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“ 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
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.
Overview
Newtonian black holes
Special and general relativity
Black hole formation
Properties of black holes
Elements of current theoretical research
Newtonian black holes
escape velocity: initial speed needed to
break free from a gravitational field
�
2GM
ve =
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
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
Correct framework: general relativity
Einstein 1907-1915
m1 m2 �
�
F =G
1r
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
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 ?
 �
x−vt
√
x
=


1−v 2 /c2


 y� = y
Lorentz transformations
�
z
=z



t−vx/c2

�
 t =√ 2 2
1−v /c
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
(x, y, z)
−→
t
(spatial distance between points
space-time
(x, y, z, ct)
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 − 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 spacetime 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
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
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
xµ , µ = 0, 1, 2, 3
space-time coordinates
ds2M in =
Minkowski space-time
�
x0 = ct,
x1 = x
ηµν dxµ dxν
µ,ν
x2 = y
x3 = z
ηµν = diag(−1, 1, 1, 1)
not invariant for an accelerated observer
more general metric
ds2 =
�
gµν (x)dxµ dxν
µ,ν
contains all the information on the gravitational field
(pseudo-)Riemannian Geometry
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 spacetime 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
geometry
1
− gµν R + Λgµν = 8πGTµν
2
Ricci tensor
energy-momentum tensor
Rµν = Rµν [g, ∂g, ∂ 2 g]
scalar curvature
matter
Tµν
R
cosmological constant Λ
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 illustration
of 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 −
Schwarzschild 1916
2M
1
2
2
2
2
2
dr
+
r
(dθ
+
sin
θdφ
)
)dt2 +
2M
r
1− r
exterior region of a star or a black hole
event horizon at
Schwarzschild
radius
r∗ = 2M
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
static observer outside:
frozen star
original model calculation of collapse based
on geodesics in Schwarzschild geometry
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
maximal analytic extension of
Schwarzschild geometry:
Kruskal diagram
Hawking & Ellis
Einstein-Rosen bridge
MTW
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 EinsteinRosen 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
angular momentum
(electric charge
J = Ma
Q )
Unique !
“ A black hole has no hair”
Wheeler
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)
Townsend
new framework: quantum field theory in curved space-time
black holes emit thermal
radiation and evaporate
(Hawking)
TH
�κ
=
2π
A
→ SBH =
4�
Bekenstein-Hawking entropy
”....
Hawking
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 spacetime. 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
but gravity is a classical field theory !
black holes for quantum gravity ≈
hydrogen atom for quantum mechanics
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 microstates 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
cosmological constant
asymptotic symmetries:
Virasoro algebra instead of
central charge
1
Λ=− 2
l
so(2, 2)
Brown & Henneaux
3l
c=
2G
ADS(3)/CFT(2) correspondence
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 nonrenormalisability 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
C.W. Misner, K.S. Thorne, and J.A. Wheeler. Gravitation. W.H. Freeman, New York, 1973.
S.W. Hawking and G.F.R. Ellis. The large scale structure of space-time. Cambridge University Press, 1973.
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.
Gary Gibbons, "The man who invented black holes", New Scientist, 28 June pp. 1101 (1979)
K.S. Thorne. Black holes and time warps: Einstein’s outrageous legacy. W.W. Norton& Company, Inc, New York, 1995.
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.
Maximo Bañados, Claudio Teitelboim, and Jorge Zanelli. The black hole in three-dimensional space-time. Phys. Rev. Lett.,
69:1849–1851, 1992.
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.
Monica Guica, Thomas Hartman, Wei Song, and Andrew Strominger. The Kerr/CFT Correspondence. Phys.
Rev., D80:124008, 2009.
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