“ 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

So thank you for the opportunity to speak here. Today I want to give you a feeling off why black holes are so central in many present day investigations in theoretical high energy physics.

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

In Newtonian physics, the concept of what would now be called a black hole goes back to the 18th century. The escape velocity of a given body, the earth say, is the initial velocity needed for an object to break free from the gravitational field of this body.

It then follows that if the body is sufficiently massive, or its radius is sufficiently small, there is a value for which the escape velocity is greater than the speed of light. So no light can reach a far-away observer in which case the body will become invisible and appears to be black.

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

Even though some qualitative features of this picture are correct, the appropriate framework for discussing black holes is general relativity. General relativity is the relativistic theory theory of the gravitational interaction and was constructed by Einstein between 1907 and 1915.

The problem he had to face is that Newton’s law for the gravitational attraction between two bodies was in contraction with his principle of special relativity. Indeed, in Newtonian mechanics, the physics is the same for all inertial observers. Such observers are related through Galilean transformations, that is to say, those transformations that describe uniform straight motions of one reference frame with respect to the other.

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

The problem with Galilean invariance was that Maxwell’s equations are not Galilean invariant. So what has to be modified, the foundation of Newtonian mechanics or Maxwell’s equations for electromagnetic phenomena ? It turns out that the laws of electromagnetism are invariant under Lorentz transformations, more general transformation than Galilean transformations, that involve the famous relativistic factor sqrt {1 - v^2/ c^2} where $c$ is the speed of light. For instance, when one uses these laws to add the velocities one finds that one cannot add them in a way to go faster than the speed of light.

Einstein’s principle of special relativity of 1905 then states that Newtonian mechanics has to be modified, physical laws should be invariant not under Galilee but under Lorentz transformations when you go form one inertial frame to the other. This principle, together with the axiom that the speed of light is constant in vacuum, has famous consequences such as the contraction of lengths, dilatation of time or the equivalence of mass and energy. The point here is however that Newtonian gravity does not fit into this framework.

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

The first step to make gravity fit is a geometrization of the problem. We see from the Lorentz transformations that, contrary to the Galilee transformations, they mix space and time. It then becomes natural to consider space and time as coordinates on a 4 dimensional entity, space-time. In Euclidean space, you measure the distance through the Pythagorean theorem: if you have two points, their squared distance is measured by the sum of squares of their coordinate differences. It turns out that in Minkoswki space-time the proper space-time interval between two events is given by the same expression, except that you have to substract the square of their time differences, multiplied by the velocity of light squared. This interval is invariant under Lorentz transformations, so it is the same for all observers. Conversely, Lorentz transformations and translations are the most general transformations that leave the Minkowskian distance invariant.

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

The next step is the equivalence principle. It states that, locally, the physics in a freely falling reference frame in the presence of a gravitational field is the same as the physics in an inertial (non accelerating frame) without gravity. In order to illustrate it, consider the standard high school physics problem of the “tir oblique”. Imagine a canon that shoots off a canon ball at some initial velocity in the gravitational field of the earth, hence giving the standard parabola for an observer, here the dog, at rest with respect to the canon. Now suppose that the cannon is at the edge of a cliff and that you take your window with when you jump off the cliff at the moment the canon ball is fired. wWhat you will see is that the ball will follow a straight line, exactly like you would if you would be at rest with respect to the canon and if there woul;d be no gravitational field.

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

In order to proceed, it is convenient to unify the space-time coordinates by denoting them by x^\mu with \mu running from 0 to 3 and x^0 being time times the velocity of light c. In more mathematical terms, the way to measure distances in Minkowski space-time is encoded in the Minkowski metric, a 4 * 4 diagonal matrix with entries -1 , 1, 1, 1, in accordance with what we have seen previously.

For an accelerated observer, the Minkowski interval is no longer invariant and how to measure distances is encoded in a more general “machine”, a metric, a symmetric non degenerate 4*4 tensor which contains all the information on the gravitational field. The correct mathematical famework to study spaces equipped with such a structure is the mathematical theory of pseudo-Riemannian geometry, the pseudo having to do with the signature, the minus sign for the time component.

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

The last step needed to complete the theory is to find the equations that this metric has to obey. For obvious reasons, they are called Einstein’s equations. In this equation R\mu\nu and R are functions of the metric and its derivatives up to second order, R\mu\nu is called the Ricci tensor, R is a derived function, the scalar curvature. These functions characterize the geometry of space-time. In addition, there can be a term involving the famous cosmological constant. In fact, this year’s Nobel prize, for the experimental discovery that the expansion of the universe is accelerating implies on the theoretical side that there should be an extremely small positive cosmological constant in the equations. The right hand side of the equations involves properties of matter encoded in a so-called energy-momentum tensor. So Einstein’s equations are a system of coupled second order partial differential equations for the metric, the gravitational field, that are sourced by the matter content. Conversely, the matter content is constrained by the geometry. For instance test particles that are so light that they do not significantly influence the energy-momentum tensor of the system are required to move along geodesics, the lines of shortest distance in the geometry described by the metric. In the famous words of Wheeler: matter tells geometry how to curve and geometry tells matter how to move. An illustrationof this which most of you have probably already seen somewhere is the analogy of a sheet which is deformed by a heavy object

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

The most important solution to Einstein’s equations in vacuum, that is to say without matter, is the spherically symmetric Schwarzschild solution. It was found by Schwarzschild just a few moths after Einstein completed his theory. It describes the exterior region of a static star or of a black hole. From the form of the metric it is clear that something special happens when r=2M as the grr component of the metric diverges there. It turns out that r= 2M is a singularity of the coordinate system, not of spacetime, a bit like the north-pole on maps of the earth. There is nothing wrong with space-time there, but what it does is that the horizon divides spacetime into two parts, once you are inside the horizon, there is no longer any way to send signals to the outside. This can be seen from the light-cones: the trajectories of both massive and massless particles like photons have to lie within the light cones. From the orientation of the light-cones in the above diagram, one sees that once one is below the Schwarzschild radius, one can no longer send signals to observers on the other side.

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

A star with less than 1.4 solar masses, the so-called Chandrasekhar limit, will end its thermonuclear evolution as a white dwarf, in this case gavitational attraction is balanced by the pressure due to electron degeneracy, more massive stars will end as neutron stars, where it is the neutron degeneracy pressure that resists gravity. Finally, for more massive stars, of a couple of solar masses, nothing can balance gravity and it will collapse through its Schwarzschild radius to form a black hole.

The relevance of the Schwarzschild metric for gravitational collapse comes from Birkhoff’s theorem: if one assumes spherical symmery, the Schwarzschild geometry is the unique solution to Einstein’s field equations, so the collapse can be modelled by using geodesics of this geometry. For a static observer, the light from the star gets more and more red-shifted, what he sees is that the collpase slows down and freezes at the Schwarzschild radius.

Astrophysical black holes are of two kinds, standard black holes of several solar masses and supermassive black holes at the center of galaxies, in particular of our own galalaxy. Today there is an overwhelming consensus due to indirect evidence that black holes do exist.

Einstein-Rosen bridge

Einstein-Rosen bridgeMTW

Hawking & Ellis

maximal analytic extension of Schwarzschild geometry: Kruskal diagram

When one considers the maximal analytic extension of the Schwarzschild geometry, that is to say when one constructs the geometry such that geodesics have no ends except at singularities, one finds the following situation. An exterior region I, where out-going light-rays, which always go at 45 degrees in these kind of diagrams, can go either to null infinity or cross the horizon and fall into the black hole. When one asks where the former come from, one finds a region II and that they originate from a past singularity, there is a second horizon into which nothing can fall into. Then the picture repeats itself on the other side. This is second region similar to the “our” region one. The Einstein-Rosen bridge is for instance the slice t=0 which connects the two “universes”. All these are just names that one gives to perfectly consistent geometries that are solution to Einstein’s equations. Nothing guarantees that extensions really exist, but a priori nothing prevents them from existing either, so people have been fascinated by such solutions and made a lot of speculations

Kerr solution & uniqueness theorem

rotating black hole

Unique !

“ A black hole has no hair”

J = Maangular momentum

Wheeler

(electric charge )Q

Chandrasekhar

Another exact solution of the vacuum Einstein equations is the Kerr solution, which describes a rotating black hole. The importance of this two parameters family solutions is that it is essentially unique, all black holes in empty space belong to this family of solutions. When one allows for electro-magnetic fields, there is an additional property, the electric charge. In this case, the family of solutions is called the Kerr-Newman family. I find the prologue from Chandrasekhar’s book on the detailed study of the Kerr geometry remarkable:

Thermodynamical properties

Townsend

A very remarkable feature of black holes is that they satisfy dynamical laws which are very similar to the laws of thermodynamics. The surface gravity kappa is a geometrical property of the horzon. The zeroth law states that it is a constant on the horizon so that it can be identified with temperature. The first law says that the variation of energy in an equilibrium process, that is to say when one goes from one stationary black hole solution to the other, is given by a law analogous to the first law of thermodynamics, provided one identifies the area of the surface of the black hole with the entropy, and kappa with the integrating factor, the temperature. Then there are work terms related to the change of the angular momentum or the electric charge that go with the appropriate potentials, omega is the angular velocity of the horizon as seen by an observer from infinity and phiH is the electric potential on the horizon. These purely ge

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�

Another remarkable property of the Kerr black hole is that one can extract energy from them. Kerr black holes have a special region outside the horizon where the norm of the time-like Killing vector that is used to measure energy as seen by an observer at infinity changes sign and becomes space-like. It then follows from energy conservation that, if a particle from infinity with a certain energy, decays into two particles in the ergo-region one of which goes to infinity and the other inside the black hole, the outgoing particle will have more energy that the ingoing one, while the energy of the black hole will have decreased. The other very important discovery about black holes is that they radiate, the emit thermal radiation with a black body spectrum and the temperature is indeed proportional to the surface gravity of the horizon. This also implies that they ultimately evaporate. When one uses this in the first law one finds that the entropy of the black hole is given by a quarter of the area of the horizon divided by Planck’s constant. One should note that this computation needs a new framework that goes beyond standard general relativity, one needs to use the theory of quantum field theory in curved space-time. This means that gravity remains purely classical and a fixed background on which one studies matter fields that are treated according to the laws of quantum mechanics. Let me profit from this occasion to show you a quote of Hawking on what he thinks about black hole physics.

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 !

There are two inter-related questions that drive a lot of current theoretical high-energy physics. The first is about the unification of the 4 known fundamental forces in nature. Three of them, the electro-magnetic, weak and strong nuclear forces belong to a common frame-work, quantum field theory and are reasonably well-unified in the context of the standard model of particle physics, whose advanced features like the Higgs sector or extensions such as super-symmetry are currently tested by the LHC at CERN. Gravity on the other hand is still a largely classical theory and a necessary step for unification is that it be made consistent with the laws of quantum mechanics. Since the early days of black hole research, there was the hope that the study of black holes and the singularities in their interior can provide a guiding principle for the construction of and thus play a similar role to the one played by the study of the hydrogen atom for the development of quantum mechanics.

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

One of the efforts in this direction is to explain the geometrical Bekenstein-Hawking entropy from a fundamental theory that allows one to directly compute the logarithm of the number of relevant micro-states. One of the first such computation concerns a very particular class of five dimensional supersymmetric black holes and states are counted in the context of string theory, a framework that some people believe is able to produce a quantum description of gravity and to unify the fundamental forces. Two years later, a simplified computation explaining the entropy of the 3 dimensional anti-de Sitter black hole has been provided.

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

The BTZ black hole is the simplest of all black holes, it is the analog of the spinning Kerr black hole in 3 space-time dimensions in the presence of a negative cosmological constant. In three dimensions there are no gravitons and none of the usual complications related to the non-renormalisability of gravity, but since there are black holes with horizons, this toy model is ideal to study properties of black holes. In particular, it turns out that the asymptotic symmetries associated with this black hole involve 2 copies of the Virasoro algebra, instead of the exact isometries of AdS space, which are described by the so(2,2) algebra and play the same role for 3d anti-de Sitter space than the Poincare algebra plays for 4d Minkowski space. With the Virasoro algebra, there is a new characteristic number for the BTZ BH, the central charge c which is given by 3 over 2 G. So this symmetry computation by Brown & Henneaux in the middle 80‘s was a predecessor to the celebrated ADS/CFT correspondence on which most research efforts in the context of string theory are now concentrated. The ppint here is that Strominger used this number in the context of conformal field theory to derive the Bekenstein-Hawking entropy in particular for the BTZ black hole. For that he uses the Cardy formula which relates the asymptotic growth of the microstates to this number, the mass and the angular momentum of the black hole.

Personal research

general mathematical framework to compute charges

M,J,Q, c

charges central charges

Let me now turn to my own research interests in this context. To compute quantities like the charges M,J,Q or the central charge c in gravitational theories can be quite complicated, so we developed a general mathematical approach to do so.

Kerr/CFT

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

astrophysical interest

From my point of view, the particular pleasant aspect of this development is that our theory has been heavily used when Strominger and his collaborators extended their derivation of the microscopic explanation of the black entropy to a four dimensional extreme Kerr black hole, which is of direct astro-physical relevance, as such black holes have been observed in the sky.

Here is a presentation that Strominger has given at the Kavli Institute in Sata Barbara where he comments on the use of uor methods. You can see that in theoretical physics, black board talks are fashionable again, as the speed in which one can give a presentation is much more adapted to the capacity of the audience. It is fun to see somebody struggling with the formulas you have developed, so here we go ....

Beyond Kerr/CFT

recent work to go beyond extremality

Penrose, Les Houches 1963

In more recent work, we have tried to go beyond the extremal case, for instance we have showed how conformal field theory arises in general asymptotically falt space-times, but I think this is a good place to stop.

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.

So what should you remember from this talk, the main message is probably that if you are interested in this type of questions, you should read Kip Thorne’s book, it is one of the best general public expositions that I have read.

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.

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