Theoretical Aspects of Black Hole Physics

Transcription

1 Les Chercheurs Luxembourgeois à l Etranger, Luxembourg-Ville, October 24, 2011 Hawking & Ellis Theoretical Aspects of Black Hole Physics Glenn Barnich Physique théorique et mathématique Université Libre de Bruxelles & International Solvay Institutes

2 Overview Newtonian black holes Special and general relativity Black hole formation Properties of black holes Elements of current theoretical research

3 Newtonian black holes escape velocity: initial speed needed to break free from a gravitational field v e = 2GM r Main idea: if the escape velocity of a body exceeds the speed of light, the body becomes invisible, black John Michell 1784 Pierre-Simon Laplace Exposition du Système du Monde 1796

4 Relativity Correct framework: general relativity Einstein F = G m 1m 2 1 r r Newton s law of gravitational interaction in contradiction with special relativity Galilean relativity : same physics for all inertial observers x = x vt y = y z = z t = t

5 Special relativity Maxwell s equation are not Galilean invariant! Puzzle: Newtonian mechanics or Maxwell s theory? x = x vt 1 v2 /c 2 y = y Lorentz transformations z = z t = t vx/c2 1 v2 /c 2 Principle of special relativity (1905) : laws of physics are invariant under Lorentz transformations Consequences: length contraction, time dilatation, E = mc 2 but no room for Newtonian gravity

6 Minkowski space-time Geometrization: Minkowski 1908 space + time space-time (x, y, z) t (x, y, z, ct) (spatial distance between points space-time interval between events ( s) 2 =(x x) 2 +(y y) 2 +(z z) 2 ( s) 2 =(x x) 2 +(y y) 2 +(z z) 2 c 2 (t t) 2 the same for all Lorentz observers

7 Equivalence principle Equivalence principle: physics in a freely falling reference frame in the presence of gravity is (locally) equivalent to special relativistic physics in an inertial frame without gravity Thorne

8 Riemannian geometry space-time coordinates Minkowski space-time x µ,µ=0, 1, 2, 3 ds 2 Min = η µν dx µ dx ν µ,ν η µν = diag( 1, 1, 1, 1) x 0 = ct, x 1 = x x 2 = y x 3 = z not invariant for an accelerated observer more general metric ds 2 = µ,ν g µν (x)dx µ dx ν contains all the information on the gravitational field (pseudo-)riemannian Geometry

9 Einstein s equations equations for the metric geometry R µν 1 2 g µνr +Λg µν =8πGT µν matter Ricci tensor R µν = R µν [g, g, 2 g] scalar curvature R energy-momentum tensor T µν cosmological constant Λ matter tells spacetime how to curve, spacetime tells matter how to move Wheeler

10 Schwarzschild metric spherically symmetric solution to vacuum equations Schwarzschild 1916 ds 2 = (1 2M r )dt M r dr 2 + r 2 (dθ 2 +sin 2 θdφ 2 ) exterior region of a star or a black hole event horizon at Schwarzschild radius r =2M Eddington-Finkelstein coordinates, light-cones! Misner, Thorne, Wheeler

11 Formation endpoints of stellar evolution of massive stars static observer outside: frozen star original model calculation of collapse based on geodesics in Schwarzschild geometry Misner, Thorne, Wheeler

12 Einstein-Rosen bridge maximal analytic extension of Schwarzschild geometry: Kruskal diagram Hawking & Ellis Einstein-Rosen bridge MTW

13 Kerr solution & uniqueness theorem rotating black hole angular momentum J = Ma (electric charge ) Q Unique! A black hole has no hair Wheeler Chandrasekhar

14 Thermodynamical properties Townsend

15 Further properties energy extraction (Penrose process) new framework: quantum field theory in curved space-time black holes emit thermal radiation and evaporate (Hawking)... T H = κ 2π Townsend S BH = A 4 Bekenstein-Hawking entropy Hawking

16 Current theoretical investigations unification of fundamental forces, quantum gravity? electro-magnetism, weak & strong nuclear forces standard model of particle physics, tested at LHC but gravity is a classical field theory! black holes for quantum gravity hydrogen atom for quantum mechanics

17 Black hole statistical mechanics explain BH entropy by counting relevant microstates in string theory for a 5d SUSY BH improved symmetry based derivation for 3d AdS BH

18 BTZ black hole and symmetries BTZ black hole cosmological constant Λ= 1 l 2 asymptotic symmetries: Virasoro algebra instead of so(2, 2) Brown & Henneaux central charge c = 3l 2G ADS(3)/CFT(2) correspondence contains information on # of microstates

19 Personal research general mathematical framework to compute charges M,J,Q, c charges central charges

20 Strominger argument extended to extreme 4d Kerr black hole of direct astrophysical interest Kerr/CFT

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22 Beyond Kerr/CFT recent work to go beyond extremality Penrose, Les Houches 1963

23 References C.W. Misner, K.S. Thorne, and J.A. Wheeler. Gravitation. W.H. Freeman, New York, S.W. Hawking and G.F.R. Ellis. The large scale structure of space-time. Cambridge University Press, R.M. Wald. General Relativity. The University of Chicago Press, Subrahmanyan Chandrasekhar. The mathematical theory of black holes. Oxford University Press, P. K. Townsend. Black holes. Lecture Notes Cambridge Mathematical Tripos Part III Gary Gibbons, "The man who invented black holes", New Scientist, 28 June pp (1979) K.S. Thorne. Black holes and time warps: Einstein s outrageous legacy. W.W. Norton& Company, Inc, New York, 1995.

24 References J. D. Brown and M. Henneaux. Central charges in the canonical realization of asymptotic symmetries: An example from three-dimensional gravity. Commun. Math. Phys., 104:207, Maximo Bañados, Claudio Teitelboim, and Jorge Zanelli. The black hole in three-dimensional space-time. Phys. Rev. Lett., 69: , Andrew Strominger and Cumrun Vafa. Microscopic origin of the Bekenstein-Hawking entropy. Phys.Lett., B379:99 104, Andrew Strominger. Black hole entropy from near-horizon microstates. JHEP, 02:009, Glenn Barnich and Friedemann Brandt. Covariant theory of asymptotic symmetries, conservation laws and central charges. Nucl. Phys., B633:3 82, Monica Guica, Thomas Hartman, Wei Song, and Andrew Strominger. The Kerr/CFT Correspondence. Phys. Rev., D80:124008, 2009.

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