Physics 776---Advanced
Gravitation Theory---Spring 2005
Black Hole Thermodynamics
Notes
Mar. 17, Thursday
+ Discussed The Four Laws of Black Hole Mechanics, as descibed e.g. in
the article of this name by J. M. Bardeen, B. Carter, S.W. Hawking,
Commun.Math.Phys.31:161-170,1973.
+ Discussed the points made by Jacob Bekenstein in Black Holes and
Entropy, Phys. Rev. D 7, 2333-2346 (1973). You can see the paper at http://prola.aps.org/pdf/PRD/v7/i8/p2333_1
from a university computer, or if you have a University ID card, you
can go in through the library's Research Port at http://www.lib.umd.edu/ You put in
your ID barcode, and your name, and then choose Electronic Journals,
find your way to PRD, and download the paper.
+ I will omit from these notes most of what I reviewed, but include
here a few of the most pertinent comments, somewhat clarified relative
to my lecture, as well as a few more remarks.
+ The interpretation of black hole entropy is proposed thusly: "It is
then natural to introduce the concept of black-hole entropy as the
measure of the INACCESSIBILITY of information (to an external observer)
as to which particular internal configuration of the black hole is
actually realized in a given case." I believe that this is a mistake,
as explained in detail in the trialog, "Black hole entropy: inside or
out?", http://arxiv.org/abs/hep-th/0501103.
I don't think the internal configuration is thermodynamically relevant,
since it is not in equilibrium, and is cut off from the exterior
forever. Rather, I think the entropy is to be located at the INTERFACE
between the exterior and the interior. I will come back to this later
in the course, in connection with the
"black hole information paradox".
+ Bekenstein proposes that the black hole entropy is a monotonic
function of area, S_bh = f(A). He has already argued that f(A) should
probably be a linear function, with the area theorem as the best
(and very good) evidence thereof. Another strong classical
argument, which he did NOT give, would have been to refer to the form
of the first law of black hole mechanics. The term \kappa/8\pi dA could
of course be written e.g. as [\kappa/8\pi n A^{n-1}] d(A^n), but that
would suggest that [\kappa/8\pi n A^{n-1}] is analogous to temperature.
Since temperature should be an INTENSIVE variable, the appearance of A
in this expression would not be natural. I conjecture that the reason
he did not argue this way is that he had not really thought much about
the first law in this way, with the coefficient of dA identified (up to
a constant) with the surface gravity. I say this because he referred to
the POSITIVITY of the surface gravity term as analogous to the
positivity of temperature, but the CONSTANCY over the horizon, which is
only meaningful when the surface gravity is recognized as an intensive
quantity that need not be constant a priori, is not even mentioned.
+ To determine the function f(A) Bekenstein estimates the minimum area
dA that can be added to a black hole consistently with quantum
mechanics, and sets dS_bh = f'(A) dA = ln2, i.e. one bit, the minimum
amount of missing information. He finds that dA is independent of the
black hole parameters, and is equal to 2 in units of the Planck length
squared, l_p^2 = hbar G (with c=1). This is independent of the
parameters of the black hole, and is just a universal constant area,
which allowed him to deduce that f'(A) = ln2/2, so that S_bh = ln2/2 A.
Since this is just an estimate, let us write it instead as S_bh = \eta
A, where the constant \eta is of order unity. In particular, this
information theoretic argument gives an independent derivation that the
entropy is proportional to the area.
+ This allows the black hole "temperature" to be identified via the
first law: the (\kappa/8\pi) dA term has the form T_bh dS_bh provided
T_bh = (\kappa/8 \pi \eta). How do we restore the dimensionful
constants hbar and G? Well in the first law one has the G dependence
(\kappa/8\pi G) dA, while the entropy is \eta A/(hbar G). Thus the
temperature is evidently T_bh = \hbar \kappa/8 \pi \eta. Note that this
is independent of Newton's constant, which has gone into the Planck
length squared that divides the area in the entropy term. The
corresponding wavelength scale is of order 1/\kappa, which is of order
the Schwarzschild radius for a nonspinning black hole. A photon of this
wavelength would have a very low energy for an astrophysical black
hole. Note also that T_bh goes to zero when hbar vanishes. Of this
temperature, Bekenstein says:
"We emphasize that one should not
regard $T_{bh}$ as THE temperature of the black hole; such an
identification can easily lead to all sorts of paradoxes, and is thus
not useful."
+ Bekenstein tests the Generalized Second Law (GSL) by shining a beam
of thermal photons into a black hole. He notes that if the temperature
of the beam is less than T_bh, then the GSL appears to be
>violated<, since dS_bh = dE/T_bh < dE/T_beam = dS_beam. In
effect, one is transferring heat from a colder to a hotter body without
doing work! So what does Bekenstein say? He notes that this only
happens when the wabelength is of order the black hole size or larger,
hence in this context fluctuations are dominant. So far so good, but
then he makes his big mistake. He says that since fluctuations dominate
one is justified in considering individual quanta, and he proceeds to
show that given some assumption about the entropy of a single quantum
the GSL is maintained at that level.
+ It is really odd that after so much clear insight, and insistence on
the second law and information theory, Bekenstein allows himeself such
a bogus escape from this violation of the GSL. It is even more odd if
we note that earlier in the paper, he has already emphasized that there
must be quantum fluctuations of the area which induce temporary small
decreases of the black hole entropy, since, the second law is
fundamentally only a statistical law. In the scenario with the thermal
photon beam impinging on the black hole, he also notes that the quantum
fluctuations are dominating the process. Hence whey would he not put
together his two observations about the role of quantum fluctuations,
and his faith in the second law, to infer that the black hole must in
fact be emitting radiation with a temperature T_bh? What are the
paradoxes to which identifying T_bh with THE temperature would lead? In
fact, quite the contrary, NOT making this identification leads to the
paradox of violation of the second law! And T_bh is proportional to
hbar, so is entirely a quantum effect, just what the doctor ordered!
For the answer we will have to ask Bekenstein himself. Perhaps he
remembers what his state of mind was. As Brendan pointed out at the end
of class, the article of Werner Israel in 300 Years of Gravitation,
eds. S. Hawking and W. Israel says that Bekenstein (just about to
finish, or just finished with his Ph.D.) was at the Les Houches summer
school in 1972 during which Bardeen, Carter, and Hawking worked out
their paper mentioned at the beginning of this leture. Hawking was
annoyed at Bekenstein's use of Hawking's area theorem to promote the
notion that black hole area is actually a measure of real entropy, as
opposed to just being ANALOGUS to entropy. Perhaps the courage to
contradict the "obvious" fact that a black hole has zero temperature
may have run out at this point under the browbeating of Hawking and
others?
Mar. 15, Tuesday
+ Hamiltonian formalism for
Maxwell theory: gauge invariance leads to a constraint term in
the Hamiltonian that generates arbitrary time-dependent gauge
transformations. The constraint is the component of the covariant field
equations that comes from varying A_0, and A_0 is the arbitrary
multiplier of the constraint term in the Hamiltonian.
+ Hamiltonian formalism for general
relativity: To get started we can choose a coordinate system
(x^0, x^i) on spacetime, and regard the Hamiltonian `time' evolution as
the evolution in x^0. General covariance leads to four constraint terms
in
the Hamiltonian generating arbitary time-dependent coordinate
transformations. The constraints are the components of the covariant
field equations that come from varying g_0a, which is the arbitrary
multiplier of the
constraint term. Since "evolution is gauge", i.e. time evolution can be
implemented by a coordinate transformation, the Hamiltonian has ONLY a
constraint term, g_0a H^a,
apart from a boundary term. Note that g_0a = g_ab N^b, where N^b =
(1,0,0,0) (in the chosen coordinate system) is the vector
field on spacetime whose flow is identified with the Hamiltonian
evolution. The boundary term is required to cancel boundary terms that
arise when varying the Hamiltonian wrt the canonical coordinates. If
N^a asymptotically generates a symmetry (e.g. time translation or
rotation), the the numerical value of the boundary term at a solution
is the corresponding conserved quantity (e.g. total energy
or angular momentum).
+ The variation of the hamiltonian has the structure dH = dH/dq dq +
dH/dp dp, which at a solution to the field eqns is -dp/dt dq + dq/dt
dp. If the solution is independent of the time coordinate then
dp/dt=0=dq/dt, so dH=0. Thus if N^a is a Killing vector of the
solution from which we vary, then dH=0.
On the other hand, since the constraint vanishes for all solutions, the
variation of H is equal to the variation of the boundary terms.
Therefore, for the variation at a stationary solution, the net
variation of the boundary terms must vanish. For a black hole, with an
inner boundary at the horizon, and with N^a taken as the horizon
generating Killing vector, the variation of the boundary terms at
infinity gives dM - Omega dJ, and the variation of the boundary term at
the horizon (most conveniently chosen at the bifurcation surface where
the Killing vector vanishes) gives -kappa/8pi dA. This is the 1st law of black hole
mechanics, as derived by Wald. If the starting point were a
different generally covariant Lagrangian, then there would be a
different expression in place of dA, so the area would no longer be the
entropy...
+ If a student provides me with notes, I'll scan them and post them
here.
Mar. 10, Thursday:
+ Event horizon and area theorem:
(For discussions of the event horizon and area theorem see Townsend's
lecture notes, the article "The Nature of Space and Time" by Hawking,
my paper "Horizon Entropy", my "Introductory Lectures on Black Hole
Thermodynamics", or for the straight dope the texts by Wald or by
Hawking and Ellis, all listed under texts at the course web page.) The
event horizon is defined as the boundary of the causal past of future
null infinity. (This assumes the asymptotic structure of the
spacetime allows future null infinity to be defined.) Alternatively, it
can be the boundary of past of a timelike curve that goes to future
timelike infinity. One could also consider the boundary of other
infinite-lifetime timelike curves, eg an accelerated one. The horizon
is a null hypersurface, whose generators may enter but never leave the
horizon since, were they to leave, the horizon would not really be the
boundary of the causal past. Hawking's original area theorem proved
that if the null energy holds, and if the horizon generators extend to
infinite affine parameter (a no-singularity condition), then the
expansion of the horizon-generating congruence cannot be negative
anywhere. The reason: the focusing theorem would imply the expansion
reaches minus infinity in a finite affine parameter, at which point
infinitesimally neighboring null geodesics cross, at which point the
horizon contains a pair of null directions, implying it is
not null but timelike, which is inconsistent with its definition as the
boundary of the causal past. A stronger version of the theorem,
assumes only that there are no singularities strictly visible from
future null infinity, i.e. it assumes no naked singularities, i.e. assumes
cosmic censorship holds. The
gist of this stronger theorem is explained
in my BH Thermo lecture notes. Everything I've said above is quite
vague. This subject requires a large amount of complicated and subtle
technicalities to be properly treated. A proof of the area theorem
that doesn't assume smoothness of the horizon was given fairly recently
by Chrusciel et al. (http://arxiv.org/abs/gr-qc/0001003).
Note
that the area theorem applies also to horizons of infinite area, since
it is really a local statement.
+ First law and heat flow:
This is discussed nicely in my paper Horizon Entropy, section 4, under
the title "physical process version" of the first law. Up to now,
we formulated the first law just as a statement about nearby Kerr
solutions, and how the variation of mass and angular momentum is
related to the variation of horizon area. In the physical process
version, we consider a weak flux of energy and angular momentum across
the horizon, which can be trated as a small perturbation of a black
hole background. Since the topic is explained in my paper, I'll only
mention here the parts that are not explained there...
++ conserved Killing currents:
Given a divergence free stress tensor T_ab and a Killing field X^a we
get a conserved current j^a = T^ab X_b. (To see it is conserved take
the divergence and use the fact that T^ab is divergence free and
Killing's equation.) This is analogous to the conserved quantity u^a
X_a along a geodesic with tangent u^a. In proving the first law
we use the horizon-generating Killing vector which yields a
combination of energy and angular momentum currents.
++ The first law involves the flux of a conserved current across the
horizon, which is a null hypersurface. This needs defining, since
unlike for spacelike or timelike surfaces, one cannot use the unit
normal and an intrinsic volume element to define the integral as \int
j^a n_a d(vol), since vol=0 and to normalize the normal co-vector n_a
one would have to divide by its norm, which vanishes. That is, we face
0/0. The usual way to deal with this is to develop the theory of
integration of differential forms, or equivalently (but formally
distinct) integration of scalar densities. However I do not want
to use the time to develop this. But I realized (and have never seen
this stated anywhere) that it is easy to define the integral in a
simple way: split the 3d null tangent space into a spacelike 2d plane
and the null direction, and define the integral as \int j_a dx^a
d(area), where area is the 2d area of the spacelike direction and dx^a
is the displacement along the null direction. I checked that this
agrees with more conventional definitions, but it would be nice to have
an explicit proof written out. The most important thing about this
definition is that with this definition, the divergence theorem holds,
so that the flux of a conserved current through a closed surface
vanishes.
++ To derive the first law we used the Einstein equation. If some other
field equation were to hold then the entropy would not be given by the
area. From the physical process approach it is not clear how to proceed
systematically starting with a different field equation to infer the
form of the entropy. However another approach, exploiting the Noether
charge associated with diffeomorphisms generated by the Killing field,
works for all generally covariant actions. This was developed primarily
by Wald.
Mar. 8, Tuesday:
+ Null geodesic congruences: (A
discussion of the following results for null geodesic congurences can
be found in many textbooks, and also in section 6.1 of the lecture notes by P.K.
Townsend posted at the course web page.) We derived the
Raychaudhuri-Sachs-Newman-Penrose eqn for the expansion of null
geodesic congruences. The expansion is now the fractional rate of
change of area rather than
volume, the shear governs the distortion of a circle into an ellipse,
and the congruence need be defined only on a 3d null hypersurface in
order for this equation to hold. The form of the equation is similar to
that for timellike congruences,
k^c theta,c = - 1/2 theta^2 - <sigma>^2 +
<omega>^2 -
R_cd k^c k^d,
where k^c is the null tangent vector to the affinely parameterized
geodesics, and the brackets on the shear <sigma_ab>
indicate that it has been projectted into a 2d spacelike subspace
orthogonal to k. A similar definition holds for <omega_ab>.
I also quoted but did not derive the equation evolving the shear:
k^c <sigma_ab>;c= - theta <sigma_ab> -
<C_acbd k^c k^d>,
where C_abcd is the Weyl curvature tensor, the tracefree part of the
Riemann tensor. More explicitly, we choose a 2d spatial subspace
orthogonal to k, and let <h_ab> be the metric on it. We also
choose a null vector l that is orthogonal to <h_ab>, and
which has l^a k_a = 1. Then the spacetime metric decomposes as g_ab =
-<h_ab> + k_a l_b + l_a k_b. Moreover, since k^a B_ab = k^b B_ab
= 0 (see Mar. 3), B_ab can be decomposed into the form B_ab =
<B_ab> + k_a v_b + w_a k_b, where the first term is the
projection of B_ab into the <h_ab> subspace, and v and w are
vectors orthogonal to k (and hence lying in the null hyperplane). We
are interested in the projection <B_ab>, which can be decomposed
as <B_ab> = -1/2 theta <h_ab> + <sigma_ab> +
<omega_ab>, where as before theta is the divergence
k^a_;a, and <sigma_ab> is symmetric and trace-free.
To obtain the null Raychaudhuri equation we take the trace of Eqn.
(B-evolution) (cf. Mar. 3). Note that B_ab B^ab = <B_ab>
<B^ab>. I leave it as an exercise to work out the <sigma>
and <omega> evolution equations.
+ Focusing theorem: For a
hypersurface orthogonal, affinely parametrized geodesic congruence the
twist vanishes in the timelike case and the spatial projection of the
twist vansihes in the null case, as shown in hw3. Hence in these cases
the right hand side of the Raychaudhuri equation is negative provided
the Ricci tensor component R_cd k^c k^d is non-negative, which by the
Einstein equation is true provided the stress tensor component T_cd k^c
k^d is non-negative. This is the null
energy condition. It is implied by the condition that the
energy density iks non-negative in all frames. It is equivalent for a
diagonalizable stress tensor T_cd to the statement that the energy
density plus any of the three principal pressures is non-negative. (In
class I think I said this wrong, with the absolute value.) For
hypersurface orthogonal null geodesic congruence in a spacetime
satisfying the null energy condition we then have k^c theta,c ≤ -1/2
theta^2, i.e. k^c (theta^-1),c ≥ 1/2. Hence if theta starts out
negative, it reaches minus infinity in a finite affine parameter less
than or equal to 2/|theta_0|. The point is that if light rays are
converging then in flat spacetime they will of course cross, and if
there is gravity then, provided the null energy condition is satisfied,
it only makes the convergence happen faster.
+ started discussing event horizons and Hawking's area theorem.
Mar. 3, Thursday:
+ Geodesic congruences, Raychaushuri
eqn: why? Our main application will be the horizon-generating
congruence of null geodesics, in particular the equation governing how
its area changes. For example this is involved in the proof of the area
theorem. Other applications are in proving the singularity
theorems, and proving that a trapped surface always lies inside a
horizon. Sometimes the Raychaudhuri equation is just a simple way to
understand implications of the Einstein equation.
+ A vector field X determines a congruence of parametrized integral curves, and from these a flow of the manifold into itself,
F_t: M->M, where F_t(x) is the point on the integral curve through x
that lies at a parameter t from x. This flow induces a linear map F'_t
(the derivative of F_t) from tangent vectors at x to vectors at F_t(x).
The image of a vector V= dy/ds tangent to a curve y(s) can be defined
as the tangent to the image of the curve, F_t(y(s)).
+ Lie bracket: a vector field
V that results from "dragging" by the flow of another vector field X
has the property that the Lie bracket [X,V]= X^a V^b,a - V^a X^b,a
vanishes. Here the comma stands for coordinate partial derivative or
any other torsion-free derivative operator (check that the symmetric
connection coefficients cancel out). This is also called the Lie derivative of V with respect to
X. It is also minus the Lie derivative of X wrt V. Thus V is dragged by
X iff X is dragged by V. We derived this by considering a 1-parameter
family of curves, y(s,t), such that X=y,t, and V=y,s. It comes down to
the commuting of partial derivatives, y,st=y,ts.
+ The infinitesimal flow is characterized by how it drags vectors, X^a
V^b;a = B^b_a V^a, where B^b_a = X^b;a. Here I use the covariant
derivative, so each side of the equation is separately a
tensor. The tensor B^b_a characterizes the linearized flow at a point.
+ Now if X is tangent to a congruence of affinely parameterized
geodesics, then B^b_a X^a = 0. If the parameter is chosen to be proper
time in the timelike case, and any affine parameter in the null case,
then X^2 is constant everywhere in the flow and therefore B^b_a X_b =
0. For timelike geodesics,
this means B is totally spatial. The
spatial metric h_ab is defined by h_ab = X_a X_b - g_ab. The
decomposition of B_ab (index lowered with g_ab) is then
B_ab = -1/3 theta h_ab + sigma_ab + omega_ab,
where the expansion theta = X^a;a is the trace of B, which generates
spherically symmetric volume changes, the shear sigma_ab = (X_(a;b) +
1/3 B h_ab) is the symmetric trace-free part, which generates volume
preserving deformations, and the twist omega_ab is the antisymmetric
part, which generates rotations. More precisely, theta is the
fractional rate of change of an infinitesimal volume,
(d(volume)/dt)/(volume).
+ Taking the covariant derivative along the flow we find
X^c B_ab;c =
-B^c_b B_ac + R_cbad X^c X^d. (B-evolution)
This is similar to the geodesic deviation
equation. Contracting with the metric g^ab we get the Raychaudhuri equation for the
derivative of the expansion along the flow:
X^c theta,c = - 1/3 theta^2 - sigma^2 + omega^2 -
R_cd X^c X^d.
+ The Raychaudhuri eqn gives an interpretation of the Ricci tensor, and
of the Einstein equation. At any point x choose a timelike direction
and launch a geodesic congruence from x arranged so that initially
theta, sigma, and omega all vanish. Then -R_cd X^c X^d gives minus the
initial rate of change of theta along the flow, which is minus the
second derivative of an infinitesimal volume, divided by the volume.
Thus in vacuum, the volume remains constant to second order in the
proper time. This simple property characterizes the vacuum Einstein
equation. In the presence of anon-zero energy-momentum tensor, the
Einstein eqn can be written R_ab = 8\pi(T_ab -1/2 T g_ab), so R_ab X^a
X^b = T_ab X^a X^b -1/2 T, the energy density rho minus half the trace.
For a perfect fluid this is (1/2)(rho + 3p). So the congruence
contracts if rho + 3p > 0. For a pedagogical presentation of the
Einstein equation based on the Raychaudhuri equation see The Meaning of Einstein's
Equation (http://arxiv.org/abs/gr-qc/0103044).
+ For null geodesics, let's
change the notation and call the vector field k. The conditions B^b_a
k^a = 0 = B^b_a k_b do not imply that B is spatial, since the subspace
orthogonal to k is not spacelike but instead is null, and contains k
itself. This means that we really only need to follow the linearized
flow in a two-dimensional spacelike subspace orthogonal to the flow.
However the choice of which 2d subspace is arbitrary. This doesn't
cause any difficulty or ambiguity however, since the quantities of
interest are independent of this choice. Next time we'll work this out
in detail.
Mar. 1, Tuesday:
+ What does it mean that the t-coordinate "goes bad" at the horizon? It
is that the gradient of t becomes an infinite co-vector. To establish
this, one could show that its (scalar) contraction with a finite vector
is infinite. This would be the fool-proof method, but it requires
knowing a finite vector. Alternatively, if the scalar g^ab t,a t,b =
g^tt diverges then t,a is infinite. (However the converse is not true,
since f,a could be infinite but null.) I illustrated this with
the Schwarzschild t-coordinate, and contrasted with the PG
T-coordinate, as well as with the Eddington-Finkelstein v coordinate.
The latter is the ingoing null coordinate, in terms of which the line
element takes the form ds^2 = (1-2M/r) dv^2 - 2dv dr -r^2 dOmega^2.
In
the case of Kerr, g^tt is not 1/g_tt since there are off-diagonal terms
g_tphi, but you can check that g^tt diverges at the horizon in BL
coordinates. I also mentioned that both PG and EF coordinates have nice
extensions to the Kerr metric. The EF coordinate analog is called
Kerr-Schild coordinates, and is related to BL coords by replacing t and
phi by new coordinates v and Phi, satisfying dv = dt + (r^2 + a^2)
dr/Delta, dPhi =
dphi + a dr/Delta, where Delta=r^2 -2Mr + a^2. Note that the
relative shift between phi and Phi diverges logarithmically at the
horizon, as does that between t and v. The constant phi lines
twist infinitely counter to the spin direction. An ingoing null ray
with constant Phi has a phi coordinate that goes to infinity as the
horizon is approached. There is a nice discussion of KS coordinates in
MTW's book, Gravitation.
A
PG coordinate analog was found fairly recently by Doran (http://arxiv.org/abs/gr-qc/9910099)
Doran's
time
coordinate is proper time on infalling timelike geodesics that begin at
rest at infinity.
+ Surface gravity: Many ways
to define this besides the one I already gave. Let X be the
horizon-generating Killing field, so X is tangent to the horizon and is
normal to it. Then X^2 is constant on the horizon, so (X^2),a is also
normal to the horizon, hence (X^2),a = -2k X_a for some k, by which I
mean kappa. This agrees with the previous definition I gave, k =
|(|X|,a)|. To see it, use (X^2),a = 2|X| |X|,a, and take the norm of
both sides. Yet another expression follows from X^2,a = 2X^b X_b;a = 2
X^b X_a;b, where Killing's eqn is used in the last step. Thus X^b X_a;b
= k X_a. Actually, we knew X was
tangent to null geodesics on the horizon, from last class. But there is
no reason why the Killing parameter should be an affine parameter, and
this eqn shows that the mismatch between Killing and affine is
characterized by the surface gravity. In fact, from this we
showed that the affine parameter s is related to the Killing parameter
v by s = a exp(kv) + b. Note this means as v goes to minus infinity,
the affine parameter has a finite range. So the Killing flow has a
fixed point. The exception is when k=0, in which case v is an affine
parameter. This is the case of an extremal black hole. Finally I
remarked that surface gravity is invariant under a change of the metric
by a conformal factor that is constant along the Killing orbits. Such a
conformal factor gives a new metric with the same Killing vector (the
components of the metric in a coordinate system adapted to the Killing
vector are constant along the Killing orbits), and one easily sees that
it gives the same surface gravity.
+ Geodesic congruences: A vector field determines a curve through each
point of spacetime. The collection of these curves is called a
congruence. The curves come with a parametrization, defined up to an
additive constant on each orbit. I introduced the Raychaudhuri eqn for
timelike geodesic congruences, and sort of explained what the
expansion, shear and twist are. I'll go back and do this better
Thursday.
Feb. 22, Tuesday:
+ two pictures of the Kerr horizon: cylinder with vertical generators,
or with vertical phi=constant lines. Interpretation of Ω as the angular
velocity of the horizon generators with respect to Killing time, d
phi/dt. There is actually a subtlety here: the Boyer-Lindquist time
goes to infinity at the horizon. We can still take the limit, as the
horizon is approached, of d phi/dt for the orbits of the Killing field.
Alternatively, we can switch to another coordinate which is regular on
the horizon, like the P-G time coordinate T in the non-rotating case.
I'll explain this a bit more on Thursday.
+ How much rotational energy can be extracted in total? Consider an
extremally rotating hole, i.e. one with a=M_i. Set its horizon area
equal to the area of a final non-rotating black hole with a mass M_f.
This yields M_f = M_i/Sqrt[2]. So 29% of the total energy can be
extracted in this limiting case.
+ In the example just discussed, the initial horizon radius is M_i, and
the final one is 2M_f = Sqrt[2]M_i > M_i. So how can the final area
be the same? Answer: "radius" is just a coordinate value. The line
element shows that, eg. in the equatorial plane, the length of an arc
of angle phi at the horizon is always ds = 2M d phi, so the initial
rotating hole indeed has a larger equatorial circumference.
+ generators of null surfaces are
geodesics. proof: Spose
N is a null surface which is the level set of a differentiable function
f with nonvanishing gradient f,a on the surface. Then n_a := f,a
is a normal 1-form, so n^a is a normal vector, tangent to the surface.
Compute: n^b n_a;b = n^b f,ab = n^b f,ba = 1/2 (n^b n_b),a. This last
expression vansihes when contracted with any vector in the surface N,
since n^b n_b = 0 everywhere on N. Hence it must be proportional to the
normal 1-form n_a, so n^b n_a;b is proportional to n_a, which is
to say that n^a is tangent to a geodesic, though not affinely
parameterized in general.
+ "look ma, no derivatives!" this is interesting: null geodesics have
been identified using just the notion of null hypersurfaces. The
definition of a null hypersurface uses the properties of the tangent
space to the hypersurface, which implicitly uses derivatives, but
derivatives of the metric components to define the connection need not
be invoked. Also, some null surfaces---and enough to capture all the
null geodesics locally---can be defined in terms of causal structure
alone: eg the boundary of the future of a point is a null hypersurface.
(The notion of "boundary" is topological, but the causal structure
determines the topology as well.) Knowing the null
geodesics---which should be differentiable curves---is enough to
*determine* the differentiable structure...provided the spacetime
dimension is greater than two, and provided a global causality
condition holds. See Brendan Foster's project paper from two years ago
for a review of this topic: http://www.physics.umd.edu/grt/jacobson/776projects/776projects.html.
It seems to me that this "relativizes" the notion of differenitable
structure in just the right way: that is not fixed a priori but is part
of the dynamics...provided the causal structure itself is somehoe more
fundamental than the differentiable structure. For a discussion of
differentiable structure and the possible role of exotic differentiable
structure in physics, see the article Exotic Smoothness on
Spacetime (http://arxiv.org/abs/gr-qc/9604048) by Carl Brans, and
references therein.
+ Moreover, the causal structure determines the light cone at every
point, which determines the metric up to an overall conformal factor
that can depend on position. Proof:
Suppose you know all the null vectors. Then fix any timelike vector t
and any other vector v. These two span a timelike plane. Write t as a
sum of two null vectors t = l + n lying in this plane. Note that t.t =
2l.n, where the dot stands for Minkoski inner product. Any other vector
in the plane is a linear combination v = al + bn. Now evaluate the
norm: v.v = ab t.t. So the norm of every vector is determined in terms
of just the norm of one fixed vector t.t and the linear combination
coefficients a,b which are not metrically determined. Finally, knowing
the norms of all vectors, we know all inner products: 2v.w
= (v+w).(v+w) - v.v - w.w. To determine the extra piece of
scaling information at a point in spacetime we can use any quantity
that is not invariant under a local rescaling of the metric. I like the
volume element, since it is required anyway to write down an action
functional for fields.
+ Causal set: a set of points with a partial ordering relation,
interpreted as causal ordering. If "locally finite", i.e.
if the intersection of the future of a and the past of b is always
finite, one can assign a notion of volume by simply counting points.
This provides the last piece of data to determine a Lorentzian metric.
So, one has a candidate discrete structure to replace the continuum,
which contains within it---in principle---all the elements of structure
needed to recover a differentiable Lorentzian manifold in some kind of
continuum limit. For references see, e.g a brief Phys. Rev. Letter, Space-time
as a causal set
(http://prola.aps.org/abstract/PRL/v59/i5/p521_1?qid=4b2c9de32607fb09&qseq=4&show=10),
by Luca Bombelli, Joohan Lee, David Meyer, and
Rafael D. Sorkin, or a set of lecture notes, Causal Sets: Discrete Gravity
(http://arxiv.org/abs/gr-qc/0309009),
by Sorkin.
+ I comment that is, as seems highly natural to me, the causal
structure is indeed more primitive than the differentiable structure,
then the fate of spacetime at a spacelike singularity should, I would
think, be to "continue" causally, albeit perhaps not to a structure
that is anymore well aproximated as a continuum.
+ It was remarked by Will Linch that perhaps the causal structure
should "dissolve" at a singularity. This is suggested by the Euclidean
continuation, motivated by Wick rotation, eg as applied by Hartle and
Hawking in the no-boundary wavefunction of the Universe. But the
underlying justification of Wick rotation is the positivity of the
energy spectrum which seems to me not likely to be meaningful under the
conditions approached at a spacelike singularity. So I prefer a
primitive causal structure being preserved. This is just a matter of
hunches, however.
Feb. 17, Thursday:
+ energy positivity: 4-momentum vector future timelike or causal, and
so its inner product with the future timelike 4-velocity of any
observer is positive. Killing energy on the other hand need not be
positive, if the Killing vector is spacelike. Example: radial outgoing
photons have negative Killing energy in Schwarzschild spacetime inside
the horizon, while radial ingoing photons have positive energy.
+ Angular-momentum of a particle is defined by L = -p.Z, where p is the
4-momentum and Z is the axial Killing vector field. The minus sign is
due to the use of signature (+ - - -).
+ What is the maximum theoretical efficiency of the Penrose process?
That is, if the bh mass changes by dM < 0 and the angular momentum
by dJ < 0, what is the maximum value of dM/dJ? The limit on
efficiency comes from energy positivity. To see how look at the point
where the particle crosses the horizon.
+ The t-translation Killing vector Y is spacelike at the horizon, and
the phi-translation KV Z is also spacelike. Some combination X=Y+ΩZ is
null, and therefore parallel to the null horizon generator. X is
therefore parallel to the null normal, hence in particular it is normal
to Y, Z, and itself. In hw2 you find Ω by requiring that X.Z=0. It
turns out that Ω is constant, i.e. independent of the polar angle
theta. This is no accident. Only if Ω is constant is X a Killing field.
Ω is called the angular velocity of
the horizon, or the angular velocity of the black hole, and is
positive if the hole is spinning in the direction of increase of phi. I
forgot to show this in class, but Ω is precisely the rate of change of
phi wrt t on the horizon generating null curves.
+ It seems there "should" be a Killing field that coincides with the
horizon generators on the horizon. Such a null surface whose generators
are tangent to a Killing field is called a Killing horizon. Hawking proved that
the event horizon of any stationary non-static black hole is
axisymmetric, and is a Killing horizon. I think he assumed analyticity
of the metric, an unphysically strong assumption. A good student
project would be to look into the theorems regarding this issue. (I
think maybe the constancy of w is easy to prove if axisymmetry is
assumed. The analyticity was probably used to show that the spacetime
must be axisymmetric if stationary but not static.)
+ When the particle crosses the horizon the inner product between its
4-momentum and X must be non-negative, p.X = E-wL ≥ 0. For a particle
with E < 0 we must therefore have L < 0, i.e. the particle adds
negative angular momentum to the hole, thus demonstrating that the
Penrose process consists of extracting rotational energy. The
inequality can be rewritten E/L ≤ w, hence we've found the upper bound
on efficiency: dM/dJ ≤ w. This limit is achieved when p.X = 0, i.e. the
particle enters the bh skimming along the null generators of the
horizon.
+ The existence of a limiting efficiency is reminiscent of
thermodynamics, and suggests looking for the analog of entropy S and
positive integrating factor temperature T such that dM - w dJ = TdS in
quasistationary processes. In the context of thermodynamics the
existence of T and S is one way of stating the second law of
thermodynamics. Here it is a mathematical identity, since the 1-form dM
- w(M,J) dJ is always integrable (its null space at a point is
one-dimensional, and these null spaces can always be integrated into
curves that are sets of constant S for some function S.). In hw2 you
show that in fact dM - w dJ = (\kappa/8\pi) dA, where \kappa>0 is
the surface gravity of the bh
and A is the horizon area. This is usually inappropriately called the first law of black hole mechanics.
The limit on efficiency is thus dA ≥ 0.
+ If charge is included, the parameter space for the bh becomes M,J,Q,
i.e. 3d. The 1-form of interest is then dM -wdJ - Phi dQ, where Phi is
the electrostatic potential difference between the horizon and
infinity. The existence of an integrating factor is no longer
identically guaranteed. But still it exists, and the 1-form is still
given by (\kappa/8\pi) dA, though of course now \kappa and A are
functions of Q as well as M and J. There is a deep reason for the
existence of this "first law" relation in general: diffeomorphism
invariance. We'll come back to this.
+ History:
Christodoulou PRL: Reversible and Irreversible Transformations of a
Black Hole, introduced the "irreducible mass" M_irr and recognized that
the constraint on efficiency is equivalent to dM_irr ≥ 0. He did not
notice that M_irr is proportional to the square root of the horizon
area. Penrose & Floyd in did realize that the constraint is dA ≥ 0.
And they said, "In fact, from general considerations one may infer that
there should be a natural tendency for the surface area of the event
horizon of a black hole to increase with time whenever the situation is
non-stationary." This seems to stop short of saying that the area
CANNOT decrease. According to Thorne's book, Hawking realized
that the area cannot decrease roughly one month before the
Penrose-Floyd paper was received, but he only published his argument a
few months later.
+ Surface gravity: first definition: norm of gradient of norm of X
evaluated at the horizon. Note X becomes timelike outside the horizon,
but then goes spacelike again further out. The norm is growing as we
move away from the horizon in a spacelike direction. In hw2 you use
this definition to compute the surface gravity of a Kerr bh. It is (r_+
- M)/(r_+^2 + a^2), which becomes 1/4M for Schwarzschild and 0 for
extremal Kerr.
+ Mathematical
interlude: covariant characterization of Killing vectors:
X_(a;b)=0. This is called Killing's
equation. Proof in hw2. Alternate proof: remember along affinely
parameterized geodesic with tangent u the quantity u.X is conserved, if
X is a Killing field. Thus 0 = (u^aX_a)_;b u^b = u^a u^b X_a;b. The
term involving the derivative of u vanishes by the geodesic equation.
Since u^a u^b is symmetric in ab, only the symmetric part of X_a;b
survives. Since this eqn holds for all possible u^a, we infer that
X_(a;b)=0. Conversely, if Killing's equation holds, then we infer that
u.X is conserved along all geodesics.
Feb. 15, Tuesday:
+ coordinate diagrams of realtion between ergosurface (stationary limit
surface) and horizon. At the axis, ergosurface convex if a <
Sqrt[3/4], concave if 1 > a > Sqrt[3/4], and conical if a=1.
+ All slices of horizon have same intrinsic 2-geometry: reason: they
are taken into each other by a symmetry and the null direction is
orthogonal to all vectors so "tilting" does not produce a change of the
metric. The horizon 2-geometry is fully embeddable in euclidean 3-space
only if a < Sqrt[3/4].
+ Not all slices of the timelike ergosurface have the same 2-geometry...
+ Showed and discussed pictures from Penrose article. One puzzle:: why
light circles at ergosurface are partlly retrograde. One student
suggested that is possible when there is an inward component of radial
velocity. I agreed, but later realized this is not correct: looking at
the Kerr metric at the ergosurface, *any* timelike futuregoing
displacement (ds^2 > 0) *must* have dphi > 0. So I still don't
understand Penrose's picture.
+ Energy in special relativity, energy-momentum 4-vector, energy
measured by an observer p_a u^a.
+ Killing energy p_a X^a, where X^a is a Killing vector. We showed this
is *conserved* along a geodesic. In the Newtonian limit it is just rest
mass + kinetic energy + gravitational potential energy as measured by a
static observer.
+ Killing energy of a static particle with mass m and 4-velocity
X/|X| is m|X|, eg m Sqrt[1-2M/r] in Schwarzschild. So we can lower a
mass to a horizon and extract 100% of its rest mass energy. Since there
are negative Killing energy configurations inside the horizon one could
extract more than the rest energy, i.e. extract some energy from the
black hole itself, except that behind the horizon there is no further
communication to the outside.
+ In the rotation case, inside the ergosphere but outside the horizon
there are negative energy configurations, so one can extract more than
the rest energy. Penrose conceived a way to do this: the rope holding
he particle must co-rotate inside the ergosurface, so he let the
scaffolding corotate as well. Then he remarked in a footnote that
perhaps one could instead do it with freely falling particles: throw a
particle in with energy E_0, arrange for it to break in two in the
ergosphere in such a way that bit 1 has negative energy E_1 < 0.
Then bit 2 will have energy E_2 greater than E_0. It will turn
out that bit 1 must carry negative angular momentum into the
hole, decreasing the spin. Thus this amounts to an extraction of
rotational kinetic energy from the hole.
Feb. 10, Thursday :
+ Null hypersurfaces (or hyperplanes): different characterizations:
- contain one null direction and all others spatial
- normal vector is null
- normal vector lies *in* the surface
- normal one-form is null
- induced metric has signature (0 - -)
+ Kerr horizon where gradient of r (the normal one-form to the constant
r surface) is null
+ Naked singularity if a >M
+ Collapse with a>M, i.e. J>M^2, must shed extra angular momentum
to become a black hole. Note at the horizon r ~ M, so J>M^2 would
mean J>Mr, which would mean the tangential speed is of order the
speed of light. So it is not implausible that there is an angular
momentum barrier which serves to shed the extra angular momentum.
+ Cosmic censorship hypothesis: no naked (i.e. visible from far away)
singularity will form from regular initial data. This does not exclude
initial singularities such as the big bang, or the white hole
singularity of the eternal Schwarzschild black hole. In fact tuned
initial data does produce a naked singularity, so the hypothesis is now
phrased with reference to *generic* initial data, i.e. an open set of
initial data (in some appropriate topology). There are various
significant bits of evidence that censorship is valid. I don't think
the censorship idea is another example of psychological aversion to the
unknown...btw, people have actually considered the possibililty that
astrophysical signals could come from naked singularities.
+ Energy extraction: from Penrose's article, "Misner's universe" of 2^N
black holes of mass m. Combine them in pairs and they become a single
hole of
mass 2m(1-K), where K is the fraction of energy radiated in
gravitational waves. Now combine the combined black holes. The *same*
fraction of energy is radiated away, since the problem involves purely
vacuum spacetime, and so is governed by the vacuum Einstein equation,
which is *scale invariant* since the Ricci tensor is invariant under a
constant scaling of the metric. Thus the third stage black holes have
mass 2(2m(1-K))(1-K). At the end we wind up with one black hole of mass
2^N m (1-K)^N. By making N very large we can extract an arbitrarily
large fraction of the energy!
Feb. 8, Tuesday:
+ Penrose diagram for collapse to a black hole.
+ Evidence for black hole in the galactic center.
- Slide show about millimeter VLBI
observations: http://www.mpifr-bonn.mpg.de/staff/hfalcke/bh/sld1.html
- Movie of stars orbiting the central mass,
from near infrared observations: http://www.mpe.mpg.de/ir/GC/index.php
+ Black hole uniqueness (a.k.a. no hair theorem): This was initially
quite surprising to many people, who thought that
somehow the multipole moments would be frozen in as an asymmetric
object collapsed through an event horizon. Instead, all gravitational
multipoles higher than monpole and dipole can and will be radiated
away. This was established by a combination of two different
approaches: (i) assume a static or stationary solution with event
horizon and prove uniqueness. (ii) show that there are no stationary
perturbations of the Kerr metric, and show how perturbations are
radiated away. We will not go into this in any detail.
+ The Kerr metric (1963), describing a rotating black hole in vacuum,
labeled by mass M and angular momentum J=aM. The angular momentum per
unit mass a is called the "spin parameter". I wrote down the line
element in Boyer-Lindquist coordinates, and started looking at its
properties. It has two Killing vectors, a time-translation symmerty and
an axial rotation. The latter becomes timelike at sufficiently small r
coordinate, yielding closed timelike curves. However, these are inside
a presumably unstable inner Cauchy horizon. The time translation
symmetry becomes null at a certain surface, and spacelike within. We
showed that this surface is NOT null, so it is not the event
horizon. More on this next time...
Feb. 3: Continuation of previous
class, focusing on diagrams for black
holes: Schwarzschild, and Reissner-Nordstrom (charged) black holes.
Instability of inner horizon of charged black holes.
Feb. 1: Introduction
to Carter-Penrose
diagrams, by Albert Roura.
Jan. 27: Historical overview of
the resistence to understanding the
nature of the Schwarzschild singularity, and of the inevitability of
formation of astrophysical black holes. Discussion largely followed
"Dark stars: the history of an idea", a chapter by W. Israel in 300 Years of Gravitation,
eds. S. Hawking and W. Israel. The main message is that many brilliant
physicists had such psychological resistence to the ideas of (a) a
non-static region of what was supposed to be a static gravitational
field, and (b) unarrested gravitational collapse, that they put forth
nonsensical arguments to try and avoid these conclusions.