Quantum Mechanics of Black Holes

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UCSBTH-94-52

hep-th/9412138

Quantum Mechanics of Black Holes

Steven B. Giddings

DepartmentofPhysics

University of California

Santa Barbara, CA 93106-9530

Abstract

These lectures give a p edagogical review of dilaton gravity,Hawking radiation, the

black hole information problem, and black hole pair creation. (Lectures presented at the

1994 Trieste Summer Scho ol in High Energy Physics and Cosmology)

Email addresses: [email protected]

1. Intro duction

Hawking's 1974 discovery[1] that black holes evap orate ushered in a new era in black

hole physics. In particular, this was the b eginning of concrete applications of quantum

mechanics in the context of black holes. But more imp ortantly, the discovery of Hawking

evap oration has raised a sharp problem whose resolution probably requires a b etter under-

standing of Planck scale physics, and which therefore may serve as a guide (or at least a

constraint) in our attempts to understand suchphysics. This problem is the information

problem.

In brief, the information problem arises when one considers a Gedanken exp eriment

where a black hole is formed in collapse of a carefully arranged pure quantum state j i,or

in terms of quantum-mechanical density matrices, = j ih j. This black hole then evap o-

rates, and according to Hawking's calculation the resulting outgoing state is approximately

thermal, and in particular is a mixed quantum state. The latter statement means that the

density matrix is of the form p j ih j, for some normalized basis states j i and

some real numb ers p more than one of which is non-zero. In the present case the p are

approximately the usual thermal probabilities. If Hawking's calculation can b e trusted,

and if the black hole do es not leave b ehind a remnant, this means that in the quantum

theory of black holes pure states can evolve to mixed. This con icts with the ordinary

laws of quantum mechanics, which always preserve purity. Comparing pure and mixed

states, we nd that there is missing phase information in the latter. A useful measure of

the missing information is the entropy, S = Tr ln = p ln p .

Hawking subsequently prop osed [2] that quantum mechanics b e mo di ed to allow

purity loss. However, as we'll see, inventing an alternative dynamics is problematical. This

has lead p eople to consider other alternatives, namely that information either escap es the

black hole or that it is left b ehind in a black hole remnant. Both of these p ossibilities also

encounter diculties, and as a result wehave the black hole information problem.

These lectures will develop these statements more fully. In the past few years a much

advanced understanding of black hole evap oration has b een obtained through investiga-

tion of two-dimensional mo dels, and b ecause of this and due to their greater simplicity

we'll start by reviewing these mo dels. Next will b e a detailed treatmentofHawking radi-

ation from the resulting two-dimensional black holes, followed by a generalization to the

derivation of four-dimensional Hawking radiation. As black holes Hawking radiate they

Other reviews include [3,4] and more recently [5,6]. 1

shrink, and a subsequent section is devoted to a semiclassical description of such black

hole evap oration. There follows a review of the black hole information problem, its var-

ious prop osed resolutions, and the problems with these prop osals. Finally we turn to a

treatment of another non-trivial asp ect of the quantum mechanics of black holes, black

hole pair pro duction. Besides its intrinsic interest, this pro cess can p otentially shed light

on the prop osed remnant resolution of the information problem, and a brief discussion of

howitdoessoisgiven.

2. Two-dimensional dilaton gravity

In 2d, formulating gravity with just a metric gives trivial dynamics; for example, the

Einstein action is a top ological invariant. Instead we consider theories with the addition

of a scalar dilaton . A particular simple theory[7] has action

2 2 2 2 2

d x ; (2:1) g e (R +4(r) +4 ) (rf) S =

where is an analogue to the cosmological constant and f is a minimally coupled mass-

less matter eld that provides a source for gravity. Note that e plays the role of the

gravitational coupling, as its inverse square app ears in front of the gravitational part of

the action.

2 a b

In two dimensions the general metric ds = g dx dx can always lo cally b e put into

conformal gauge,

2 2 +

ds = e dx dx ; (2:2)

0 1

with the convention x = x x . The equations resulting from the action (2.1) are most

easily analyzed in this gauge. The matter equations are

@ @ f =0 ; (2:3)

with general solution f = f (x )+f (x ). Next, the relation

i +

gR = 2 (2:4)

allows rewriting of the gravitational part of the action,

n o

2 2 2 2()

S = d x 2r( )re +4 e : (2:5)

grav

2 2

The equation of motion for is therefore that of a free eld, with solution

= w (x )+w (x ) : (2:6)

The equation for then easily gives

Z Z

x x

2 2 + w w

e = u + u dx e dx e (2:7)

where u (x ) are also free elds. Finally,varying the action (2.1) with resp ect to g ,

g gives the constraint equations,

++ 2 2

@ f@ f g : G e 4@ @ 2@ =

+ + ++ + +

(2:8)

2 2

g : G e 4@ @ 2@ = @ f@ f:

These determine u in terms of f and w :

Z Z

w w

u = dx e dx e @ f@ f; (2:9)

where M is an integration constant. In the following we will cho ose units so that =1.

The theory is therefore completely soluble at the classical level. The unsp eci ed

functions w result from the un xed remaining gauge freedom; conformal gauge (2.2)

is unchanged by a reparametrization x = x ( ). This freedom may b e used to set

w + w = , for example. In this gauge the general vacuum solution is[8,9]

d d

ds =

1+Me

(2:10)

: = `n M + e

The case M = 0 corresp onds to the ground state,

2 +

ds = d d

(2:11)

= :

This is the dilaton-gravity analogue of at Minkowski space, and is called the linear dilaton

vacuum.

+ +

The solutions for M>0 are asymptotically at as !1.At !1

they are apparently singular, but regularity is restored by the co ordinate transformation

x = e ;x = e : (2:12)

A true curvature singularity app ears at x x = M , and x = 0 is the horizon. The

corresp onding Penrose diagram is shown in Fig. 1; the solution is a black hole and M is its

.For M<0 the solution is a naked mass. Notice the imp ortant relation e j =

horizon

singularity. 3 Singularity

Horizon

Singularity

Fig. 1: Shown is the Penrose diagram for a vacuum two-dimensional dilatonic

black hole.

Singularity

+ Black I hole Linear oo Infalling matter oo Dilaton

Horizon -ln ∆

- Vacuum σ- I σ+

-oo - oo

Fig. 2: The Penrose diagram for a collapsing black hole formed from a left-

moving matter distribution.

Next consider sending infalling matter, f = F (x ), into the linear dilaton vacuum.

This will form a black hole, as shown in Fig. 2. Before the matter infall the solution is

given by (2.11). Afterwards it is found by using (2.6)-(2.9),

e = M + e e

(2:13)

d d

ds =

e 1+Me

where one can easily show

M = d T

(2:14)

= d e T ;

++ 4

and where

(@ F ) (2:15) T =

+ ++

is the stress tensor. The co ordinate transformation

+ +

= `n e ; = (2:16)

returns the metric (2.13) to the asymptotically at form

d d

ds = ; (2:17)

1+Me

showing that indeed wehave formed a black hole.

3. Hawking radiation in two dimensions

With a collapsing black hole in hand, we can study its Hawking radiation. The

quickest derivation of the Hawking ux arises by computing the exp ectation value of the

matter stress tensor. Consider the stress tensor for right-movers; b efore the hole forms

they are in their vacuum, and

hT i = lim h0j @ f (^ )@ f ( )j0i

^ !

(3:1)

; = lim

4( ^ ) ^ !

where the second line uses the 2d Green function,

+ +

h0jf (^ )f ()j0i = ln(^ )+ln(^ ) : (3:2)

As usual, one removes the in nite vacuum energy by normal-ordering:

1 1

: T : = T + : (3:3)

4 (^ )

The formula (3.1) can also b e applied at I , but the at co ordinates are now . Therefore

to compare the stress tensor to that of the outgoing vacuum on I ,we should subtract

the vacuum energy computed in the co ordinates,

: T : = T + : (3:4)

4( )

For other references see [1,10,11]. 5

The corresp onding exp ectation value is

1 1

h: T : i = lim h0j @ f ( ) @ f ( ) j0i +

! 4( )

(3:5)

1 ^ 1

= @ @ ln ( ) ( ) + : lim

4( )

Next one expands ( ) ab out , and in a few lines nds

2 3

000 00

1 3

4 5

h: T : i = (3:6)

0 0

24 2

where prime denotes derivative with resp ect to .

Using the relation (2.16) b etween the two co ordinate systems gives the outgoing stress

tensor from the black hole,

1 1

1 : (3:7) h: T : i =

(1 + e )

This exhibits transitory b ehavior on the scale ln , but as !1 it asymptotes

to a constantvalue 1=48. As will b e seen shortly, this corresp onds to the thermal Hawking

ux at a temp erature T =1=2.

A more detailed understanding of the Hawking radiation arises from quantizing the

scalar eld. Recall the basic steps of canonical quantization:

1. Find a complete orthonormal basis of solutions to the eld equations.

2. Separate these solutions into orthogonal sets of p ositive and negative frequency.

3. Expand the general eld in terms of the basis functions with annihilation op erators

as co ecients of p ositive frequency solutions and creation op erators for negative fre-

quency.

4. Use the canonical commutation relations to determine the commutators of these ladder

op erators.

5. De ne the vacuum as the state annihilated by the annihilation op erators, and build

the other states by acting on it with creation op erators.

In curved spacetime general co ordinate invariance implies that step twoisambiguous:

what is p ositive frequency in one frame is not in another. Consequently the vacuum state

is not uniquely de ned. These two observations are at the heart of the description of

particle creation in curved spacetime. This ambiguitywas resp onsible for the di erent

normal-ordering prescriptions in our preceding derivation. 6

Now follow this recip e. Begin by noting that the equations of motion imply existence

of the conserved Klein-Gordon inner pro duct,

(f; g)=i d fr g (3:8)

for arbitrary Cauchy surface .

Steps 1 and 2: A convenient basis for right-moving mo des in the \in" region near I

are

1 1

i! i!

p p

u = e ;u = e : (3:9)

2! 2!

These have b een normalized so that

(u ;u )=2(! ! )=(u ;u ) ; (u ;u )=0 : (3:10)

0 0

! ! !

Furthermore, note that they are naturally separated according to p ositive or negative

frequency with resp ect to the time variable .

Step 3: The eld f has expansion in terms of annihilation and creation op erators

(in) : (3:11) f = d! a u + a u

! !

Step 4: The canonical commutation relations

0 0 1 01

0 00 0 00

[f (x);@ f (x )] = [f (x);@ f(x )] = i(x x ) (3:12)

0 0

x =x x =x

together with the inner pro ducts (3.10) imply that the op erators a satisfy the usual

commutators,

y y

0 y

[a ;a ]= (!! ) ; [a ;a ]=0 ; [a ;a ]= 0 : (3:13)

0 0

! ! !

! !

Step 5: The in vacuum is de ned by

a j0i =0 ; (3:14)

! in

other states are built on it by acting with the a 's.

To describ e states in the future regions it is convenient to follow these steps with a

di erent set of mo des. Mo des are needed b oth in the \out" region near I and near the

singularity. The former are the obvious analogues to (3.9),

1 1

i! i!

p p

e ;v = e : (3:15) v =

2! 2! 7

The latter are somewhat arbitrary as the region near the singularity is highly curved. A

convenient co ordinate near the singularity proves to b e

=ln( e ) ; (3:16)

and corresp onding mo des bv and vb are given by a formula just like (3.15).

In terms of these mo des, f is written

1 h i

y y

b b

f = d! b v + b v + b vb + b vb (out + internal) : (3:17)

! ! ! !

These mo des are normalized as in (3.10), and the corresp onding eld op erators ob ey com-

mutators as in (3.13). The vacua j0i and j0i are also de ned analogously to

out internal

(3.14).

The non-trivial relation (2.16) b etween the natural timelike co ordinates in the in

and out regions { or equivalently b etween the symmetry directions with resp ect to which

p ositive and negative frequencies are de ned { imply that a p ositive frequency solution in

one region is a mixture of p ositive and negative frequency in another region. This mixing

implies particle creation. For example, p ositive frequency out mo des can b e expressed in

terms of the in mo des,

0 0 0

v = d! [ u + u ] : (3:18)

!! ! !!

0 0

The Fourier co ecients , are called Bogoliub ov co ecients, and they can b e

!! !!

calculated byinverting the Fourier transform, or equivalently from

1 1

0 0 0

(v ;u (v ;u ) : (3:19) ) ; =

! ! ! !! !!

2 2

In the present mo del they can easily b e found in closed form in terms of incomplete b eta

functions[11]

1 !

i! 0

= B ( i! + i! ; 1+i! )

2 !

(3:20)

! 1

i! 0

B ( i! i! ; 1+i! ) : =

2 !

Although we will not use these formulas directly they are exhibited for completeness.

Toinvestigate the thermal b ehavior at late times, ln , 'ln , we

could examine the asymptotic b ehavior of (3.20), but a shortcut is to use the asymptotic

form of (2.16) found by expanding around = ln ,

e =e ' ( +ln) ~ : (3:21) 8

Note that this is the same as the relation b etween Rindler and Minkowski co ordinates in

the context of accelerated motion. Likewise one nds

' ~ : (3:22) e

Next, a trick[12] can b e used to nd the approximate form for the outgoing state. Notice

that functions that are p ositive frequency in ~ are analytic in the lower half complex ~

plane. Therefore the functions

i! !

u / (~ ) / v + e vb

1;! !

(3:23)

i! !

u / (~ ) / bv + e v

2;! !

are p ositive frequency. This means that the corresp onding eld op erators a and a

1;! 2;!

must annihilate the in vacuum. The inverse of the transformation (3.23) gives the relation

between eld op erators, as can easily b e seen by reexpanding (3.17) in the u 's and a 's:

i;! i!

! y

a / b e b

1! !

(3:24)

! y

a / b e b :

2! !

These then determine the vacuum, since it ob eys

y y

0=(a a a a )j0i

1;! 2;!

y y

^ ^

/ (b b b b )j0i (3:25)

! !

/ (N N )j0i ;

! !

where N , N are the numb er op erators for the resp ective mo des. The latter equation

! !

implies that

j0i = c (fn g) jfn gijfn gi (3:26)

! ! !

fn g

for some numb ers c (fn g). These numb ers are determined up to an overall constant from

the recursion relation following from the equation a j0i =0:

c(fn g)=c(f0g) exp d!!n : (3:27)

! !

Thus the state takes the form

d!!n

j0i = c (f0g) e jfn gijfn gi : (3:28)

! !

fn g

! 9

It is clear from this relation that the state inside the black hole is strongly correlated

with the state outside the black hole. Observers outside the hole cannot measure the state

inside, and so summarize their exp eriments by the density matrix obtained by tracing over

all p ossible internal states,

2 d!!n

out 2

=Tr j0ih0j = jc (f0g) j e jfn gihfn gj : (3:29)

inside ! !

fn g

This is an exactly thermal density matrix with temp erature T =1=2. The corresp onding

energy density is that of a rightmoving scalar eld,

! d! 1

= T = ; (3:30) E =

!=T

2 e 1 12 48

which agrees with (3.7) if we account for the unconventional normalization of the stress

tensor used in [7],

T : (3:31) E =

Both the total entropy and energy of this density matrix are in nite, but that is simply

b ecause wehave not yet included backreaction which causes the black hole to shrink as

it evap orates. Before considering these issues, however, we will extend these results to

four-dimensional black holes.

4. Hawking radiation in four dimensions

Fig. 3 shows the Penrose diagram for a collapsing four-dimensional black hole; this

picture is analogous to g. 2. Making the assumption of radial symmetry implies the metric

takes the form

2 2 2 2 2

ds = g dt + g dr + R (r;t)d (4:1)

tt rr

is the line element on the two-sphere. where g ;g , and R are functions of r and t and d

tt rr

Notice that as b efore reparametrizations of r and t allow us to rewrite (4.1) in \conformal

gauge",

2 2 0 2 1 2 2 2

ds = e (dx ) +(dx ) + R (x)d : (4:2)

In particular, outside the black hole the metric is time indep endentby Birkho 's theorem,

and can b e written

2 2 2 2 2

ds = g dt + dr + r d (4:3)

2 10 - ξ =const + I

horizon

- σ ξ+ =const R

Fig. 3: The Penrose diagram for a collapsing four-dimensional black hole.

Also indicated are lines of constant (solid) and lines of constant

(dashed), as well as the ray R describ ed in the text.

where the tortoise co ordinate r is de ned by

dr g

= : (4:4)

dr g

(For the Schwarzschild black hole r = r +2M`n(r 2M).)

Now consider propagation of a free scalar eld,

2 1 4

g (rf ) : (4:5) S = d x

The solutions take the separated form

u(r;t)

f = Y (; ) (4:6)

in terms of spherical harmonics and two-dimensional solutions u(r;t). To compare with

our two-dimensional calculation, rst rewrite (4.5) in two-dimensional form. For example,

in the outside co ordinates (4.3),

Z Z

4 2 2 2 2

d x g (rf ) / dr dt (@ u) (@ u) V (r )u (4:7)

t r 11

where the e ective p otential is

`(` +1) @ r

V (r )=g + : (4:8)

r r

The latter term vanishes at r = r = 1 as , and b oth terms vanish as r !1.For

Schwarzschild the maximum of the p otential is of order `(` +1)=M , and o ccurs where

r M . These features are sketched in g. 4.

2 ~l(l+1)/M ~plane ~plane waves waves

~M r

r2M *

Fig. 4: Asketch of the e ective p otential as a function of tortoise co ordinate

r . (Shown is the case of the Schwarzschild black hole.)

In the limits r !1the solutions are therefore plane waves. However, an outgoing

plane wave from r = 1 is only partially transmitted. As we'll see, the Hawking radiation

can b e thought of as thermally p opulating the outgoing mo des, and these energy-dep endent

\gray b o dy" transmission factors lead to deviations from a pure black b o dy sp ectrum.

These factors are, however, negligible for ! V , where for Schwarzschild V

max max

`(` +1)=M .

The Hawking e ect is again found by relating the in mo des at I to the out mo des

at I . The go o d asymptotically at co ordinates outside the collapsing b o dy were given

in (4.3):

2 + 2 2

ds = g d d + r d ; = t r : (4:9)

These co ordinates are suitable for the in or out regions. We will also use conformal co or-

dinates for the interior of the collapsing b o dy,

2 2 + 2 2

ds = ( ) d d + r d : (4:10)

In particular, we can lab el the slices so that near the p oint where the surface of the

collapsing matter crosses the horizon

= r (4:11) 12

and so that j = 0, for some time co ordinate . These co ordinates pass continuously

horizon

through the horizon, as shown in g. 3.

As in two dimensions, the state is de ned so that the in mo des are in their vacuum,

and we wish to compare the resulting state to the out vacuum. The in co ordinate with

+ +

resp ect to which the vacuum is de ned is . This is related to , then to , whichis

then related to . Let's follow this chain backwards. First, as !1, that is near the

horizon, ( ) is given by

d @ @ @ @ @r

1 1

= : (4:12) = +

2 2

d @t @r @t @r @r

This is simpli ed using

+ + +

d @ @ @ @ @r

0= / = (4:13)

d @t @r @t @r @r

and we nd

@r d

= : (4:14)

d @r

Expanding ab out r = r then gives

horizon

d @r @ @r

' : (4:15) r = r =2 '

horizon

d @r @r @r

horizon

The quantity

@ @r

(4:16) =

@r @r

horizon

is the usual surface gravity, and (4.15) integrates to

(4:17) = Be

for some constant B . This formula exhibits the exp ected exp onentially increasing redshift

+ + +

near the horizon. The co ordinate is matched to at r = 0 and then to at the

+ + +

surface of the b o dy.For large we only need to relate to and to in the

vicinity of the ray R shown in g. 3:

d d

' + const : (4:18)

+ +

d d

R 13

The factor gives a constant blueshift, and simply mo di es B :

+ 0

' B e : (4:19)

This formula is nearly identical to (3.21). The remaining derivation of the Hawking

radiation follows as in two-dimensions, with the replacement ! ! !=. Therefore the

temp erature is

T = ; (4:20)

for a Schwarzschild black hole this gives the familiar

T = : (4:21)

The outgoing state is again thermal, except for the grey b o dy factors which are negligible

for all but low-frequency mo des. Up to a numerical constant, the entropy of the Hawking

radiation is the same as that of the black hole, which is easily computed:

=8 M dM ; (4:22) dS =

S =4M = ; (4:23)

where A is the black hole area. This is the famous Bekenstein-Hawking [13,1] entropy.

We can also easily estimate the lifetime of the black hole. The energy densityper

out mo de is !=2; this is multiplied by the thermal factor, the velo city(c= 1), and the

transmission co ecient through the barrier, then summed over mo des:

!;`

dM d!

!;`

= ! : (4:24) (2` +1)

8M !

dt 2 e 1

The transmission factor can b e roughly approximated by no transmission b elow the barrier

and unit transmission ab ove,

(a! M `) (4:25)

!;`

for some constant a. M scales out of the integral (4.24), and we nd

dM 1

/ : (4:26)

dt M 14

This gives a lifetime

t ; (4:27)

which is comparable to the age of the universe for M 10 M .

sun

Finally, note that as in two dimensions there are correlations b etween mo des on either

side of the horizon, and these mean missing information from the p ersp ective of the outside

observer. Also, note that although the calculation refers to ultrahigh frequencies, in essence

all that is b eing used is that the infalling state near the horizon is approximately the

vacuum at high frequencies.

5. Semiclassical treatment of the backreaction

So far black hole shrinkage from Hawking emission has b een neglected. This sec-

tion will treat this backreaction in two dimensions and semiclassically; for attempts at

construction of a full quantum theory of 2d black hole evap oration see e.g. [14-17].

The e ect of the Hawking radiation on the geometry is determined by its stress tensor.

Recall that the asymptotic stress tensor was computed in sec. 3 by relating the normal

ordering prescriptions in the two di erent co ordinate systems. The metric in the out region

asymptoted to

2 + 2 +

ds = d d = e d d : (5:1)

where

d d

: (5:2) e =

d d

With a little e ort, (3.6) can b e rewritten in terms of .Working in co ordinates (but

keeping track of which co ordinates are used for normal ordering) gives

2 2

: T : =: T : + @ (@ ) : T : +t (5:3)

where we de ne t to b e the di erence b etween the two normal-ordered stress tensors.

This formula is valid for an arbitrary Weyl rescaling , and allows us to determine the

full stress tensor given the stress tensor in co ordinates. In particular, supp ose that the

initial state satis es

: T : =0 (5:4)

in accord with conformal invariance of the scalar eld. Then the conservation law

r T + r T =0 ; (5:5)

+ + 15

which should hold for the stress tensor in any generally co ordinate invariant regulation

scheme, implies

2 2

0= @ @ (@ ) + @ t t ; (5:6)

+ + +

where wehave used = =0. The Christo el symbol is easily computed,

+ +

giving

= g =2@ : (5:7)

Eq. (5.6) then integrates to

@ @ ; (5:8) t =

+ +

: T : = @ @ : (5:9)

+ +

The right side is prop ortional to the curvature R, and wehave

R: (5:10) hTi=

This is the famous conformal anomaly: the regulated trace of the stress tensor varies with

the metric used to de ne the regulator.

Eq. (5.9) can b e integrated to nd the quantum e ective action of the scalar eld,

using

1 2

(rf)

hT i = `n D fe p

i g g

(5:11)

= S :

We nd

Z Z

1 1

2 2

S = d @ @ = d x g ; (5:12)

e +

6 24

or, using the relation R = 2 and the Green function for ,

Z Z

2 2 0 0

S = d x g d x g R(x) (x; x ) R(x )

: (5:13)

This expression is known as the Polyakov-Liouville action[18].

The quantum mechanics of the evap orating black hole is describ ed by the functional

integral

Z Z

i i

S S

grav f

h h

D fe () : (5:14) D g D e 16

Here we reinstate h, and the ellipses denote matter sources that create the black hole. In

particular, if these are taken to b e classical, (5.14) b ecomes

i i

S + S +iS

grav PL

h h

D g D e : (5:15)

The Hawking radiation and its backreaction is enco ded in S , which corrects the func-

tional integral at one lo op. However, there are other one lo op corrections arising from

and g , and the Hawking radiation gets lost among them. Toavoid this, consider instead

N matter elds f , so (5.15) b ecomes

i i i

S + S + (N h)S

grav PL

h h h

: (5:16) D g D e

This has a semiclassical limit exhibiting Hawking radiation for large N : N h must b e xed

as h ! 0. The semiclassical equations of motion are

2 2 c`

@ (@ ) G = T +

+ ++

(5:17)

2 2 2

e 2@ @ 4@ @ e G = @ @

+ + + +

and likewise for G ; the dilaton equation is unmo di ed.

The equations (5.17) are no longer soluble. They have b een studied numerically [19-

21], but instead we'll take a di erent approach. Quantization of (5.16) requires addition of

counterterms, and sp eci c choices of these exist [23-25] that magically restore solubility!

These choices of counterterms don't sp oil the essential features of the evap oration, and the

resulting theories are thus soluble mo dels for evap orating black holes.

We will fo cus on the prescription of Russo, Susskind, and Thorlacius[25], who add

counterterms that ensure the current @ ( ) is conserved at the quantum level. To see

how to accomplish this, add (2.5) and (5.12) to get the action including Hawking radiation

e ects,

i h

1 N h

2 2() 2

S = + 2r( ) r e +4e r r : (5:18) d x

2 12

Conservation of @ ( ) can clearly b e reinstated by adding the counterterm

Z Z

hN 1 Nh

2 2

d x gR= d x r r; (5:19)

48 2 12

For a general discussion of physical constraints on such counterterms see [22,4]. 17

and the action b ecomes

n o

2 2 2()

S = d x 2r( ) r e + +4e : (5:20)

RST

2 2

where we de ne = N h=12.

Eq. (5.20) is the same as (2.5) with the replacement

2 2

e ! e + =2 ; (5:21)

so the solutions are found directly from (2.6) and (2.13),

( ) =

(5:22)

e + = M + e (e ) :

Note in particular that

2 +

) (e e + = M + e ( ) : (5:23)

2 4

The left hand side has a global minimum at

= `n(=4) ; (5:24)

so the solution b ecomes singular where the right hand side falls b elow this. Here additional

quantum e ects should b ecome imp ortant. A Kruskal diagram for the formation and

evap oration is sketched in g. 5.

A sensible de nition of the apparent horizon is the place where lines of constant

b ecome null, @ = 0 since if these lines are spacelike it means one is inevitably dragged

to stronger coupling as in the classical black hole. Di erentiating (5.23), the equation for

the horizon is

=+ e : (5:25) e

The singularity b ecomes naked where it and the horizon meet,

= `n : (5:26)

4M=

1 e

With the singularity revealed, future evolution outside the black hole can no longer b e

determined without a complete quantum treatment of the theory: the semiclassical ap-

proximation has failed. The last lightray to escap e b efore this happ ens is called the

e ective horizon. These features are also shown in g. 6. 18 Effective horizon Singularity

Apparent horizon

-ln ∆ Infalling matter

Fig. 5: Shown is a Kruskal diagram for collapse and evap oration of a 2d

dilatonic black hole. The matter turns the singularityat spacelike, and

forms an apparent horizon. The black hole then evap orates until the singu-

larity and apparent horizon collide. The e ective horizon is just outside the

line = ln .

The Hawking ux is computed as b efore; as !1, the metric asymptotes to the

previous form (2.17). The ux h: T :i is therefore still given by (3.7), and asymptotes

to as !1. Asacheck, we can compute the mass lost up to the time (5.26) where

our approximations fail:

constant

: (5:27) d h: T :i = M

As anticipated, the semiclassical approximation breaks down when the black hole reaches

the analogue of the Planck scale, here M 1=.Furthermore, as in the preceding section

it app ears that there are correlations b etween the Hawking radiation and the internal state

of the black hole, and that the Hawking radiation is approximately thermal. An estimate

of the missing information comes from the thermo dynamic relation

dS = ; (5:28)

T 19 ? ? ?

effective horizon

Fig. 6: The Penrose diagram for the evap orating two-dimensional black hole.

The lower part of the diagram has b een truncated at , and the upp er

part where the semiclassi c al approximation fails due to app earance of large

curvatures.

which gives

S =2M : (5:29)

It was mentioned that attempts have b een made to go b eyond the semiclassical approx-

imation and de ne a complete quantum theory bycho osing a b oundary condition[14-17],

for example re ecting, at . These attempts have met with various obstacles, for example

of instability to unending evap oration. It is p erhaps not surprising that a simple re ecting

b oundary condition has had diculty in summarizing the non-trivial dynamics of strong

coupling. Such a b oundary condition presumes that there are no degrees of freedom at

strong coupling, yet if one thinks of the connection of these solutions to four-dimensional

extremal black holes, it seems quite p ossible that there are either a large numb er of degrees

of freedom or very complicated b oundary interactions.

6. The Information Problem

6.1. Introduction

The preceding sections have outlined a semiclassical argument that the Hawking radi-

ation is missing information that was present in the initial state. We'd like to know what 20

happ ens to this information. One p ossibilitywas prop osed byHawking: the black hole

disapp ears at the end of evap oration and the information is simply lost. Although on the

face of it this is the most conservative solution, it is really quite radical; information loss

is equivalent to a breakdown of quantum mechanics.

This prompts us to investigate other alternatives. A second p ossibility is that the

information is returned in the Hawking radiation. This could result from a mistakein

our semiclassical argumentinvolving a fundamental breakdown of lo cality and causality,

as advo cated in [26,15,17,27]. Another p ossibility is that the information is radiated after

the black hole reaches M m and the semiclassical approximation fails. Here ordinary

causality no longer applies to the interior of the black hole, and it's quite plausible that the information escap es.

t=t

evap

Fig. 7: APenrose diagram appropriate to a long-lived remnant scenario. The

singularity is replaced by a planckian region near r =0.

However, the amount of information to b e radiated is, in four dimensions, given by

S M , and the amountofavailable energy is just the remaining black hole mass,

M m . The only way that the outgoing radiation can contain such a large amount

of information with such a small energy is if it is made up of a huge number of very

soft particles, for example photons. Each photon can carry approximately one bit of

information, and so M photons are required. Their individual energies are therefore 21

E 1=M , and the decay time to emit one such photon is b ounded by the uncertainty

principle,

M : (6:1)

It then takes a time

t (6:2)

rem pl

to radiate all of the information in M photons[28,29]. For example, this approximates the

lifetime of the universe for a black hole whose initial mass is that of a typical building. This

implies the third alternative: that the black hole leaves a long-lived, or p erhaps stable,

remnant. An example is exhibited in the Penrose diagram of g. 7: after the Hawking

radiation ceases, a remnant is left at the origin. A qualitative picture of such an ob ject

is shown in g. 8, which shows the geometry of the time slice t in g 7. The remnant

evap

is a long planckian b er with radius r , and the infalling matter concentrated at its

tip. Describing such a con guration certainly requires planckian physics, and it is quite

plausible that this physics allows the remnant to slowly decay.

~l pl

t=t

evap

Fig. 8: A late time slice through g. 7. shows a long Planck sized b er

attached to an asymptotically at geometry.

Each of these three scenarios has staunch advo cates. Each also app ears to violate some

crucial principle in low-energy physics. The resulting con ict is the black hole information

problem. Let's investigate this in more detail.

6.2. Information Loss

Information loss violates quantum mechanics, but worse, it app ears to violate energy

conservation, and this is a disaster. The heuristic explanation for this is that information 22

transmission requires energy, so information loss implies energy non-conservation. Supp ose

for example that we attempt to summarize the formation and evap oration pro cess by giving

a map from the initial density matrix to the nal outgoing density matrix,

=6 S ; (6:3)

f i

where 6 S is a linear op erator. Such op erators, if not quantum mechanical,

6= S S ; (6:4)

f i

typically either violate lo cality or energy conservation [30].

A useful way of explaining the connection b etween information and energy is to

mo del the black hole formation and evap oration pro cess as the interaction of two Hilb ert

spaces[32], H ; H . The rst includes the states of the outside world, and the second the

o h

internal states of the black hole which are thought of as \lost". Supp ose these interact via

a conserved hamiltonian

H = H + H + H (6:5)

o i h

where H acts only on H , H only on H , and the interaction hamiltonian H acts on

o o h h i

b oth. H summarizes the physics that transfers information from the outside Hilb ert space

to the hidden one.

Now, the black hole formation and evap oration pro cess involves loss of information

during a time M . On larger time scales the pro cess can b e rep eated at the same

lo cation; that is, another indep endent black hole formation and evap oration can take place.

In general, wesay that the information loss is repeatable if the information loss in n such

exp eriments is n times the information loss in a single exp eriment

S = nS : (6:6)

n 0

One can plausibly argue [32] if not prove that rep eatable information loss on a time scale

t indicates energy losses

E / 1=t (6:7)

from the observable Hilb ert space (our world) to the hidden one (the interior of the black

hole). Supp ose, for example, that there is no energy loss, H = 0. A simple example for

the hidden Hilb ert space is the states of a particle on a line, H = fjxig, and an example

hamiltonian is

H = O x^ (6:8)

i 23

for some op erator O acting on the observable space. The wavefunction j i of the coupled

system ob eys the Schrodinger equation with hamiltonian (6.5), and the outside density

matrix is de ned as

= tr j ih j : (6:9)

o h

One can readily show [33] that in n scattering exp eriments, the information loss p er ex-

p eriment declines:

=0 : (6:10) lim

n!1

This is not rep eatable information loss.

The explanation for this is familiar from wormhole physics [33,34]. If we started in an

x^ eigenstate,

j i = j ijxi ; (6:11)

then the sole observable e ect of the interaction is a mo di cation of the external hamilto-

nian:

H = H + O x: (6:12)

The eigenvalue x simply plays the role of a coupling constant. If we instead started in a

sup erp osition ofx ^ eigenstates, then we simply have a probability distribution for coupling

constants. For example, supp ose we b egin in the sup erp osition

jx i + jx i : (6:13)

1 2

The hamiltonian do es not change this internal state; there is a sup erselection rule. If we

compute the exp ectation value of some lo cal op erators O ; ;O in this state, it gives

1 n

2 2

j j hx jO O jx i + j j hx jO O jx i : (6:14)

1 1 n 1 2 1 n 2

Performing rep eated exp eriments simply correlates the outside state with the value of the

e ective coupling constant x, that is, measures the coupling constant. Once these are

determined there is no further loss of information.

These arguments generalize to more interesting Hilb ert spaces H , and more general

interaction hamiltonians,

H = O I (6:15)

where O and I act on H and H resp ectively. Finally, note that if H 6= 0 but

o h h

the op erators I only act b etween energy levels with energy separation E , then the

. This indicates a basic information loss will not b e rep eatable on time scales t

connection b etween information loss and energy non-conservation. 24 hard virtual ∆t particle

black hole

Fig. 9: Shown is the contribution of a virtual black hole to two b o dy scat-

tering; information and energy loss are exp ected to contaminate arbitrary

pro cesses through such diagrams.

Worse still, by basic quantum principles, if information and energy loss is allowed,

it will take place in virtual pro cesses. An example is shown in g. 9. For such a virtual

pro cess on time scale t one exp ects energy loss E 1=t.Furthermore, there is no

obvious source for suppression from such pro cesses o ccurring at a rate of one p er Planck

volume p er Planck time with t m . Allowing such energy uctuations is analogous

to putting the world in contact with a heat bath at temp erature T T : this is in gross

contradiction with exp eriment.

Indeed, [30] argued that a general class of lo cal S matrices mimic such thermal uc-

tuations. It can b e shown [32] that these corresp ond to a thermal distribution at in nite

temp erature.

Attempts to turn Hawking's picture of information loss into a consistent scenario are

found in [35] and very recently in [36], where a theory of long-lived remnants emerges (see

section 6.4).

6.3. Information return

Our semiclassical description, together with the assumption that the black hole dis-

app ears, clearly led to information loss | the Hawking radiation at any given time was

thermal with no correlations with either the infalling matter or with earlier Hawking ra-

diation. Furthermore, the semiclassical approximation app ears valid up until the time

when the black hole reaches the Planck mass; b efore this time curvatures are everywhere

small. In the two-dimensional mo del of section 2, the rate of mass loss of the black hole 25

is prop ortional to N , and using the relation (5.29) b etween information and energy, one

nds that if the information is to b e radiated b efore the Planck scale it must b e at a rate

/ N (6:16)

by the time the black hole has reached a fraction of its original mass, say M=10. No such

information return at this order in the large-N expansion is seen.

In fact, there is another argument that the Hawking radiation do esn't contain infor-

mation. For it to do so, the outgoing state would have to b e some mo di cation of the

Hawking state. If weevolve such a state backwards in time until it reaches the vicinityof

the horizon, this state will di er from the lo cal vacuum, and b ecause of the large redshift

the di erence will b e in very high energy mo des. An infalling observer would encounter

these violentvariations from the vacuum at the horizon. This con icts with our b elief that

there is no lo cal way for a freely falling observer to detect a horizon: we could all b e falling

through the horizon of a very large black hole at this very moment, and may not knowit

until we approach the singularity.

y .x . S

infalling

observer

Fig. 10: Lo cality in eld theory implies that observations made at x and y

with spacelike separations must commute.

For a related argument that information return must b egin by this time see [37]. 26

That information is not returned is a consequence of lo cality and causality.To see this,

consider the spacelike slice S of g. 10. Lo cality/causality in eld theory is the statement

that spacelike-separated lo cal observables commute, so

[O (x); O (y )]=0 : (6:17)

This means that up to exp onentially small tails, the state on the slice S can b e decomp osed

\=" j i j i (6:18)

in out

where j i , j i are states with supp ort inside or outside the black hole. As we've

in out

said, infalling observers don't encounter any particular diculty at the horizon, and so

their measurements reveal di erentinternal states j i dep ending on the details of the

collapsing matter, etc. Thus when we trace over internal states to nd the density matrix

relevant to the outside observer, it has missing information, I M . The only obvious

way one could avoid correlations leading to this impurity is if the internal state of the black

hole is unique, i.e.

= j i j i ; (6:19)

0 in out

and in particular is indep endent of the infalling matter. In this case an infalling observer

would b e \bleached" of all information when crossing the horizon. Since there should b e

no way to discern a horizon lo cally this app ears imp ossible.

A p ossible weakness of this argument is the notorious problem of de ning observables

in quantum gravity; Perhaps there are no truly lo cal observables. However, within regions

where the semiclassical approximation is valid one exp ects the observables of quantum

gravity to reduce to ordinary eld theoretic observables, up to small corrections.

An extreme example of what information return in the Hawking radiation entails

comes from the observation that the vicinity of the horizon for a large mass black hole is

well approximated by at space. Indeed, in the limit jr 2M j << 2M , the Schwarzschild

metric

dr 2M

2 2 2 2

(6:20) + r d dt + ds = 1

is well approximated by

2 2 2 2 2

ds = dt + dx +4M d ; (6:21)

16M

For related discussion in two-dimensional mo dels, see [17]. 27

2 2 2

as can b e shown using the substitution x =8M(r2M). For large M ,4M d '

2 2

dy + dz , and the line element (6.21) is that of at Minkowski space as seen by a family

of accelerated observers. This is known as Rindler space, and is shown in g. 11. If the

information from the infalling matter is in the Hawking radiation, it must b e accessible to

observers outside the horizon, and this should also hold true as M !1. Then information

ab out states in the entire left half of Minkowski space is accessible to observers in the right

wedge of Rindler space. If I walk past the horizon and keep going for a billion light-years

to p oint x, the observer at y still has full access to myinternal state. This represents a

gross violation of the lo cality/causality, (6.17).

9 10 light years

accelerated observer

freely falling observer

horizon

Fig. 11: The rightwedge of Minkowski space corresp onds to Rindler space;

it is the region observable by uniformly accelerated observers. Taking the

M !1 limit of the black hole nonlo cality arguments shows that information

at an arbitrary p oint x is fully accessible in the vicinity of the Rindler horizon.

This hyp othetical \holographic" prop erty of spacetime, that all information in a three-

dimensional region is enco ded on the surface of that region, requires drastically new physics

and has b een advo cated by 't Ho oft[26] and more recently by Susskind, Thorlacius, and

Uglum [27]. 't Ho oft e orts involve the hyp othesis that the world is at a fundamental

level similar to a cellular automaton[38]. Susskind et. al.'s attempts instead rely on string

theory. 28

Indeed, unlike p oint particles, strings are intrinsically non-lo cal ob jects. Measure-

ments made in the vicinity of p oints x and y of g. 11 in string theory are exp ected not to

commute. But these violations of lo cality are exp ected to generically fall o exp onentially

fast on the string scale, which is approximately the Planck length,

2 2

(xy ) =`

[O (x); O (y )] e ; (6:22)

and as such are to o small to b e relevant to the information problem. However, the Rindler

observers are moving at huge velo cities relative to those freely-falling, and Lowe, Susskind,

and Uglum [39] argue that these enormous relative b o osts comp ensate for the exp onential

fall o and make

[O (x); O (y )] 1 (6:23)

for suitable ultra high-energy observations at y .

So in string theory, it is p ossible that in some sense the information from the left half

of the world is enco ded in the right half. In fact, all of the information in the Universe

might b e exp ected to b e enco ded in the surface of a piece of chalk, a p ost-mo dern take

on Huxley[40]! However, an op en question is whether it is enco ded in any realistically

accessible fashion | (6.23) would hold only for particular ultra high-energy observations,

and it is not clear that the information available from such unrealistic observations is

imprinted on the Hawking radiation. At present no detailed mechanism to transmit the

information to the Hawking radiation has b een found. In fact, a logical p ossibility is that

the information never escap es from the horizon, and this is consistent with a remnant

picture.

In conclusion, information return requires violations of lo cality/causality. Since lo cal-

ity and causality are at b est p o orly understo o d in string theory, this op ens the p ossibility

of information return. However, an explicit quantitative picture of how this happ ens is yet

to b e pro duced.

6.4. Remnants

If black hole remnants, either stable or long lived, resolve the information problem,

then there must b e an in nite numb er of remnant sp ecies to store the information from

an arbitrarily large initial black hole. For neutral black holes Hawking's calculation fails

at the Planck mass, so this means an in nite sp ecies of Planck-mass particles. These are

exp ected to b e in nitely pair pro duced in generic physical pro cesses | a clear disaster. 29

This statement is most easily illustrated if we imagine that remnants carry electric

charge; we return to the neutral case momentarily. Stating that a remnantischarged is

equivalent to assuming that there is a non-zero minimal-couplingtolow-frequency photons,

as shown in g. 12. By crossing symmetry, this coupling implies that pairs will b e pro duced

bySchwinger pro duction[41] in a background electric eld. The total pro duction rate is

m =q E

Ne (6:24)

vac

where m is the remnant mass, E is the eld strength, and N is the numb er of sp ecies.

Although the exp onential may b e small it is overwhelmed in the case of in nite sp ecies.

Fig. 12: Using crossing symmetry, coupling of a particle to the electromag-

netic eld implies Schwinger pair pro duction.

Neutral remnants have the same problem. For example, the gravitational analog of

Schwinger pro duction is Hawking radiation, for whichwe'd nd a rate

dM 1

/ N (6:25)

dt M

for a black hole to decay. N = 1 gives an in nite rate. Likewise pairs could b e pro duced

in other everydayphysical pro cesses that have sucientavailable energy. Although such

pro duction is highly suppressed by small form factors, this suppression is overcomp ensated

by the in nite states.

6.5. Summary

prop osal principles violated

information loss unitarity, energy conservation

information radiated lo cality/causality

remnants crossing symmetry 30

Table 1

A summary of the prop osed fates of information is shown in Table 1, along with the

corresp onding ob jections. Each of these ob jections can b e phrased solely in terms of low-

energy e ectivephysics, and is apparently indep endentofanyhyp othesized Planck-scale

physics. For this reason this con ict has b een called the information paradox. The matter

would b e reduced to merely a problem if a convincing waywere found whereby Planck-scale

physics mightevade one of these ob jections.

7. Black hole pair pro duction

An asp ect of the quantum mechanics of black holes that is of interest in its own

right is the pair pro duction of black holes. This phenomenon is also of direct relevance to

the information problem. Indeed, let us assume the validity of unitarity (no information

destruction) and lo cality (no information return in Hawking radiation). We've seen that

these imply evap orating black holes leave an in nite variety of neutral remnants. The same

reasoning in the charged sector implies that there are an in nite number of internal states

of an extremal, M = Q, Reissner-Nordstrom black hole. Indeed, assume we b egin with

an extremal hole; wemay then feed it a huge amount of information, say,by dropping

in the planet Earth. The black hole then Hawking radiates backtoM =Qand leaves

an extremal black hole with the extra information enco ded in its internal states. This

pro cess maybecontinued inde nitely, implying an in nite number of internal states of

Reissner-Nordstrom black hole.

In nite states suggest unphysical in nite Schwinger pair pro duction. There are two

p ossibilities. The rst is that a careful calculation of the pro duction rate gives a nite

answer. In this case Reissner-Nordstrom black holes would provide an example of how

to avoid the in nite states/in nite pro duction connection that could likely b e translated

into a theory of neutral remnants with nite pro duction. Alternatively, the assumption

of in nite states may indeed imply in nite pro duction. This then would app ear to rule

out either unitarity or lo cality, and remove the raison d'^etre for neutral remnants.

An instanton exists describing such pair pro duction, but let us rst discuss this pro cess

following Schwinger's original arguments. Since Reissner-Nordstrom black holes | or

neutral remnants | are lo calized ob jects and must have a Lorentz invariant, lo cal and

Dilaton black holes have also b een investigated in this context[42,43]. 31

causal, and quantum mechanical description, they should b e describ ed by an e ective

quantum eld[44,32]. This eld should have a sp ecies lab el,

(x) ; (7:1)

and let these have masses m .We will take these to b e electrically charged with charge

q , although the case of magnetic monop ole pro duction in a magnetic eld is equivalentby

electromagnetic duality. This means that there should b e a minimal coupling interaction

with the electromagnetic eld, plus higher dimension op erators:

4 2 2 2

S = d x jD j m j j + (7:2)

e a a

with D = @ + iq A .

The decay rate into sp ecies a of the vacuum consisting of a background electric eld

with vector p otential A is given by the vacuum-to-vacuum amplitude,

iS A +A;

[ ]

eff 0 a

D A D e

VTiE T

= h0j0i = ; (7:3) e

iS A;

[ ]

eff a

D A D e

where V is the volume, and where wehave separated o electromagnetic uctuations A.

To leading order in an expansion in the charge, this gives

2 2

D + m

0 a

VT = 2Re `n D et

2 2

@ + m

(7:4)

2 2

D + m

0 a

= Re T r `n :

2 2

@ + m a

Fig. 13: Ab ove the slice S (dotted) is shown the lorentzian tra jectories of

a pair of opp ositely charged particles in an electric eld. Below S is the

euclidean continuation of this solution. Matching the euclidean and lorentzian

solutions smo othly along S gives a picture of Schwinger pro duction followed

by subsequentevolution of the pair of created particles.

See, e.g., [45]. 32

The trace can b e rewritten in terms of a single-particle amplitude:

Z Z

4 iH (A )t iH (0)t

VT =Re d x hxje jxihxje jxi (7:5)

with

x R

i d +iq A x_ +

2 2

0 iH (A)t

hx je jxi = D xe : (7:6)

This functional integral has a dominant euclidean saddlep oint corresp onding to circular

motion of the particle in the eld. This instanton is the euclidean continuation of a solution

where a pair of charged particles started at rest run hyp erb olically to opp osite ends of the

background eld, as shown in g. 13. Then if the euclidean tra jectory is cut on the dotted

line, where the velo cities vanish, it smo othly matches to the lorentzian solution: pairs are

created and run to in nity.

The action of the instanton is easily found to b e

; (7:7) S =

giving an approximate decay rate into sp ecies a,

m =q E

e + : (7:8)

Corrections arise b oth from the higher order terms in (7.2) and from subleading quantum

corrections. These are typically subleading in an expansion in qE . The total decay rate is

m =q E

e : (7:9)

An in nite sp ectrum of remnants with nearly degenerate masses clearly gives an in nite

decay rate. Note in particular that if the sp ectrum is of the form m = m +m , where

a 0 a

m m are small mass splittings, then the decay rate is prop ortional to the partition

a 0

function for internal states:

m =q E m

e Tr e ; (7:10)

with =2m =q E .

0 33

Turning now to black holes, the remarkable fact is that solutions analogous to those

in g. 13 are explicitly known [46-50] for charged black holes! Although the solutions

are known for arbitrary strength coupling to a dilatonic eld (as in string theory), here

we'll consider only the simpler case without a dilaton, and for magnetic black holes in a

background magnetic eld. The solutions are (don't worry ab out how they were found!)

G(x)

2 2 1 2 1 2 2

ds = G(y )dt G (y )dy + G (x)dx d ; +

2 2 2 2 2

A (x y ) A (x y )

(7:11)

A = Bqx + k: 1+

Here

2 3 2

G( )=(1 r A )(1 + r A ) r r A ( ) ; (7:12)

+ + i

i=1

with ordered zeros

= ;; ; ; (7:13)

1 2 3 4

r A

and

B G(x)

(x; y ) = (1 + qBx) + : (7:14)

2 2

4A (x y)

Choice of k corresp onds to convention for lo cation of Dirac string singularities. A; B ; r ;r ,

and q are parameters, to b e thought of roughly as the acceleration, the magnetic eld, the

inner and outer horizon radii, and the charge. These identi cations b ecome exact in the

limit qB ! 0. The latter three are related by

q = r r (7:15)

as for a free Reissner-Nordstrom black hole. Furthermore, the charge, magnetic eld, mass

and acceleration should b e related by an expression that simply reduces to Newton's law

mA = qB at low accelerations, B 1. Without this relation the solution has a string-like

singularity connecting the black holes. 34 t

3 y= y= 2 x= x= 4 3 z

Fig. 14: Shown is the lorentzian Ernst solution for an accelerating mag-

netically- charged black hole. The co ordinates used in (7.16) cover only the

unshaded p ortion of the gure.

The solution (7.11) app ears rather complicated, but its structure can b e deduced by

examining various limits. As x ! y the solution asymptotes to

! !

2 2

2 2 2 2

~ ~

B B

2 2 2 2 2 2

ds = 1+ + 1+ dt +dz + d d ;

4 4

(7:16)

B 1

A = ;

2 2

~ ~

where B = B (r ;r ;B) ' B, and where a change of co ordinates has b een made. This

is the Melvin solution[51] | due to the energy density of the magnetic eld, this is the

closest one can come to a uniform magnetic eld in general relativity. The solution (7.11)

naturally yields a Rindler parametrization, and only covers half of the Melvin solution. The

rest is found by continuation. The region near the black hole corresp onds to y ' , and

in this limit the solution asymptotes to the Reissner-Nordstrom solution. The black hole

follows a tra jectory with asymptotes y = , corresp onding to the acceleration horizon.

These features are illustrated in g. 14.

For more details, see e.g. [50]. 35

Analytic continuation, t = i , gives a black hole moving on a circular tra jectory,asin

the lower half of g. 13. Regularity of the solution at the acceleration horizon requires a

sp eci c p erio dicity for as in standard treatments of Rindler space. However, regularity

of the solution at the black hole horizon y = also requires p erio dic identi cation of ,in

general with a di erent p erio d. Demanding that these identi cations match gives another

condition on the parameters,

+ =0 : (7:17)

1 2 3 4

This can b e thought of as a condition matching the Hawking temp erature of the black hole

and the acceleration temp erature from its motion, which is necessary to nd a stationary

solution with the black hole in thermo dynamic equilibrium.

Fig. 15: Asketch of the temp erature vs. mass curve for a Reissner-Nordstrom

black hole.

Fig. 16: The spatial geometry of the time-symme tri c slice through the worm-

hole instanton.

Asketch of the temp erature vs. mass curveisin g. 15. The solution[48,49] to

T = T is given by taking M slightly larger than Q. On the symmetric slice S

BH accel

of g. 13, the three geometry is that of g. 16 {near the black hole it corresp onds to a

symmetric slice through the black hole horizon, as in g. 17. Thus it's geometry is that of

a Wheeler wormhole with ends of charge Q . Another solution[47,50] takes advantage of 36

the ambiguity in the extremal limit M ! Q: here the \throat" near the horizon b ecomes

in nitely long, and the p erio dic identi cation is no longer xed at the horizon. The

geometry of the symmetric slice is sketched in g. 18.

Fig. 17: A p ortion of the Penrose diagram for an M>QReissner-Nordstrom

black hole. Near the black hole, the geometry of g. 16 is similar to that on

the slice S .

Fig. 18: The geometry of the symmetric spatial slice through the extremal

instanton.

Necessary conditions for the validity of the semiclassical expansion are q 1 (sup er-

planckian black holes) and qB 1(weak magnetic elds). To leading order in this

expansion the pro duction rate is given by the instanton action,

inst

: (7:18) / e

One can then show [48,52] that in terms of the physical charge Q and eld B ,

S = (1 + O (QB )); (7:19)

inst

in agreement with (7.7). However, as ab ove one would exp ect the rate to also b e prop or-

tional to the numb er of black hole internal states, whichwehave argued is in nite. 37 late-time slice

- Hawking radiation r=r

horizon

+ r=r

infalling matter

M=Q black hole

Fig. 19: A p ortion of the Penrose diagram of an extremal Reissner-Nordstrom

black hole, into which some matter has b een dropp ed, and which subsequently

evap orates back to extremali ty.

In order to see how the in nite states contribute, it's useful to think ab out their

description. In particular, supp ose we start with an extremal black hole, throw some

matter into it, and let it evap orate back to extremality[53]. The Penrose diagram for this

is shown in g. 19. In describing the evolution we are allowed to cho ose any time slicing,

and from our earlier discussion of the instanton we exp ect a slicing with spatial slices that

stay outside the horizon to b e a useful choice. At long times compared to the evap oration

time scale Q , the mass excess from the infalling matter will have b een re-radiated in

the form of Hawking radiation, and this will asymptote to in nity. The state inside a

nite radius R is therefore that of an extremal black hole until the slice nears the incoming

matter; see g. 20. As t!1, this matter is in nitely far down the black hole throat, and

is in nitely blueshifted. Clearly a description of it using our time slicing requires planckian

physics.

blueshifted matter + gravitational dressing M=Q vacuum solution r =R Hawking radiation

r=0

Fig. 20: Aschematic picture of the state of the black hole of g. 19 as

describ ed on a late time slice that stays outside the horizon. 38

The contribution of such states to the pair-pro duction rate should come from the

functional integral ab out the instanton, or, at linear level, from the uctuation determi-

nant. The reason is that the instanton only describ es tunneling to the classical turning

p oint, but in computing the full tunneling rate we should consider tunneling to the nearby

con gurations with gravitational and matter p erturbations, as describ ed ab ove. Tunneling

to nearby states is via paths near to the instanton, and it can b e shown that contributions

of such paths gives the uctuation determinant at the linear level[54], or the full functional

integral if interactions are included.

The necessity for the outside observer to use Planckphysics in describing these states

hints that p erhaps Planckphysics could play an imp ortant role in computing the pro duc-

tion rate[32], and in particular one might hop e for it to b e nite.

However, a careful examination of the instanton reveals that for weak elds, QB <<1,

the instanton closely approximates the euclidean solution for a free black hole near the

horizon. The contribution to the functional integral from the vicinity of the black hole

should therefore b e essentially the same as that to Tr e from a free black hole, where

the trace is over black hole internal states, H is the hamiltonian, and is given by the

acceleration temp erature. The app earance of such a factor is in agreement with the e ective

eld theory computation, (7.10). If the numb er of black hole internal states is literally

in nite, this factor would b e exp ected to b e in nite as would b e the pro duction rate.

These issues are still b eing investigated [55], and a de nitiveverdict on remnants is not

in. Despite the apparent need for a planckian description of these ob jects, the argument

for in nite pro duction has endangered the viability of remnants. It may b e that black

holes are only in nitely pro duced if they truly have a nite number of internal states.

8. Conclusions

The quantum mechanics of black holes will no doubt remain a fascinating op en window

on Planck scale physics for some time to come. In particular, we can hop e to sharp en our

knowledge of the prop erties of quantum gravity in our continued confrontation with the

information problem. A diversity of opinions on its ultimate resolution ab ounds, and this

suggests the answer might b e quite interesting when nally discovered.

Black hole pair pro duction should either tell us how to kill the remnant scenario or how

to save it. In either case, it remains a rich and interesting pro cess, nontrivially combining 39

the phenomena of Hawking and acceleration radiation as well as other asp ects of quantum

gravity.

The quantum mechanics of black holes has much left to teach us.

Acknowledgments

I wish to thank the organizers for inviting me to Trieste to participate in this most

interesting scho ol. I have b ene ted from discussions with T. Banks, G. Horowitz, D. Lowe,

W. Nelson, M. Srednicki, A. Strominger, L. Susskind, E. Verlinde, H. Verlinde, and many

others. This work was supp orted in part by DOE grant DOE-91ER40618 and by NSF PYI

grant PHY-9157463. 40

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