Chapters of Advanced General Relativity Notes for the Amsterdam-Brussels-Geneva-Paris doctoral school 2014 & 2016 In preparation Glenn Barnich Physique Théorique et Mathématique Université Libre de Bruxelles and International Solvay Institutes Campus Plaine C.P. 231, B-1050 Bruxelles, Belgium E-mail: gbarnich@ulb.ac.be Abstract. The major aim of the course is to provide technical background material needed for standard computations in general relativity and its extensions. The material covered includes the Cartan formulation, variational principles of gravitational theories, the Newman-Penrose formalism and theoretical aspects of black hole physics. The choice of chapters is to a large extent idiosyncratic.
Contents 1 Elements of differential geometry 4 1.1 Manifolds and functions............................. 4 1.2 Tangent vectors and vector fields........................ 4 1.3 Covectors..................................... 4 1.4 Tensor algebra.................................. 4 1.5 Metrics...................................... 4 1.6 Differential forms................................. 4 1.7 Maps and Lie derivative............................. 4 1.8 Affine connection and covariant derivative................... 4 1.9 Matter couplings................................. 4 1.10 Variations..................................... 4 1.11 Exercices..................................... 4 2 Variational principles for Einstein s equations 5 2.1 Auxiliary fields and symmetries......................... 5 2.2 Gauge symmetries................................ 5 2.3 Metric formulation................................ 5 2.4 Palatini formulation............................... 5 2.5 Cartan formulation................................ 5 2.6 Einstein-Dirac theory.............................. 5 2.7 3d gravity as a Chern-Simons theory...................... 5 3 Newman-Penrose formalism 6 3.1 Null basis and spin coefficients......................... 6 3.2 Weyl, Ricci and Riemann tensor........................ 6 3.3 Structure constants................................ 6 3.4 Bianchi identities................................. 6 3.5 Lorentz transformations and gauge fixation.................. 6 3.6 Optical scalars.................................. 6 3.7 Petrov classification............................... 6 3.8 Goldberg-Sachs theorem............................. 6 4 Black holes 7 4.1 Killing horizons.................................. 7 4.2 Schwarzschild black hole and Rindler spacetime................ 7 4.3 Kerr black hole.................................. 7 4.4 The first law of black hole mechanics...................... 7 2
5 Asymptotics 8 5.1 Asymptotic symmetries............................. 8 6 Acknowledgements 9 References 9
1 Elements of differential geometry This section mostly follows [1]. Other references I have found useful are [2], [3], [4]. Missing proofs of Frobenius and Stokes theorem can be found in [5]. 1.1 Manifolds and functions 1.2 Tangent vectors and vector fields 1.3 Covectors 1.4 Tensor algebra 1.5 Metrics 1.6 Differential forms 1.7 Maps and Lie derivative 1.8 Affine connection and covariant derivative 1.9 Matter couplings 1.10 Variations 1.11 Exercices 4
2 Variational principles for Einstein s equations This part follows exercises 1.4 and 3.17 of [6], chapter 21 of [7]. The application to 3d gravity follows [8] (see also [9]). The Einstein-Dirac application is taken from [10]. The discussion of Noether charges, identities and conserved n 2 forms is adapted from [11]. 2.1 Auxiliary fields and symmetries 2.2 Gauge symmetries 2.3 Metric formulation 2.4 Palatini formulation 2.5 Cartan formulation 2.6 Einstein-Dirac theory 2.7 3d gravity as a Chern-Simons theory 5
3 Newman-Penrose formalism This part follows closely [12] with some elements from [13]. A useful summary of the main results of this and the first chapter can be found in [14]. 3.1 Null basis and spin coefficients 3.2 Weyl, Ricci and Riemann tensor 3.3 Structure constants 3.4 Bianchi identities 3.5 Lorentz transformations and gauge fixation 3.6 Optical scalars 3.7 Petrov classification 3.8 Goldberg-Sachs theorem 6
4 Black holes This part follows mainly [15]. See also chapter 20 of [7], [16], [17], [18], [19], [20], [11], [21]. 4.1 Killing horizons 4.2 Schwarzschild black hole and Rindler spacetime 4.3 Kerr black hole 4.4 The first law of black hole mechanics 7
5 Asymptotics Some elementary results on symmetries of asymptotically flat and AdS spacetimes in d dimensions are presented. Original literature is [22], [23], [24], [25]. The presentation here is based on [26], [27]. 5.1 Asymptotic symmetries 8
6 Acknowledgements This work is supported in part by the Fund for Scientific Research-FNRS (Belgium) and by IISN-Belgium. References [1] G. Gibbons, Part III: Applications of Differential Geometry to Physics. Unpublished lecture notes, 2006. [2] T. Eguchi, P. B. Gilkey, and A. J. Hanson, Gravitation, gauge theories and differential geometry, Phys. Rept. 66 (1980) 213. [3] P. Spindel, Gravity before supergravity, in Supersymmetry, pp. 455 533. Springer, 1985. [4] B. Felsager, Geometry, particles, and fields. Springer Science & Business Media, 2012. [5] B. Schutz, Geometrical methods of mathematical physics. Cambridge University Press, 1980. [6] M. Henneaux and C. Teitelboim, Quantization of Gauge Systems. Princeton University Press, 1992. [7] C. Misner, K. Thorne, and J. Wheeler, Gravitation. W.H. Freeman, New York, 1973. [8] E. Witten, (2+1)-dimensional Gravity As An Exactly Soluble System, Nucl. Phys. B311 (1988) 46. [9] A. Achucarro and P. K. Townsend, A Chern-Simons action for three-dimensional anti-de Sitter supergravity theories, Phys. Lett. B180 (1986) 89. [10] M. Henneaux, Sur la géometrodynamique avec champs spinoriels. PhD thesis, Université Libre de Bruxelles, 1980. [11] G. Barnich and F. Brandt, Covariant theory of asymptotic symmetries, conservation laws and central charges, Nucl. Phys. B633 (2002) 3 82, hep-th/0111246. [12] S. Chandrasekhar, The mathematical theory of black holes. Oxford University Press, 1998. 9
[13] J. Stewart, Advanced General Relativity. Cambridge University Press, 1991. [14] H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt, Exact solutions of Einstein s field equations. Cambridge University Press, 2003. [15] P. K. Townsend, Black holes, gr-qc/9707012. [16] J. M. Bardeen, B. Carter, and S. W. Hawking, The four laws of black hole mechanics, Commun. Math. Phys. 31 (1973) 161 170. [17] T. Regge and C. Teitelboim, Role of surface integrals in the Hamiltonian formulation of general relativity, Ann. Phys. 88 (1974) 286. [18] R. M. Wald, Black hole entropy is Noether charge, Phys. Rev. D48 (1993) 3427 3431, gr-qc/9307038. [19] V. Iyer and R. M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D50 (1994) 846 864, gr-qc/9403028. [20] V. Iyer and R. M. Wald, A comparison of Noether charge and Euclidean methods for computing the entropy of stationary black holes, Phys. Rev. D52 (1995) 4430 4439, gr-qc/9503052. [21] E. Poisson, A Relativist s Toolkit. The Mathematics of Black-Hole Mechanics. Cambridge University Press, 2004. [22] R. K. Sachs, Asymptotic symmetries in gravitational theory, Phys. Rev. 128 (1962) 2851 2864. [23] M. Henneaux and C. Teitelboim, Asymptotically anti-de Sitter spaces, Commun. Math. Phys. 98 (1985) 391. [24] J. D. Brown and M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: An example from three-dimensional gravity, Commun. Math. Phys. 104 (1986) 207. [25] M. Henneaux, Asymptotically anti-de Sitter universes in d = 3, 4 and higher dimensions, in Proceedings of the Fourth Marcel Grossmann Meeting on General Relativity, Rome 1985, R. Ruffini, ed., pp. 959 966. Elsevier Science Publishers B.V., 1986. [26] G. Barnich and C. Troessaert, Aspects of the BMS/CFT correspondence, JHEP 05 (2010) 062, 1001.1541. 10
[27] G. Barnich and P.-H. Lambert, Einstein-Yang-Mills theory: Asymptotic symmetries, Phys.Rev. D88 (2013) 103006, 1310.2698. 11