrees of freedom. But again, having these coe¢ cient …xed places very limiting conditions on them if we want a conserved energy-momentum tensor. To retain the general solutions, we must instead take them as dynamic objects. Energy-momentum conservation is then a result of the di¤eomorphism invariant action. If the new tensors also have a potential that gives them a non-zero vacuum expectation value, Lorentz symmetry is spontaneously broken. Our main interest will be one such theory, most often called Einstein-aether theory. 38 6. Einstein-aether theory Einstein-aether theory is a theory of gravity with dynamic preferred frame [35]. It introduces a single new dynamic timelike vector …eld u, called the aether. By encoding the Lorentz violation with this aether …eld and the metric, the theory preserves rotational invariance. The most general action that is di¤eomorphism invariant and quadratic in derivatives of the aether and metric is S = 1 16G Z d4x pg R +Kruru + V (uu) (6.1) K  c1gg + c2 + c3 + c4uug The ci coe¢ cients are dimensionless constants2. Compared with the SME gravity action, here q =  = 0 and integrating by parts the s term gives the c2 and c3 terms. The potential V imposes a non-zero vacuum expectation value. However, the theory is unstable (negative energies associated with u0) unless the potential is a strict constraint of the form  (uu  v2) where  is a Lagrange multiplier, or in the special case c1 + c2 + c3 = 0 [36]. The constraint removes a degree of freedom, leaving a positive de…nite Hamiltonian but also making the theory non-renormalizable and valid only semi-classically (at tree level). Since the length of u is extra information not needed to specify a preferred frame, we will …x v2 = 1 by rescaling u. Then small Lorentz violation at low energies translates to small 2Note that all cited work in which these coe¢ cients appear used the signature (+;;;) instead of (;+;+;+) that we use here so there will be sign di¤erences when comparing with our calculations. In particular, since the c4 term includes the metric, our c4 is the opposite of theirs. Unfortunately, there are many other sign di¤erences whose sources are not as easy to isolate. We chose to copy the constraints in section 5.3 and other relations of these parameters (e.g. 6.2) as they appear in the cited work, so again we cannot compare these directly with our results. 39 kinetic coe¢ cients ci. We could instead rescale u such that the coe¢ cients are of order 1, then it is the size of the vacuum expectation value for u that would be constrained. We will derive the …eld equations by varying with respect to the metric, the aether and the Lagrange multiplier. Note that we choose the most traditional (Einstein-Hilbert) action for general relativity, where the fundamental variable is the metric which uniquely de…nes the torsion-free connection. However, there are many other actions for general relativity that use di¤erent fundamental variables and that are equivalent at the classical level [37]. They would not all be equivalent in the case of Einstein-aether theory. For example, in the Palatini formulation, the spin connection is varied independently from the tetrad. In general relativity its equation of motion gives zero torsion, but here the covariant derivatives of the aether would introduce additional terms, giving non-zero torsion and indicating a di¤erent theory. Such a theory has not yet been investigated. It has been noted [38, 39] that if the parameters satisfy c1 + c4 = c1 + c2 + c3 = 0 (6.2) c3 =  p c1 (c1  2) c1  0 then the theory in vacuum is equivalent to general relativity by a …eld rede…nition of the form g0 = g + (1B)uu (6.3) where B > 0 in order for g0 to have Lorentzian signature. It is also possible to rede…ne the …elds such that c1 + c3 = 0. Combined with the previous result, the theory in vacuum would then be equivalent to general relativity if all ci’s vanish [35]. These results have been obtained by considering a speci…c type of …eld rede…nition depending on two parameters and there might be other ways to relate the theory to general relativity. Thus the claim 40 that non-zero coe¢ cients when c1 + c3 = 0 ensure true deviation from general relativity [35] isn’t obviously true. In any case, once we consider the theory with matter, we are no longer free to rede…ne the metric (to which the matter couples) and Lorentz violating e¤ects are present. 6.1 Field equations We now derive the …eld equations from the following action: S = 1 16G Z d4x pg (R + Lu) (6.4) Lu = Kruru +  (uu + 1) K  c1gg + c2 + c3 + c4uug First, varying the Lagrange parameter enforces the constraint: uu = 1. Next, varying the aether we get Lu u = 2u + 2c4u;u  ;u   2 Ku;; 0 = u + c4  u;u  ;  2u;(u;)  u Ku; (6.5)  = Kuu; + c4uu  u[;]u  ;  u;u;  As in general relativity (e.g. [20] chapter 21), we can carry the variation with respect to the metric in a frame where  = 0 and then substitute covariant derivatives for ordinary derivatives to recover the general formula. The variation of the scalar curvature is then R = gR + g R = g R + 2g [;] (6.6) Integrating by parts the  term will give it a zero coe¢ cient (in the form of covariant deriv- atives of the metric and its determinant). We then obtain the familiar form for Einstein’s 41 equation, with an aether dependant energy-momentum tensor. G = T = 1 2 gLu  Lu g (6.7) In the presence of other matter …elds , we get instead G  T (u) = 8GT () (6.8) The variation of the aether Lagrangian with respect to the metric is Lu g = uu + c1gu;u;   c1g  + c4u u  u;u; (6.9) +K  u;  g + u;  g ! u where the minus signs come from g = ggg . Since K is invariant under the exchange of ;  with ; , both  terms are equal. Using again a frame where the connection coe¢ cients vanish, we …nd  = 1 2 g       +           g; (6.10) = 1 2  g     +   g    ggg  (g); Then integrating by parts, the K factor becomes Ku  ;u   ; =  c4  u;u  + uu;  u;u  +K  u;u  + u;u  ;  (6.11) Expanding everything and keeping only the symmetric part in  for the …eld equation (since the metric is symmetric), we get T = 1 2 gK  u  ;u  ;  c1 h u(;)u   u;(u)  ; + gu;u; i (6.12) c2 h g  u;u   ;  u;(u) i  c3 h u(;)u   ; + u;(u);  g  u(u);  ; i c4 h u;(u)u  ;u  +  u(u);u u  uuuu;  ; i where we have substituted for  using the other …eld equations. 42 6.2 Wave modes Next we …nd the linearized …eld equations about a ‡at background  = (1; 1; 1; 1) and constant aether w = (1; 0; 0; 0): g =  + h (6.13) u = w + v To …rst order in the perturbations h and v, the  …eld equations becomes 2w v + gw w = 1 (6.14) 2v0 + h00 = 0 2v0 = h00 We …nd that  is …rst order in the perturbations,  = c1 u0;  c2u;0  c3u;0 + c4u0;00 (6.15) so the uncontracted form of the aether …eld equation (6.5) becomes 0 = V  w + c1au; + c2u; + c3u; + c4u;00 (6.16) Finally, the metric equations are G  G(1)  T (1) (6.17) = 2hxp;yq + hx;y +c1  u(;)0  u;(w)  + c2  u  ;0  u;(w)  +c3  u(;)0  w(u);  + c4  w(u);00  wwu;0  = 2hxp;yq + hx;y +w( c1u;)  c2u;)  c3u); + c4u);00 +c2u  ;0 + (c1 + c3)u(;)0  c4wwu;0 43 where the second derivatives of the aether are taken to …rst order:  u; (1) =  v; +   w   ; (6.18) = v; + 1 2 
; + h
;  h;
)w = 1 2 
; + h
0; + h
;0  h0;
) We now have 15 unknowns (10 components of h, 4 of v and ) and 15 …eld equations relating them, but the Bianchi identity (and energy-momentum conservation) reduces the number of independent relations by 4 and we have the usual gauge freedom. Also, we …nd that the expressions for 2Gi; and Vi;0 are equal. The relation 2Gi; = Vi;0 (6.19) is obvious since the left side is zero by the Bianchi identities and energy-momentum con- servation and the right side by the aether …eld equations, but since the actual expressions are equal, this means that the aether …eld equations are redundant. Having no interest in the value of , we also disregard the equation V0 = 0 and we are left with the 11 equations (6.17) and (6.14). In previous work [40], the wave modes were found using the following gauge which was shown to be reachable by a coordinate transformation. h0i = 0 (6.20) vi;i = 0 Instead, since we are only interested here in the gravitational waves, we will use the preferred frame picked by the aether …eld, i.e. we impose the conditions v = 0 (6.21) 44 Note that there is residual gauge freedom. Any constant coordinate transformation ((3.4) with ";0 = 0) is still allowed. With equation (6.14), this gauge choice implies h00 = 0, leaving 10 equations for the remaining 9 components of h. In this gauge and in vacuum, (6.17) becomes 0 = 2G (6.22) = haa; + 2  ha(;)a  h0(;)0  h;aa +  (haa;bb  haa;00  hab;ba + 2ha0;0a) +w(  2 (c3 + c4)h)0;00  (c1 + c3)h)a;a0 + (c1  c3)  h)0;aa  h0a;a)  c2haa;0) 2c4wwh0a;a0 + c2haa;00 + (1 + c1 + c3)h;00 The equation for each component can be further simpli…ed: 2G00 = 2 (c1  c4)h0a;a0  haa;bb + hab;ba (6.23) 4G0i = 2 (c1  c4)hi0;00 + (c2  2)haa;0i +(2 + c1 + c3)hia;a0 + (2 + c1  c3) (h0a;ai  hi0;aa) 2Gij = (1 + c1 + c3)hij;00 + ij [ (1 + c2)haa;00 + haa;bb  hab;ba + 2ha0;0a] haa;ij + haj;ia  hij;aa + hia;aj  2h0(i;j)0 Then assuming a plane wave solution h = e ikx (6.24) k = (k0  sk3; 0; 0; k3) 45 we …nd the following set of equations G12 : 0 = (1 + c1 + c3) 12s2  12 (6.25) G11  G22 : 0 = (1 + c1 + c3) (11  22) s2  (11  22) G0I : 0 = 2 (c1  c4) 0Is2 + (2 + c1 + c3) I3s (2 + c1  c3) 0I GI3 : 0 = (1 + c1 + c3) I3s 0I GII : 0 = (c1 + c2 + c3) IIs2  (1 + c2) (II + 233) s2 + II + 403s G33 : 0 = (c1 + c2 + c3) 33  (1 + c2) II G00 : 0 = 2 (c1  c4) 03s II where I takes the values 1 and 2. As expected since we had an overdetermined system, we …nd that one more equation is redundant: the expression for 2G03 is the same as G33s1+G00s. From this system we …nd 5 independent wave modes with distinct polarizations. The two usual transverse traceless (spin 2) modes are still there, but their speed s is no longer the speed of light, although it is in the limit of small coe¢ cients ci: 12 (6.26) 11 = 22 s2 = 1 1 + c1 + c3 ! 1 Then there are two traceless (spin 1) modes involving both the propagation direction and time: I0 = (1 + c1 + c3) sI3 (6.27) s2 = (2c1 + c 2 1  c23) 2 (c1  c4) (1 + c1 + c3) ! c1 c1  c4 46 And …nally, a trace (spin 0) mode: 11 = 22 = c1 + c2 + c3 2 (1 c2) 33 = (c1  c4) s03 (6.28) s2 = 2 c1c4 + 1 (1 + c1 + c3)  2(1c2) c1+c2+c3  1  ! c1 + c2 + c3 c1  c4 Comparing with the modes found in [40], the aether there replaces the polarization compo- nents 0i here, but otherwise the polarizations and speeds match. 6.3 Constraints on the aether Lagrangian parameters Looking at the wave modes3, we can put bounds on the aether parameters by requiring stability, positive energy, causality and the absence of ghosts [35]. First, we want positive squared speeds to avoid exponentially growing modes. In the limit of small parameters (always assumed in the following), this means c1 c1 + c4  0; c1 + c2 + c3 c1 + c4  0 (6.29) Requiring that the waves have positive energy yields [41] c1 > 0; c1 + c4 > 0 (6.30) This last condition also ensures that the aether modes aren’t "ghost-like" [36]. Finally, the question of causality leads us to consider whether or not superluminal speeds should be rejected. This would imply c1 c1 + c4  1; c1 + c2 + c3 c1 + c4  1; c1 + c3  0 (6.31) 3We remind the reader that the constraints in this section come from works which use a di¤erent metric signature. They cannot be directly compared with the speeds obtained in the previous section because of sign di¤erences. 47 In the presence of small Lorentz violation, we would see only slightly superluminal speeds in some frames and physics would be local and causal [34]. With a dynamic aether however, some con…gurations can lead to energy-momentum ‡owing around closed curves [42]. Still, the formation of closed timelike curves is possible even in general relativity so this is not a …rm objection to superluminal speeds. The parameters are also constrained by observations. Cosmological considerations [43] give bounds of order 101. Parameterized post-Newtonian parameters would suggest a bound of order 107 for the generic case [44] but if the parameters satisfy c2 = c23  c1c3  2c21 3c1 ; c4 = c 2 3 c1 (6.32) then all PPN parameters are the same as in general relativity [45] and the aether modes are superluminal. In the case of subluminal propagation, a study of the possible emission of metric-aether µCerenkov radiation by high energy cosmic rays [36] gives a constraint jcij < 1015, except in the special case c4 = 0; c1 + c3 = 0; c2 = c1 1 2c1 (6.33) where all the modes propagate exactly at the speed of light. 6.4 Other developments Energy in this theory has been investigated using Einstein and Weinberg pseudotensors [41] and the Noether charge method [46]. Although the linearized modes have positive energy when the parameters satisfy the conditions previously mentioned, the question of positivity of energy in the full nonlinear theory remains unresolved. The total energy in an asymptotically ‡at spacetime is found to be E = r0 2G  1 c1 + c4 2  (6.34) 48 Compared to the value obtained in general relativity, r0 2GN , we see that the aether e¤ectively renormalizes Newton’s constant: GN = G  1 1 2 (c1 + c4) 1 (6.35) This is con…rmed by studying the Newtonian limit of the theory [43]. Note however that in a cosmological setting the e¤ective Newton’s constant receives a di¤erent correction which reduces the expansion rate of the Universe. Finally, it is worth mentioning the recent study of time-independent spherically symmet- ric solutions [47, 48]. A three parameter family of vacuum solutions was found. Adding asymptotic ‡atness removes one parameter. Pure aether stars do not exist, but solutions are found for regular asymptotically ‡at perfect ‡uid stars and black holes. 49 7. Waves from a cusp in Einstein-aether theory The procedure to …nd the gravitational waves emitted by the source here is more involved than in general relativity (section 3). There, each component of the metric perturbation had a separate source and all shared the same Green’s function, so we only had to solve equation (3.15) once. Here, we have a system of 9 equations that mixes 9 components of h (remembering that h00 = 0 by our gauge choice and the equation for G03 is redundant): G (h) = 8GT (7.1) To simplify the system, we go to Fourier space and we again pick the direction k = (k0; 0; 0; k3), obtaining a system equivalent to (6.25) but without assuming a plane wave solution. 2G12 =  k23  (1 + c1 + c3) k20  h12 (7.2) 2 (G11  G22) =  k23  (1 + c1 + c3) k20  (h11  h22) 2GI3 = k3k0h0I  (1 + c1 + c3)hI3k20 4G0I =  (2 + c1  c3) k23  2 (c1  c4) k20  h0I  (2 + c1 + c3) k3k0hI3 2GII =   (c1 + 2c2 + c3  1) k20 + k23  hII + 2 (1 c2) k20h33  4k3k0h03 2G33 = (1 c2) k20hII  (c1 + c2 + c3) k20h33 2G00 = k23hII  2 (c1  c4) k3k0h03 We then have to treat each mode separately. 50 The components for the spin 2 modes appear in 1 equation each and are ready to be solved as before. 16GT12 =  k23  (1 + c1 + c3) k20  h12 (7.3) 16G (T11  T22) =  k23  (1 + c1 + c3) k20  (h11  h22) From these, we can read the appropriate Green’s function. H (k) = 2  k23  (1 + c1 + c3) k20 1 (7.4) We will treat them in detail after looking at the other modes. On the other hand, the components for the spin 1 modes have 2 equations each (the third and fourth lines in (7.2)). However, a combination of these 2 equations vanishes by conservation of T and allows us to relate the two components of that mode: 0 = 4 (GI3k3  G0Ik0) (7.5) hI3 = 2 (c1  c4) s (c1  c3) s1 c1 + c3 h0I where s  k0 k3 as before. Substituting hI3 in one of the original equations we …nd 16GT0I =  k23 +  1 + 1 c1 + c3  (c1  c3) k23  2 (c1  c4) k20  h0I (7.6) which gives us the Green’s function for that mode. Finally, there remains 3 equations for the spin 0 mode. However, there is only one combination that vanishes, 0 = 2 G33k23  G00k20 (7.7) h03 = 1 2 (c1  c4) s [c2hII + (c1 + c2 + c3)h33] 51 apparently leaving 2 degrees of freedom. 16GT33 = (1 c2) k20hII  (c1 + c2 + c3) k20h33 (7.8) 16GTII =   (c1 + 2c2 + c3  1) k20 +  1 + 2c2 (c1  c4)  k23  hII +2  (1 c2) k20  (c1 + c2 + c3) (c1  c4) k 2 3  h33 This is due to the remaining gauge freedom, and …xing it should remove one of them. We won’t treat the spin 0 and spin 1 modes further. Going back to the spin 2 mode, we now need to take the inverse Fourier transform of the Green’s function. To do so, we substitute back k instead of k3. This is justi…ed because G has rotational symmetry and picking a direction helped us …nd a simple expression. That being said, our results are still only valid on the x3 axis so the axes have to be oriented accordingly. Proceeding as in section 3, we …nd H (x) = 2 Z d3k (2)4 eikx Z 1 1 d! ei!t k2  (1 + c1 + c3)!2 (7.9) =  (t) 2 jxj  tp 1 + c1 + c3  jxj  As expected the delta function enforces the altered wave speed s we found in (6.26). h12 (x) = 4G Z d4y 1 s jx yj  x0  y0  jx yj s1  T12 (y) (7.10) = 4G s Z d3y T12 (x0  jx yj s1;y) jx yj = 4G s Z d! 2 Z d3y ei!(x0jxyjs 1)T12 (!;y) jx yj To account for the wave speed, we rede…ne the retarded time and wave-vector in the direction of the observer. tR = t r s (7.11) kx = !  1; x r s1  52 Thus we …nd a very similar expression to (3.26) for the wave. h12 (x) = 2G s Z d! ei!tR r Z d3y eikxyT12 (!;y) (7.12) = 2G rs Z d! ei!tRT12 (kx) At …rst sight, it might seem that the only di¤erences are a slight change in amplitude (due to the factor s1) and a later time of arrival, but as we will see, the small change in the wave vector kx will also change how the source points are distributed according to retarded time, with signi…cant e¤ects on the wave pro…le. 7.1 Spectrum Because of the di¤erent wave vector, we need to revise our expression for the string energy- momentum tensor (2.34). After we use the cusp Taylor expansion in the integrals I  Z 1 1 d  _X0 + X  0  e i 2 kX (7.13) we …nd an extra term in the phase compared to (4.3): kx X =  1; s1 _X0    _X0 + 1 2 X02 + 1 6 ... X03  (7.14) =  s1  1  1 6 s1 X2 3  Since we only need T12, the term with _X  0 doesn’t contribute. Also, with the proper rotation (as explained below (4.11)), we …nd X(1+0 X 2) 0 = 1 2  X+0  X0. Then if we generalize our normalized variables (4.15) by including appropriate speed factors, ~h (~!;x)  rs G  X+0  X0h12 (!;x) (7.15) ~!   X+0  X0 12s ! 53 the wave spectrum is found to be ~h (~!;x) = ~! 4 3K (a~!)K  a1~!  (7.16) K (w) = Z w 13 w 13 du u eiw 2 3 c a ueiu 3 where we introduced a new speed parameter4, c  6 (1 s) X+0  X0 (7.17) Note that c will be positive for subluminal speeds and negative for superluminal speeds. We did not …nd an expression in terms of known functions for the integral K. We used numerical integration to plot the spectrum for various values of c (…gure 7.1). For any non-zero value of c, we …nd that the spectrum has points where the amplitude is zero. We plotted the absolute value of the spectrum, but it is purely real as it was in the general relativity case, so these points correspond to phase inversions (the phase changes by ). For positive values of c, we also see that the spectrum is much less regular and can have a much greater amplitude at higher frequencies. For negative values of c on the other hand, the opposite is true and the power seems to be limited to low frequencies. These features will be easier to understand by looking at the wave pro…les. 7.2 Wave pro…le Again, to be consistent with the normalized frequency, we generalize the normalized retarded time by adding a factor of s. ~tR = 12stR X+0  X0 (7.18) 4The choice of the symbol c seemed appropriate at the time, since it includes the aether Lagrangian parameters ci. It should not be confused with the speed of light. 54 10 Ω Ž 0.02 h Ž a = 1.8, c a = 0.3 10 Ω Ž 0.02 h Ž a = 1.8, c a = 1 10 Ω Ž 0.02 h Ž a = 1.8, c a = -0.3 5 Ω Ž 0.05 h Ž a = 1.8, c a = -1 Figure 7.1: Gravitational wave spectra from a string cusp in Einstein-aether theory for both subluminal (c > 0) and superluminal (c < 0) waves. c is a parameter combining wave speed, cusp acceleration and cusp length. 55 Note that ~h  ~tR  requires no speed factor and remains as before (4.19), i.e. ~h  ~tR  = r 12G h (tR;x) (7.19) A survey of wave pro…les for various values of the 2 parameters a and c reveals a diversity of shapes, but as was the case before, many features are artefacts of the edges of the source patch. We …rst observe (…gure 7.2) that for negative values of c the waves tend to smooth out, which is consistent with the spectrum having high frequencies suppressed. On the other 1 tR Ž 0.1 h Ž c a = -0.3 2 tR Ž0.05 h Ž c a = -1 2 tR Ž 0.05 h Ž c a = -2.5 Figure 7.2: Pro…le of superluminal gravitational waves emitted by a string cusp in Einstein- aether theory, for various values of our speed parameter c, with the ratio of the + and  cusp acceleration amplitudes a = 1:8. hand, …gure 7.3 shows that positive values of c introduce extra structure at the center of the wave which tends to widen and eventually smooth out for larger values of c. Two notable features of this structure are …rst, a discontinuity and second, a doubling of the central peak when a 6= 1. The discontinuity and sharpness of the peaks are consistent with the spectrum having a relatively large amplitude at high frequencies. To verify if these features are physical, and to better understand them, we again look at how the source is distributed in terms of retarded time. We will look at the cases a = 1:8 with c = 1 and the case a = c = 1. First, …gure 7.4 shows the wave and source for a 56 0.5 0.2 a = 1 c a = 0.3 0.5 0.1 a = 1.8 0.5 0.05 a = 3.2 0.25 0.5 c a = 1 0.5 0.2 0.5 0.1 1 1 c a = 2.5 0.5 0.5 1 0.2 10 0.02 c a = 10 5 0.025 2 0.05 h ŽHtRŽ L Figure 7.3: Pro…le of subluminal gravitational waves emitted by a string cusp in Einstein- aether theory, for various values of our speed parameter c and of a, the ratio of the + and  cusp acceleration amplitudes. 57 superluminal wave. The distribution of the source is not much di¤erent than our general relativity reference (c = 0, …gure 4.4), but there seems to be less distortion at the origin. The smoothing of the peak is not hard to understand. Since the cusp tip no longer reaches the wave speed, there is less "piling up" of energy. There is more to say about the subluminal cases. First, let’s observe the case a 6= 1 (…gure 7.5). We chose c = 1 because the extra structure of the peak is wide enough to be clearly seen and it has not started to smooth out. The doubling of the peak can be understood intuitively if we think of the source as a single point that accelerates up to the speed of light and slows down afterwards. Just like a plane reaching the speed of sound encounters the sound barrier, when the source reaches the speed of the wave energy accumulates. For speeds lower than the speed of light, this will occur twice. Of course, the situation is more complicated since only quadrupolar motion transverse to the direction of observation produces gravitational waves, but with the appropriate transverse motion, the analogy applies. In the …gure of the worldsheet patch, we drew a line at points where the string moves at the speed of the wave. We can see that this line crosses 2 saddle points and these correspond to the two peaks. Looking at the 3d worldsheet, we see that at the moment of the cusp (see …gure 2.3 to recall it’s location) the points that contribute to the peaks are slightly below the tip, where the string speed is very close to the wave speed. A closer examination would be required to verify that the longitudinal component (in the direction of the observer) of the string speed matches the wave speed. The other feature we observed is the discontinuity on each side of the double peak. Again looking at the worldsheet, starting from the left, we observe a single source line for each time, but at the time of the discontinuity, an extra point appears, which expands in a loop later. Furthermore, this extra point seems to be very close to the gray line (but not exactly on it), meaning that the string at that point moves with a longitudinal speed equal to the speed of the wave. So we can conclude that both the double peak and the discontinuities 58 tΣ 1 tR Ž 0.05 h Ž Figure 7.4: Comparison of the wave pro…le and its source, with a = 1:8; ca = 1. Each color corresponds to a speci…c retarded time, allowing us to locate the region of the source on the worldsheet (in space: upper right, or in worldsheet coordinates: left) that produces a point of the wave (lower right). 59 0.5 tR Ž 0.2 h Ž t Σ Figure 7.5: Comparison of the wave pro…le and its source, with a = 1:8; ca = 1. 60 are real e¤ects, in the sense that they are not due to the edges of the worldsheet or our approximations. Of course, we would expect them to be smoother and possibly modi…ed in some other way in more realistic models, for example for strings with a non-zero width. We can also understand by looking at this …gure why the extra wave structure tends to widen and eventually smooth out for larger values of the speed parameter. There are 2 ways in which c can become larger. Either decrease the speed or the acceleration at the cusp. In both cases, this has the e¤ect of stretching the source line pattern which is easily understood by thinking only of the gray line where string and wave speeds are equal. Obviously, lowering the wave speed will stretch this loop, but similarly, lowering the cusp acceleration will mean a larger area around it will have a speed higher than the wave’s. Since our patch has …nite size, as the pattern stretches, more and more of it will be cut o¤, eventually loosing the …rst appearance of the second source line (smoothing out the discontinuity) and also the saddle points (reducing the sharpness of the double peak). The last case we will observe is a = c = 1 (…gure 7.6). In this case, the extra symmetry of the worldsheet results in both saddle points contributing to the same retarded time, namely tR = 0. The entire string at the moment of the cusp also contributes to tR = 0, the cusp in this case being a single straight line along X0. Thus there is a single peak in this case. The discontinuity is also visible here. 61 0.2 tR Ž 0.5 h Ž t Σ Figure 7.6: Comparison of the wave pro…le and its source, with a = c = 1. 62 8. Conclusion We studied the gravitational waves emitted by a string cusp in the context of Einstein-aether theory. The dynamic timelike aether …eld picks a preferred frame at each point in spacetime, thus spontaneously breaking Lorentz symmetry. This gives rise in the linearized theory to …ve wave modes. We found the spectrum and wave pro…le of the usual transverse traceless mode for waves produced by an idealized string cusp. The string is taken to have zero width and represented by a Taylor expansion around the cusp, whose tip moves at the speed of light in the direction of the observer. The source is a square patch of the string worldsheet in null worldsheet coordinates. The spectrum contained integrals that had to be computed numerically and the wave pro…le was obtained by sampling the spectrum and performing a discrete Fourier transform. The results were plotted and analyzed for various values of two free parameters: the ratio a of the norm of the + and  (left and right movers) string acceleration components at the cusp (  X0) and a speed parameter c that depends on these acceleration components (in units of the string length l) and the wave speed s. We found distinctive features compared to the waves obtained in pure general relativity. All the observed di¤erences between the waves with and without the aether …eld stem from the modi…ed wave speed s = 1p 1+c1+c2 . It can be both lower or higher than the speed of light, although the former is much more constrained by observations than the latter. If the wave is superluminal, then no point of the string, in particular the cusp, reaches this speed and the gravitational wave burst is not as intense. The power is concentrated in lower frequencies and the pro…le is smoother than without the aether. If the wave is subluminal, the opposite is true. There is more power at high frequencies and some extra structure 63 appears in the wave pro…le. If a 6= 1, the central peak splits in two and in all cases, a discontinuity appears on both sides of the peak. Other non-physical features were observed to be the result of the edges of the source patch. For cosmic strings with high enough tensions, gravitational wave bursts from cusps could be detected by the gravitational wave detectors LISA and Advanced LIGO planned for operation in 2015 and 2013 respectively. It remains to be seen if the distinctive features of the waves in Einstein-aether theory could be observed. To do so, we need to examine the amplitude of the spectrum at the frequencies the detectors are sensitive to. This may be di¢ cult since the main parameter that determines the strength of the aether e¤ects c depends not only on the aether Lagrangian parameters ci which are somewhat constrained, but also on the cusp acceleration amplitudes  X+0  X0. We need more speci…c models to determine possible values for these. If we assume that they are of order 1, then c would be small and we would need relatively high frequencies to observe the Lorentz violating e¤ects. One could also look at the other modes that are not usually present without Lorentz violation. These could also provide a detectable signature of Lorentz violation. The spin 1 mode has a similar Green’s function (7.6) and the analysis should not be much more complicated than it was for the spin 2 mode. For the spin 0 mode however, we need to specify further gauge constraints to remove one of the apparent two degrees of freedom in the remaining system (7.8) and the only freedom left is rigid (constant in time) coordinate transformations. Once this is realized, it is probable that the analysis would be similar to the other two modes. This work could be complimented by studying other aspects of the waves or more realistic models for the strings. Firstly, given that the wave burst is an event of short duration we should be concerned that the probability of being in the right direction at the right time might be very small. We could study the waves at small angles from the cusp speed direction, 64 as was previously done in general relativity [21] to see if the aether signature is still present. We should also consider the e¤ects of the cosmic string width. Obviously, our zero-width approximation would break down at points where the string radius of curvature becomes comparable to the string width, namely at the cusp. Cosmic superstrings do not require this assumption, but for D-strings, there are other e¤ects to consider. For example near the tip of the cusp, fundamental strings attached to the D-string can stretch between the two branches of the cusp, providing extra tension which probably prevents the tip from forming. Finally the self gravity of the string at the cusp might be signi…cant. These possible causes of "smoothing" might signi…cantly reduce not only the wave burst intensity, but also the distinctive wave features in Einstein-aether theory. 65 Bibliography [1] T. Jacobson, S. Liberati, and D. Mattingly, “Quantum gravity phenomenology and Lorentz violation,”Springer Proc. Phys. 98 (2005) 83–98, gr-qc/0404067. Also in: Particle Physics and the Universe (Dubrovnik 2003). [2] T. W. B. 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Grav. 23 (2006) 5643–5660, gr-qc/0604088. [49] S. M. Carroll, Spacetime and Geometry: An Introduction to General Relativity. Addison Wesley, San Francisco, 2004. 71 A. Coordinate transformations and di¤eomorphisms Here, we review how in…nitesimal coordinate transformations lead to the use of the Lie derivative [49, Appendices A and B] and we detail some aspects of the passive and active viewpoints that can be confusing. Let us …rst establish some notation. We will work with a manifold M and coordinates x :M ! Rn (A.1) p 7! x (p) More precisely, the coordinates are de…ned on a patch U M , but we can neglect this in our discussion. We will often suppress indices, especially on the coordinates. x (p) will always refer to a speci…c coordinate system, functions on M . On the other hand, when x is taken as an argument, as in f (x), it should be thought of simply as values; the coordinate system they represent depends on the function. Finally, [   ] (x) means the expression in brackets is evaluated at x. Consider an in…nitesimal change of coordinates given by a vector …eld  and in…nitesimal parameter t, y (p) = x (p) + t (p) (A.2) and the corresponding Jacobian matrix J  @y @x =  + t  ; (A.3) 72 Then given a function f (x), de…ne a "transformed" function f 0 that takes the new coordi- nates as argument: f 0 (y (p))  f (x (p)) (A.4) f 0 (x) = f (x t)  f (x) t@f (x) = f (x) Ltf (x) Again, here f 0 takes x as new coordinates and f as old ones, so f 0 (x) and f (x) are at di¤erent points. Applying this formula to the original coordinates, we …nd an expression that merits a few comments: x0 (x) = x (x) t (A.5) First, according to how we de…ned f 0, x0 (x) represents the old coordinates as a function of the new ones, not the other way around. That is why we didn’t call the new coordinates x0 (as is often the case) but y, otherwise the notation is inconsistent: on all functions and other objects the prime means the argument is new coordinates, and x is the name of the old coordinates. Second, by de…nition x (x) is just the identity function (as is y (x)) so on the right x (x) just gives back the value of the new coordinates we plug in on the left. Things look less confusing if we write this as x0 (y) = y  t. A vector …eld X transforms in two ways. Like functions, X 0 (x) = X (x t), but also, X (p) = [X@] (p) (A.6) =  XJ @ @y  (p) Together, we …nd X 0 @ @y  (x) =  XJ @ @y  (x t) (A.7)   X   + t  ;  @ @y  (x)  tX; @ @y  (x) 73 So to …rst order in t, the components satisfy X 0 (x) =  X + t;X   tX;  (x) (A.8) = X (x) LtX (x) Let’s now turn to so-called active transformations. Given a manifold, we can de…ne maps from M to other spaces (sections of various bundles over M) and think of the images of these maps as objects living on the manifold. For example, functions map M onto (M)R and vector …elds mapM ! TM the tangent bundle. Then, given a di¤eomorphism  :M !M , we can take its composition with these maps and think of the resulting images as the transformed objects. To understand how the composition is realized on di¤erent objects, it is clearer if we take  to be an invertible map to a di¤erent manifold  :M ! N . For example, we de…ne the push-forward  of a function as the "transformed" function on N: f :M ! R (A.9) p 7! f (p) f : N ! R p 7!  [f (p)]  (f) (p) =  f  1 (p) = f (p) We see that we must compose with 1 to get from N toM (and then to R). We also de…ne the pull-back  as the inverse operation, i.e. given g : N ! R, g = 1 g. Note that if  is not invertible, which is never the case for a coordinate change, then only the pull-back would exist for functions and covectors, only the push-forward for vectors and neither for general tensors. In practice we use functions of coordinates, which are themselves functions over M . f : Rn ! R (A.10) x 7! f x1 (x) 74 It would seem natural to de…ne the transformed function in terms of the new coordinates x, but this would not correspond to our passive change of coordinates. Indeed, since (x) (p) = x (p), we would …nd that the function doesn’t change: f : Rn ! R (A.11) (x) (p) 7! (f) ((x) (p)) = f (x (p)) x 7! (f) (x) = f (x) The last result applied to the coordinates themselves may again seem a bit odd (x) (x) = x (x) but by de…nition both functions are just the identity. Now if we wanted to see how an object is transformed at a …xed point, then we should compare (f) (x 0 (p)) with f (x (p)), but this still would not correspond to the passive case. To make it correspond we need to express the transformed function in terms of the old coordinates (obviously this is only possible if M = N). To avoid confusion, call this new function ~f . ~f : Rn ! R (A.12) x (p) 7!  [f (x (p))]  (f) (x (p)) = f ((x) (p)) = f (x (p)) x 7! (f) (x) = f (x) We want to …nd transformation rules equivalent to (A.4) and (A.8) so introduce a vector …eld  and construct a di¤eomorphism t de…ned by moving each point an in…nitesimal distance t along the integral curves of . To …rst order in t, x (tp)  x (p)+t (p) and the transformed 75 function becomes ~f (x (p)) = f ((tx) (p)) (A.13) = f  x  1t p  = f  x  tp   f (x (p)) + t  @f (x) @x @ @t  x  tp  t=0 = f (x (p)) t@f (x (p))  (x (p)) ~f (x)  f (x) Ltf (x) (A.14) A vector …eld X acts on a function as a directional derivative Xf  hdf;Xi = X@f , giving another function. If g is a function on N , then from (A.9) we must have X (p) (g) (p) =  [X (p) ( g) (p)] = (X) (p) g (p) (A.15) so  [X (p)] = (X) (p) = X (p)  . Again, if we de…ne X in terms of the new coordinates, we …nd that nothing changes: (X)  @ @ (x)  g ((x) (p)) = X  (x (p)) @  [g ((x) (p))] (A.16) = X@g (x (p)) = X @x (p) @x (p) @ @ (x)  g ((x) (p)) = X@ 0 g ((x) (p)) (X)  (x) = X (x) (A.17) Though we are comparing components at two di¤erent points. In terms of the old coordinates on the other hand, ~X (x (p)) @ @x (p) g (x (p)) = X (x (p)) @  [g (x (p))] (A.18) = X (x (p)) @g (x (p)) = X (x (p)) J @ @x (p) g (x (p)) 76 where J  @x (tp) @x(p) =  + t  ; . To …rst order in t, we get ~X (x (p))  X x 1t p+ t;X x 1t p (A.19)  X (x (p)) + t @X  (x) @x @x  tp  @t  t=0 + t;X  (x (p)) =  X  t@X; + t;X  (x (p)) ~X (x)  X (x) LtX (x) (A.20) This carries on to general tensors: thinking of them as maps from vectors and covectors to functions we …nd (T ) = T  where the pull-back is applied to the vectors and covectors. As a function of the new coordinates the components don’t change (T )   (x) = T   (x) (A.21) but in terms of the old coordinates, they pick up a Lie derivative. 77 B. Calculation of the wave shape The calculation of the gravitational wave shape can be done explicitly by starting with equation (3.23) and without going to Fourier space. h (x) = 4G Z d3y T  (t jy  xj ;y) jy  xj (B.1) = 4G r Z d3y T   t r + x r  y;y  = 4G r Z d3y Z d2 (3) (y X)   t r + x r  y X0  _X _X X 0X 0  = 4G r Z d2   k̂x  (X  x)  _X _X X 0X 0  Where k̂x = (1; xr ) is again the direction of the observer. We next go to null worldsheet coordinates. h (x) = 2G r Z d+d   t r + k̂x  1 2 (X+ +X)  _X ( + _X )  (B.2)   t r + k̂x  1 2 (X+ +X)  =  (+  +) k̂x  12 _X+ (+) (B.3) The value + in the delta function will be complicated in general, but in the gravitational wave burst direction, k̂x = _X0 and we have h  t; r _X0  (B.4) = G r e  X+0  X0 Z d+d +tR  1 12  X2+0 3 + + X20 3   The delta function will be zero unless jtRj < 1 12 l3 8  X2+0 + X 2 0  (B.5) 78 This is the range in which the e¤ects of our worldsheet are felt. Now let X< and X> be the smallest and largest respectively of  X+0 and  X0. We do the > integral …rst assuming the delta function is not zero.   tR  1 12  X2> 3 > + X2< 3 <  =   >   12 X2> tR  X 2 < X2> 3<  1 3  3 12 X2>  12 X2> tR  X 2 < X2> 3<  1 3 2 (B.6) h  t; r _X0  = 4G r e X< X> Z l 2  l 2 d< <  12 X2> tR  X2< X2> 3< ! 1 3 (B.7) = 2Gl 3r e  X< X> ! 1 3 T 1 3 Z 1 1 ds s 1 3  1 s T  1 3 We have introduced the normalized variables s = 8 l3 3< ; T = 8 l3 12 X2< jtRj 2 " 0; X2> + X2< X2< # (B.8) We must now ensure the limits of the > integral are respected. The requirement is X2< X2> jT  sj < 1. Combining this with jsj < 1, the range of integration is now max  1; T  X2> X2< ! < s < 1 (B.9) The lower bound will change when X2> X2<  T = 1 (B.10) jtRj = 1 12 l3 8  X2>  X2<  Things get a bit complicated now and we must calculate the integral in 5 cases. 79 1. T < X2> X2<  1 and T < 1 h  t; r _X0  = Gl r e  X< X> ! 1 3  2 p 3  T 1 3  2 3 3  4 3 (1 T ) 23 (B.11) 2 3 (T + 1) 2F1  1 3 ; 2 3 ; 5 3 ;T   T 13 2F1  1 3 ; 2 3 ; 5 3 ; 1 T  2. 1 < T < X2> X2<  1. This happens if X2> > 2 X2<. h  t; r _X0  (B.12) = Gl r e  X< X> ! 1 3 T 1 3  2F1  1 3 ; 2 3 ; 5 3 ; 1 T   2F1  1 3 ; 2 3 ; 5 3 ; 1 T  3. X2> X2<  1 < T < 1: Here the lower bound increases with T , from 1 to +1. We de…ne V = X2> X2<  T . h  t; r _X0  (B.13) = 2Gl 3r e  X< X> ! 1 3 T 1 3 Z 1 0 ds s 1 3  1 s T  1 3  Z V 0 ds s 1 3  1 + s T  1 3  = 2Gl 3r e  X< X> ! 1 3 T 1 3  "Z T 0 + Z 1 T  ds s 1 3  1 s T  1 3  V 23 Z 1 0 dq q 1 3  1 + V T q  1 3 # = Gl r e  X< X> ! 1 3 " 2 p 3  T 1 3  2 3 3  4 3 (1 T ) 23 2 3 (T + 1) 2F1  1 3 ; 2 3 ; 5 3 ;T   T 13V 23 2F1  1 3 ; 2 3 ; 5 3 ;V T  80 4. 1 < T and X2> X2<  1 < T < X2>X2< h  t; r _X0  = Gl r e  X< X> ! 1 3  2F1  1 3 ; 2 3 ; 5 3 ; 1 T   4 3 (1 T ) 23 (B.14) 2 3 (T + 1) 2F1  1 3 ; 2 3 ; 5 3 ;T   T 13V 23 2F1  1 3 ; 2 3 ; 5 3 ;V T  5. X2> X2< < T < X2> X2< + 1 h  t; r _X0  = Gl r e  X< X> ! 1 3 T 1 3  2F1  1 3 ; 2 3 ; 5 3 ; 1 T   V 23 2F1  1 3 ; 2 3 ; 5 3 ;V T  (B.15) The simplest case is when  X+ =  X. Then we only need two pieces (3 and 5) for T 2 [0; 1] and [1; 2]. This is the …rst wave shown in …gure 4.3 (a = 1). We can also get the value at tR = 0: h  r _X0  = 4Gl r e  X< X> ! 1 3 (B.16) 81


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