Gravity in Two Spacetime Dimensions Der Fakult¨at f¨ur Mathematik, Informatik und Naturwissenschaften der Rheinisch–Westf¨alischen Technischen Hochschule Aachen vorgelegte Habilitationsschrift zur Erlangung der venia legendi arXiv:hep-th/0011240v1 27 Nov 2000 von Dipl.–Ing. Dr. Thomas Strobl aus Wien Contents 1 Introduction 7 2 2d geometry and gravitational actions 11 2.1 Fromfourtolowerdimensions . 11 2.2 Purelygeometrical2dactions . 14 2.3 Einstein–Cartanvariables . 17 2.4 Extension to theories with nontrivial torsion . 19 2.5 WhysomanydifferentLagrangians . 22 2.6 Why consider low–dimensional, topological models . 24 2.7 Role of the dilaton within the present chapter . 27 3 Generalized dilaton theories and matter actions 29 3.1 Generalized dilaton action replacing the Einstein–Hilbert term . 29 3.1.1 Spherically reduced gravity . 29 3.1.2 String inspired dilaton gravity . 32 3.1.3 Mostgeneraldilatontheory . 36 3.1.4 Two possible routes to a simplified, equivalent model . 36 3.1.4.1 Conformal rescalings of the metric . 37 3.1.4.2 Equivalence to theories with torsion . 40 3.2 Matteractions ............................ 42 3.3 Generalfieldequations ........................ 45 3.4 More on dimensionally reduced theories . 48 3.5 Extensionto2dsupergravity . 51 4 2d gravity–Yang–Mills systems in terms of Poisson σ–models 55 4.1 Poissonmanifolds .......................... 56 4.2 Defining the σ–model ........................ 60 4.3 Generalized dilaton gravity as Poisson σ–model .......... 64 4.4 Extensiontoinclude2dsupergravity . 66 3 4 4.5 Includingmatterfields ........................ 71 4.5.1 Additional Yang–Mills fields . 71 4.5.2 Fermionicmatter ....................... 72 4.5.3 Scalarfields .......................... 73 5 Classical solutions on a local level 75 5.1 Pure 2d Gravity–Yang–Mills models . 75 5.1.1 Gaugefixing method, torsion–freetheories . 77 5.1.1.1 Attainability of certain gauge conditions . 77 5.1.1.2 Metrical variables . 79 5.1.1.3 Transforming back to original dilaton variables . 80 5.1.1.4 In Einstein–Cartan variables . 82 5.1.2 Targetspacemethod,thegeneralcase . 83 5.1.2.1 Torsion–free theories . 83 5.1.2.2 Theories with general torsion dependence . 84 5.1.2.3 Solution in the neighborhood of saddle-points . 88 5.1.2.4 Inclusion of Yang–Mills fields . 90 5.1.2.5 The model (2.21) . 91 5.1.3 Supergravity ......................... 92 5.2 Dilaton theories with matter . 96 5.2.1 Solvable generalizations of the CGHS model . 96 5.2.1.1 No gauge fields (α0 =0):.............. 98 5.2.1.2 Inclusion of gauge fields (arbitrary α0) ...... 101 5.2.2 Generalized dilaton gravity with chiral matter . 103 6 Classical solutions on a global level 107 6.1 Maximal Extensions for all Gravity–Yang–Mills systems . 109 6.1.1 Generalities on maximal extensions . 109 6.1.2 Some remarks onCarter-Penrose diagrams . 110 6.1.2.1 Further preliminaries . 113 6.1.3 Extremals ........................... 114 6.1.4 Construction of the Building Blocks, Carter-Penrose Dia- grams ............................. 116 6.1.5 Maximal Extension and Saddle-Point Charts . 123 6.1.6 Recipeandexamples . 129 6.1.7 Remarksonglobalcoordinates . 138 6.2 All global solutions of (all 2d) Gravity–Yang–Mills models . 139 5 6.2.1 Introductoryremarks . 139 6.2.1.1 Motivation . 140 6.2.1.2 Organization . 141 6.2.1.3 Anticipation of some of the results . 142 6.2.2 Cut-and-paste technique for the KV model . 143 6.2.2.1 The moduli space for M Σ IR ........ 147 ∼ × 6.2.3 A global σ-modelapproach. 148 6.2.4 Systematic grouptheoreticalapproach . 152 6.2.4.1 Preliminaries . 152 6.2.4.2 Thesymmetrygroup . 155 6.2.4.3 Subgroupsandfactorspaces . 162 6.2.4.4 Morethanonegenerator . 171 6.2.4.5 Remarks on the Constant Curvature Case . 180 6.2.5 Remarksonkinks ...................... 182 7 Towards quantum gravity 187 7.1 Hamiltonian formulation . 187 7.1.1 For a cylindrical spacetime . 188 7.1.2 Modifications for open spacetimes . 191 7.1.3 Abelianization and the reduced phase space . 195 7.1.3.1 Remarks on the variable p ............. 195 7.1.3.2 Global abelianization for Schwarzschild . 197 7.2 Dirac quantization of gravity–Yang–Mills models . 198 7.2.1 Physicalstates ........................ 198 7.2.1.1 The case of simply connected symplectic leaves . 201 7.2.1.2 Multiply connected leaves . 202 7.2.1.3 Lorentzian signature of the theory . 203 7.2.1.4 Comparison of classical and quantum results . 205 7.2.1.5 The missing kink number . 206 7.2.2 Furtherremarks ....................... 207 7.2.2.1 Comparing the Lorentzian and Euclidean theories 207 7.2.2.2 Modifications for open spacetimes . 208 7.2.2.3 The inner product and the issue of time . 209 7.3 Brief outlook on Hawking radiation . 210 Bibliography 213 6 Chapter 1 Introduction There is no doubt that, at the fundamental level, theoretical physics is in the need of reconciling General Relativity and quantum theory. This becomes obvious already from Einstein’s equations 8πG G = T . (1.1) µν c2 µν Here Gµν is the Einstein tensor, determined solely in terms of the metric g, which governs the ‘gravitational force’, and Tµν is the energy momentum tensor of matter fields. We know that matter fields are governed by a quantum field theory, the standard model of strong and electroweak interactions. Equations (1.1), however, contain purely classical, commuting quantities. On the other hand, Einstein’s equations show that, in the presence of matter, T = 0, the µν 6 metric g can not be strictly Minkowskian (since then G = 0); it can be so at µν 6 best to some approximation. A Minkowski background metric is used, however, in standard quantum field theory of elementary particle physics. Certainly, no current experiments in our laboratories come only close to ex- plicitly display the above contradiction experimentally. Still, from the point of view of a unified and mathematically consistent theory of all interactions, a new theory is needed. There is a variety of (at least apparently) different contemporary approaches to improve on this situation: It ranges from noncommutative geometry [1], over twistors [2], over the attempt to quantize Einstein gravity nonperturbatively in a canonical fashion [3], to the advent of string theory [4], to name just the most prominent directions. The at the moment probably most promising of these attempts is the string approach. But also string theory is surely not formulated in a fully satisfactory and self-consistent way yet. 7 8 Black holes appear to provide an important link between gravity and quan- tum theory. First, near a black hole (and in particular near the singularity) the gravitational field cannot be ignored or approximated sufficiently well by a flat, Minkowskian metric. Second, the discovery of Hawking radiation [5] demon- strated that quantum effects cannot be ignored for black holes; rather they lead to a (slow, but still nonvanishing) evaporation of the classically indissoluble ob- jects. Moreover, theoretical considerations allow to establish a striking analogy between the laws of thermodynamics and the laws governing black holes in classi- cal general relativity [6]. Further Gedankenexperiments [7] show that the second law of thermodynamics should be altered to include an additive contribution to the entropy, assigned to the black hole. Already on dimensional grounds, the 3 Planck length lP = ~G/c enters the generalized second law of thermodynam- ics postulated in thisp way. It is felt by many contemporary theoreticians that black holes will be the gate to some fundamentally new picture of our world, bringing together seemingly separate subjects such as gravity, quantum theory, and thermodynamics. As a side-remark let me mention that by now certainly black holes are no more purely theoretical objects. There is already overwhelming experimental evidence for the existence of black holes in the center of many galaxies, including ours (cf., e.g., [8]). Also black holes have been found as partners in binary systems [9]. It is one of the famous results of General Relativity, called the no–hair theorem [10], that within Einstein–Maxwell theory all asymptotically flat, stationary black holes are determined uniquely by specification of their mass m, their charge q, and their angular momentum J. This scenario extends to some other matter couplings such as to the case of additional scalar fields (with convex potential). Still, the ‘no–hair conjecture’ does not hold in full generality: counter examples have been found recently (cf references in [10]) such as, e.g., in the Einstein–Yang– Mills theory with gauge group SU(2) (note that these solutions are unstable, however). Thus, (in those cases where the no–hair theorem or conjecture holds) the space of (stationary) black hole solutions (modulo gauge transformations) is finite dimensional only. The formulation of gravity in two spacetime dimensions allows to isolate pre- cisely these most interesting objects of General Relativity: Upon definition of ap- propriate two–dimensional gravitational actions, cf. chapters 2 and 3 below, the space of their classical solutions does not only contain (two–dimensional) black holes, it consists only of such spacetimes, except possibly for naked singularity 9 solutions and a vacuum ground state. Many of the considerations performed for their four–dimensional analogs, moreover, like the prediction of Hawking radia- tion as well as the analogy to thermodynamics, may be transferred also into the two–dimensional setting. Finally, also here the original theory is invariant with respect to diffeomorphisms (the group of coordinate transformations), such that a transition to the quantum theory is plagued by similar conceptual difficulties as in four spacetime dimensions (like the ‘problem of time’ [11] to name just one of them). During about the last decade, two–dimensional gravity models thus turned out as an interesting and promising theoretical laboratory to test ideas on how to approach quantization of gravity.