“ Les Chercheurs Luxembourgeois à l’Etranger”, Luxembourg-Ville, October 24, 2011

Glenn Barnich

Physique théorique et mathématique

Université Libre de Bruxelles &International Solvay Institutes

Theoretical Aspects of Black Hole Physics

Hawking & Ellis

Newtonian black holes

Special and general relativity

Overview

Elements of current theoretical research

Black hole formation

Properties of black holes

Newtonian black holes

Main idea: if the escape velocity of a body exceeds the speed of light, the body becomes invisible, “black”

John Michell 1784

escape velocity: initial speed needed to break free from a gravitational field

ve =

�2GM

r

Pierre-Simon Laplace Exposition du Système du Monde 1796

Relativity

Einstein 1907-1915

Newton’s law of gravitational interaction in contradiction with special relativity

Galilean relativity : same physics for all inertial observers

Correct framework: general relativity

�F = Gm1m2

r�1r

x� = x− vty� = yz� = zt� = t

Special relativity

Maxwell’s equation are not Galilean invariant !

Puzzle: Newtonian mechanics or Maxwell’s theory ?

Lorentz transformations

x� = x−vt√1−v2/c2

y� = yz� = z

t� = t−vx/c2√1−v2/c2

Principle of special relativity (1905) : laws of physics are invariant under Lorentz transformations

Consequences: length contraction, time dilatation, E = mc2

but no room for Newtonian gravity

Minkowski space-time

Geometrization: Minkowski 1908

space + time −→ space-time

(x, y, z) t (x, y, z, ct)

(∆s)2 = (x� − x)2 + (y� − y)2 + (z� − z)2

(spatial distance between points space-time interval between events

(∆s)2 = (x� − x)2 + (y� − y)2 + (z� − z)2 − c2(t� − t)2

the same for all Lorentz observers

Equivalence principle

Thorne

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

Riemannian geometry

(pseudo-)Riemannian Geometry

not invariant for an accelerated observer

x0 = ct,

x1 = x

x2 = y

x3 = z

xµ, µ = 0, 1, 2, 3

ds2Min =�

µ,ν

ηµνdxµdxν

ηµν = diag(−1, 1, 1, 1)

space-time coordinates

Minkowski space-time

more general metric ds2 =�

µ,ν

gµν(x)dxµdxν

contains all the information on the gravitational field

Einstein’s equations

equations for the metric

Rµν − 1

2gµνR+ Λgµν = 8πGTµν

Rµν = Rµν [g, ∂g, ∂2g]

Ricci tensor

geometry

scalar curvature R

cosmological constant Λ

energy-momentum tensor

Tµν

matter

matter tells spacetime how to curve, spacetime tells matter how to move

Wheeler

Schwarzschild metric

spherically symmetric solution to vacuum equations

ds2 = −(1− 2M

r)dt2 +

1

1− 2Mr

dr2 + r2(dθ2 + sin2 θdφ2)

Schwarzschild 1916

exterior region of a star or a black hole

r∗ = 2Mevent horizon at Schwarzschild

radius

Eddington-Finkelstein coordinates, light-cones !

Misner, Thorne, Wheeler

Formation

endpoints of stellar evolution of massive stars

original model calculation of collapse based on geodesics in Schwarzschild geometry

static observer outside:frozen star

Misner, Thorne, Wheeler

Einstein-Rosen bridge

Einstein-Rosen bridgeMTW

Hawking & Ellis

maximal analytic extension of Schwarzschild geometry: Kruskal diagram

Kerr solution & uniqueness theorem

rotating black hole

Unique !

“ A black hole has no hair”

J = Maangular momentum

Wheeler

(electric charge )Q

Chandrasekhar

Thermodynamical properties

Townsend

Further properties

energy extraction (Penrose process)

black holes emit thermal radiation and evaporate (Hawking)

Townsend

TH =�κ2π

new framework: quantum field theory in curved space-time

”....

Hawking

Bekenstein-Hawking entropy

→ SBH =A

4�

Current theoretical investigations

unification of fundamental forces, quantum gravity ?

electro-magnetism, weak & strong nuclear forces

−→standard model of particle physics, tested at LHC

black holes for quantum gravity ≈ hydrogen atom for quantum mechanics

but gravity is a classical field theory !

Black hole statistical mechanics

explain BH entropy by counting relevant micro-states in string theory for a 5d SUSY BH

improved symmetry based derivation for 3d AdS BH

BTZ black hole and symmetries

BTZ black hole

asymptotic symmetries: Virasoro algebra instead of so(2, 2)

ADS(3)/CFT(2) correspondence

cosmological constant Λ = − 1

l2

central charge c =3l

2G

Brown & Henneaux

contains information on # of microstates

Personal research

general mathematical framework to compute charges

M,J,Q, c

charges central charges

Kerr/CFT

Strominger argument extended to extreme 4d Kerr black hole of direct

astrophysical interest

Beyond Kerr/CFT

recent work to go beyond extremality

Penrose, Les Houches 1963

References

S.W. Hawking and G.F.R. Ellis. The large scale structure of space-time. Cambridge University Press, 1973.

Gary Gibbons, "The man who invented black holes", New Scientist, 28 June pp. 1101 (1979)

C.W. Misner, K.S. Thorne, and J.A. Wheeler. Gravitation. W.H. Freeman, New York, 1973.

K.S. Thorne. Black holes and time warps: Einstein’s outrageous legacy. W.W. Norton& Company, Inc, New York, 1995.

R.M. Wald. General Relativity. The University of Chicago Press, 1984.

Subrahmanyan Chandrasekhar. The mathematical theory of black holes. Oxford University Press, 1998.

P. K. Townsend. Black holes. Lecture Notes Cambridge Mathematical Tripos Part III. 1997.

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, 1986.

Monica Guica, Thomas Hartman, Wei Song, and Andrew Strominger. The Kerr/CFT Correspondence. Phys. Rev., D80:124008, 2009.

Andrew Strominger and Cumrun Vafa. Microscopic origin of the Bekenstein-Hawking entropy. Phys.Lett., B379:99–104, 1996.

Andrew Strominger. Black hole entropy from near-horizon microstates. JHEP, 02:009, 1998.

Glenn Barnich and Friedemann Brandt. Covariant theory of asymptotic symmetries, conservation laws and central charges. Nucl. Phys., B633:3–82, 2002.

Maximo Bañados, Claudio Teitelboim, and Jorge Zanelli. The black hole in three-dimensional space-time. Phys. Rev. Lett., 69:1849–1851, 1992.

Announcement

Announcement