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GRAVITATIONAL MODELS IN 2+1 DIMENSIONS WITH TOPOLOGICAL TERMS AND THERMOFIELD DYNAMICS OF BLACK HOLES By BETTINA E. KESZTHELYI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA ACKNOWLEDGEMENTS would like to acknowledge all the professors I have worked with over the years, especially my present advisor, Professor Pierre Ramond. He was responsible for introducing me to the study of dual solutions in 2+1 dimensions. He also made valuable suggestions to me on the topic of thermofield dynamics. I would like to thank Professor Stanley Deser for his help and hospitality during my stay at Brandeis University. on topologically massive gravity. I owe him special thanks for his suggestions I would also like to thank Gary Kleppe for a productive collaboration and for sharing his insights on nonlocal regularization. Finally, I must thank Alexios Polychronakos for his invaluable advice early in my career. Digitized by the Internet Archive in 2011 with funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation TABLE OF CONTENTS ACKNOWLEDGEMENTS ABSTRACT I I I I I I I S a 11 . . I S S S S V . I . a S 5 5 5 S I I 1 DUAL SOLUTIONS IN 2+1 DIMENSIONS .1 Introduction .2 Weyl Theory in 2+1 Dimensions . .3 Stationary Solutions . . .4 Solutions with Magnetic Field . .5 Solutions with no Electromagnetic Field: 2.6 Stationary Axial Symmetric Solutions RENORMALIZABILITY = 3 TMG 8 15 S 18 ity . 21 . 23 Einstein Grav 32 3.1 Introduction 3.2 PowerCounting Renormalizability and Gravity 3.3 Topologically Massive Gravity . 3.4 Nonlocal Regularization . . 3.5 Nonlocal Feynman Rules . . 3.6 Renormalizability. . THERMOFIELD DYNAMICS OF BLACK HOL * I I . * S f S . * a * * S S ES . 33 . 34 38 . 41 . 43 . 48 4.1 Introduction 4.2 Massless Scalar Particles on the Schwarzschild Background 4.3 Many Black Holes . 4.4 Neutrinos on the Schwarzschild Background CONCLUSIONS 51 . .S55 S . . *66 APPENDIX A APPENDIX B FEYNMAN RULES MEA FOR TMG 69 SURE FACTOR FEYNMAN RULE FOR TMG REFERENCES * I I 5 . 5 5 5 5 S S 5 *I7 6 BIOGRAPHICAL SKETCH S S S S S S I 78 INTRODUCTION PaRe Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy GRAVITATIONAL MODELS IN 2+1 DIMENSIONS WITH TOPOLOGICAL TERMS AND THERMOFIELD DYNAMICS OF BLACK HOLES By BETTINA E. KESZTHELYI December 1993 Chairman: Pierre I Major Department: lamond Physics We consider two extensions of Einstein gravity in 2+1 dimensions. First we study some consequences of duality in three dimensions. In the case of pure gravity, a dual Ansatz is shown to lead to pure gauge configurations, and in a Weyl invariant gravitational theory, duality arises as an equation of motion. Its solutions obey the Liouville equation and describe a rotating ChernSimons fluid in a gravitational field. Next we analyze the theory of topologically massive gravity in three space time dimensions, conjectured to be renormalizable by Deser and Yang, using the nonlocal regularization. The validity of this technique, however, depends on the existence of a gaugeinvariant measure for the nonlocal theory. that such a measure exists, Assuming we show that the possible obstacle to renormaliz ability found by Deser and Yang does not appear. Finally thermofield dynamics is used to derive the Hawking radiation of black holes emitting massless scalar particles or spin one half fermions. also show how to generalize this method to particle creation in spacetimes  1 I I *, 1, 1 1 1 CHAPTER 1 INTRODUCTION work we study two 2+1 dimensional gravitational models, Weyl gravity and topologically massive gravity. We study the consequences of duality the first case and renormalizability in the second. We also consider the Hawking radiation of black holes in 3+1 dimensions. Field theories in lower spacetime dimensions have recently attracted at tention as tractable toy models for realistic 3+1 dimensional systems and also in their own right for their unusual topological properties. For example, one can hope to find 2+1 dimensional models involving gravity which have better short distance behavior than those in 3+1 dimensions. The latter are known to be nonrenormalizable due to the presence of the gravitational coupling which has negative mass dimension. Indeed, one may be able to discover 2+1 dimen sional theories which are renormalizable and may help one to better understand the quantum behavior of the higher dimensional theory. these theories are also interesting. Classical solutions of These can be used to study the nature of gravitational singularities or can represent physical axially symmetric solutions such as cosmic strings. We consider two models addressing the above questions. Since they both involve gravity, we review the main features of 2+1 dimensional pure Einstein gravity. These features are unique, but restrictive. This is why it is I I .I I *1 S  mi n 1 j_ ..  f9 be derived from the Einstein equation which is given by G y, + Ag = TWV (1.1) where Gpy = Rl v jRgwv is the Einstein tensor. Expressed in terms of the Ricci tensor we have R.V = 2Agt, + s(Tyv gtVT) , (1.2) where A is the cosmological constant and the coupling t has mass dimension The curvature tensor and Ricci tensor both have independent components and one can write the curvature in terms of the Ricci tensor, gIXRup + g pRX  g9pRyx gvARx p (1.3) 3R(g\Agvp gYpgYA) From (1.2) and (1.3) one can see that the curvature tensor is completely de fined by the energymomentum tensor Tay and A. When Ty, = 0, the scalar curvature is R = 6A so that any curvature effects produced by matter do not propagate through spacetime and there are no dynamical degrees of freedom. This could simply be seen by counting degrees of freedom. the sign of A, the spacetime is either locally flat, Depending upon de Sitter or antide Sitter. Although local curvature in sourcefree regions is unaffected by matter, it can still produce nontrivial global effects. Even in the simplest cases such global effects are present. For example, for a point mass and A = 0 spacetime is flat except along the world line of the particle. In the static case coordinates can be chosen so that the constant time surfaces are conical [1]. This conical spacetime is obtained by removing a wedge from Minkowski space and identi Ap = 3 source is also conical, but in this case, the points that are identified across the deleted wedge differ in their time coordinate value by an amount proportional to the angular momentum of the source. Thus in the presence of a spinning source the spacetime has a "helical structure," a rotation about the source is accompanied by a shift in time [2]. This conicalhelical geometry characterizes the spacetime outside of more general compact matter distributions, because matter cannot affect the local curvature in source free regions. Despite being locally flat except along the particle's world line, these space times have interesting global geometric properties. For example, there is an analogue of the AharanovBohm effect in that a vector parallel transported around a loop surrounding the source experiences a nontrivial rotation, even though the loop lies entirely within flat regions of the spacetime [3 Similarly, two geodesics passing of opposite sides of the source may intersect twice. This effect also arises in 3+1 dimensional gravity in the context of cosmic strings [4]. Here we will consider two extensions of the Einstein theory in 2+1 dimen sions, Weyl gravity and topologically massive gravity (TMG). The outline of this work is as follows: Chapter we show that Weyl gravity has selfdual solutions, and in certain cases exact solutions can be solutions, as described in Sect. 2.1 [ Our motivation for seeking such 5], is the important role they play in 3+1 dimensional theories. For example in four Euclidian dimensions selfduality of the YangMills field strength leads to instanton solutions similar condition imposed on the gravitational connection leads to the EguchiHanson gravitational instantons [7]. In Sect. 2.2 we review Wevl theory in d+ 1 snacetime dimensions. Then we gravity, a Weyl gauge field and a real scalar field. We find some interesting features of the field equations, obtained from the Lagrangian of this theory. These are useful to predict some of the general features of the solutions. example we show that our solutions correspond to a special type of rotating fluid immersed in a 2+1 dimensional gravitational field. This indicates that such solutions might have interesting applications in fluid mechanics. back to this analogy throughout the following subsections of Chapter 2. We refer We also show that the field equation of the Weyl gauge field in a specific gauge imposes a selfduality condition between the field strength and the gauge potential. In Sect. 2.3 we discuss stationary solutions in this gauge. We show that they can be classified by the nonvanishing elements of the field strength tensor. We find that the case with no electric or magnetic field (Ei = 0, to Einstein gravity in flat or in de Sitter space [2] B = 0) reduces The purely magnetic case 0) and the case with nonzero electric and magnetic field (Ei B f 0) are more interesting. In the former case we find all the static solutions, but not in the latter. From here we proceed by solving the field equations for the E, = 0, B case in Sect 2.4, and for the Ei tinue with studying our solutions. 0 case in Sect 2.5. In Sect. 2.6 we con We find that in the axisymmetric case they correspond to known 3+1 dimensional spacetimes. It is also interesting that in the proper coordinates our solutions have the same conicalhelical geometry characteristic to a large class of known solutions of 2+1 gravity as mentioned above. Weyl gravity is one of the natural generalizations of Einstein gravity if nne Wi;hss tn Qtiim1 t"h^ niiinrnfm bnTrr hon^ancn ;+ b.r0 hno ++nr 0br.4 rla e.a n (Ei = 0, behavior arising from conformal invariance [8]Q1) In 2+1 dimensions there is another possible choice, topologically massive gravity [9]. Chapter 3 we study the renormalizability of TMG by using nonlocal regularization In Sect. we describe why it is a good candidate for a renormalizable theory with the symmetries of gravity. There are several arguments [9] including ours [10], indicating that TMG is renormalizable, but none of them can be considered a strict proof. The importance of the question that if these arguments are proven to be true, TMG would be the only known theory with such properties. In Sect. 3.2 we use general power counting arguments to show why certain gravity theories are not renormalizable. In Sect. 3.3 we review the main features of TMG, and we show that it is power counting renormalizable. However, in order to conclude that the the ory is renormalizable, one also has to show that the gauge invariance may be maintained in the regulated version of the theory without giving up the de sirable power counting behavior. In other words, without the use of a gauge invariant regulator, additional terms might be required to cancel the gauge transformation of the effective action, and these might spoil powercounting renormalizability. For these reasons we use gauge invariant nonlocal regular ization. In Sects. 3.4 and 3.5 we review the method of nonlocal regularization and the nonlocal Feynman rules, respectively [11]. In Sect. 3.6 we apply these rules for TMG. Unfortunately, to find a proper gauge invariant measure factor for the nonlocalized theory, or at least prove its (1) It is necessary to consider a generalized theory, because Einstein theory in existence, is an extremely difficult problem. Only a perturbative method exists which allows us to calculate it to any desired order in the fields. In theories without gauge anomalies (like TMG), it is reasonable to assume it s existence. We show that if this holds, the only anomaly term, found in ref. 9, does not appear, and the theory is renormalizable. In Chapter 4 we consider another interesting problem of general relativity, namely particle creation from the vacuum in spacetimes with causally discon nected regions. It is known that the quantization of fields on such spacetimes leads to particle creation from the vacuum as a consequence of the information loss associated with the presence of the event horizon(s) In stationary spacetimes with a simply connected event horizon (such as a stationary black hole, an accelerated observer in Minkowski spacetime, or de Sitter type cos mologies), the emitted particles have a thermal spectrum [13]. This result has been first obtained by Hawking 14] for black holes. He has shown, that the effective temperature of this radiation is TH = K, the Hawking temperature, where i is the surface gravity of the black hole (units are chosen throughout such that k = 1). These results have been confirmed and de rived in a number of ways, and several attempts have been made to gain a better understanding of the features of the Hawking process and the physical role of the event horizon [14,15,16]. A particularly interesting approach into this direction is the one used Israel [16], who considered the problem of particle creation on the Schwarzschild background. The idea is to quantize the fields in the full analytically extended Schwarzschild spacetime (known as the Kruskal extension) in order to keep track of particle states on the hidden side of the horizon as well. The same idea allows us to annlv a auantumstatistical 7 Here we follow Israel's approach, and first we rederive the wellknown re suits of Hawking radiation of a black hole. dard results. This method reproduces the stan We look for possible extensions to the case of many black holes. The organization of this chapter is as follows: In Sect. 4.1 we briefly review the canonical formulation of thermofield dynamics for free fields [17] and show how it can be generalized to describe particle creation in general relativity [16]. In Sect. 4.2 we review Israel's paper [16] . The quantization of a massless scalar field on the Schwarzschild background is considered by using thermo field dynamics. We conclude that the results obtained this way are equivalent to the earlier ones. In the rest of the paper we describe the possible generaliza tions of this approach. In Sect. 4.3 an approximate multiblack hole solution is considered as an example to demonstrate how to extend the method to space times with many causally disconnected regions. Finally in Sect. 4.4 we derive the Hawking radiation of a black hole emitting neutrinos and antineutrinos. CHAPTER 2 DUAL SOLUTIONS IN 2+1 DIMENSIONS 2.1 Introduction In four Euclidean dimensions, selfduality of YangMills gauge fields leads to the classical instanton solutions. A similar condition imposed on the gravi national connection leads to the EguchiHanson gravitational instantons. Mo tivated by the recent interest in theories of lesser number of dimensions, we investigate analogs of selfduality in the classical solutions of some theories in 2+1 dimensions. In the case of YangMills gauge fields, selfduality cannot be imposed on the field strengths, as in four dimensions, but only between the gauge poten tial and the field strength. In order to find such a solution, we investigate a simple theory involving gravity and an Abelian gauge potential, linked by the requirement of Weyl invariance. duality (in a special gauge). We stu The classical equations of motion demand idy stationary solutions of these equations. We first solve the equations of motion in the pure gauge case, when our theory reduces to Einstein gravity with nonzero cosmological constant. Then we find the general solution when the magnetic field does not vanish for these the conformal factor of the two dimensional space satisfies a Liouville equation. The physical situation corresponds to a special type of fluid immersed in a 241 gravitational field. The fluid is a rotating perfect fluid with velocity corre duality condition and a massive KleinGordon equation. When specializing to axisymmetric solutions, we recover either the 2+1 dimensional G6del solution or the boundary of 4 dimensional TaubNUT spacetime solutions. Weyl Theory in 2+1 Dimensions order to consider gauge fields, we use as a principle Weyl's original theory which links gravity to electromagnetism through a "gauge" principle For the reader who may be unfamiliar with this type of theory, we start with a brief review. In the second order formalism of general relativity, the fundamental fields are the metric field addition to the usual diffeomorphism of general relativity, Weyl requires invariance under conformal rescaling or "gauge" transformation gpv(x) Q g111(2) + 4 x)gpv(x 2(x)g{"(x (2.2.1) (2.2.2) x)  In Einstein Ap(x) + 2, In l (xa theory the spacetime is described (2.2.3) Riemannian geometry, where one has a metric connection, the covariant derivative of the metric is zero, gv = 0 . (2.2.4) This implies that lengths and angles are preserved under parallel transport. In order to get a locally scale invariant theory Weyl weakened this condition, and required only that ( 9 f\ 7 7n... 4 \ n.... where A4 is a vector field. That is, in Weyl's theory only the angles, but not the lengths, are preserved under parallel transport. If we assume that the torsion is zero, then Eq. (2.2.5) can be solved for the connection in terms of gpy and A The result is = { 2A +6aA pyAa), / zl (2.2.6) where { i} is the usual Einstein connection. Notice that Viv a is gauge in variant and symmetric. From Fyw a we can construct the conformally invariant curvature tensor: Rpva = Dpa  Ov1 pa + va  Fpa 7v7Y (2.2.7) The conformally invariant Ricci tensor and the scalar curvature are given by contracting (2.2.7) with the metric. In d + 1 dimensions the Ricci tensor is 1 <(d DD1 A, D A, + gp D AC ) RV = Rpvy (2.2.8)  1 (A A, A gy AaA ) and the scalar curvature is g= VRV = RE + d DAC Aa A0 (2.2.9) where RE and RE are the usual Einstein Ricci tensor and scalar curvature, respectively, and Dp is the covariant derivative with respect to { a, }. Notice that the Ricci tensor is not symmetric, since it contains the antisymmetric field strength tensor F0e. From Eq. (2.2.8) its antisymmetric part is given by d1 F 2 rL (2.2.10) nC 1'1 1 l 7Fy SRE  Ul Rp R,? nnrl 7? .. e c ol oA 11 because the connection is symmetric. The covariant derivative with respect to the connection given by Eq. (2.2.6) acts on a scalar field y and on a vector field Vp of Weyl weight w in the following manner , (2.2.2.12) w v, 8pV, I' Va + A V, ^v ir/ ^ v'1" 1^r (2.2.13) A field 2 is of (Weyl) weight w if it transforms under conformal rescaling as 0w (2.2.14) Weyl invariance limits the form of the gravitational interaction. construct a purely gravitational action, One cannot first order in the curvature, and reproduce Einstein's theory (except in the "trivial" 1+1 dimensional when it is a surface term). case, Mathematically this means that, under a conformal rescaling +4 d+1 VHT (2.2.15) where g = det gp,, and (2.2.16) the Einstein action is not invariant: 16W7G d+lx  R 167 16irG dd+lx gR Q d1 (2.2.17) where G is Newton's constant. The action IE is invariant only if that is in 1+1 dimensions where it is the Euler characteristic. In 3+1 dimensions the Einstein action is not gauge invariant (an invariant gravitational action can be fnrmedrl bit i hno n nhr minrlrntirin 7? (n *hP nthhor nnrt IVlnvxM^ll' c nrfi;nn VkIf up = 9P + Apup R  Weyl invariant. This is not surprising, because electromagnetism does not require any dimensional constant. In order to reproduce Einstein's theory, many authors [8,19] have intro duced a scalar field of unit weight, that is (2.2.19) corresponding to its canonical dimension. Then the action d4x igRy(2 is indeed invariant. (2.2.20) 9 acquires a nonzero value o0 in the vacuum, it yields the Einstein action with 16r = 2. Weyl invariance in 3+1 dimensions allows for a kinetic term for 9 as well as a potential term, thus making it a dynamical field. However, it is not clear how to generate such a vacuum value in a theory that is not only without a scale but also not renormalizable. In 2+1 dimensions one can also introduce a scalar field with Weyl weight 1 (not its canonical dimension) in order to construct a gauge invariant action, given by d1 (gRy+cpFA, d3x( ll  +)A (2.2.21) in this case, Weyl invariance allows us to write a cubic potential term for t, but not a kinetic term. Thus we do not expect 2 to correspond to a dynamical degree of freedom. We further note that the action contains only a ChernSimons term for the vector potential and no Maxwell term. The equations of motion obtained from varying the action with respect to the fields and the metric are the following: _? .+ lcp (2.2.23) 1 (RV 9RgV)9 + {p ** v} = (V, VV gpvV!Va) 2 + ~ gpv + {I <+ v} (2.2.24) It is interesting to note that, even derivatives of the scalar field, p still though our Lagrangian does not contain 1 obeys a KleinGordon equation, with the source as the Chern imons density 1 eD FapA A,  i1 EC/?9fl (2.2.25) using Eq. 2.2.23 ) and assuming the Bianchi identity on Fy, (absence of a magnetic monopole). Actually 9 is a derived field which obeys a first order differential equation of the form "9 = (2.2.26) where Vuj is a vector field chosen such that V Fa pfalY =0 . (2.2.27) This means that 9 is covariantly constant along a direction locally determined by the electric and magnetic fields. Before looking for solutions of these equa tions, we make the following simplifying observations. First, not all of the above three equations are independent. The trace of Eq. (2.2.24) is equivalent to Eqs. (2.2.22) and (2.2.23). This allows us to use only the last two equations when seeking solutions. Second, Weyl vector terms can be incorporated into the energymomentum tensor of the matter fields. As a result, Eq. (2.2.24) n r' V.,,,: 1.,,n a stct I 1 /' ,P.. 1 r* lvi 4t % r .64 4 n ri C, rc t t ti. a F, = gTVp 2g Vg y^ ) nr /yv r* v f n r% f~^r r^ // where the energymomentum tensor Tpn is given by 1 (Dcov gDa,) + Avpp + ApaO gpv A a +A 2 +( '6  AaAa)gsv + 4 (2.2.29) To proceed, we use the Weyl invariance to go into a gauge where (2.2.30) which is allowed as long as y does not vanish anywhere. There, Eq. ( 2.2.23 rewritten 1 P Fap = 0 A7 1 (2.2.2.31) from which it follows that DoAn =0. From Eq. (2.2.31) one also finds that A, satisfies D"'DaA Av + Aa R = 4 the field equation of a massive vector field. (2.2.32) A similar result has been found for the Abelian ChernSimons theory on Minkowski spacetime in refs. 20 and 21. In this gauge the energymomentum tensor of Eq. (2.2.29) simplifies to S 4A A0 )gv + ~04 (2.2.33) a form reminiscent of a fluid. It is not quite perfect since the pressure p and the density p depend on AaAC as well as on 2)0. Furthermore the fluid has a velocity vector proportional to Ap will study the properties of this flui( which obeys the constraint (2.2.31). d in greater detail in the context of exact solutions. If the electromagnetic field is zero, i.e. E = B = 0, we find from Eq. (2.2.23) . I A 6 Tv= AUA, c = Q , AA , /~ L II 11 15 and =the Einstein equation reduces to and the Einstein equation reduces to (2.2.35) Clearly, A2 +2). when the Weyl invariance is gauge fixed to a constant (2.2.36) we obtain Gpy = gplV6 (2.2.37) reducing the space to an Einstein space with cosmological constant. following, In the we discuss stationary solutions. Stationary Solutions the stationary case the most general form of the line element in component notation is = N2(dt + Kidxz)2 + yijdxzdx3 (2.3.1) where N , Ki and Yij depend only on the spatial coordinates, x and i, That is, the metric components are , goi = , gij = 7ij N2KiK (2.3.2) with inverse components 1 N2 + "y Ki ij vg= =N , 7 = det 7ij Sgi , g1 >0. =7 7 (2.3.3) (2.3.4) The remaining reparametrization auge freedom, t * t + A(r), can be fixed by setting DiKz = 0, where D; is the two dimensional covariant derivative with T I I 1 1 r~ 1 S . 1 1 A G, = 4(A A, + D A, + D A ) + ggy(2Da A goo00 = N 731 = ys^.Ki J J 1 2N 1 (D = N2KikKik  Dj(N3Kij) , (2.3.6) 1 k _ 2NyirKkJlKkt (2.3.7) where Kij = DiK1 = OiKj jKi, and 2R is the two dimensional curvature. We choose spatial conformal coordinates. This can be done locally without loss of generality in a two dimensional space: dij = dij dl2 = (dx >0), (2.3.8) + dy In these coordinates the DZ Ki = 0 condition reduces to OiKi = 0. This allows us to write Ki = eij V. with 12 = where eij is the two dimensional LeviCivita tensor E21 = With this form of the metric, the field equations (2.2.22), (2.2.23) (2.2.24) in the p = p0 gauge become .ij iAj   N0 2 N2 + Kj(Aj  AoKj)) (2.3.9) eijQjAo = N(A 2J  AoKi) , (2.3.10) A 1 3N2 2 AaAa )_212 2N2 (2.3.11) N3 Oj( eijAV) = Ao(Ai (2.3.12) 1 1N(DiON  A 2 =(   6jAN) ij(iV)2 23 43  AaA a)gij  (2.3.13) AiAj, 2 where A is the two dimensional flat Laplacian. solutions can be characterized by the nonvanishing components of  iJD2)N  DjlKi , (  AoKi) ? magnetic field, Ei = Foi. B = 2%e~Efij, and the space components of the electric field, We have the following cases. a) B  0, E;= We have already seen that this case reduces to Einstein gravity in flat or in de Sitter space [2]. b) B This means that A0 is constant and from Eqs. (2.3.10) we have Ai = AKi. . Further, since B is different from zero, A0 itself cannot vanish. Thus Ai = AoeijjV allows us to rewrite Eq. (2.3.9) as (2.3.14) 2N which by comparing with Eq (2.3.12) leads to = NO = constant. (2.3.15) Note that our gauge condition (2.2.30) still allows us to make constant gauge transformations to rescale N0 (and 0).o We fix it choosing NO = 1. this purely magnetic case we were able to find all the solutions of the field equations. c) E; , B = anything. In this case, when the electric field is not zero, we did not find any static solutions. We can say however that if only component of the electric field is nonzero, say El then the solutions depend on only one spatial coordinate x1. In the next two sections we solve the field equations in the purely magnetic case with B 0 (Sect. 2.4) and with B = 0 (Sect. 2.5). In the latter case, when our theory reduces to the Einstein case, the equations of motion. we find the general solution of In Sect. 2.6 we continue with studying our solutions, obtained for the pure magnetic case. We show that in the axisymmetric case one E;= 2.4 Solutions with Magnetic Field In this section we solve Eqs. (2.3.9)(2.3.13) in the purely magnetic case At the end of the previous section we showed that in this case Ag = constant, Ai = AgKi and NO = constant. Using these results we find that Eq. (2.3.10) is trivially solved and the remaining equations are PoAo 1 Alnb + a (2.4.1) 3 AV 2 ( (2.4.2) 3 AV 0, 8a = 0 , (2.4.3) ( ) where we have set NO = 1. 2 + A, (2.4.4) From the above equations one finds that 4 satisfies a Liouville equation In 4 = /3' where (2.4.5) A. All the other quantities can be expressed in terms of the solutions of this equation and the constants 0 and A as follows A2 = 0n 1 4 (2.4.6) 3 ) , (2.4.7) (2.4.8) Note that because AO is nonnegative only the A g values are allowed. This leads to 3 < that is, it can be either negative or positive. From the above equations one finds that the magnetic field, AO B = V = _A0 (2.4.9) 0, Ei = 0). 1 = 2 Ai = Aoeij jV e3ijiA  the length of the Weyl gauge field, SA0A  A2,  "0' (2.4.10) and the Einstein scalar curvature, 2R + 4KijKij AIln + 22(AV)2 2 S8 (2.4.11) 16A 3 )' 3 are constant. We note that in this case, the perfect fluid analogy mentioned in Sect. 2.2 is complete, because as follows from Eqs. (2.2.30) and (2.4.10) the energymomentum tensor is just =16 2 gA AA (2.4.12 ) The normalized velocity is then Up = (2.4.13) Ao IAO that is, uP = 1 and ui = ij jV The equation of state relating the density {8P3 p =O0 where the p = 0 case corresponds to dust. (2.4.14) p to the pressure p is given by (2.4.15) otherwise, We note that the (weak and domi nant) energy conditions [23], stemming from demanding causality, lead to the condition p which gives further restriction on /: either ,3 < 0 and p or 0 < / 3 In addition to this equation of state, the velocity vector up obevs the further equation > Ip 20 which is indicative of rotation. In particular this means that the velocity obeys a massive KleinGordon equation, and that the fluid is incompressible. Depending on how we choose the value of A, R can be positive or nega tive. Positive R corresponds to a compact spacetime manifold (i. e. "closed universe"). In this case the solutions can be characterized by topological in variants of the manifold. For negative R, that is, for noncompact manifolds (or "open universe" ), one can define the energy and the angular momentum of the solution. We also find that our solution is conformally flat. three dimensional space the Weyl tensor is always zero in the absence of matter. However there is another tensor, the Cotton tensor [24], =(1 EaflVfRi + eaflV^R ) , (2.4.17) which plays the same role as the Weyl tensor in higher dimensional spaces. It is symmetric, covariantly conserved and vanishes if and only if the spacetime is conformally flat. For example all the vacuum solutions as well as the point particle and rotating solutions of Deser et al. [2] are conformally flat. In our case C~" is vanishing; that is, our solution is also conformally flat. general solution Liouville equation is given terms of complex functions z) and g(z), _2 S(f( z ff g(z) (2.4.18) where f( z) and g( ) are such that they give real positive values to 4. Explicit forms of f and g, that satisfy this requirement, are known [25] case. As the simplest we will consider axial symmetric solutions But first we tirli R the R = g(z))2 2.5 Solutions with no Electromagnetic Field: Einstein Gravity Although we have already shown that this case reduces to Einstein gravity with a cosmological term, it is instructive to elaborate on the form of the static solutions. The equations become 1 Aln4 1 4) 3N2 2 (AV2 4, A2 3= (2.5.1) di (N3%V)= (2.5.2) 1 (D N NA(D lN  siAN) 4 (6 Sij = si? (2.5.3) To solve these equations, we have to consider the N = constant and $ constant cases separately. a) N = constant: Without loss of generality we can set N = 1, then from Eqs. (2.5.1) and (2.5.2) we obtain cp = /3 (2.5.4) ^AV) 84,) (2.5.5) (A') (2.5.6) Notice that these equations are similar to the ones we obtained for the pure magnetic case. The field 4 satisfies a Liouville equation (Eq. (2.5.4)), and, from Eqs. (2.5.5) and (2.5.6), can be obtained in terms of the solutions of that equation. However there are differences between the two cases. The above equa tions have solutions only for nonnegative A as follows from Eq. (2.5.6) (in the mnornetic caspe we have A This imnlies (PFn (21..4V that, 13 has to he A ln = (a '06ifJ depending on the sign of ft. Here we note that because the B = 0 case corre spends to flat spacetime solutions, i.e., Aln$ = 0 and AV = 0, we do not consider it here. Instead, we discuss the more interesting /3 > 0 and 0 cases. We can say that because 3 > 0 in this case, solutions while in the magnetic case we obtain only one class of we have both classes. # constant: To solve our equations in the N constant case we follow similar tech niques used in ref. 2 to obtain multiparticle solutions for the Einstein equations with nonzero cosmological constant. Note that our solutions are more general, since the solutions of ref. case in our notation), wl We start with separating Eq 2 correspond to nonrotating sources (a = N3 = 0 while ours describe rotating sources as well. (2.5.3) into the spatial trace AN + N a2 NQ 22N4 A 2 3' (2.5.7) and into the traceless part j kMk = 0 , 2.5.8 where Mi = 1iN a = NA = constant, (2.5.9) as follows from Eq. ( Note that if we define a complex function M MI +iM2 = for M, whici 'aN then Eq . (2.5 = M( I is solved by .8) becomes the Cauchy Riemann equation z). Equations (2.5.1) and (2.5.7) become SA 2 4 5^ 2N4 0 ,2 3 (2.5.10) b) N 9iM + ijM After multiplication by 9tN and integration with respect to Z, (2.5.11) becomes z)zN ( N2 6 b + 2 2) e(z) 4 2 (2.5.12) where e( z) is an arbitrary integration "constant" If we introduce a real pa rameter z 1/ 2 \ dw M(w) + (2.5.13) then Eq. (2.5.12) becomes 8 N = N32 (A o0 (2.5.14) This is a first order ordinary differential equation for N(() the solutions are real only if e is a real constant, and they are given by standard integrals through N(() N(c~o)  CO (2.5.15) The solution for the spatial conformal factor 4 is given in terms of the solutions of Eq. (2.5.15), z) and the constant parameters A, (o0, a and ( 2N2 2M( as one can z)M( see from Eq. (2.5.1 Thus in the N 2 a 22 (2.5.16) and the definition of M( constant case, the solution is given by Eqs. (2.5.9), (2.5.15) and (2.5.16) in terms of an arbitrary holomorphic function, M( z). Once M( is specified, the explicit forms of N(x), V(x) and x) are obtained by the above equations. 2.6 Stationary Axial Symmetric Solutions SN N2 Maw) M(w}) N+2 2+e + E given by Eq. (2.3.1 with N = 1 and 7ij = bnij and the problem reduces to the solution of a Liouville equation (Eqs. (2.4.5) and (2.5.4)) for the spatial conformal factor 4 . This means that the spatial part of the spacetime is of constant curvature (negative if /3 > 0 and positive if Q <0). In order to sim plify our discussions, we consider only axial symmetric solutions. The most general such solutions are given in terms of two real parameters, a and v the radial coordinate r = q/(x2 + y2) as follows [25] 2 2v a + a2V)2 8v2a2Y , =8 22v  a2)2 <0, (2.6.1) >0, (2.6.2) case would correspond to the flat solution, A4 = AV = 0. The parameter a can be absorbed into r by introducing ( The other parameter, v, has to be nonzero; otherwise b would be zero. Because of the invariance of 5 under inversion, r  1 r , it is enough to consider the v case. the magnetic case, spacetime components Ki = eij jV of the metric are given by Eqs. (2.4.4) and (2.4.6): f\2 T , ^^ VI  2vr2V r2v a2v (2.6.3) In the B=0 case, as one can see from Eq. (2.5.6), Ki is given by the same expression, constant b in Eq. set b= (v 1). with different numerical factors. We choose the integration (2.6.3) such that IKi is nonsingular at the origin; that is, Then, the line element in spherical coordinates reads n Ei = 1. I  Note the metric is singular at  00 because 4 vanishes there. the f3 > 0 case it is also singular at r = a. The latter is very much like the case of solutions of Einstein's equations that describe a rotating fluid. also that because the diffeomorphism invariant quantities such as th """" ~""~~r"~"` "~"" r"~*'1 Notice curvature and the length of the Weyl vector are nonsingular everywhere (they are constants) these are only coordinate singularities. Since we have an explicit solution, we can complete our perfect fluid analogy discussed previously by calculating the normalized velocity. From Eq. (2.4.13) we find that in spherical coordinates U0 = ur = 0 and U, = 79 (2.6.5) a2v where we have used that AgKi = Ai in the purely magnetic case. Thus our solution corresponds to circular flow with vorticity, va = .ap 72vy: Wl~l ' 2 = 50 2 that is Be: , vi=O , , only the time component is nonzero, and it is constant. fore we proceed with the discussion of the spacetime structure of our solution, we note that the v case always can be brought into the form of the v = case by rescaling the radial coordinate. As we shall see, the only difference that the range of the angular coordinate will change. To see this let us define a new radial coordinate, (= a In terms of the line element becomes e scalar where the new angular coordinate 4" = v4 ranges from 0 to 2ruv (if 0 27r); that is, the points with Ib and 4/ + 2rrv are identified. In the following we consider the positive and negative 3S cases separately. We consider the v = 1 case to simplify our discussion. a) >0: The solution of the Liouville equation is given by Eq. (2.6.2). The metric in this case is given by (2.6.5) with the lower sign, and it is singular at =1. ce, we have to consider the ( < 1 and C cases separately. < 1, the change of the radial coordinate  C2) gives the following line element = (dt  2 2 d 2dg2) (2.6.7) where  00 27 and 0 A final, "hyperbolic," formula is obtained by defining F = sinha, = (dt (cosh /3 a  1)dw) 2 2 / ( + sinh2 ad 2) , (2.6.8) and 0 Similarly for E > 1, the change of the radial coordinate  1) gives + 1 + 1)d, ) db2) , (2.6.9) 2 d(2 + {( + 3 i2+ 1 where oo < t oo and 0 . The final " hyperbolic " formula is obtained by defining ( = sinh a: = (dt + (cosh + 1)d0}) /3 + (d2  + sinh2 ad4' (2.6.10) where 0 < a < oo. We note that this is the metric of thpe flel univprsp [9Rl 0 = (dt + / nI < J> ( = ( 2(/(1 27 We have already mentioned that our solution is conformally flat (the Cotton tensor is vanishing). Here we show that one can find a set of coordinates, in terms of which time. In the v , the form of the metric reduces to the flat solution, with periodic = 1 case, the metric is given by = (dt + Kidxi)2 + $dxdxzi (2.6.11) where Ki and 4 are given by Eqs. (2.6.3) and (2.6.1)(2.6.2), respectively. us introduce new coordinates, = r2 and denote the corresponding angular coordinate X. of these coordinates is (2.6.12) The line element in terms = (dt + + (dp + p2 dx (2.6.13) where one has the lower sign if r < a, and the upper sign if r Let us make one more coordinate transformation: (2.6.14) Then the metric reduces to the familiar form = (dr z dx) + a(d2 'IP  o a(dpi2 +y(^ + p'2dx (2.6.15) + p2dx2) where we have introduced a new periodic time coordinate (2.6.16) a 2 , = dt'2 < oo, but the coordinates cover only the 0 r < a or the a < r part of the spacetime. Let us make the following coordinate transformation: sin2 2 2 (2+1 t (2.6.17) ,0 < < 00  00 < 00 . (2.6.18) In these coordinates the line element has the form = 4 (dT _+ (dO2 sin2 + sin 2d2) . (2.6.19) As we have already mentioned the spatial part of the metric is a two dimen sional sphere. And because the curvature of this sphere is 21R = Aln,/4 = IiI, the factor in front of the spatial part, 2// 1, is the square of the radius. The Euler characteristic, j f d 2xJ 2R = 2, is that of the sphere. The metric is regular everywhere, except at 0 = it, string type singularity. where it has a Dirac One can remove this singularity by introducing a new time coordinate 1 + ^r W~C (2.6.20) The metric then becomes  cos La' + ~ (dB2 + sin2 Od 2). (2.6.21) This is regular at 0 = x, but not at 0 = 0. 0, 4) to cover the northern hemisphere (0 9, ', ) at the southern hemisphere (2 < 0 One can therefore use the coordinates _< ), and the coordinates . Because i is an angular b) 0 29 on the sphere and the curvature (or radius) of the sphere. Namely, in order for the field to be regular, single valued with time dependence eiwT  2~ integer, , the equality, (2.6.22) has to hold. It is interesting to notice that the form of our metric is the 3+1 dimen sional TaubNUT solution for a fixed radial coordinate [28 Because we can choose r * o00, we can think of our solution as the boundary of the TaubNUT solution. The topology of the boundary (and of any r = constant surface) is locally , but globally t is that of a deformed sphere in the following sense. Killing vector field where the defines is parametrized by 6 and 4 a nontrivial Hopf fibration: and the fibres are circles.  S2 Thus the topolgy is a "twisted product" xS2 . Thus the solution can be characterized with the Hopf invariant of the mapping from the compact three dimensional spacetime manifold to the two dimensional spatial part, and with the Euler characteristic of the latter. As in the positive If case the solution is not only conformally flat, but also can be brought into flat form, In the v = 1 with unconventional range of the coordinates. case the metric is given by = (dt + Kidx + 4dx'dxt (2.6.23) where Ki and 4 are given by Eqs. (2.6.3) and (2.6.1)(2.6.2), respectively. us introduce new coordinates, (2.6.24) = r2 +a2 and denote the corresponding angular coordinate X. The line element in terms of these coordinates is = (dt + 2od)2 (4 1 __ on (dp2 + P2dx (2.6.25) Let us make one more coordinate transformation (2.6.26) The metric then reduces to the familiar form 2= (dt + ) + (dp2 + p'2dx (2.6.27) where we have introduced a new periodic time coordinate, =*+ 20x (2.6.28) The metric given by Eq. (2.6.20) is flat, but again the range of the coordinates is unusual, t and t + integer x o are identified, and 0 p' < 1/a. Notice both cases, metric can transformed into Minkowski form, if we introduce a new periodic time coordinate. this feature, Because of we suspect that our theory is equivalent to a finite temperature one. We have discussed our stationary solutions in the axial symmetric case. We found that in the positive and negative / cases the solutions have different properties. In the Einstein case ( correspond to 2+1 Ei = B = 0) one has only the solutions that dimensional Godel universes, because /3 In the case of nonzero magnetic field (B , Ei = 0) however, one can have the solutions = dt'2 Ott (dp12 + T^ + p'2dx2) , 31 We have also observed in Sect. 2.4, that in the latter case our solution is similar to that for a rotating "ChernSimons" fluid. As we have shown, this solution is causal only if p the TaubNUT 3 KTm This means that the fluid analogy holds for case, and for the GSdel case with /  , but in the latter the pressure is negative. CHAPTER 3 RENORMALIZABILITY OF D = 3 TMG 3.1 Introduction By now it is wellknown that perturbative quantum gravity in four space time dimensions suffers from the problem of nonrenormalizability. This may be cured by going to lower dimensions, but in this case the theory is much less interesting, because gravity in D the absence of matter has no dynamical degrees of freedom. topologically massive gravity Recently Deser and Yang [9] have shown that three dimensions has the possibility being renormalizable. Because this theory is massive, it does possess dynamics even in three dimensions. Although such a three dimensional theory clearly does not describe the universe in which we live, it would be of great theoretical interest to find such a renormalizable theory with the symmetries of gravity. Deser and Yang have shown, by using an unusual parametrization of the metric, that TMG has power counting behavior consistent with renormaliz ability. This by itself does not establish the result, because one needs to show that the theory may be regulated in such a way to preserve both the theory's gauge invariance and the desirable power counting behavior. the newly discovered nonlocal regularization [11] to this theory. We will apply We will show that using this regulator, the possible obstacle to renormalizability discussed Deser and Yang does not appear, and that if this technique is valid, 33 technique depends on the existence of a functional integration measure which is invariant under the nonlocally generalized gauge symmetry, at this time been proven. 3.2 PowerCounting Renormalizability and Gravity which has not Most gravity theories are not powercounting renormalizable due in part to the presence of a coupling with negative mass dimension. To determine whether any theory of gravity has the hope of being renormalizable we look at the generic ultraviolet behavior of Lloop diagrams in d spacetime dimensions. First, we note that n all geometrical gravity theories, the propagator and vertex have reciprocal power behavior. For example, in Einstein theory the propagator A p 2 and the vertex V ~ p2 in any dimension. Higher derivative terms suc different as R2 and R3 behavior. can be added to the Einstein action with somewhat Adding an R2 term introduces p dependence into the propagator which improves the properties of the theory, however, such a theory is either not unitary or not causal or both. Adding higher powers of R does not affect the propagator but worsens the UV divergences because the vertices contain higher powers of moment. Assuming this generic reciprocal behavior , the divergence of a one loop npoint function is proportional to ddp(AV)" , Ad Because of the topo logical relation = N Nv +1, (3.2.1) where NI and NV are the number of internal lines and vertices, higher loops have one more power of propagator than vertex. Each respectively, loop also has I i i I i A I, 1 d number of counterterms we must have d r Because unitarity forbids r > 4 and there are no propagating degrees of freedom in a pure gravity theory in d= the only possibility is d = r = 3. We will see that TMG has this property. 3.3 Topologically Massive Gravity The action for TMG is given by SE + where the Einstein and Chern Simons terms are respectively d3 xv/ R , (3.3.1a) Scs = L  d3x ePI ,(P 8(, pF, ) t,Aqt1pv ir )r vp) (3.3.1b) The field equations are third order in derivatives of the metric, and they are given by  1 Cpv =0, (3.3.2) where G^P and CI"' are the Einstein and Cotton tensors, respectively. Eq (3.3.2) can be split into a trace R = 6A , (3.3.3) and a tracefree part, 1 R) (3.3.4) Just in case of Einstein gravity the solutions of Eqs (3.3.3) and (3.3.4) are spaces with constant curvature, that is, de Sitter (A > 0), antide Sitter < 0), or flat (A = 0). But unlike three dimensional Einstein gravity, TMG has a single dynamical mode, a graviton with mass m = UK. If, as usual, one expands the metric about the flat background. 2 g SE = K = (RI1 one finds that the h propagator has both p and p components and one cannot apply the simple power counting arguments discussed in the previous section. Instead , following Deser and Yang 1) we parametrize the metric ac cording to (3.3.6) gYu = + + hy1,) = 4 g$ y . where h satisfies h2 = 0!2) The action becomes d3 x x/ [8g9VQ9^ + 2R()) + ( (3.3.7) Since the ChernSimons action is conformally invariant, it is unaltered by this rescaling. Gaugefixing is performed by setting h"V , 1 hP" this neces states the introduction of a Lagrange multiplier B1, and ghosts b6, ci . The resulting gaugefixed action is SGF = S + S+ where B1 1, h V (3.3.8) is the gaugefixing term and SG = + D!"c  39' (qa^g D(c )] (3.3.9) is the ghost action. Here D is the covariant derivative with respect to g. total action is then invariant under the BRST transformation S = #(c" cAy)  c A,A (3.3.10a) Sht" +D" = CV,"I + cC," A + hV"c A h"Ac (3.3.10b)  3 3 + hYi)(c ,A + hafa,a) 2 la Dc Ir SF = 1'Scs(g) . ,M + h " =D^ c" + h c = c c1 Sba = (3.3.10c) (3.3.10d) Bo = 0 . (3.3.10e) The propagators are 1 4" < h Vhaf 4pT(PeYa pZV + evf pm) , (3.3.11a) (3.3.11b) 2 1 p (?77pv P 2pppv) (3.3.11c) where = ?r7v pjppV Thus with parametrization (3.3.6), propagator has the desired p 3 behavior. The vertices may be found by expanding the action desired order Appendix they clude terms of the form 4) n > fields ,3(Viph)n+l and bO2(c/ihn1) where 2. Note that the vertices from the Einstein term contain exactly two 4 . while the other vertices contain none. We now notice that negative powers of m (equivalently, positive powers of n) may never appear in any Feynman diagram, since they do not appear in any vertices or propagators, hence K is a superrenormalizable coupling. expansion parameter p i The true s dimensionless, providing the first indication that the theory may b e renormalizable. To prove the naive powercounting argument we determine the highest degree of divergence D of any Lloop diagram in the theory. Let Nf and Nt, be the number of internal lines of species x and the number of vertices of degree 9Y , respectively. The degree of divergence is then + vcY pIPfl + e37 Pva >=z > = z BQ , < $ < bpc, 8 (W)n1 Using (3.2.1) in (3.3.12) one obtains (3.3.13) where Nuh V/ and N? are the number of ghost and vertices respectively. Since any vertex has at most two ghost or two $ fields, the terms in parentheses are nonnegative and thus the degree of UV divergence is always These diver gences can be absorbed into the coefficients of the Einstein term of dimension 1, the ChernSimons term of dimension 0 and possibly a cosmological term of dimension thus the theory is powercounting renormalizable. The loophole in the above argument is the assumption that gauge invari ance may be maintained in a regulated version of the theory without giving up the desirable power counting behavior mentioned above. Without the use of such a gauge invariant regulator, there is the possibility that additional terms might be required to cancel the gauge transformation of the effective action, and that these terms might contain negative powers of m. To see if quantum corrections to the theory violate the BRST invariance through such a term, we look for nontrivial solutions to the BRST cohomology problem as follows. Let A = QF be the possible violation of BRST symmetry, where Q is the BRST transformation and P is the effective action to some loop order. We consider general solutions to the cohomology problem QA=0, =0 (3.3.14) If the solution is trivial A= Qr' we can add Fi as a counterterm to P to cancel the anomaly. If the solution is nontrivial and is indeed generated D = 3 (Ngh  (N$ ) . I I~ determine, through such an analysis, that there is one such possibility: term ,a=  arises from + higher order terms, the BRST transformation of the effective action, terterm necessary to cancel this term (3.3.15) then the coun will add negative powers of the mass and hence ruin renormalizability. Deser and Yang showed that to one loop in dimensional regularization this term does not appear. Unfortunately, since the / function for p vanishes to one loop, one cannot apply the AdlerBardeen theorems [29,30,31] to conclude that it cannot occur at higher loops. to determine whether or not this Thus, term arises, one must use a suitable gauge invariant regularization. 3.4 Nonlocal Regularization In this section we review the method of nonlocal regularization. Details may be found in ref. 11. Consider a generic action in d spacetime dimensions which can be written as a free part plus an interacting part: S[i] = ddx iFijfij + I[f] (3.4.1) where qij are fields of any type, and Fij course contains derivatives. define the nonlocal smearing operator Sexp I where A is the regularization parameter. (3.4.2) The local limit is obtained by taking + oo limit. Our convention is the derivatives in an $2 act on everything to the right, unless otherwise specified. For each field qSi, we introduce an auxiliary field 4'i of the same type, and construct the regulated action ddx (i ~j i~iz51 +IMt+ 1 (3.4.3) where 1) (3.4.4) It is to be understood that i are auxiliary fields which are to be eliminated using their equations of motion: =0 (3.4.5) Multiplying (3.4.5) by O we obtain the unique solution for 4 as a functional ti[]W = QOij (3.4.6) Equation (3.4.6) can be solved iteratively for t.i The solution for 4i has a convenient graphical expression: the unregulated theory, 0'i is given by with a factor of 2 evaluating tree amplitudes of  1 on each propagator (see ref. 11 for details). Substituting this solution into (3.4.3) gives the nonlocalized action for the 6 fields, S([] = S[,()]. (3.4.7) Suppose that S[] is invariant under any symmetry Ti[ef] (3.4.8) Let T consist of a linear part plus a nonlinear part, T , then S[j, ] as defined by (3.4.3) will be invariant under the new symmetry (3.4.9a) b6i = = Tt 84;= [, ] = / 3  bl[# + ^}] +rTn Tyl + E2T"i[ + 1], where Kl1 1 . 3 (3.4.9c) 62M~b In order to obtain (3.4.9b ) one must use the equation of motion (3.4.6) for the & field. Note that the nonlocalized symmetry transformations can be chosen such linear part is independent of the auxiliary fields. Generally, must be gaugefixed in order that we may solve for W []; then symmetry T represents the BRS symmetry of the gaugefixed theory. Classically, the nonlocal action Sf[] is equivalent to the original S[4] former being obtainable from the latter by some field redefinition. ence arises upon quantization. The differ The old functional measure does not exist in the new basis due to ultraviolet divergences. To quantize the theory a new measure must be constructed which is welldefined in the new basis, is analytic in the moment, and obeys the symmetries of the theory. The invariance of the quantum theory under the nonlocalized symmetry requires the invariance of the functional integral Di([Di] (o]Gauge fixing) exp (iS[]) Although (3.4.10) the full action including gaugefixing terms is invariant under the symmetry transformations, a measure factor [])= eiSM[] , must be introduced to insure invariance of the functional measure: ([D4o][w]) =0 . (3.4.11) (3.4.12) S[ ,] 41 The condition of Eq. (3.4.12) relates the variation of the measure factor to the Jacobian of the transformation via SSM [ 4 661 i s & m J (3.4.13) Tr{ tJ + w] Ojj Kim + 4 t IZJ 9kL3'L where the second equality uses (3.4.9) and the trace is over spacetime coor dinates. We can use (3.4.13) to solve for the measure factor order by order, resulting in a completely invariant theory. In practice, this is difficult to do for higher order terms, and it is hoped that further study of nonlocal theories will reveal easier ways of generating the measure factor. We must also note that it has not yet been proven that it is always possible to construct an appropriate measure factor to all orders. If such a measure factor does not exist for the theory then a local symmetry is potentially anomalous. For our arguments concerning TMG, we will be assuming that an appropriate measure does exist. 3.5 Nonlocal Feynman Rules We have described how to obtain the nonlocal action (3.4.7) by solving the auxiliary field equation of motion. However the Feynman rules for Green functions theory are inconvenient for calculations due to all interactions induced when the auxiliary field is eliminated. Instead we will work with the Feynman rules derived from the action S[#, 4 which are closer to those of the original theory and enforce the condition that satisfy its classical field equation by requiring that there are no closed loops consisting of only 4 lines (4 must be onshell in any diagram). Since one is interested in amplitudes involving the physical field no auxiliary fields appear as external 42 The general Feynman rules in the theory in terms of S[4, 4'] are as follows. The 6 and 4 propagators are exp ex(p (r A2 (3.5.1) io= _2 2 exp F (3.5.2) respectively. The field is indeed an auxiliary field which should not appear on any external legs, as its propagator, from Eq. (3.5.2), has no pole. The vertices are of the same form as in the local theory. The higher induced vertices in the S[#] theory are obtained graphically from the S[(4, ] theory: they are the connected tree diagrams which follow from using the local interaction vertices but with propagators replaced by iO (i lines). There are also vertices from the measure factor which will be connected only to 4 lines. reducibility of Feynman diagrams in the theory in terms of Questions such as [4i, f] are resolved as in the S[ ] theory with the additional requirement that 4 lines cannot be cut. Feynman rules for nonlocal TMG are collected in Appendix A. We have shown that there should exist an appropriate measure factor in order to have a welldefined anomaly free quantum theory. In most cases extremely difficult construct it, however it can be computed perturbatively to any order in the coupling of the theory. If we expand Eq. (3.4.13) we can obtain a set of Feynman rules for calculating the variation of the measure factor under the nonlocal symmetry. Writing K = Oik (Skj Okl51 / 62 1 = Oik(6kj + kl (3.5.3) + Okl Ji m 0(f~i0(f, 621 Omn C~rmTT7~T iS2 F+ie f00 =i = i 43 and inserting this in (3.4.13), one obtains SSM ] = Tr S% 6 r c []Sk 62I + Oki kSl 16m . .)**[*> (3.5.4) + Okl 6 np S pkm 0 bpm We may read diagrammatic rules from this expression by writing it in momentum space. Since there is only one trace over spacetime coordinates, we need only look at one loop diagrams. Each diagram has a single vertex factor coming from The remaining vertices are arbitrary in number and are the same as those discussed above. The first type of vertex always connects to an internal line with "propagator" The other vertices connect either to two internal lines with "propagator" 0 or to one internal line of each type. external legs correspond to either or fields. TMG are given in Appendix B. These diagrammatic rules for By computing all one loop npoint diagrams of this type one obtains a perturbative expression for SSM which must be inverted to get the measure factor SM. 3.6 Renormalizabilitv We now apply this method to TMG. We associate auxiliary fields P with the fields 4 , hlv, cP respectively. In this field basis each field is massless, so the smearing operator is simply = exp (2 /A2) (3.6.1) The gauge transformation laws for the fields then become nonlocalized accord ing to (3.4.9a ): I/1 >T., [ \ IL f IL\PL'I.( ,\ kty, EC <2 bhP" = cV, + CP , ,a + 2 (h+k)(c (3.6.2b) iP(h + k) p(c + d)Q, (h + k)"V(c + d) , (h + k)0"(h + k)ap(c + d)aP] = E2(c + d),(c (3.6.2c) Ba , (3.6.2d) 6Ba= 0 . (3.6.2e) We see that nonlocal regularization gives a regulated theory which is au tomatically BRS invariant, but it must be checked that the desirable power counting behavior of the unregulated theory still persists. Negative powers of m could be generated either in the measure factor, or by the loop integration themselves. We will examine each of these possibilities. We now examine (3.4.13) to determine which terms could possibly a contribution of the form (3.3.15). Tr cannot contribute to this term. This is because and b, c each couple only to h, so every tree graph which con tributes to 4' or dp must include at least one h. By the same token, each term in k" includes either an h, a pair of ghosts, or a pair of b's with derivatives on them, none of which is what we are looking for. So the 4 term does not contribute to the possible anomaly. for the same reasons. c ghost term also does not contribute, Since b and B do not contribute to the measure factor at all, the only possible contribution is from Tr We find from (3.6.1b), that this contribution comes from the diagram shown in Figure 1. Since its 6b,= +d)" rnUca  (c + d) (h + k)r" + (h + k)" (c + d)" + d)g , contracted into after the internal momentum has been integrated over. is necessarily zero by Lorentz invariance. This Therefore there are no contributions to the measure factor of the form (3.3.15). C a11111 Contribution to the measure factor of the form of Eq. (3.3.15). Here a wavy line with a bar corresponds to the k prop agator where k is the auxiliary field field for h. A wavy line with a dot corresponds to a "propagator" given by the smearing operator 2 as described in Sect. 3.5. the remaining we show that such powers do not arise from the loop integration themselves. This situation would correspond to divergences in the limit where m *0, A remains finite. Since nonlocal regularization regulates all int egrals at p * , unregulated divergences can only occur for p + 0, it i.e. in the infrared, which is not at all affected by this procedure. contrast to ultraviolet divergences, which are determined by the net effect of all the propagators around an entire loop, infrared singularities are determined by a single propagator, or any group of propagators, whose mo mentum goes to zero. If we let all the loop moment b e independent and keep r nm avr C art tn n r, nri r' n' A1 4. n C r,^ nn ,l, nfl .i a tar4: nnn an ii an ..rn nr1a 4 Vt r 4. * Figure 1. ghost propagators actually help matters because they go like . P The only possible divergences of this type will be when one or more of the momentum conserving delta functions give 6 3(0). going into the vertex go to zero. He This can happen if all of the moment iwever, each vertex contains derivatives, which will in momentum space give powers of the moment which will neces sarily soften these singularities. Specifically, each vertex either contains three derivatives, which softens the singularity to at most logarithmic, or two deriva tives plus 4 or ghost propagators or positive powers of m, or both. there is no possibility of a power law singularity for m  0. In any case In higher oops, there is the possibility that a 6(0) singularity might generated not by one vertex but by a combination of two or more. This can happen when some subgraph is imbedded inside another graph. For m  0 this graph may be singular when the momentum on the line connecting the subgraph to the other graph is zero. If the momentum factors associated with the vertices lie on the subgraph then they will not cancel the singularity. In this case we must argue by induction against the possibility of any problem being caused. Suppose that the theory is proved to be renormalizable to N 1 loops, and that this renormalization has been carried out. diagram of the type described above. Then consider an N loop The subgraph of this graph has fewer than N loops, so by assumption the theory must at this point contain counterterms to make this subgraph finite. But by simple dimensional analysis, the subgraph must have dimension 3 n/ where n is the number of 4 or ghost (not h) lines coming out of the subgraph. Since the subgraph plus counterterms must be finite, the graph cannot achieve this dimension by being proportional to the lPI .. 1 1 1 ., * nn / I r^ i . A the singularity from the delta functions to be at most logarithmic. Note that logarithmic infrared singularities are not a problem as they merely indicate the presence of In A/mn terms. We therefore see that infrared powerlaw divergences do not arise as m * 0, so that negative powers of m do not arise from loop integration. Thus, assuming that a gauge invariant measure exists for the nonlocal theory, theory is indeed anomaly free and hence renormalizable. Our result will still hold if this assumption is false, if the violation of gauge invariance is such that the noninvariant terms in the effective action vanish to all loop orders in the local limit, A > 00. Since TMG has no actual gauge anomalies, this is a reasonable assumption, but is by no means a foregone conclusion. Thus at present we have discussed but one approach which gives strong support to the conjecture of Deser and Yang. Our result cannot be considered a proof until the existence of the appropriate measure factor is established. CHAPTER 4 THERMOFIELD DYNAMICS OF BLACK HOLES 4.1 Introduction It is known that the quantization of fields on spacetimes with causally dis connected regions leads to particle creation from the vacuum as a consequence of the information loss associated with the presence of the event horizon(s) In stationary as a stationary black hole, spacetimes with a simply connected event horizon (such an accelerated observer in Minkowski spacetime, or de Sitter type cosmologies), the emitted particles have a thermal spectrum This result has been first obtained by Hawking (14] for black holes. has shown, Hawking temperature, the effective temperature of this radiation is I= _ K where K is the surface gravity of the black hole (units are chosen throughout such that k =h= c=G=1 These results have been confirmed and derived in a number of ways, and several attempts have been made to gain a better understanding of the features of the Hawking process and the physical role of the event horizon [14,15,16 A particul approach into this direction is the one used by Israel [16], who problem of particle creation on the Schwarzschild background. early interesting considered the The idea is to quantize the fields in the full analytically extended Schwarzschild spacetime (known as the Kruskal extension) in order to keep track of particle states on the hidden side of the horizon as well. The same idea allows us to apply a 49 Here we briefly review the canonical formulation of thermofield dynamics for free fields. This will provide the main ideas and all the technical tools we need (for more general and detailed discussion see e.g., ref. 17). The central idea of thermofield dynamics is to express the statistical average of any operator 0 as a single vacuum expectation value (o) = (o() 0(/)> (4.1.1) where p is the inverse temperature. This can be achieved by augmenting the physical Fock space F by a fictitious, dual Fock space F That is, for each op erator O(a ,,aj ) and each state vector I 1 anflj , vnj we introduce a dual operator O(a tw ) and a dual state vector n>2) W= FWI = v",. i ao where a . ' ., a and a j U)] LWIJ are creation and annihilation operators of the wj modes (j labels the degeneracy of the energy level w) with the usual com mutation (anticommutation) relations. Namely, for bosons the only nonzero commutators, and for fermions the only nonzero anticommutators are a j,aj,] ]= [a j,aW,,j] = 6jj,8(w w (4.1.2) {a~wj, ,aj,} = {ij, wj } = 6j'(w w (4.1.3) respectively. The states I0), and 0) are the vacuum states annihilated by awj and a.j respectively. the direct product Fock space , spanned the state vectors n,m) =I n the temperature dependent vacuum state 0(p)) in (4.1.1) is given by a Bogoliubov transformation of 10,0) .. 4 I I  n = j ), with the 0, parameters defined by sinh2 8 = sin2 = = Ce 1 1 for bosons, (4.1.5) for fermions. We also introduce the operators ) e iG t such they satisfy same commutation (anticommutation) relations (4.1.2) and (4.1.3). The state 0(3)) is annihilated by the operators amj(i) and a and the entire Fock space can be constructed successively from 0(3)) using the creation operators a4j(j) and a4 Using the above construction of 0(3)) the statistical average of any phys ical operator (a functional of a4j and atj only) can be expressed */ *i expectation value of the form (4.1.1). as a vacuum In particular, as it is easily seen from Eqs. (4.1.4) and (4.1.5), the average number of the wj modes are given by the familiar Fermi and Bose distributions. The formalism described above can easily be generalized to black holes by identifying the physical Fock sp zon, and the tilde space ace with particle states outside with particle states inside the horizon. hori The above decomposition of the whole Fock space into the direct product space O E corresponds to the conventional definition of positive frequency modes (parti cle states). It is known [13,14,15,16] that this definition with respect to the Schwarzschild time coordinate t (which is defined by a timelike Killing vector field 8/Ot everywhere outside the horizon) leads to a mode expansion of the aj(p) = e a eiG wi at i "'*( 0 = e j(). 51 mode expansion. Note that the latter corresponds to positive frequency modes with respect to Kruskal time coordinate, which defines a Killing vector field only on the horizon. This linear combination is generated by a Bogoli ubov transformation, which is uniquely determined by the requirement that the fields be analytic on the horizon [15,16 The formalism described above can be applied to the problem of particle creation by black holes, and as we shall see, far from the black hole it leads to a thermal distribution with the Hawking temperature. We begin by reviewing of Israel's paper [16 The quantization of a massless scalar field on the Schwarzschild background is considered by using thermo field dynamics. We conclude that the results obtained this way are equivalent to the earlier ones. In the rest of the paper we describe the possible generaliza tions of this approach. In Sect. 4.3 an approximate multiblack hole solution is considered as an example to demonstrate how to extend the method to space times with many causally disconnected regions. Finally in Sect. 4.4 we derive the Hawking radiation of a black hole emitting neutrinos and antineutrinos. 4.2 Massless Scalar Particles on the Schwarzschild Background We first consider the creation of massless scalar particles in the gravita tional field of a black hole. We consider the Schwarzschild metric = (1 2M)dt2 V  (1 2M) dr2 r2 (i r~  r2 d2 (4.2.1) as an example, and look for solutions of the massless scalar field equation, 1( J( ',CL g (g) 2),y = 0 , (4.2.2) 1 .1 P 1 r r 1 Psi r* fwl obeys the equation: d ~dr*T dr* d dr* (4.2.4) S2M ( + 1) (r) r r2 / r*=r + 2MlnI2 1 p2 (4.2.5) Let us consider the solutions of Eq. (4.2.4) which correspond to outgoing modes at the past horizon I" Near the horizon these solutions behave like fwlm e ZWtU (4.2.6) where w > 0 and u = t r* This is the usual definition of positive frequency states with respect to the Schwarzschild time. Everywhere outside the horizon, 4 is a timelike Killing vector. It is also possible to define positive frequency modes with respect to the Kruskal time coordinate U. In Kruskal coordinates [15], = 4Me U 4M = 4Me4M (4.2.7) S= t r* the Schwarzschild metric has the form = 2Mr r 2M dUdV  r2 (4.2.8) On the past horizon TC , 9/OU is a null Killing vector. Note that fwim defined by Eq. (4.2.6), fwlm e zwun iw lnlU (4.2.9) is also a complete set of positive frequency solutions (w the KrniTkal time T_ > 0) with respect to Here v=t +r* Since 71 divides spacetime into two causally disconnected regions, the one outside the horizon (region I) and the other inside the horizon (region II), two "Schwarzschild" modes can be associated with any given solution fwimn: 1 Ym9 2i riwI  "lnjU A*, 'r outside the horizon inside the horizon, (4.2.10a) F O =< 7^^~ outside the horizon ,p)e+ln U inside the horizon. (4.2.10b) Note however that are singular because U goes to zero on the future horizon. On the other hand, the following linear combinations: Ht/+ = F+)cosh, + F( sinh0u , (4.2.11a)  F(+) sinh 8 are analytic in the lower half complex U cosh 0, plane if tanh 6, = e (4.2.11b) The modes are positive (negative) frequency "Kruskal" modes. Both F ) Wj and H( Wi are complete sets of modes, satisfying the orthonormality conditions (F F))=Sjji6(ww ') (4.2.12a) (H'I ) = 6jj,'S(w w (4.2.12b) with respect to the KleinGordon scalar product, ( F,2)= i (FrO* F2 F2&pFf) ndZ , (4.2.13) where nP is a future directed unit vector orthogonal to the Cauchy surface and dE is the volume element in E. Ti rinn+;'7n rtb0 1 crnClnr fn1A di ; +ormc F f /!/ nirP nYvnn~n , p)e FW+) F + F, ^n\ where a , Uwj and a() are creation and annihilation operators, respectively, obeying the usual commutation relations, that is the only nonzero commutators are ,a, =t 6j6(w w'). (4.2.15) The alternative expansion of (c(x) in terms of H ) wj (+t,)(t)W a +a t(K)HI +h.c.) (4.2.16) The operators a(:t (,) a .(c) WI also satisfy the commutation relations .15), and are given by the Bogoliubov transformation: a(() = exp(iG)a exp(iG) (4.2.17) = cosh9 a()  sinh9 a where the hermitian operator G is defined by i (+)t ()t ^^J Wj The physical vacuum state near the black hole, the a ( a ~wI w (4.2.18) "Kruskal" vacuum, is de termined by requiring that freely falling observers encounter no singularities as they pass through the horizon. annihilated by the operators a), wn 0) denotes the "Schwarzschild" vacuum then the "Kruskal" vacuum: 0(K)) = exp(iG) 10) (4.2.19) is annihilated by the operators a(t) (n). Far from the black hole, at infinity, the observable quantities are the vac uum expectation values of the operators of the form O(a4 the Kruskal vacuum. In particular the average number of a i) calculated in wj modes at infinity, given a thermal distribution with temperature equal to the Hawking temperature TH. 4.3 Many Black Holes The method can be extended to spacetimes with more than two causally disconnected regions. Because such a solution is not known we consider the idealized case of N wellseparated black holes as an example. Our assumptions are as follows: (1) they can be considered static; (2) far from the black holes the metric is approximately Minkowski; (3) the radiation emitted by the individual black holes is uncorrelated. scalar field equation By approximation (1) and (2) the solution of the . eiw(tr) and near the horizon of the i'th black hole the metric is approximately Schwarzschild. Thus, fwim, ~ e i where i=1,2...N and ti are the surface gravities of the horizons. With these assumptions a linear combination of the "Schwarzschild" modes can be found which is analytic everywhere. The creation, annihilation opera tors and the corresponding vacuum state are given by a Boguliubov transfor mation. The new vacuum state will not appear to be empty for a stationary observer far from the black holes. The spectrum of the emitted particles (in this idealized case) will be the sum of the individual black hole's thermal spectra. We first consider the N=2 case. The solution of the field equation is such fwim e C :iw(tr) T hiU^ lt/'* far from the black holes (4.3.1) at the horizon of the i'th black hole. In order to define a complete set of modes we divide the space into two cells, 1 l l .. 1 1 .1 1 1 1_ 1 mi_.~ _  i  A fwmn are such that F /) zw3 are nonzero in the i'th cell only, and they are given by 0 X EIW7'm( \y Z7;i 1 r m(9  tInlUil K . inside i'th horizon (4.3.2) elsewhere, outside i' th horizon (4.3.3) elsewhere, with the normalization, F!) (F.w vtwa ,F!) = 6i,6j6(w w'). (4.3.4) Analogous to the one black hole case we consider another set of modes such that they are given by a linear combination of the above defined modes and are analytic everywhere. "Schwarzschild" These are given by H(lw = F) cosh O1, cos F(+) cosh 02w sin ,j (4.3.5a) FI sinh01, cos H(+) (+) cosh 02w sin 2wj 2wj sb  F(! sinh 02 sin 2w3 + Fl) cosh 01, + lw~j cos C, + F(! sinh 02w cos ,w + Fl( cosh 02w sin w H({ = F(I) sinh 01 cos q, F) sinh 02 sin ~ w 2w wj  cosh 901 cos .1 H( = Fj) sinh 02w + F2] cos cos w (4.3.5b) (4.3.5c) F() cosh 02w sin w , 2wj + F(+) sinh 1,w sin ,w (4.3.5d) ;h 01, cos 4w + F{W) cosh 01w sin , Note first the solutions are matched on the horizons of the individual black holes independently by choosing the 0, parameters such that HX) !(region I of the i'th black hole) /,_t 1 4 9 (+) F. w SW} , )e )  .A 0, t'\ 57 which is exactly the analyticity condition for the modes in the case of one black hole. The remaining freedom is used to match the solution between the black holes, that is on the "wall" separating the two cells. To do this we set the d, parameters to be tan 4w = cosh 81w cosh 22w 1 exp( 1 exp( _27wU _2xw K1 (4.3.7) ) 1/2 H +) are also normalized according to z 3 (H H ) It=66= w To quantize we expand in terms of F ) Tow (4.3.8) (+) (+) ( ()t F.iwj wa t +f Fa j + h.c.) (4.3.9) w,WJ where a!+ satisfy the commutation relations twj ,a i, ] = ,i'6jfj'6(w ) ', the other commutators being zero. (4.3.10) The vacuum state is defined by S ) =0o. (4.3.11) But we can also expand in terms of Hif: (H+a) (+ ) +H)a(n ( 1iwj tuij + JiwH zj uj ) + h.c) (4.3.12) where a+ () also satisfy the commutation relations, ZWJ ! )' ) !+*,)t .()] = tii'6 Sjj16(w w') , a ,a (4.3.13) 1 I 1 1  ni i 1 1 1I 58 The operators ao (ti,) are given by the Bogoliubov transformation .(+) a(+) (+) cosh 0 sin g lwj( ) = alwj Cosh 1o COSqw + a2w cosh2 sin (4.3.15a)  a() sinh 1i, sin a sinh 2, sin a2j.( )= a2j cosh2w sin > = (+) cosh 01, "t alw cos w + a2. sinh 02 sin w a(lwj cosh 92w sin a(+). sinh 01, cos 4, a2 t sinh 02w sin awjs1`82 sn 3 (4.3.15b) (4.3.15c) + ac lwj cosh (i^ cos 4' +  a(3 sinh 02w a2wj cos d, (+) 2wj cosh01io . cosh 02w sin + awj sinh w1, sin f,0 cos {w) a () cosh 61w sin If we introduce the hermitian matrices ~I 1,2 2=l,2 Sa 3 Wj ) (4.3.16) 0 (taw+ia t [^I^W zo3 jo 0= exp(i&, cos p  sin C, COS Aw cos 'N  sin d.,, cos ~ (4.3.17) where E2=i O 20 and 72 oa2 1 ,then Eqs. (4.3.15a)  (4.3.15d) can be written in a compact form: (+) ( j ) a2w <'(li) to)f 2L( =O1 G1 =_ GI 1 (+) alwj 1 (+) a2wj a 3) (4.3.18) I * r . (4.3.15d) aI ( a2w3 ( 59 one black hole only, such that the corresponding expansion functions Hiwj are analytic on the horizon of that black hole. The matrix 0 is a two dimensional rotation between the operators near different black holes. parameter, It depends on one to be chosen such that the modes be continuous in the region between the two black holes. The vacuum states are given by 0) = 0 (4.3.19) 0())= 0 , (4.3.20) where 0(K)) = 01G1 The expectation value of the number of wj modes is , (o ) (+)t (+) , ) (() a aw, j (+) t (+) + a2wj a2wi O(K)) (4.3.21) = sinh2 G1 + sinh2 02 = 2jrw K1 _ 2xrw K2 Note that our result does not depend on t, only on the Oi, other words parameters. , it depends only on how we match the solutions on the horizon. Generalization to arbitrary N is straightforward. N cells, each of them containing only one black hole. We divide the space into We define normal modes which are nonvanishing in one cell only, and are given Eqs. (4.3.2) and (4.3.3) except now i goes from to N. A linear combination of these modes can be found that is analytic everywhere, by first matching the solutions on the horizons of the individual black holes, then in the region between the black holes, that is on the "walls" of the cells. Again, the corresponding Bogoliubov za3 S() the components of a 2N component vector. G is a product of N transforma tions of the form given in Eq. (4.3.19) with i=l...N. The N parameters Q0 are chosen such that the new modes Hiwj analytic on the horizon. Now O is an N dimensional rotation, mixing the particle states near the horizons of different black holes. The N(N  1)/2 parameters (4, in the N=2 case) are to be chosen such that the solution is continuous in the region between the black holes. expectation value of the wj modes is again unaffected by these rotations, and, similar to Eq. (4.3.21), we obtain )  sinh2 i = (4.3.22) S2rw ,i The spectrum of the created particles is the sum of the thermal spectra of the individual black holes. 4.4 Neutrinos on the Schwarzschild Background Now we examine massless spin one half fermions, that is neutrinos. the scalar case, As in we have to start with finding the normal mode expansion of the Dirac equation near the horizon. We will use the vierbein formalism. shall see that, with a suitable choice of the vierbein fields, the Dirac equation is separable [32] The metric tensor in this formalism is related to the flat metric through vierbein which satisfies orthonormality, Vp V, = 6pa  CgN and completeness, VP(x)V f(x)1, a gpv conditions. In particular, Schwarzschild case the latter is g9, = V a(x) V1 19( = f1 2M 1r r sin 9 (1 2 1 (4.4.1) where and the indices a, /3 mean local frame indices and /i, v are spacetime indices. We can choose the vierbein such that its nonzero components are 2M )1/2 r =(1 2M 1/2 r (4.4.2) = r sin O9 rsin 0 2M 1/2 F The massless Dirac equation in curved spacetime becomes (4.4.3) where tb is a Dirac spinor field with (1  75 The gamma matrices are given by =V a (4.4.4) and they are the curved space counterparts of the usual flat space Dirac ma trices, a(l1 Y They clearly satisfy the anticommutation relations: V} = 2g= l (4.4.5) which are the curved space generalization of the flat space anticommutation relations, (4.4.6) ,7 = 217 The spin connection, FT. is given by rF(x ) AKh ,a]Vva(x (4.4.7) We are looking for solutions of the form \,  x  r T+) = 0o, S = (1 =(1 V29 :M_1/2 (1 ) = 0. VV/(x I 62 This leads to the following coupled first order differential equations for R1, R2, S1, and 2M 1O r i zwr R1 = kR2 , 1 2M 1   1 2M 1 BrR2 zUr 1 2M 1 r r O9 Sl + sin s2 m sin 9 R2 = kRi , S1 = kS2 , S2 = kS1 (4.4.9) where the separation constant k is to be chosen such that Si(09) and 52(0) are regular at i9 = 0 and 9 = Ir. We are interested in the solution of the radial equations at the past horizon. We find that R1 ~" exp( R2 ~ exp( near the horizon. (4.4.10) iwu izwv (4.4.11) Thus, for neutrinos, the solution which corresponds to out going waves at the past horizon is I ~e V, t1m7 (1> iwu S1(9) o)0 (4.4.12) For antineutrinos the solution of Dirac equation is given charge conjugation (C = i,2 0 2202 77* '* 1 "'7 * 92 r/2 '4f (4.4.13) where tr4 ~ exp(iwu) represents outgoing negative frequency waves at the past horizon. Hence the solution corresponding to outgoing antineutrinos at the past horizon is given by n\ T=C Iltu = ~ 63 To quantize we expand the field 4 + h.c.) (4.4.15) where at and b, represent creation operators for neutrinos and antineutrinos respectively, while aw and bt are the corresponding annihilation operators b 10)=0, satisfying the following anticommutation relations: at,} = {b, (the other anticomnmutators are zero). ,} = 6(w w') The spinors un and (4.4.16) form a complete orthonormal set, ue) = (, ,) = (v, ,v ) = ( , ) = S( w') (4.4.17) (others are zero), and near the horizon they are given by UwC= e zwu e imtp S1(0i) 1 O 1 0 ~ e  IlnlUII K 1 0 1 0 (4.4.18) 5,= eiwu Vwd  Sr(o) ln Uj 0  1 0 (4.4.19) us consider only neutrinos and define a complete set of positive fre quency modes on the whole extended Schwarzschild spacetime: _ lnlU mx / 1 0 1 0 outside the horizon (4.4.20) inside the horizon, (1 eimp (a, u + bLvi The field 4 can be expanded in terms of complete set of modes F) wj S(F(+ a(+) FaI )a()t WI + h.c.) a(+)(t) + H()a(.)t(a) + h.c. Uw 3 j (4.4.22) where the modes H(I U)1 are defined by H(+ w3 w3(~ = F() cosw 0 F( = F+) sin + F (4.4.23) (4.4.24) Both F( w3 satisfy the orthonormality conditions (F Cd) F(j). (()3 Cdj) ~ WJ ) = 6jjf6, (w w) . (4.4.25) As in the bosonic case H are positive (negative) frequency modes, which are analytic on I if tan 9, The corresponding creation and annihilation operators a ( ) () are given by the Bogoliubov transformations: w3r o '(+) = a .( )= a t cos, + a) sin 0 a I  exp( G)a' exp(G) , (4.4.26) ( (+) sin + ()t cos a w())= a sm +a cOS = exp G)a j exp(G) , (4.4.27) where =E S,(+)ta ()t ^^j aw  a(+) a(). (4.4.28) The vacuum annihilated by a (t) is given by "J I0( K( = exn(G10\ (4.4.29) or H +F^ * F(? = ( Ssin 0 , cos 6, 65 The number of the wj modes detected by an observer at infinity is given as a vacuum exceptation value (n"j)u = (O(la) a.)a 10())> = sin2 0a = For antineutrinos we have the same result, (4.4.30) ,j) = sin 2 = 1)'/7 sin2 O^  (4.4.31) This is, as expected, a thermal distribution of fermions with effective temper ature equal to the Hawking temperature. The result obtained above can easily be extended to the case of many black holes. One should follow essentially the same steps as we have in Sect 4.3 for the bosonic case, and find similar results. In particular, one finds that spectrum of the created fermions is the superposition of the thermal spectra of fermions created independently by the individual black holes. Summarizing our results, we have found that, in agreement with previous calculations, the thermofield approach lead to particle creation in a spacetime with causally disconnected regions. the case of a single black hole spectrum of the emitted particles is thermal with effective temperature equal to the Hawking temperature. the case of wellseparated black holes the spectrum is the superposition of individual black hole spectra. Furthermore our approach suggests that these results mainly depend on the analytic behavior of the fields on the horizon, but not on the statistics of the particles. This been demonstrated by quantizing both a massless boson and a fermion field. Consequently, we hope that similar studies will lead to a better understanding CHAPTER 5 CONCLUSIONS We have investigated two extensions of Einstein gravity in 2+1 dimensions, Weyl gravity and topologically massive gravity. We have also considerated the applications of thermofield dynamics to particle creation by black holes. In the case of Weyl gravity we considered the consequences of duality in the context of Weyl theory in three dimensions. We constructed a theory of gravity with Weyl invariance and a noncanonical scalar auxiliary field, as a lab oratory to study duality between the gauge field and its field strength. There it appears as an equation of motion. We have studied the classical solutions, and found that they can be classified by the nonvanishing components of the field strength. There are stationary solutions only if the electric field is van fishing. If the magnetic field is vanishing as well, ., in the pure gauge case, our theory reduces to Einstein gravity in flat or de Sitter space. solution was found. The general In the case when only the magnetic field is nonvanishing, the problem reduces to the solution of a Liouville equation. We studied the axial symmetric solutions in more detail. the helicalconical structure, they are 2+1 We found characteristic to dimensional analogs of the known 3+1 that the solutions have dimensional gravity, dimensional GSdel and TaubNUT type solutions. Interestingly, the "matter part" was described by a rotating ChernSimons fluid with intriguing properties. Consequently this work mirht, have interest.inip annlicationns in fluid mechanics. Next we studied the renormalizability of TMG by using nonlocal regulariza tion. We found that the theory is renormalizable under a certain assumption, namely when the nonlocal measure factor exists. Although we cannot give a general proof of its existence, it can be constructed perturbatively, and its existence and gauge invariance can be checked to any order. We showed that a possible anomaly which could spoil its power counting renormalizability does not occur. If our assumption is valid, topologically massive gravity is the only known example of a renormalizable and dynamical theory of gravity. Finally, we have used thermofield dynamics to study particle creation in causally disconnected spacetimes. We have chosen 3+1 dimensional black hole spacetimes, because these are the best known examples with the above prop erty. We have found that our results are consistent with those obtained by different methods. In particular the thermal character of the vacuum has been derived for the emission of massless scalar particles and for the emission of neutrinos. We also discussed how to generalize the method to space times with many disconnected regions. For definiteness we considered the example of many wellseparated black holes, and found that the spectrum is the super position of the individual black hole thermal spectra. Approximations were necessary in order to obtain the multiblack hole metric, but only because we do not know any exact solutions of the Einstein equation with the above prop erty, not because of the failure of our formalism in a more accurate case. the same time our example clearly shows the basic ideas. This method not only provides a new technical tool to discuss the quanti zation in such spacetimes, but it also helps us to understand the features of the 68 we have found that the analytic behavior of fields on event horizons is crucial to the derivation of the spectrum of the created particles. depend on the statistics of the particles. The effect does not This has been demonstrated explicitly by quantizing a massless scalar field and a neutrino field on the Schwarzschild background using thermofield dynamics. In both cases the spectrum of the produced particles is thermal with effective temperature equal to the Hawking temperature. As in earlier works, we also have found that the particle creation process is due to the presence of the event horizon. To see this one should note that the physical observables are the (temperature dependent) vacuum expectation values, they contain information only about particle states outside the horizon(s) (only these states have nonzero contribution). But we have learned more than that by realizing that the spectrum of the radiation depends on the properties of the event horizon, namely on the number of disconnected pieces, and on the behavior of the fields near the horizon. APPENDIX A FEYNMAN RULES FOR TMG The Feynman rules for TMG follow from the action described in Eqs. (3.3.1), (3.3.7), (3.3.8) and (3.3.9) in the usual way. The corresponding rules of the nonlocalized theory can be obtained by applying the general rules of Sect. 3.5 to TMG. Associated with each field of the local theory is an auxiliary field. in the nonlocal theory auxiliary fields 4, Thus kny and dP are associated with the original fields 4, hpy and c . The ghost field b1 does not require an auxiliary field because its BRS variation is linear. In the nonlocalized theory the form of the vertices are unchanged except that now the lines represent both phys ical and auxiliary lines. The propagators of the local theory are replaced by smeared ones in the nonlocal theory. The original propagators are multiplied by 2 defined in (3.6.1), while the auxiliary field propagators are 1 E multi plied by the original ones. In constructing Feynman diagrams in the nonlocal theory, one does not include loops involving only auxiliary fields or diagrams with auxiliary fields on external lines. Details can be found ects. 3.4 and Below we list the Feynman rules for theory are described above. TMG where the rules for the nonlocal An auxiliary field line is represented by putting a bar on the corresponding physical field line. First we list the propagators. The $ propagator is C The h propagator is fiv ap ,, t~txx~(/xlx + TE lPpva + eLfr P/Ia) Finally, the ghost propagator is It S P( p2 Pv) where rply = diag(1, 1, 1), Epya is the 2+1 dimensional LeviCiviti tensor and the projection operator Pr" is = p2 (A.1 Next we give the vertices. Since the vertices involve all orders in the h field we shall only give vertices at most cubic 4 point vertices. in the h field and no higher than The vertices arising from the Einstein and kinetic terms of (3.3.7) are quadratic in the ) field and contain two derivatives. The lowest order vertices are ~kV k i/Cl{ap {tp (8k e+ 3p +q2))  16(k3fl + kpc v) 2pvqp} 1p4r(ap" 4p4 + va p1l3 16i ppqv , Sq + 2(p2 71 The vertices from the ChernSimons term contain only h fields and three deriva tives. 'I q The lowest order vertices coming from this term are  : X/7jE7r( pyckKqp?7frpaa?77Tv + PpkpkK1 va]7 + pvkaktrlprlvprl 7*O~ Ta TO' pv k^ kp rtpaQ P 7a7ra P kk Tkra'nprlglrra + gp ka qprp Ea rirv * qkpriaprt 7KUrTVr) + permutations, (YaKrP(fk * qp, + k Sqp) qp(k + aq9(k 1( Kp  k1Y]p)) A7v ( 2k7 (7ap rloagKQep  rlpprcnqag/p) 1p.( taK q I q7 Er) KZICo  + 4tina(r, QK e  + 3y p~'ilva(o 'l 191 plra K 'J~v} + permutations. The vertices arising from the ghost part of the action in (3.3.9) are  AP1 2qan p  iep) * ^  e, r p(Qa qarLu ns(,a(17pWW^ .7,5, a6 kQ rla r1p 77pTa i (ppkvqrlap + pflk r1va  ppqlrlav q a\ Ji^(puklaAT?, j+ PkpTJalAVI) Here it is understood that one must symmetrize each pair of indices on an external h line and take permutations of the various identical h fields arising from a vertex as indicated in the figures. 2 2pkp va 4pai, kv) APPENDIX B MEASURE FACTOR FEYNMAN RULES FOR TMG The Feynman rules for the nonlocal BRST variation of the measure were discussed in Sect. 3.5 and were derived from Eq. (3.5.4). There are two types of vertices, those from the nonlocal theory as described in Appendix A the other type connects to internal lines, one of which has Such internal lines are represented with a dot on their legs. "propagator" All the diagrams contributing to the variation of the measure factor are one loop diagrams and contain only one of second type of vertex and an arbitrary number of vertices from the original theory. cT in the vertices that follow. T All dashed lines correspond to a ghost field hose diagrams coming from the contribution of the variation of the 4 field are 1  Qv + pv 0  'P ' p 1 3 pv $qv 74  ,V/pq + mp qi) , sp vJ4('7p/pPv + ?lvppj'})  n (p+ 0vpp) vertices arising from the contribution of the variation of h are P (i 1 p + Vrrp 771Qflp  Urp7(pp + P X rl^r^p?) 75 j2 (api + eup7C.3 e .4 *'p ,k p 2 3J~ft(Pfih;'crp 'lv 7l~7?A77 + pA 7Kp Finally, the vertex coming from the contribution of the variation of the ghost field p __ ___ 'P Pp'7ay + qy raB p\ p \ REFERENCES A. Staruszkiewicz, Acta Phys. Polon. 24, 734 (1963). S. Deser, R. Jackiw and G. S. Deser and R. Jackiw 't Hooft, Ann. ibid. 153 Phys. (N.Y.) 152 0 (1984); ,405 (1984). J. C. Buges, Phys. Rev. D32 504 (1985); V. B. 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She then travelled to the United States to study in the graduate program in astrophysics at the University of Chicago from 1984 to 1986. She decided to change her research direction to the area of particle physics and subsequently went to the University of Florida from 1987 to 1993. She spent much of the 1992 to 1993 academic year on leave studying field theory at Brandeis University. She is currently an Honorary Fellow at the University of Wisconsin. I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Pierr .C Ramond, Chairman Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Pierre Sikivie Professor of Physics I certify that I have read this study and that in my opinion it acceptable standards of scholarly presentationI and is fully adequo and quality, as a dissertation for the degree of1 Dqdctr df Philosop ^L, , forms to in scope I1 *" <^ ^ ^ Jame, R. Ipser Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. sk N. Fry ciate Professor of Physics I certify that I have read th acceptable and quality standards of scholarly study and that in my opinion it conforms to presentation and is fully adequate, in scope as a dissertation for the degree of Doctor of Philosophy. SrCri Louis S. Block Professor of Mathematics This dissertation was submit ted to the Graduate Faculty of the Department of Physics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December 1993 Dean, Graduate School 1 ,_... , UNIVERSITY OF FLORIDA IIIll ll 1 1 11811 11 IIII I 3 1262 08556 8342 