Carlip S. Quantum Gravity in 2+1 Dimensions

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196 12 The (2+l)-dimensional black hole

properties of black holes in terms of fundamental quantum states.* More-

over, several important questions remain completely open, most notably

the 'information loss paradox'

[126]:

what happens to the information

carried by a quantum field in a pure state which collapses into a black

hole and then disperses as Hawking radiation in, presumably, a mixed

state? It might be hoped that (2+l)-dimensional gravity, which permits

an exact quantum mechanical description of black holes, could suggest

answers to some of these questions.

12.2

The

Lorentzian black hole

wish to understand the quantum mechanics of the (2+l)-dimensional

black hole, an obvious preliminary question is whether Hawking radiation

occurs. This is a semiclassical question, one that depends on the proper-

ties of quantum fields in a black hole background but does not require

a full treatment of quantum gravity. The starting point for the quan-

tum field theoretical computation is an appropriate two-point function

G(x,x

) = (0|(/)(x)(/)(x

)|0}, from which such quantities as the expectation

values (OIT^vIO) can be derived. In particular, a Green's function that is

periodic in imaginary time with period /3 is a thermal Green's function

corresponding to a local inverse temperature /?, and such a periodicity can

be interpreted as an indication of Hawking radiation.

The properties of the Green's function G(x,x

) can be computed by

brute force, but it is simpler and more instructive to take advantage of a

peculiar property of black holes in 2+1 dimensions, the fact that they are

described by spaces of constant curvature. As we saw in chapter 2, the

(2+l)-dimensional black hole can be represented by the metric

-N

+ N+dif

(12.3)

with

— M A

I N =

' 2r

where M and J are the mass and angular momentum. But as a space of

constant negative curvature, the BTZ black hole is also locally isometric to

anti-de Sitter (adS) space. In the language of chapter 4, the black hole has

an anti-de Sitter geometric structure, and should be expressible as a set

of adS patches glued together by suitable

<SX(2,

R) x

SL(2,

R) holonomies.

This means that the Green's function G(x,x

) can be described in terms

of the much simpler Green's function in adS space.

There

has

been some very recent progress

finding

string theoretical description

black hole statistical mechanics

[151,

246].

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12.2 The Lorentzian black hole

197

To derive the geometric structure of the BTZ black hole, recall from

chapter 4 that anti-de Sitter space can be obtained from flat

2>2

with

coordinates

(X\,X

T\, T

) and metric

= dX\ + dX

- dT\ -

dT%,

(12.5)

by restricting to the submanifold

+ X\- T

- T

= -/

(12.6)

with the_induced metric. The relevant region of the universal covering

space adS of anti-de Sitter space may be covered by an infinite set of

Kruskal coordinate patches of three types, corresponding to the regions

of the Penrose diagram of figure 3.2 [22]:

I. (r > r+)

±t/>

- ^ , X

^<j)

^tj

II. (r-<r <

±4>

-j^ , X

-/VT^si

= /7Scosh , T

= -^Vl-

III.

(0 < r < r_)

^(j) - y t\ , T

= -

(12.7)

where

It is straightforward to show that the standard adS metric dS

then

transforms to the BTZ metric (12.3)—(12.4) in each patch. The 'angle'

(j>

equation (12.7) has infinite range, however; to make it into a true angular

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198 12 The (2+l)-dimensional black hole

variable, we must identify

<\>

with

(f>

+ 2n. This identification is an isometry

of anti-de Sitter space - it is a boost in the X\-T\ and X2-T2 planes

- and as in chapter 4, it corresponds to a calculable element

(PL,PR) of

<SL(2,R) x SL(2,R)/Z2. Indeed, a simple computation shows that if we set

(12.9)

and define X as in equation (4.18), then the transformation X

—

corresponds to the shift

</)

—>(/>

+ 2nn.

The BTZ black hole is thus a quotient space adS

/((PL,PR)), where

{(PL,PR)}

denotes the group generated by (PL,PR)- Note that as in the

cosmological solutions discussed earlier, the holonomies

{PL->PR) may be

identified with the holonomies of a suitable Chern-Simons connection A,

where from equation (2.77), the relevant

SX(2,

R) x

SL(2,

R) connection is

(±)a

= (D

(m())

The group elements (12.9) were first found, in a somewhat more compli-

cated conjugated form, by Cangemi et a/., who pointed out that the mass

and angular momentum M and J have a natural interpretation in terms

of the two quadratic Casimir operators of SL(2,R) x SL(2,R) [47]. Steif

has investigated a similar quotient construction for the supersymmetric

(2+l)-dimensional black hole

[245].

Now_let GA(X,X') be a two-point function on the universal covering

space adS of anti-de Sitter space. The corresponding two-point function

for the BTZ black hole can then be expressed by the method of images as

x'l (12.11)

where Ax

denotes the action of the group element (12.9) on x

'. The

phase 3 is zero for ordinary ('untwisted') fields, but may in principle be

arbitrary, corresponding to boundary conditions 0(Ax) = e~

(j)(x); the

choice 5 = n leads to conventional 'twisted' fields. Our problem has thus

been effectively reduced tqjhe comparatively simple task of understanding

quantum field theory on adS.

While quantum field theory on anti-de Sitter space is fairly straight-

forward, it is by no means trivial. The main difficulty comes from the

fact that neither anti-de Sitter space nor its universal covering space are

globally hyperbolic. As is evident from the Penrose diagrams of figure 3.2,

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12.2 The Lorentzian black hole 199

spatial infinity is timelike, and information may enter or exit from the

'boundary' at infinity. One must consequently impose boundary conditions

at infinity to formulate a sensible field theory.

This problem has been analyzed carefully in 3+1 dimensions by Avis,

Isham, and Storey, who show that there are three reasonable boundary

conditions for a scalar field at spatial infinity: Dirichlet (D), Neumann

(N),

and 'transparent' (T) boundary conditions [19]. The last - essentially

a linear combination of Dirichlet and Neumann conditions - are most

easily obtained by viewing adS as half of an Einstein static universe;

physically they correspond to a particular recirculation of momentum

and angular momentum at spatial infinity. The same choices exist in 2+1

dimensions. In particular, for a massless, conformally coupled scalar field,

described by the action

(

^) (12.12)

the adS Green's functions are

Here o(x,x

) is the square of the distance between x and x

in the

embedding space R

described above; in the coordinates of equation

(12.7),

o(x,x

)

[(Xi

- X[f -

(T,

- T[f +

)

- Ttf]

(12.14)

These expressions should be supplemented with appropriate factors of ie

that determine which Green's functions they represent; for the Feynman

propagator, for instance, a should be interpreted as a +

ie.

A more general

Green's function with 'mixed' boundary conditions,

(12.15)

may also be considered. The corresponding BTZ Green's functions are

then obtained from the sum (12.11).

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200 12 The (2+l)-dimensional black hole

The crucial observation is that the functions

GBTZ(X,X') are periodic in

imaginary time, with a period /? given by

(

1216

)

P-'o "

2 _

2_ -—'

(

1216

)

where K is the surface gravity (3.53). To see this, let us consider the

simple case of the static (r_ = 0) black hole, and let x = (t = 0, r,

4>)

and x' = (£,r,</>) in the 'Schwarzschild' coordinates (12.3). Recall from the

derivation above that the group element A of equation (12.9) corresponds

to a shift

(j) —• (j)

+ In. A simple computation shows that in region I

(j(x, Ax ) = £

— 1 + -y cosh — y-

111

cosh

with similar results in region II. It is clear by inspection that

<T(X,

A"X

;

)

periodic under t

—>

t + z'/J, and hence that each term in the sum (12.11)

has the same periodicity. Strictly speaking, one must be careful of the

ie factors and the location of the branch cut needed to define cr

. This

takes a bit more work, but is not too complicated; see reference

[177].

The periodicity implies, in turn, that the Green's functions in a BTZ

black hole background are thermal, with temperature To = jS"

. Indeed,

a typical thermal Green's function in flat spacetime takes the form

Gp(x, x ) =

Tre

-pH '

and the Hamiltonian H is the generator of time translations, so

~~ Tre~P

where x = (t +

*/?,

cf)).

A free conformally coupled scalar field in a

static (2+l)-dimensional black hole background thus behaves as if it were

in a thermal bath of temperature To with respect to the Hamiltonian

generating translations in the adS time t of equation (12.3). Note that the

relationship (12.16) of To and

is precisely the same as the corresponding

result (12.1) in 3+1 dimensions.

A bit of caution is needed in interpreting this temperature. There is no

obvious 'natural' scale for the anti-de Sitter time t - the scale is defined by

the condition that the metric (12.3) asymptotically approach the standard

anti-de Sitter metric,

= (-1 + ^)dt

d4>

- (-1 + ^r'dr

, (12.20)

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12.2 The Lorentzian black hole 201

but the physical significance of such an asymptotic condition is not entirely

clear. Rather, the temperature To should be understood as follows. For

an observer at a fixed value of r and </>, the proper time is Nt. A Green's

function is thus periodic in imaginary proper time with period N/To, and

this will determine the local temperature measured by such an observer.

The factor g^ correcting To is the standard Tolman red shift factor for

the temperature in a gravitational field.

When r_

=f=

0, the computation is slightly more complicated, but still

straightforward. One finds that a is now periodic under the transformation

(t,0)->(t + #,0 +

iO>),

(12.21)

where /? is again given by (12.16) and

This periodicity is the sign of a thermal Green's function with an inverse

temperature /? and a rotational chemical potential

li = -*//* = ~ = OH, (12.23)

where QH is the angular velocity of the horizon. Indeed, a typical Green's

function in the grand canonical ensemble takes the form

where J is the generator of rotations, and the same derivation that led to

equation (12.19) now yields a periodicity under (t, $) —• (t +

i$,(\> —

ifl/a).

Alternatively, if one defines a new angular coordinate

# = 0-£H

(12.25)

it is straightforward to show that the Green's functions are again periodic

under t

—>

t +

i/3.

The coordinate

rotates with the horizon, and it is not

hard to see that the shift function N$ vanishes at r = r+.

By studying the analyticity properties of the Green's functions (12.11),

Lifschytz and Ortiz argue that the relevant vacuum state is the Hartle-

Hawking vacuum

[177].

We may obtain further evidence for Hawking

radiation by computing the Bogoliubov coefficients that transform between

anti-de Sitter 'in' states in the far past and BTZ 'out' states in the far

future. The relevant computations have been performed for the rotating

black hole with transparent boundary conditions by Hyun, Lee, and Yee

Cambridge Books Online © Cambridge University Press, 2009

202 12 The (2+l)-dimensional black hole

[154],

who show that the expectation value of the 'out' number operator

in the 'in' vacuum state is

The thermal form of equation (12.26) is again an indication of Hawking

radiation; the dependence on

— mfi# rather than co is a sign that

the rotating BTZ black hole, like the Kerr black hole, has super-radiant

modes. Further computations of thermodynamic behavior have been

performed for a variety of field theories in black hole backgrounds [155,

156,

184, 240, 241, 244].

The semiclassical computations of this section do not directly yield a

value for the entropy of the (2+l)-dimensional black hole. However, an

expression for the entropy can be obtained by integrating the first law of

thermodynamics,

dM=

TdS-iidJ.

(12.27)

Inserting the expressions (3.46), (12.16), and (12.23) for M, J, T, and //,

we find that

dS = 4ndr+, (12.28)

or restoring factors of G,

S = ^±. (12.29)

Once again the results agree with the (3+l)-dimensional expression (12.2)

for the Bekenstein-Hawking entropy: the entropy is one-fourth of the

horizon size in Planck units.

12.3 The Euclidean black hole

The thermodynamic properties of black holes can be derived, as we have

just seen, by considering the behavior of quantum fields in a classical

black hole background. But as Gibbons and Hawking first pointed out,

using techniques later refined by Brown and York, these properties can

also be obtained from a suitable approximation of quantum gravity itself

[44,

125].

In the absence of a complete quantum theory of gravity, of course, one

cannot hope to compute the thermodynamic behavior of a black hole

exactly. But even without a complete theory it is possible to write down

a formal path integral for the gravitational partition function, which can

then be approximated by the method of steepest descents described in

123 The

Euclidean

black hole 203

chapter 10. In 3+1 dimensions, the leading saddle point may be shown to

be the Euclidean black hole, which exists only when the Euclidean time

has an appropriate periodicity. This periodicity, in turn, corresponds

to an inverse temperature identical to that derived from quantum field

theoretical calculations in a black hole background. Moreover, the saddle

point contribution to the partition function gives the correct Bekenstein-

Hawking entropy. In this section, we shall see that a similar computation

is possible in 2+1 dimensions, and that we can even compute the one-loop

correction to the entropy.

The starting point for the Gibbons-Hawking approach to black hole

thermodynamics is the path integral

Z\fi] = Tre~

= f[dg]e~

lE{p

\ (12.30)

where IE(P) is the Euclidean gravitational action for a space time peri-

odic in (imaginary) time with period /?. As usual in the path integral

formulation of thermodynamics, the quantity Z\fi\ is to be interpreted

as the canonical partition function at inverse temperature /?. Brown and

York have argued that the canonical formalism is not, strictly speaking, a

consistent approach - black holes have negative heat capacities, making

the partition function ill-defined - and that one should instead start with

a microcanonical path integral for a system in a box with a finite radius

[44].

For the (2+l)-dimensional black hole, however, the two methods

ultimately lead to identical expressions for temperature and entropy, so for

simplicity I will concentrate on the original Gibbons-Hawking approach.

(See reference [83] for the application of the Brown-York formalism in

2+1 dimensions.)

Let us evaluate the path integral (12.30) in the saddle point approxima-

tion, in the topological sector that includes the Euclidean version of the

BTZ black hole. The relevant metric may be obtained from (12.3)—(12.4)

by letting t = h and J =

UE,

yielding

+ ffdr

+ r

(d(j)

+ N^dzf (12.31)

with

= -M + ^-^L ,

(12.32)

Note that the continuation of J to imaginary values, necessary for the

metric (12.31) to be real, is physically sensible, since the angular velocity

is now a rate of change of a real angle with respect to imaginary time.

204 12 The (2+l)-dimensional

black

hole

The Euclidean lapse function now has roots

—i\r-\

(12.33)

The metric (12.31) is

Euclidean (positive definite) metric of constant

negative curvature, and the spacetime

therefore locally isometric

hyperbolic three-space H

. This isometry

most easily exhibited by

means of the Euclidean analog of the coordinate transformation (3.59),

which is now globally valid [71],

y =

z =

1/2

In these coordinates, the metric becomes that

the standard upper

half-space representation of hyperbolic three-space,

ds\

= -^(

+ dy

+ dz

)

fdR

, ,

sin

/ (12.35)

where

(R,

are spherical coordinates for the upper half-space

> 0,

(x,

(R cos 6 cos

R sin

9 cos

sin

x)-

(12.36)

Just

periodicity

the Lorentzian solution

the Schwarzschild

angular coordinate

led to the identifications (3.61), periodicity of the

Euclidean solution requires that we identify

(12.37)

Equation (12.37) is the Euclidean version of the identifications {(PR,PL))

of section 2. A fundamental region for these identifications is the space

12.3 The Euclidean black hole

205

Fig. 12.1. In the upper half-space representation of H

, a fundamental region

for the Euclidean black hole is the region between two hemispheres, with points

on the inner and outer surfaces identified along lines such as L. (The outer

hemisphere has been cut open here to show the inner hemisphere.)

between two hemispheres R = 1 and R =

2nr+

with the inside and

outside boundaries identified by a translation along a radial line followed

by a 2n\r-\/£ rotation around the z axis (see figure 12.1). Topologically,

the resulting manifold is a solid torus; for /

7^=

TC/2,

each slice of fixed x is

an ordinary two-torus, with circumferences parametrized by the periodic

coordinates inR and 0, while the degenerate surface x =

TT/2

is a circle at

the core of the solid torus. Physically, R is an angular coordinate, related

to the Schwarzschild coordinate

<j>\

the azimuthal angle 6 measures time;

and x is a radial coordinate.

Now, for the coordinate transformation (12.34) to be nonsingular at

the z axis r = r+, we must require periodicity in the arguments of the

trigonometric functions. That is, we must identify

where

(12.38)

(12.39)

If this requirement of periodicity is lifted, the Euclidean black hole acquires