Date: Nov 4, 1997 9:02 PM Author: john baez Subject: This Week's Finds in Mathematical Physics (Week 112)

This Week's Finds in Mathematical Physics - Week 112

John Baez

This week I will talk about attempts to compute the entropy of a

black hole by counting its quantum states, using the spin network

approach to quantum gravity.

But first, before the going gets tough and readers start dropping like

flies, I should mention the following science fiction novel:

1) Greg Egan, Distress, HarperCollins, 1995.

I haven't been keeping up with science fiction too carefully lately, so

I'm not really the best judge. But as far as I can tell, Egan is one of

the few practitioners these days who bites off serious chunks of reality

--- who really tries to face up to the universe and its possibilies in

their full strangeness. Reality is outpacing our imagination so fast

that most attempts to imagine the future come across as miserably

unambitious. Many have a deliberately "retro" feel to them --- space

operas set in Galactic empires suspiciously similar to ancient Rome,

cyberpunk stories set in dark urban environments borrowed straight from

film noire, complete with cynical voiceovers... is science fiction

doomed to be an essentially *nostalgic* form of literature?

Perhaps we are becoming too wise, having seen how our wildest

imaginations of the future always fall short of the reality, blindly

extrapolating the current trends while missing out on the really

interesting twists. But still, science fiction writers have to try to

imagine the unimaginable, right? If they don't, who will?

But how do we *dare* imagine what things will be like in, say, a

century, or a millenium? Vernor Vinge gave apt expression to this

problem in his novel featuring the marooned survivors of a "singularity"

at which the rate of technological advance became, momentarily,

*infinite*, and most of civilization inexplicably... disappeared. Those

who failed to catch the bus were left wondering just where it went.

Somewhere unimaginable, that's all they know.

"Distress" doesn't look too far ahead, just to 2053. Asexuality is

catching on bigtime... as are the "ultramale" and "ultrafemale" options,

for those who don't like this gender ambiguity business. Voluntary

Autists are playing around with eliminating empathy. And some scary

radical secessionists are redoing their genetic code entirely, replacing

good old ATCG by base pairs of their own devising. Fundamental physics,

thank god, has little new to offer in the way of new technology. For

decades, it's drifted off introspectively into more and more abstract

and mathematical theories, with few new experiments to guide it. But

this is the year of the Einstein Centenary Conference! Nobel laureate

Violet Masala will unveil her new work on a Theory of Everything. And

rumors have it that she may have finally cracked the problem, and found

--- yes, that's right --- the final, correct and true theory of physics.

As science reporter Andrew Worth tries to bone up for his interviews

with Masala, he finds it's not so easy to follow the details of the

various "All-Topology Models" that have been proposed to explain the

10-dimensionality of spacetime in the Standard Unified Field Theory. In

one of the most realistic passages of imagined mathematical prose I've

ever seen in science fiction, he reads "At least two conflicting

generalized measures can be applied to T, the space of all topological

spaces with countable basis. Perrini's measure [Perrini, 2012] and

Saupe's measure [Saupe, 2017] are both defined for all bounded subsets

of T, and are equivalent when restricted to M - the space of

n-dimensional paracompact Hausdorff manifolds - but they yield

contradictory results for sets of more exotic spaces. However, the

physical significance (if any) of this discrepancy remains obscure...."

But, being a hardy soul and a good reporter, Worth is eventually able to

explain to us readers what's at stake here, and *why* Masala's new work

has everyone abuzz. But that's really just the beginning. For in

addition to this respectable work on All-Topology Models, there is a lot

of somewhat cranky stuff going on in "anthrocosmology", involving

sophisticated and twisted offshoots of the anthropic principle. Some

argue that when the correct Theory of Everything is found, a kind of

cosmic self-referential feedback loop will be closed. And then there's

no telling *what* will happen!

Well, I won't give away any more. It's fun: it made me want to run out

and do a lot more mathematical physics. And it raises a lot of deep

issues. At the end it gets a bit too "action-packed" for my taste, but

then, my idea of excitement is lying in bed thinking about n-categories.

Now for the black holes.

In "week112", I left off with a puzzle. In a quantum theory of gravity,

the entropy of a black hole should be the logarithm of the number of its

microstates. This should be proportional to the area of the event

horizon. But what *are* the microstates? String theory has one answer

to this, but I'll focus on the loop representation of quantum gravity.

This approach to quantum gravity is very geometrical, which suggests

thinking of the black hole microstates as "quantum geometries" of the

black hole event horizon. But how are these related to the description

of the geometry of the surrounding space in terms of spin networks?

Starting in 1995, Smolin, Krasnov, and Rovelli proposed some answers to

these puzzles, which I have already mentioned in "week56", "week57", and

"week87". The ideas I'm going to talk about now are a further

development of this earlier work, but instead of presenting everything

historically, I'll just present the picture as I see it now. For more

details, try the following paper:

2) Abhay Ashtekar, John Baez, Alejandro Corichi and Kirill Krasnov,

Quantum geometry and black hole entropy, preprint available as

gr-qc/9710007.

This is a summary of what will eventually be a longer paper with two

parts, one on the "black hole sector" of classical general relativity,

and one on the quantization of this sector. Let me first say a bit

about the classical aspects, and then the quantum aspects.

One way to get a quantum theory of a black hole is to figure out what a

black hole is classically, get some phase space of classical states, and

then quantize *that*. For this, we need some way of saying which

solutions of general relativity correspond to black holes. This is

actually not so easy. The characteristic property of a black hole is

the presence of an event horizon --- a surface such that once you pass

it you can never get back out without going faster than light. This

makes it tempting to find "boundary conditions" which say "this surface

is an event horizon", and use those to pick out solutions corresponding

to black holes.

But the event horizon is not a local concept. That is, you can't tell

just by looking at a small patch of spacetime if it has an event horizon

in it, since your ability to "eventually get back out" after crossing a

surface depends on what happens to the geometry of spacetime in the

future. This is bad, technically speaking. It's a royal pain to deal

with nonlocal boundary conditions, especially boundary conditions that

depend on *solving the equations of motion to see what's going to happen

in the future just to see if the boundary conditions hold*.

Luckily, there is a purely local concept which is a reasonable

substitute for the concept of event horizon, namely the concept of

"outer marginally trapped surface". This is a bit technical --- and my

speciality is not this classical general relativity stuff, just the

quantum side of things, so I'm no expert on it! --- but basically it

works like this.

First consider an ordinary sphere in ordinary flat space. Imagine light

being emitted outwards, the rays coming out normal to the surface of the

sphere. Clearly the cross-section of each little imagined circular ray

will *expand* as it emanates outwards. This is measured quantitatively

in general relativity by a quantity called... the expansion parameter!

Now suppose your sphere surrounds a spherically symmetric black hole.

If the sphere is huge compared to the size of the black hole, the above

picture is still pretty accurate, since the light leaving the sphere is

very far from the black hole, and gravitational effects are small. But

now imagine shrinking the sphere, making its radius closer and closer to

the Schwarzschild radius (the radius of the event horizon). When the

sphere is just a little bigger than the Schwarzschild radius, the

expansion of light rays going out from the sphere is very small. This

might seem paradoxical --- how can the outgoing light rays not expand?

But remember, spacetime is seriously warped near the event horizon, so

your usual flat spacetime intuitions no longer apply. As we approach

the event horizon itself, the expansion parameter goes to zero!

That's roughly the definition of an "outer marginally trapped surface".

A more mathematical but still rough definition is: "an outer marginally

trapped surface is the boundary S of some region of space such that the

expansion of the outgoing family of null geodesics normal to S is

everywhere less than or equal to zero."

We require that our space have some sphere S in it which is an outer

marginally trapped surface. We also require other boundary conditions

to hold on this surface. I won't explain them in detail. Instead, I'll

say two important extra features they have: they say the black hole is

nonrotating, and they disallow gravitational waves falling into S. The

first condition here is a simplifying assumption: we are only studying

black holes of zero angular momentum in this paper! The second

condition is only meant to hold for the time during which we are

studying the black hole. It does not rule out gravitational waves far

from the black hole, waves that might *eventually* hit the black hole.

These should not affect the entropy calculation.

Now, in addition to their physical significance, the boundary conditions

we use also have an interesting *mathematical* meaning. Like most other

field theories, general relativity is defined by an action principle,

meaning roughly that one integrates some quantity called the Lagrangian

over spacetime to get an "action", and finds solutions of the field

equations by looking for minima of this action. But when one studies

field theories on a region with boundary, and imposes boundary

conditions, one often needs to "add an extra boundary term to the

action" --- some sort of integral over the boundary --- to get things to

work out right. There is a whole yoga of finding the right boundary

term to go along with the boundary conditions... an arcane little art...

just one of those things theoretical physicists do, that for some reason

never find their way into the popular press.

But in this case the boundary term is all-important, because it's...

THE CHERN-SIMONS ACTION!

(Yes, I can see people world-wide, peering into their screens, thinking

"Eh? Am I supposed to remember what that is? What's he getting so

excited about now?" And a few cognoscenti thinking "Oh, *now* I get it.

All this fussing about boundary conditions was just an elaborate ruse to

get a topological quantum field theory on the event horizon!")

So far we've been studying general relativity in honest 4-dimensional

spacetime. Chern-Simons theory is a closely related field theory one

dimension down, in 3-dimensional spacetime. As time passes, the surface

of the black hole traces out a 3-dimensional submanifold of our

4-dimensional spacetime. When we quantize our classical theory of

gravity with our chosen boundary conditions, the Chern-Simons term will

give rise to a "Chern-Simons field theory" living on the surface of the

black hole. This field theory will describe the geometry of the surface

of the black hole, and how it changes as time passes.

Well, let's not just talk about it, let's do it! We quantize our theory

using standard spin network techniques *outside* the black hole, and

Chern-Simons theory *on the event horizon*, and here is what we get.

States look like this. Outside the black hole, they are described by

spin networks (see "week110"). The spin network edges are labelled by

spins j = 0, 1/2, 1, and so on. Spin network edges can puncture the

black hole surface, giving it area. Each spin-j edge contributes

an area proportional to sqrt(j(j+1)). The total area is the sum of

these contributions.

Any choice of punctures labelled by spins determines a Hilbert space of

states for Chern-Simons theory. States in this space describe the

intrinsic curvature of the black hole surface. The curvature is zero

except at the punctures, so that *classically*, near any puncture, you

can visualize the surface as a cone with its tip at the puncture. The

curvature is concentrated at the tip. At the *quantum* level, where the

puncture is labelled with a spin j, the curvature at the puncture is

described by a number j_z ranging -j to j in integer steps.

Now we ask the following question: "given a black hole whose area is

within epsilon of A, what is the logarithm of the number of microstates

compatible with this area?" This should be the entropy of the black

hole. To figure it out, first we work out all the ways to label

punctures by spins j so that the total area comes within epsilon of A.

For any way to do this, we then count the allowed ways to pick numbers

j_z describing the intrinsic curvature of the black hole surface. Then

we sum these up and take the logarithm.

That's roughly what we do, anyway, and for black holes much bigger than

the Planck scale we find that the entropy is proportional to the area.

How does this compare with the result of Bekenstein and Hawking,

described in "week111"? Remember, they computed that

S = A/4

where S is the entropy and A is the area, measured in units where c =

hbar = G = k = 1. What we get is

S = (ln 2 / 4 pi gamma sqrt(3)) A

To compare these results, you need to know what is that mysterious

"gamma" factor in the second equation! It's called the Immirzi

parameter, since it was first discovered by Giorgio Immirzi in the

following paper:

3) Giorgio Immirzi, Quantum gravity and Regge calculus, in

Nucl. Phys. Proc. Suppl. 57 (1997) 65-72, preprint available as

gr-qc/9701052.

It's an annoying unavoidable arbitrary dimensionless parameter that

appears in the loop representation, which nobody had noticed until

Immirzi came along --- people had been unwittingly setting it to a

particular value for no good reason. It's still rather mysterious. But

it works a bit like this. In ordinary quantum mechanics we turn the

position q into an operator, namely multiplication by x, and also turn

the momentum p into an operator, namely -i d/dx. The important thing is

the canonical commutation relations: pq - qp = -i. But we could also

get the canonical commutation relations to hold by defining

p = -i gamma d/dx

q = x/gamma

since the gammas cancel out! In this case, putting in a gamma factor

doesn't affect the physics. One gets "equivalent representations of the

canonical commutation relations". In the loop representation, however,

the analogous trick *does* affect the physics --- different choices of

the Immirzi parameter give different physics! For more details try:

4) Carlo Rovelli and Thomas Thiemann, The Immirzi parameter in quantum

general relativity, preprint available as gr-qc/9705059.

How does the Immirzi parameter affect the physics? It *determines the

quantization of area*. You may notice how I keep saying "each spin-j

edge of a spin network contributes an area proportional to sqrt(j(j+1))

to any surface it punctures"... without ever saying what the constant

of proportionality is! Well, the constant is

8 pi gamma

Before the Immirzi parameter was noticed, everyone went around saying

the constant was 1. (As for the factor of 8pi, I'm no good at these

things, but apparently at least some people were getting that wrong,

too!) Now Krasnov claims to have gotten these damned factors

straightened out once and for all:

5) Kirill Krasnov, On the constant that fixes the area spectrum in

canonical quantum gravity, preprint available as gr-qc/9709058.

So: it seems we can't determine the constant of proportionality in the

entropy-area relation, because of this arbitrariness in the Immirzi

parameter. But we can, of course, use the Bekenstein-Hawking formula

together with our formula for black hole entropy to determine gamma,

obtaining

gamma = ln(2) / sqrt(3) pi

This may seem like cheating, but right now it's the best we can do. All

we can say is this: we have a theory of the microstates of a black hole,

which predicts that entropy is proportional to area for largish black

holes, and which taken together with the Bekenstein-Hawking calculation

allows us to determine the Immirzi parameter.

What do the funny constants in the formula

S = (ln 2 / 4 pi gamma sqrt(3)) A

mean? It's actually simple. The states that contribute most to the

entropy of a black hole are those where nearly all spin network edges

puncturing its surface are labelled by spin 1/2. Each spin-1/2 puncture

can have either j_z = 1/2 or j_z = -1/2, so it contributes ln(2) to the

entropy. On the other hand, each spin-1/2 edge contributes 4 pi gamma

sqrt(3) to the area of the black hole. Just to be dramatic, we can call

ln 2 the "quantum of entropy" since it's the entropy (or information)

contained in a single bit. Similarly, we can call 4 pi gamma sqrt(3)

the "quantum of area" since it's the area contributed by a spin-1/2

edge. These terms are a bit misleading since neither entropy nor area

need come in *integral* multiples of this minimal amount. But anyway,

we have

S = (quantum of entropy / quantum of area) A

What next? Well, one thing is to try to use these ideas to study

Hawking radiation. That's hard, because we don't understand

*Hamiltonians* very well in quantum gravity, but Krasnov has made some

progress....

6) Kirill Krasnov, Quantum geometry and thermal radiation from black holes,

preprint available as gr-qc/9710006.

Let me just quote the abstract:

"A quantum mechanical description of black hole states proposed recently

within the approach known as loop quantum gravity is used to study the

radiation spectrum of a Schwarzschild black hole. We assume the

existence of a Hamiltonian operator causing transitions between

different quantum states of the black hole and use Fermi's golden rule

to find the emission line intensities. Under certain assumptions on the

Hamiltonian we find that, although the emission spectrum consists of

distinct lines, the curve enveloping the spectrum is close to the Planck

thermal distribution with temperature given by the thermodynamical

temperature of the black hole as defined by the derivative of the

entropy with respect to the black hole mass. We discuss possible

implications of this result for the issue of the Immirzi gamma-ambiguity

in loop quantum gravity."

This is interesting, because Bekenstein and Mukhanov have recently

noted that if the area of a quantum black hole is quantized

in *evenly spaced steps*, there will be large deviations from the Planck

distribution of thermal radiation:

7) Jacob D. Bekenstein and V. F. Mukhanov, Spectroscopy of the quantum

black hole, preprint available as gr-qc/9505012.

However, in the loop representation the area is not quantized in evenly

spaced steps: the area A can be any sum of quantities like 8 pi gamma

sqrt(j(j+1)), and such sums become very densely packed for large A.

Let me conclude with a few technical comments about how Chern-Simons

theory shows up here. For a long time I've been studying the "ladder of

dimensions" relating field theories in dimensions 2, 3, and 4, in part

because this gives some clues as to how n-categories are related to

topological quantum field theory, and in part because it relates quantum

gravity in spacetime dimension 4, which is mysterious, to Chern-Simons

theory in spacetime dimension 3, which is well-understood. It's neat

that one can now use this ladder to study black hole entropy. It's

worth comparing Carlip's calculation of black hole entropy in spacetime

dimension 3 using a 2-dimensional field theory (the Wess-Zumino-Witten

model) on the surface traced out by the black hole event horizon --- see

"week41". Both the theories we use and those Carlip uses, are all part

of the same big ladder of theories! Something interesting is going on

here.

But there's a twist in our calculation which really took me by surprise.

We do not use SU(2) Chern-Simons theory on the black hole surface, we

use U(1) Chern-Simons theory! The reason is simple. The boundary

conditions we use, which say the black hole surface is "marginally outer

trapped", also say that its extrinsic curvature is zero. Thus the

curvature tensor reduces, at the black hole surface, to the intrinsic

curvature. Curvature on a 3-dimensional space is so(3)-valued, but the

intrinsic curvature on the surface S is so(2)-valued. Since so(3) =

su(2), general relativity has a lot to do with SU(2) gauge theory. But

since so(2) = u(1), the field theory on the black hole surface can be

thought of as a U(1) gauge theory.

(Experts will know that U(1) is a subgroup of SU(2) and this is why we

look at all values of j_z going from -j to j: we are decomposing

representations of SU(2) into representations of this U(1) subgroup.)

Now U(1) Chern-Simons theory is a lot less exciting than SU(2)

Chern-Simons theory so mathematically this is a bit of a disappointment.

But U(1) Chern-Simons theory is not utterly boring. When we are

studying U(1) Chern-Simons theory on a punctured surface, we are

studying flat U(1) connections modulo gauge transformations. The space

of these is called a "Jacobian variety". When we quantize U(1)

Chern-Simons theory using geometric quantization, we are looking for

holomorphic sections of a certain line bundle on this Jacobian variety.

These are called "theta functions". Theta functions have been

intensively studied by string theorists and number theorists, who use

them do all sorts of wonderful things beyond my ken. All I know about

theta functions can be found in the beginning of the following two

books:

8) Jun-ichi Igusa, Theta Functions, Springer-Verlag, Berlin, 1972.

9) David Mumford, Tata Lectures on Theta, 3 volumes, Birkhauser, Boston,

1983-1991.

Theta functions are nice, so it's fun to see them describing states of a

quantum black hole!

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