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Commun. math. Phys. 43, 199--220 1975)

@ by Springer-Verlag 1975

Particle Crea tion by lack H ole s

S. W. Hawking

Department of Applied Mathemat ics and Theoretical Physics, University of Cambridge,

Cambridge, England

Received April 12, 1975

Abstract In the classical theory black holes can only absorb and not emit particles. However it

is shown that quantum mechanical effects cause black holes to create and emit particles as if they

were hot bodies with temperature ~ ~ 10 where ~ is the surface gravity of the black

hole. This thermal emission leads to a slow decrease in the mass of the black hole and to its eventual

disappearance: any primordial black hole of mass less than about 10 is g would have evaporated by

now. Although these quantum effects violate the classical law that the area of the event horizon of a

black hole cannot decrease, there remains a Generalized Second Law: S Anever decreases where S

is the entropy of matter outs ide black holes and A is the sum of the surface areas of the event horizons.

This shows that gravitational collapse converts the baryons and leptons in the collapsing body into

entropy. It is tempting to speculate that this might be the reason why the Universe contains so much

entropy per baryon.

t.

Although there has been a lot of work in the last fifteen years see [1, 2] for

recent reviews), I think it would be fair to say that we do not yet have a fully

satisfactory and consistent quantum theory of gravity. At the moment classical

General Relativity still provides the most successful description of gravity. In

classical General Relativity one has a classical metric which obeys the Einstein

equations, the right hand side of which is supposed to be the energy momentum

tensor of the classical matter fields. However, although it may be reasonable to

ignore quantum gravitational effects on the grounds that these are likely to be

small, we know that quantum mechanics plays a vital role in the behaviour of

the matter fields. One therefore has the problem of defining a consistent scheme

in which the space-time metric is treated classically but is coupled to the matter

fields which are treated quantum mechanically. Presumably such a scheme would

be only an approximation to a deeper theory still to be found) in which space-

time itself was quantized. However one would hope that it would be a very good

approximation for most purposes except near space-time singularities.

The approximation I shall use in this paper is that the matter fields, such as

scalar, electro-magnetic, or neutrino fields, obey the usual wave equations with

the Minkowski metric replaced by a classical space-time metric g,b. This metric

satisfies the Einstein equations where the source on the right hand side is taken

to be the expectation value of some suitably defined energy momentum operator

for the matter fields. In this theory of quantum mechanics in curved space-time

there is a problem in interpreting the field operators in terms of annihilation and

creation operators. In flat space-time the standard procedure is to decompose

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200 S.W. Hawking

the field into po sitive a nd negative freq uenc y co mp one nts. Fo r example, if ~b is

a massless Hermitian scalar f ield obeying the equation dp;abrfab=0 one expresses

~b as

= + al} 1.1)

where the {f~} are a compl ete o rth ono rm al family of comple x valued solutions

of the wave equat ion

f~;,brlab= 0

which contain only positive frequencies with

respect to the usual M ink ows ki t ime coordinate. The opera tors a i and a~ are

interpreted as the annihilation and creation operators respectively for particles

in the i th state. The vacu um state t0) is defined to be the state from which one

cannot annihilate any particles, i.e.

ailO) --0 for all i.

In curved space-time one can also consider a Hermitian scalar f ield operator

~b which obeys the covariant wave e quat ion q~;,bg b=0. How ever one c anno t

decompose into i ts positive and negative frequency parts as positive and negative

frequencies have no invariant meaning in curved space-time. One could sti l l

require that the {f } and the {j~} together formed a complete basis for solutions

of the wave equations with

iSs(fifj; ~-

~ f~;,)d Z = bij (1.2)

where S is a suitable surface. However condition (1.2) does not uniquely fix the

subspace o f the space of all solutions which is spanne d by the {~} and therefore

does not determ ine the split t ing of the oper ator q~ into annihilation and crea tion

parts . In a region of space-time which was flat or asymptotically flat , the appro-

priate criterion for choosing the {f/} is that they should contain only positive

frequencies with respect to the Mi nkow ski t ime coordinate. How ever if one has

a space-time which contains an initial fiat region (1) followed by a region of

curv atur e (2) then a final flat region (3), the bas is { fli} which contai ns only posi tive

frequencies on region (1) will not be the same as the basis {fal} which contains

only positive frequencies on region (3). This m eans that the init ial va cuum state

T01) the state which satisfies al i l 01 )= 0 for each initial annihilation oper ator

ali,

will not be the same as the final vacuum state f03) i.e. a3~101):#0. One can

interpret this as implying that the t ime dependent metric or gravitational f ield

has cause d the creatio n of a certain nu mb er of particles of the scalar field.

Alth ough it is obvio us w hat the subs pace spann ed by the {f~} is for an asy mpto -

tically flat region, it is no t u niqu ely defined for a general poin t of a curv ed spac e-

time. Consider an observer with velocity vector

v

at a point p. Let B be the least

upper bound R~bcat in any or thonormal te t rad whose t imel ike vector coincides

with v . In a nei gh bo ur ho od U of p the obser ver ca n set up a local inertial co-

ordinate system (such as norm al coordinates) with coordina te radius of the order

of B - L He can then c hoose a family {f } which satisfy equat ion (1.2) and w hich

in the neighbourhood U are approximately positive frequency with respect to

the t ime coord inate in U. For mod es fi whos e characteristic frequency co is high

co mp ar ed to B ~, this leaves an ind eter min acy betw een .f~ and its com plex con-

jugate ~ of the order o f the exponential of some multiple of - co B -~. The indeter-

minacy betwe en the annihilation oper ator a~ and the creation oper ator

a[

for the

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Particle Creation by Black Holes

201

mode is thus exponentially small. However, the ambiguity between the a~ and

the a[ is virtually complete for modes for which co < B~. This ambiguity introduces

an uncertainty of + in the number operator a~ a~ for the mode. The density of

modes per unit volume in the frequency interval co to co + dco is of the order of

coZdco

for co greater than the rest mass m of the field in question. Thus the un-

certainty in the local energy density caused by the ambiguity in defining modes

of wavelength longer than the local radius of curvature B -~, is of order B 2 in

units in which G= c= h= 1. Because the ambiguity is exponentially small for

wavelengths short compared to the radius of curvature B- ~, the total uncertainty

in the local energy density is of order B z This uncertainty can be thought of as

corresponding to the local energy density of particles created by the gravitational

field. The uncertainty in the curvature produced via the Einstein equations by

this uncertainty in the energy density is small compared to the total curvature

of space-time provided that B is small compared to one, i.e. the radius of curvature

B -~ is large compared to the Planck length 10 .33 cm. One would therefore

expect that the scheme of treating the matter fields quantum mechanically on a

classical curved space-time background would be a good approximation, except

in regions where the curvature was comparable to the Planck value of 1066 cm 2.

From the classical singularity theorems [3-6], one would expect such high cur-

vatures to occur in collapsing stars and, in the past, at the beginning of the present

expansion phase of the universe. In the former case, one would expect the regions

of high curvature to be hidden from us by an event horizon [7]. Thus, as far as

we are concerned, the classical geometry-quan tum matter treatment should be

valid apart from the first 10 .43 s of the universe. The view is sometimes expressed

that this treatment will break down when the radius of curvature is comparable

to the Compton wavelength ~10-13cm of an elementary particle such as a

proton. However the Compton wavelength of a zero rest mass particle such as

a photon or a neutrino is infinite, but we do not have any problem in dealing

with electromagnetic or neutrino radiation in curved space-time. All that hap-

pens when the radius of curvature of space-time is smaller than the Compton

wavelength of a given species of particle is that one gets an indeterminacy in the

particle number or, in other words, particle creation. However, as was shown

above, the energy density of the created particles is small locally compared to the

curvature which created them.

Even though the effects of particle creation may be negligible locally, .I shall

show in this paper that they can add up to hove a significant influence on black

holes over the lifetime of the universe ~1017s or 106o units of Planck time.

It seems that the gravitational field of a black hole will create particles and emit

them to infinity at just the rate that one would expect if the black hole were an

ordinary body with a temperature in geometric units of ~/27c, where ~ is the

surface gravity of the black hole [8]. In ordinary units this temperature is of

the order of 1026M - 1 K, where M is the mass, in grams of the black hole. For

a black hole of solar mass (1033 g) this temperature is much lower than the 3 K

temperature of the cosmic microwave background. Thus black holes of this size

would be absorbing radiation faster than they emitted it and would be increasing

in mass. However, in addition to black holes formed by stellar collapse, there

might also be much smaller black holes which were formed by density fluctua-

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202 S W Hawking

tions in the early universe [9, 101. These small black holes, being at a higher

temperature, would radiate more than they absorbed. They would therefore pre-

sumably decrease in mass. As they got smaller, they would get hotter and so

would radiate faster. As the temperature rose, it would exceed the rest mass of

particles such as the electron and the muon and the black hole would begin to

emit them also. When the temperature got up to about 1012 K or when the mass

got down to about 1014 g the number of different species of particles being emitted

might be so great [11] that the black hole radiated away all its remaining rest

mass on a strong interaction time scale of the order of 10 -23 s. This would pro-

duce an explosion with an energy of 1035 ergs. Even if the number of species of

particle emitted did not increase very much, the black hole would radiate away

all its mass in the order of 10-28M 3 s. In the last tenth of a second the energy

released would be of the order of 103o ergs.

As the mass of the black hole decreased, the area of the event horizon would

have to go down, thus violating the law that, classically, the area cannot decrease

[-7, 12]. This violation must, presumably, be caused by a flux of negative energy

across the event horizon which balances the positive energy flux emitted to

infinity. One might picture this negative energy flux in the following way. Just

outside the event horizon there will be virtual pairs of particles, one with negative

energy and one with positive energy. The negative particle is in a region which

is classically forbidden but it can tunnel through the event horizon to the region

inside the black hole where the Killing vector which represents time translations

is spacelike. In this region the particle can exist as a real particle with a timelike

momentum vector even though its energy relative to infinity as measured by the

time translation Killing vector is negative. The other particle of the pair, having

a positive energy, can escape to infinity where it constitutes a part of the thermal

emission described above. The probability of the negative energy particle tun-

nelling through the horizon is governed by the surface gravity ~c since this quantity

measures the gradient of the magnitude of the Killing vector or, in other words,

how fast the Killing vector is becoming spacelike. Instead of thinking of negative

energy particles tunnelling through the horizon in the positive sense of time one

could regard them as positive energy particles crossing the horizon on past-

directed world-lines and then being scattered on to future-directed world-lines by

the gravitational field. It should be emphasized that these pictures of the mecha-

nism responsible for the thermal emission and area decrease are heuristic only

and should not be taken too literally. It should not be thought unreasonable that

a black hole, which is an excited state of the gravitational field, should decay

quantum mechanically and that, because of quantum fluctuation of the metric,

energy should be able to tunnel out of the potential well of a black hole. This

particle creation is directly analogous to that caused by a deep potential well in

flat space-time [18]. The real justification of the thermal emission is the mathe-

matical derivation given in Section 2) for the case of an uncharged non-rotating

black hole. The effects of angular momentum and charge are considered in

Section 3). In Section 4) it is shown that any renormalization of the energy-

momentum tensor with suitable properties must give a negative energy flow

down the black hole and consequent decrease in the area of the event horizon.

This negative energy flow is non-observable locally.

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Particle Creation by Black Holes

203

The dec rease in a rea o f t he even t ho r i zon is caused by a v io l a t ion o f t he weak

energy cond i t ion [5 7, 12] wh ich a r ise s f rom the inde te rmina cy o f pa r t i c le num-

ber and ene rgy dens i ty in a cu rved space - time . H oweve r , a s was show n above ,

thi s i nde te rm inacy i s sma ll , be ing o f t he o rde r o f B 2 where B is t he m agn i tu de

of the cu rvat ure tensor . Th us i t can have a d iv erg ing effec t ion a nul l surface like

the even t ho r i z on wh ich has ve ry sma l l convergence o r d ive rgence bu t i t can no t

un t rap a s t rong ly converg ing t rapped su r face un ti l B becomes o f t he o rde r o f

one . There fo re one wou ld no t expec t t he nega t ive ene rgy dens i ty to cause a

b re akd ow n o f t he c l ass ica l singu la r it y t heo rem s un ti l t he rad ius o f cu rva tu re o f

space- t im e beca m e 10- 33 cm.

Perhaps the s t rongest reason for be l iev ing tha t b lack holes can crea te and

em it par t ic les a t a s teady ra te is tha t the pre dic ted ra te i s jus t tha t o f the therm al

emiss ion o f a b ody wi th the t emp era tu re /2rc. T here a re i ndependen t , t he rm o-

dynamic , g rounds fo r rega rd ing som e m ul t ip l e o f t he su r face g rav i ty a s hav ing

a c lose re l a t ion to t emp era tu re . The re i s an ob v ious ana logy wi th the second l aw

of the rm odyn am ics in t he l aw tha t , c l ass ica lly, t he a rea o f t he even t ho r i zon can

never dec rease and tha t when two b lack ho les co l l i de and merge toge the r , t he

area o f the f ina l event ho rizo n i s grea te r than the sum of the areas o f the two

orig inal hor iz ons [7 , 12]. The re i s a lso an ana logy to the f i rs t law of ther mo -

dynamics in t he re su l t t ha t two ne ighbour ing b l ack ho le equ i l i b r ium s t a t e s a re

re l a t ed by [8 ]

dM = i~ dA f2dJ

where M, f2 , and J a re respec t ive ly the :mass, angular ve loci ty and angula r mo -

me n tum o f t he b l ack ho le and A is the a rea o f t he even t ho r izon . Co mp ar ing thi s t o

d U = T d S p d V

one sees tha t i f some mul t ip le of A is reg arde d as be ing analo gous to en t ropy ,

then som e m ul t ip le o f ~c is analogo us to tem pera ture . Th e surface gravi ty is a lso

ana logous to t empera tu re in t ha t i t i s cons t an t ove r t he even t ho r i zon in equ i -

l ibr ium. B eckenste in [19] sug gested tha t A a nd ~c were not m ere ly analo gous to

en t r opy and t em pera tu re re spec t ive ly bu t t ha t , i n some sense , t hey ac tua l ly were

the en t rop y and t em pera tu re o f t he b lack ho le. A l though the o rd ina ry second

law o f t he rm ody nam ics i s t ranscend ed in tha t en t rop y can be lo s t down b lack

ho les , t he f low o f en t ro py ac ross t he even t ho r i zo n w ou ld a lways cause some

increase in the area of the horizo n . B eckenste in theref ore suggested [20] a Gen-

era l ized Second Law : E nt ro py + some m ul t ip le unspecif ied) of A never decreases .

However he d id not suggest tha t a b lack hole could emi t par t ic les as wel l as

abso rb them. Wi thou t such emiss ion the Genera l i zed Second Law wou ld be

v io l a ted by fo r example , a b l ack ho le imm ersed in b l ack b ody rad ia t ion a t a l ower

t em pera tu re than tha t o f t he b l ack ho le . On the o the r hand , i f one accep ts t ha t

b lack holes do em i t par t ic les a t a s teady ra te , the ident i f ica t ion of ~/2rc wi th tem -

pera tu re and A wi th en t ropy i s e s t ab l i shed and a Genera l i zed Second Law

confi rmed.

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2 Gravitational Collapse

It is now generally believed that, according to classical theory, a gravitational

collapse will produce a black hole which will settle down rapidly to a stationary

axisymmetric equilibrium state characterized by its mass, angular momentum

and electric charge [-7, 13]. The Kerr-Newman solution represent one such family

of black hole equilibrium states and it seems unlikdy that there are any others.

It has therefore become a common practice to ignore the collapse phase and to

represent a black hole simply by one of these solutions. Because these solutions

are stationary there will not be any mixing of positive and negative frequencies

and so one would not expect to obtain any particle creation. However there is

a classical phenomenon called superradiance [.14-17] in which waves incident

in certain modes on a rotating or charged black hole are scattered with increased

amplitude [see Section (3)]. On a particle description this amplification must cor-

respond to an increase in the number of particles and therefore to stimulated

emission of particles. One would therefore expect on general grounds that there

would also be a steady rate of spontaneous emission in these superradiant modes

which would tend to carry away the angular momentum or charge of the black

hole [.16]. To understand how the particle creation can arise from mixing of

positive and negative frequencies, it is essential to consider not only the quasi-

stationary final state of the black hole but also the time-dependent formation

phase. One would hope that, in the spirit of the no hair theorems, the rate of

emission would not depend on details of the collapse process except through the

mass, angular momentum and charge of the resulting black hole. I shall show

that this is indeed the case but that, in addition to the emission in the super-

radiant modes, there is a steady rate of emission in all modes at the rate one

would expect if the black hole were an ordinary body with temperature ~/2~z.

I shall consider first of all the simplest case of a non-rotating uncharged black

hole. The final stationary state for such a black hole is represented by the

Schwarzschild solution with metric

ds2= - 1 - 2~Mr dtZ + 1 - 2~Mr - l dr2 + r2 dO2+ sin2 Od4)2) (2.1)

As is now well known, the apparent singularities at r= 2M are fictitious, arising

merely from a bad choice of coordinates. The global structure of the analytically

extended Schwarzschild solution can be described in a simple manner by a

Penrose diagram of the r-t plane (Fig. 1) [6, 13]. In this diagram null geodesics

in the r-t plane are at +_45 to the vertical. Each point of the diagram represents

a 2-sphere of a r e a 4g r 2 A conformal transformation has been applied to bring

infinity to a finite distance: infinity is represented by the two diagonal lines (really

null surfaces) labelled J + and J - , and the points I +, I - , and I . The two hori-

zontal lines r= 0 are curvature singularities and the two diagonal lines r= 2M

(really null surfaces) are the future and past event horizons which divide the

solution up ihto regions from which one cannot escape to J + and J- . On the

left of the diagram there is another infinity and asymptotically flat region.

Most of the Penrose diagram is not in fact relevant to a black hole formed

by gravitational collapse since the metric is that of the Schwarzchild solution

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Part icle Crea t ion by Black Holes

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r =0 s ingu lar i ty i +

l o

r = 0 s in gu lar i ty 17

Fig. 1. The Penro se diagra m for the analytically extended Sch warzschild solutio n

r=O s ing ular i ty I * sur face of

~ . - col lap sed body

/ / / / / ~ ~ ]

1 ~ ~ l -

r gion not appl icable to

a grav i ta t ional co l lapse

Fig 2. Only the region of the Schwarzschild solut ion outside the col lapsing body is relevant for a

black hole formed by gravi tat ional col lapse. Inside the body the solut ion is completely different

s ingu la r i t y ~ even t hor i zon

r=O

o r i g i n o f /

coordinates

\col laps ing body

Fig 3. The Penrose diagram of a spherical ly symmetric col lapsing body p roducing a black hole. The

vert ical dot ted l ine on the left represents the non-sing ular centre of the bod y

only in the region outside the collapsing ma tter an d only in the asym ptotic future.

In the case of exactly spherical collapse, which I shall co nsider for simplicity, the

metric is exactly the Schwarzchild metric ev erywhere outside the surface of the

collapsing object which is represented by a timelike geodesic in the Penrose

diag ram Fig. 2). Inside the object the m etric is comple tely different, the past

event horizon, the past r = 0 singulari ty and the oth er asymptotically flat region

do no t exist and are replaced by a time-like curve representing the origin of polar

coordinates. The appro priate Penrose diagram is shown in Fig. 3 where the con-

formal freedom has been u sed to m ake the origin of polar coordinates into a

vertical line.

In this space-time consider again for simplicity) a massless He rmitia n scalar

field operator ~ obeying the wave equation

~;~bgab=0. 2.2)

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206 S.W. Hawking

(The resul ts obta i ned w ould be the same i f one used the c onform ally invar ian t

wave equa t ion :

~b .b + ~ R

,abg ~ p = 0 .)

Th e o pe rat or 4~ can be expressed as

O= Z,{ f , , , , f j i } . 2.3)

The solu t ions {f} of the wave eq uat i on f~;abg b = 0 can be chose n so tha t on past

nu ll in f in ity J - they fo rm a comple te fami ly sa t is fy ing the o r tho norm al i ty con-

di t ions (1.2) where the surface S is J - and so tha t they conta in only posi t ive

frequencies with respect to the canonical a ff ine para me ter on J - . (This las t con-

di t ion of posi t ive f requency can be un iquel y def ined despi te the exis tence of

super t rans la t ions in the Bond i -M etzner -Sachs a sym pto t ic sym me try g roup

[21, 22] . ) The opera tors a l and a~ have the natura l in terpre ta t ion as the annihi-

la t ion and crea t ion opera tors for ingoing par t ic les i .e . for par t ic les a t pas t nul l

in f in i ty J - . Because mass less f ie lds a re comple te ly de te rmined by the ir da ta on

J - , the ope ra to r 4~ can be expressed in the form (2 .3) everywhere . In the region

outs ide the event hor izon one can a lso determ ine massless fie lds by the ir da ta on

the event hor izo n and on fu ture nul l inf in ity J + . Th us one can a lso express q~

in the form

= 2 i {Pib i + Pi b~i + qi ci

+ c~c/*}. (2.4)

Her e the {p~} are so lutions of the wa ve equ ati on wh ich are pu rely o utgo ing, i.e .

they have zero Cau ch y data on the event hor izo n an d the {q~} are so lu t ions which

con ta in no ou tgo ing componen t , i . e . they have ze ro Cauchy da ta on J+ . The

{Pi} and {q~} are required to be com plete famil ies sa t is fy ing the o r th on orm ali ty

cond i t ions (1.2) where the surface S is take n to be J + and the event hor i zon

respectively. In add itio n the {Pi} are requ ired to co nta in o nly positive frequencies

with respect to the cano nical a ff ine par am eter a long the nul l geodesic genera to rs

o f ~+. W ith the posi t ive f requency con di t io n on {pi} , the ope ra tors {bi} and {b~}

can be in terpre ted as the annihi la t ion and crea t ion opera tors for outgoing par-

ticles, i.e. for particles on o+. I t i s not c lear whether one should impose some

posi t ive f requency cond i t ion on the {qi} and i f so with respect to w hat . The choice

of the {qi} does no t affect the ca lcula t io n of the emiss ion of par t ic les to J + . I shal l

return to the question in Section (4).

Because massless f ie lds are com plete ly determ ined by the ir da ta on J- one

can express {Pi} and {qi} as l inear c om bin ati on s of the {f~} an d {~}:

P i = Z j ~ i j f j + f i i j ~ ) , (2.5)

q i = ~ _ , ~ T i J j+ r h j L ) .

(2.6)

These re la t ions lead to corresponding re la t ions between the opera tors

b i = 2j ~i j a j - - ~i j

aJ ,

(2.7)

ci = ~_,i ~ij a i - Viii a~j).

(2.8)

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Particle Creation by Black Holes 207

The in i t ia l vacu um sta te ]01 , the s ta te c onta in ing n o incom ing par tic les , i.e .

no par t ic les on J - , is def ined by

a i l 0 ) = 0 for a l l i . (2.9)

Ho wev er, because the coefficients/~ij wilt no t be z ero in general, the init ial vacu um

sta te wil l not appear to be a vacuum sta te to an observer a t J+. Ins tead he wil l

f ind tha t the expec ta t ion va lue o f the num ber opera to r fo r the i th ou tgo in g mode i s

(0 _ [bf b, 10_ ) = Yj[31j[ 2. (2.10)

Thus in o rde r to de te rm ine the nu mb er o f pa r tic le s c rea ted by the g rav i ta t iona l

f ie ld and emit ted to inf in i ty one s imply has to ca lcula te the coeff ic ients

flij.

O n e

would expect th is ca lcula t ion to be very messy and to depend on the deta i led

natur e of the gravi ta t io nal col lapse . How ever , as I shal l show, one can der ive an

asym ptot ic form for the fl~i which d epends only on the surface gravi ty of the

resul t ing b lack hole . There will be a cer ta in f in ite am ou nt of par t ic le crea t ion

which depends on the deta i ls of the col lapse . These par t ic les will d isperse an d a t

la te re tarded t imes on J + there wil l be a s teady f lux of par t icles de termined b y

the as ym ptot i c form of flij.

In o rde r to ca lcu la te thi s a sym pto t ic fo rm i t is more conven ien t to decompose

the ingo ing and ou tgo ing so lu t ions o f the wave equa t ion in to the i r Four ie r com-

ponen ts wi th re spec t to advanced o r re ta rded t ime and use the con t inuum nor -

maliza t ion . The f in i te normaliza t ion so lu t ions can then be recovered by adding

Fourier components to form wave packets . Because the space- t ime is spher ica l ly

symmetr ic , one can a l so decompose the incoming and ou tgo ing so lu t ions in to

spher ica l harmonics . Thus , in the region outs ide the col lapsing body, one can

wr i te the incoming and ou tgo in g so lu t ions a s

f~',m = (2u) - r -

l(co')-}I~;,(r)eiY~Ytm(O, ),

(2.11)

po~t,,=(2~)-~r- lco-P~o(r)ei~UY~m(O (o),

(2.12)

where v and u are the usual advanced and re tarded coordinates def ined by

21-~M- 1 , (2.13)

= t + r + 2 M l o g

u = t - r - 2 M

log 2M - 1

(2.14)

Each so lu t ion

Po~m

can be expressed as an integral with respect to co ' over solu-

tion s fo~,zm an d f~'lm w ith the sam e value s of l an d Lmi (from no w on I sh all dr op

the suffices I, m):

P~ = [. ~(c oy foy + flo~,L,)dco' .

(2.151

To calc ulate the coefficients c ~ , an d flo~o,, con sid er a so lut ion p~ pro pa ga tin g

backw ards f rom J + with zero Ca uc hy data on the event hor izon. A par t p(o} of

the so lu t ion

P,o

wilt be scattered by the static Schwarzchitd field outside the col-

lapsing bod y and will end up on J - with the same frequ ency co. This will g ive

a c~(co'-co) term in c ~,. Th e r em ai nd er p(21 of p~ will en ter the colla psin g bo dy

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2 08 S . W . H a w k i n g

event horizon

surfaces of

Jco n s ta n t p h a se

V V

F i g . 4 . T h e s o l u t i o n p ~ o f t h e w a v e e q u a t i o n h a s a n i n f i n i te n u m b e r o f c y c l e s n e a r t h e e v e n t h o r i z o n

an d n ea r t h e s u r f ace v = v o

where i t wil l be par t ly sca t tere d and par tlY ref lec ted thr oug h the cent re , eventu al ly

emer ging to J - . I t i s th is par t p~) which produ ces the in teres t ing effec ts . Because

the re t a rde d t ime coor d ina t e u goes to i n f in ity on the even t ho r i zon , t he su r faces

of con stan t phase of the so lu t ion Po, wi ll p ile up near the eve nt ho rizo n (F ig . 4).

To an obse rve r on the co l laps ing bod y the wave wou ld seem to have a ve ry l a rge

blue-sh i ft . Because i t s e ffec tive f requ ency was v ery h igh, the wave w ould pro pa -

g a te b y g e o me t r i c o p t ic s t h ro u g h t h e c e n t re o f t h e b o d y a n d o u t o n J - . O n

~-p~2) wou ld h ave an inf in i te num be r of cycles jus t before the ad van ced t ime

v= Vo where Vo is the la tes t t ime tha t a nul l geodesic could leave J- , pa ss th ro ug h

t h e c e n tr e o f t h e b o d y a n d e s c ap e t o J + b e fo re be i n g tr a p p e d b y t h e e v e n t

ho r i zon . O ne can e s t ima te the fo rm o f _(2)

' ,o on J - nea r v= Vo in the fo l lowing

way . Le t x be a po in t on the even t ho r i zon ou t s ide the ma t t e r and l e t 1 be ' a nu l l

vector tang ent to the hor izon . Le t n a be the fu tu re-d i r ec ted nul l vec tor a t x wh ich

i s d i rec t ed rad ia l ly inwards and no rmal i zed so tha t I n , = - t . T h e v e c t o r -~ n

(e sma l l and pos it ive ) w il l conne c t t he po in t x o n the eve n t ho r i zo n wi th a nea rby

nu ll su r face o f cons t an t re t a rde d t ime u and the re fo re wi th a su r face o f cons t an t

phase of the so lu t ion p~). I f the vectors l and n a are para l le l ly t rans por ted a long

the nu l l geodes i c ~ th ro ugh x w h ich gene ra te s the ho r i zon , t he vec to r -e n will

a lways conn ect the event ho rizo n wi th the sam e surface of con stan t phase ~r ~(2)

vl [ ~o

T o see wh at the re lat io n betw een e and the p has e of p(o ) is , ima gine in Fig. 2 tha t

the co ll aps ing bod y d id no t ex i s t bu t one an a ly t i ca lly con t inued the emp ty space

Schwarzch i ld so lu t ion back to cover t he who le Pen rose d i ag ram. One cou ld then

t ran spo rt the pa i r (P, n ) back a long to the poin t wh ere fu ture an d pas t eve nt

ho r i zons in t e rsec t ed . Th e vec to r - en~ wou ld then l ie a long the pas t even t ho r i zon .

Le t 2 be the a f f ine pa ramete r a long the pas t even t ho r i zon wh ich i s such tha t a t

dx ,~

t he po in t o f in t e rsec t ion o f the two ho r i zons , 2 = 0 a nd -d ~- = n . The a ff ine pa r -

amete r 2 i s re l a t ed to the re t a rded t ime u on the pas t ho r i zon by

2 = - Ce- (2.16)

where C i s constan t and ~c is the surface gra vi ty of the b lack hole def ined by

K ,b K b= - - xK on the ho r i zon where K ~ i s t he t ime t rans l a t ion Ki l l ing vec to r.

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P ar t i c l e Crea t ion by B lack H o les

209

1)

For a Schwarzchi ld b lack hole K=~-M . I t fo l lows f rom th is tha t the vector

- e n a

connects the fu ture event hor izon with the surfaoe of constant phase

~ (loge - log C) of the solu tio n p~Y). Th is res ult w ill also h ol d in the real space-

t ime ( inc luding the col lapsing body) in the region outs ide the body. Near the

event hor izon the solu t ion p~) wil l obey the geometr ic opt ics approximation as

i t passes thro ugh the b ody because i ts effec t ive f requency will be very h igh. This

mea ns th a t i f one extends the nul l geodesic 7 back past the end-p oint o f the event

hor izon and ou t on to J - a t v = vo and para l le l ly t rans ports n ~ a long 7, the vector

en

~ will s ti ll connect 7 to a surface of consta nt phase of the solu t ion

. 2)

O n

to~

J - n a

will be parallel to the Kill in g vector K ~ whic h is tang en t to the null geodesic

genera to rs o f d - :

n ~ = D K ~

Thus on J - fo r v o - v smal l and posi tive, the phase o f the so lu tion wil l be

co ( log(v o - v ) - l o g D - logC ) . (2 .17)

K

-- (2)

Th us on p~ will be zero for v > Vo an d for v < v0

C D

~

(2r0 co r Po~exp lo g (2.18)

where Pg ~ Po~(2M) is the va lue o f the ra dial func tion for P~ o n the past event

hor izo n in the analyt ica l ly con t inu ed Schw arzchi ld solu t ion . The express ion

(2.18) for p~) is valid onl y for Vo - v small a nd positive. At earlier adv anc ed times

the amp li tude will be d if ferent and the f requen cy meas ured with respect to v, will

approach the original frequency c0.

By Fourier t ransforming p~) one can evaluate i ts contr lbut ions to ~o~, and

fi,o~,. Fo r large values of co these will be determ ine d b y the asy m pto tic form

(2.18). T hu s for l arge co

1 ~- ~ C O tO ~ - 1 +- ~-

~(2) ~(2 ~) - P ~ ( C D ) ex p(t (co - co )vo F 1 - - i w ) (2 .19)

t OXo

fl(2) ~ _ i ~ ) ,~ ) (2.20)

OCO ~ (- -

The solu t io n p~) is zero on J - for la rge values ofv . This mean s tha t i ts Fou rier

t ransfo rm is analy t ic in the upper ha lf co p lane and tha t p~) will be correc t ly

represented by a Fou rier in tegra l in which the co ntou r has been d isplaced in to the

upper ha l f co p lane. The Fou r ie r t r ans fo rm of p~) con ta ins a fac to r ( -

i c o ' ) - ~ + ~ -

whic h ha s a loga rith mi c sing ulari ty at c o = 0. To obt ain R(2) from ~/(2) by (2.20)

J ~0(,0 ~ ~O3O ~

on e ha s to an al yt ic al ly co n ti n ue ~o~ (z)o~,ant ic lockwise round th is s ingular i ty , This

means tha t

~,o,, ex rcco (2)

(2.21)

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210 S W t t a wk ing

Actually, t he fact that p(o ) is not given b y (2.18) at ear ly a dva nce d times mea ns

that the singularity in c~,o,, occur s at co'=co and not at c# = 0. How eVer the rela-

tion (2.2t) is still valid for large co'.

The expectation value of the total nu mbe r of created particles at J + in the

frequency range co to co+dco is dcoS~ Ifio,~,,lZdco. Be ca us e lfi~,,,,l goe s like (co') -}

at large co' this integral diverges. This infinite tota l n um be r of creat ed particle s

corre spon ds to a finite steady rate of emission continuing for an infinite t ime as

can be seen by bui ld ing up a complete or tho norm al family of wave packets f rom

the Fourier com pon ents p~,. Let

PJ, = e- ~ ~5~+1)~e- 2rclne- ~O, o,dco (2.22)

where j and n are integers, j > 0 , e> 0. Fo r e small these wave packets will have

frequency je and wil l be peaked aro und re tarded t ime u= 2tone-1 with width e-1.

On e can e xpa nd {pj,} in terms of the {fo,}

PJ, = S~ (c~,o, L + fij ,o ,'L ') dco' (2.23)

where

o~jn~,=e-SJJ+*)~e-i='ne-'o:~o,dco etc. (2.24)

F or j>>e, n>>e

i c o ~

[c~jno~,l -= (272)- 1P~ co-~ F (1 - ~ -) g - ~(co ) -~

~j+l)~ logco')dco

~ exp ico (- 2tone- a + to- 1

= 7r-IP~,CO-~F(1- ~ ) ~-~ (co ')-~ z- 1 sinez (2.25)

where co=j e and z= -1 logco ' -2rcne - I . For w ave-packets which reach J + a t

late retarded times, i.e. those with large values of n, the main contribution to

c~j.,,, an d fia,o,' co m e f ro m ve ry h igh fre qu enc ies co' of the or de r of exp(2rcn~ae- 1).

This means that these coefficients are governed only by the asymptotic forms

(2.19, 2.20) for high co' whic h are inde pen dent of the details of the collapse.

The expectatio n value of the num ber of particles created and emitted to

infinity J + in the wave -packet mo de pj, is

oo 2 t

So l/3mo'l dco . (2.26)

One can evaluate this as follows. Consider the wave-packet pj , propagating

backwards f rom J+ . A fract ion 1 - F ; , of the wave-packet wil l be sca ttered by

the static Schwarzchild field and a fraction F;, will enter the collapsing body.

2

,,W,,,o', -Ifi),~ ,,[ )do) (2.27)

wh ere eJ]~,, an d fl}])~,, are ca lcu lat ed using (2.19, 2.20) fr om the pa rt p}2) of the

wave -packe t which enters the star. The m inus sign in front of the second term on

~2) make

he right of (2.27) occurs b ecause the negative frequenc y co mp onen ts ot vj ,

a negative c ont ribu tion to the flux into the collapsing body . By (2.21)

~ 2) - 1 2)

~a,o,,1 = exp(rcco~c )[fi),,o,[ , (2.28)

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Particle Creation by Black Holes 2

Thu s the to ta l n um ber of par t ic les crea ted in the mod e p j , is

F]. exp 2rccotc- 1)_ 1)- 1.

2.29)

But for wave-packets a t la te re tarded t imes , the f rac t ion Fjn which enters the

col lapsing body is a lmo st the same as the f rac t ion of the wave-packet tha t wo uld

have crossed the pas t event hor izon had the col lapsing body not been there but

the exter ior Schwarzchi ld so lu t ion had been a naly t ica l ly cont inued. Thus th is

fac tor F j , i s a lso the same as the f rac t ion of a s imilar wave-packet coming from

J - wh ich wou ld have c rossed the fu tu re even t ho r izon and have been absorbed

by the b lack hole . The re la t ion between emiss ion and absorpt ion cross-sec t ion is

therefore exactly that for a body with a temperature, in geometric units , of tc/2rc.

Similar resul ts hold for the e lec trom agnet ic a nd l inear ised gravi ta t iona l f ie lds.

The fields p rodu ced on J - by pos it ive frequency waves f rom J + have the same

asym ptot ic form as 2 .18) but with an extra b lue sh ift fac tor in the ampli tude .

This extra fac tor cancels out in the def in i t ion of the sca lar pro duc t so tha t the

asy m pto tic form s of the coefficients e an d fi are the same as in the Eqs. 2.19) and

2.20). Thu s one would expect the b lack hole a lso to radia te ph oton s and gravi tons

thermally . Fo r massless fermions such as neutr inos one again gets s imilar resul ts

except tha t the negat ive f requency com pon ents g iven by the coeff icients f i now

make a posi t ive contr ibut ion to the probabi l i ty f lux in to the col lapsing body.

Th is means tha t the te rm I ~12 in 2.27) no w has the opp osite sign. F ro m this i t

fo llows tha t the n umb er o f pa rt icle s emi t ted in any ou tgo ing wave packe t m ode

is exp 2rccotc-1)+ 1)-1 t imes the frac tion of tha t wave pac ket th at w ould ha ve

been absorbe d by the b lack hole had i t been inc ident f rom J - . This is again

exact ly what one would expect for thermal emiss ion of par t ic les obeying Fermi-

Dirac statist ics.

F ie lds o f non-ze ro re s t mass do no t reach J - and J + . One there fore has to

descr ibe ingoing and outg oing s ta tes for these f ie lds in te rms of some concept such

as the projec t ive inf in i ty of Eard ley and Sachs [23] and Schm idt [24] . However ,

i f the in i t ia l and f inal sta tes are a sym ptot ica l ly Schwarzchi ld or Ker r so lu t ions ,

one can descr ibe the ingoing and outgoing s ta tes in a s imple manner by separa-

t ion of var iables and one can def ine posi t ive f requencies with respect to the t ime

transla t ion Kil l ing vectors o f these in i tia l and f ina l asym ptot ic space- t imes . In the

asym ptot ic fu ture there will be no b ou nd s ta tes: a ny par t ic le will e ither fa ll thro ugh

the event hor izo n or escape to inf in i ty . Thu s the u nb ou nd outg oing s ta tes and the

event hor iz on s ta tes togethe r form a complete bas is for so lu t ions of the wave

equat ion in the region outs ide the event hor izon. In the asymptot ic pas t there

could be bou nd s ta tes i f the bod y tha t co l lapses had had a b ou nde d radius for

an inf in i te t ime. However one could equal ly wel l assume that the body had col-

lapsed f rom an inf in i te radius in which case there would be no bound s ta tes . The

possible existence of bo un d states in the past does n ot affect the rate o f particle

emiss ion in the asym ptot ic fu ture wh ich will again be tha t of a bo dy with tem-

perat ure ~/2rc. Th e onl y difference from the zero rest m ass case is that the fre-

qu en cy co in the th er m al fa cto r exp 27rco~c-1)_T_ 1)-1 no w in clu des t he re st m ass

energy of the par t icle . Thus there wil l not be m uch emiss ion of par tic les of res t

mass m unless the tem per atu re ~c/27r is greater th an m.

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212 S.W. Hawking

One can show tha t t hese re su l t s on the rmal emiss ion do no t depend on

spher i ca l symmet ry . Cons ide r an a symmet r i c co l l apse wh ich p roduced a b l ack

ho le wh ich se t t l ed to a non- ro t a t ing uncharged Schwarzch i ld so lu t ion (angu la r

momentum and cha rge wi l l be cons ide red in the nex t sec t ion ) . The fac t t ha t t he

f ina l s ta te i s asymptot ica l ly quasi -s ta t ionary means tha t there i s a preferred

B o n d i c o o rd i n a t e sy s te m [2 5 ] o n J + w i t h r e s pe c t t o w h ic h o n e c a n d e c o m p o s e

the Cauchy da ta fo r t he ou tgo ing s t a t e s i n to pos i t i ve f requenc ies and spher i ca l

h a rmo n i c s. O n J - t h e re ma y o r ma y n o t be a p r e f e r re d c o o rd i n a t e s y st e m b u t

i f t he re is no t one can p i ck an a r b i t ra ry Bond i co ord in a t e sys t em and de com pose

the Cauchy da ta fo r t he ingo ing s t a t e s i n a s imi l a r manner . Now cons ide r one

of the J+ s ta t es P ,o~,, p rop aga t in g back war ds th rou gh thi s space - t ime in to the

co l l apsing bod y and ou t aga in on t o J - . Tak e a nu l l geodes i c gene ra to r 7 o f t he

even t ho r i zon and ex tend i t backwards beyond i t s pas t end-po in t t o i n t e rsec t J -

a t a poin t y on a nul l geodesic gen era tor ,~ of J - . Ch oos e a pa i r of nul l vec tors

(l a, fia) at y with 1 tan ge nt to 7 an d fia tan ge nt to 2. Para l lel ly p ro pa ga te P, h a

a long 7 to a poin t x in the region o f space- t im e where the m et r ic is a lm ost th a t

of the f ina l Sc hwa rzchi ld so lu t ion . At x ria wi l l be some l inear combinat ion of P

and the rad ial i nward d i rec t ed nu l l vec to r n a. Th i s m eans tha t t he vec to r -e f t

w il l conne c t x to a su r face o f phase -co /~c ( lo g e - l ogE) o f the so lu t ion

P~am

where E is some cons t an t . As be fore , by the geom et r i c op ti c s appro x im a t ion , t he

ve cto r - e f t ~ at y will c on ne ct y to a surf ace of p has e -co/~c (log~ - log E) o~ vot,~(2)

where p ~ i s the par t of P~t,~ which enters the col lapsing body. Thu s o n the nul l

ge od es ic g e n e ra to r 2 o f J - , th e ph as e of~ ~,o,L,, (z)will be

ico

- - - ( log(v 0 - v ) - logH ) (2.30)

K

where v is an a n n e pa ra m ete r on 2 wi th va lue v0 a t y and H is a co nstant . By the

geom etr ica l opt ics ap pro xim at io n , the va lue -f ,,(2)-~ e,oz,, o n 2 will be

for v 0 - v~ smal l and posi t ive an d zero for v > v o whe re L is a constant . On each

null geodesic generator of J-vot,~ (2~wil l have the form (2.31) with different values

of L, vo, and H. Th e lack of spherica l s ym m etr y dur ing the col lapse wi ll cause

p 2 )

o,Zm on J - t o con ta in com pon en t s o f sphe r i ca l ha rmon ic s w i th indices ( l', m ')

d i fferent f rom (l, m). This me ans tha t one now has to exp ress Vo,~m'(2)n the fo rm

P(t~2l)m E l 'm ' ~o~ {O~((2l)mce,,m, L 'l 'm -F fi(co2l)m~,l,m L , Vm ,}dco' . (2.32)

Bec ause of (2.31), the coefficients ~(2) an d fl~2) wil l hav e the sam e co' de pe nd en ce

as in (2.19) and (2.20). Thus one st i l l has the same relat ion as (2.21):

2) - 1 / ? 2 ) (2.33)

Ic ~l ~ ,v , , , l = ex p( ~tc~:

) t'~olm~'Vm

AS before , for each (l, m), one can m ake up wave packets Ps,~,: T h e n u m b e r o f

pa r t ic l e s emi t t ed in such a wave pac ke t m od e i s

Zt ' ,m ' ~ I fl jn l,no 'vm'] 2d co ' .

(2.34)

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Particle Crea tion by Black Hotes 213

Similarly, the fraction Fj,I,~ of the wave packet that enters the collapsing body is

~,,~j.,,.o,,,.,.,~ -I/~5.,,.~,,~.,..I

} d o o

2.35)

Again, Fj,~,, is equal to the fraction of a similar wave packet coming from o-

that would have been absorbed by the black hole. Thus, using (2.33), one finds

that the emission is just that of a body of temperature

~c/2n:

the emission at late

retarded times depends only on the final quasi-stationary state of the black hole

and not on the details of the gravitational collapse,

3 Angular Momentum and Charge

If the collapsing body was rotating or electrically charged, the resulting black

hole would settle down to a stationary state which was described, not by the

Schwarzchild solution, but by a charged Kerr solution characterised by the

mass M, the angular momentum J, and the charge Q. As these solutions are

stationary and axisymmetric, one can separate solutions of the wave equations

in them into a factor e ~u or e ~v times e- i'~ times a function of r and 0. In the case

of the scalar wave equation one can separate this last expression into a function

of r times a function of 0 [26]. One can also completely separate any wave equa-

tion in the non-rotating charged case and Teukolsky [27] has obtained com-

pletely separable wave equations for neutrino, electromagnetic and linearised

gravitational fields in the uncharged rotating case.

Consider a wave packet of a classical field of charge e with frequency co and

axial quantum number m incident from infinity on a Kerr black hole. The change

in mass d M of the black hole caused by the partial absorption of the wave packet

will be related to the change in area, angular momentum and charge by the

classical first law of black holes:

dM = ~ dA + ~ d J + ~d~2 3.1)

where f2 and q~ are the angular frequency and electrostatic potential respectively

of the black hole [13]. The fluxes of energy, angular momentum and charge in

the wave packet wilt be in the ratio ~ o : m : e . Thus the changes in the mass, angular

momentum and charge of the black hole will also be in this ratio. Therefore

d M 1 - f2rnc o - 1 _ eq~co - 1 ) = _ ~ d A . (3.2)

A wave packet of a classical Boson field will obey the weak energy condition: the

local energy density for any observer is non-negative. It follows from this [7, 12]

that the change in area d A induced by the wave-packet will be non-negative.

Thus if

co < m f2 + eq~ (3.3)

the change in mass d M of the black hole must be negative. In other words, the

black hole will lose energy to the wave packet which will therefore be scattered

with the same frequency but increased amplitude. This is the phenomenon known

as superradiance .

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214 S.W. Hawking

Fo r c lass ica l f ields of ha lf - in teger sp in, de ta i led ca lcula t ions [28] show that

there is no superradiance . The reason for th is is tha t the sca lar product for ha lf -

integer spin fields is positive definite unlike that for integer spins. This means

that the probabi l i ty f lux across the event hor izon is posi t ive and therefore , by

conserv at ion of probabi l i ty , the pro babi l i ty f lux in the sca t tered wave packet m ust

be less than tha t in the inc ident wave packet. The reas on tha t the above ar gum ent

based on the f ir st l aw b reaks do wn is tha t the ene rgy -mo me ntum tensor fo r a

c lass ica l ha lf - in teger sp in f ie ld does not obey the weak energy condi t ion . On a

quan tum , pa r tic le l evel one can unders tan d the absence o f super rad iance fo r

fermion f ie lds as a consequence of the fac t tha t the Exclus ion Pr incip le does not

a l low more than one pa r t ic le in each ou tgo ing wave packe t mode and the re fo re

does not a l low the sca t tered wave-packet to be s t ronger than the inc ident wave-

packet.

Pass ing now to the q ua nt um theory , consider fi rs t the case of an un chang ed,

ro ta t ing b lack ho le . One can as be fo re p ick an a rb i t ra ry Bond i coord ina te f ram e

on J - and dec ompose the opera to r k in te rms o f a fami ly { f~ ,,} o f incoming

solutions where the indices co, l , and m refer to the advanced time and angular

dependen ce of f on J- in the g iven coor dina te sys tem. On J + the f inal quasi-

s ta t io nary s ta te of the b lack hole def ines a preferred Bond i coord inate sys tem

using wh ich one can define a fam ily {P,ol,,} of outg oin g so lutions. Th e index l in

th is case labels the sphero idal har mo nics in te rms of which the wave equat ion is

separable . On e p roceeds as before to ca lcula te the a sym ptot ic form of p~],~ on

J - . The on ly d i ffe rence i s tha t because the hor izon i s ro ta t ing wi th angu la r

veloci ty f2 with respect to J + , the effec tive f requency near a genera tor o f the

event hor izon is not co but co-m~2. This m eans tha t the nu m ber of par t ic les

em itted in the wav e-pac ket m od e Pj,~m is

{exp 2rc~c- l co---mO)) -T- 1} - 1/~j,zm. 3.4)

The effec t of th is is to cause the ra te of emiss ion of par t ic les with posi t ive angu lar

mo m ent um m to be h igher tha n tha t o f pa r tic les w i th the same f requency co and

q u a n tu m n u m b e r l b u t w i th n e g at ive a n g u la r m o m e n tu m - m . Th u s t h e p a rt ic le

emiss ion tends to ca r ry away the angu la r momentum. For Boson f ie lds , the

factor in c urly bra ckets in 3.4) is negativ e for co M bu t no t fo r those wi th 0 )< M .

Becau se the {h,o}, unlike the {P,o}, are c on tin uo us across the e vent h orizo n, th ey

will a lso be cont inu ous on J - . I t i s the d iscont i nui ty in the {Po,} on J - a t v = Vo

which is responsib le for crea t ing an inf in i te to ta l nu mb er of par t icles in each mode.

Po, by p rod uc ing an (co )-1 tail in the Fo ur ie r tra nsf orm s o f the {P,o} at large

negat ive f requencies 0) . On the o ther hand , the {h,o} for 0) > M will have very

smal l nega t ive f requency com ponen ts on J - . Th is means tha t the obse rve r a t

the eve nt hor izon wil l see few part ic les with co > M. He will not be able to de tec t

par t ic les with co < M because they w ill have a wave length b igger tha n h is par t ic le

detec tor which must be smaller than M. As descr ibed in the in troduct ion , there

wi ll be an inde te rm inacy in the ene rgy dens i ty o f o rde r M -4 co r respond ing to

the inde te rm inacy in the pa r t ic le num ber fo r these modes .

The above d iscuss ion shows tha t the par t ic le crea t ion is rea l ly a g lobal process

and is no t localised in the col lapse: an observe r fa l l ing thro ugh the event hor izon

wou ld no t see an infin i te num ber of par t ic les com ing ou t f rom the col lapsing

body. Because i t i s a non-local process , i t i s probably not reasonable to expect

to be ab le to fo rm a loca l ene rg y-m om entum tensor to desc r ibe the back- reac t ion

of the par t ic le crea t ion on the metr ic . Rath er , the negat ive energy densi ty needed

to acc oun t for the decrease in the area o f the hor izon , should be thou ght o f as

a r i sing f rom the inde te rm inacy o f o rde r o f M - 4 o f the loca l ene rgy densi ty a t the

horizon. Equivalent ly , one can th ink o f the area decrease as resul ting f rom the

fact tha t q ua ntu m f luctuat ion s of the metr ic will cause the posi t ion an d the very

concep t o f the even t hor izon to be somewha t inde te rmina te .

Al tho ugh i t i s p robab ly no t mean ingfu l to ta lk ab ou t the local ene rgy-mom en-

tum of the crea ted par t ic les , one m ay s ti ll be able to def ine the to ta l energy f lux

over a su i tably large surface . Th e p robl em is ra ther ana logo us to th a t o f def in ing

gravi ta t io nal energy in c lass ica l genera l re la t iv i ty : there are a num ber of d if ferent

ene rgy-momentum pseudo- tensors , none o f which have any inva r ian t loca l s ig -

nif icance , but which a l l agree w hen in tegra te d over a suffic ient ly large surface .

In the par t ic le case there are s imilar ly a num ber of d if ferent express ions one can

u s e f o r t h e r e n o r m a l i s e d e n e r g y - m o m e n tu m t e n s o r . Th e e n e r g y - m o m e n tu m

ten sor for a classical field ~b is

Tab= q~.~b;b-I ~ag,bg qS;c4~;a (4.2)

If one takes th is express ion over in to the q ua nt um theo ry an d rega rds the ~b s as

opera tors one obta ins a d ivergent resul t because there is a crea t ion opera tor for

each mode to the r igh t o f an ann ih i la t ion opera to r . One the refo re has to sub t rac t

ou t the d ive rgence in some way . Var ious me thods have been p roposed fo r th i s

(e .g . [30]) bu t th ey a ll seem a b i t ad hoe . Howev er , on the ana logy of the pseudo-

tensor , one would hope tha t the d if ferent renormalisa t ions would a l l g ive the

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Particle Creation by Black Holes 217

same in te gra ted f luxes . This is indeed the case in the f ina l quasi-s ta t ionar y region:

a ll r enormal i sed en e rgy -m om entu m opera to rs Tab which obey the conse rva t ion

equat ions T;~b=0, which are s ta t ionary i .e . which have zero Lie der ivat ive with

respect to the t ime t ran sla t ion Kil l ing vector K a and which agree near J + will

g ive the same f luxes of energy and angu lar m om en tu m over any surface of con-

s tant r o uts ide the e vent hor izon. I t i s therefore suff ic ient to evaluate the energy

f lux near J + : by the co nserva t ion equ at ions th is wil l be equal to the energy f lux

ou t f rom the even t ho r izon . Near J + the obv ious way to renormal i se the ene rgy-

mo m ent um opera to r i s to no rm al o rde r the express ion (4 .2 ) wi th re spec t to

posi t ive and negat ive f requencies def ined by the t ime-transla t ion Kil l ing vector

K a of the f inal quasi-s ta t ionar y s ta te. Ne ar the event ho r izon norm al order ing

wi th re spec t to K canno t be the co r rec t way to renormal i se the ene rgy-m ome ntum

opera to r s ince the normal -o rde red opera to r d iverges a t the ho r izon . However i t

s t il l g ives the sam e energy ou tf low across any surface of con stan t r . A re norm alised

opera tor which was regular a t the hor izon would have to v io la te the weak energy

condi t ion by having negat ive energy densi ty . This negat ive energy densi ty is not

observable locally .

In o rde r to eva lua te the normal o rde red opera to r one wan ts to choose the

{qi} which describe waves crossing the event horizon, to be positive frequency

with respect to the t ime p aram eter def ined by K a a long the gen era tors o f the

horiz on in the f ina l quas i-s ta t io nary s ta te . The cond i t ion on the {q~} in the t ime-

depe nden t col lapse phase is no t de te rmin ed b ut th is should no t affec t wave

packets o n the ho r izon a t la te times . I f one m akes up w ave-packets {qj,} l ike the

{Pi,} , one f inds tha t a f rac t ion Fj , pen etra tes thr oug h the poten t ia l barr ier a ro un d

the b lack hole and gets ou t to J - with the same frequency co tha t i t had on the

horiz on. T his pro duc es a c5(~o-c0') beh avi ou r in 7~,~, . The rem aini ng fractio n

1-F j

of the wave-packet is ref lec ted back by the potent ia l barr ier and passes

th rough the co l laps ing body and ou t on to J - . Here i t wi ll have a s imi la r fo rm

to p}2). Th us for larg e & ,

~)j 2 ) = - 1 2)

,~ ,, exp (rcc o~ : )lt/),,o,,[. (4.3)

By a s imilar argument to tha t used in Sect ion (2) one would conclude tha t the

num ber of par t ic les cross ing the event hor izon in a wave-p acket mod e peaked a t

la te t imes wo uld be

(1

--Fj,

{exp (2~zco~c-1) _ t} - 1 (4.4)

Fo r a given freque ncy ~o, i.e. a given value of j , the abs orp tion fractio n Fj, goes

to zero as the angu lar qu an tum num ber 1 increases because of the centr ifugal

barr ier . Thu s a t f i rst s ight it m ight seem th at each wav e-packet mo de of h igh 1

va lue wou ld con ta in

{exp (2rco)~c- 1 )_ 1} --1

par t icles and th a t the to ta l ra te o f par t ic les and energy cross ing the event hor izon

would be inf in i te . This ca lcula t ion would , of course , be inconsis tent with the

resul t obta ined above tha t an observer cross ing the event hor izon would see only

a f in i te small energy densi ty of order M - 4. The reason for th is d iscrepancy seems

to be tha t the wave-packets

{pj,}

a n d

{qj,}

provide a comp lete bas is for so lu t ions

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218 S.W. Hawkin g

of the wave equatio n only in the region outs ide the event horizon and not actu-

ally on the event horizon itself. In order to calculate the particle flux over the

horizon one therefore has to calculate the flux over some surface just outs ide the

horizon and take the l imit as the surface approaches the horizon.

To perform this calculation it is convenient to define new wave-packets

X - - n 2)-4-n 2)

j , - t , j , -u j , wh ich rep resen t the pa r t of Pjn and q j , which passes through the

collapsing body and jj,v.= ,-j,'(l)4-- -~jna l)which represents the part of pj , and q , which

propagates out to J - through the quasi-s ta t ionary metr ic of the f inal b lack hole .

In the initial vac uum state the {y~,} mod es will not contain any particles but

each x~ mo de will conta in {exp(2~zcox-1) - 1} -1 particles. T hese particles will

appear to leave the collapsing bo dy just outside the event horizon and will prop a-

gate radially outwards. A fraction Fin will penetrate through the potential barrier

peaked a t r = 3M and wil l escape to J+ where they will const i tu te the thermal

emission of the black hole. The remaining fraction 1-Fj, will be reflected back by

the potential barrier and will cross the event horizon. Thus the net particle flux

across a surface of const ant r just outside the h orizon will be Fj, directed outw ards.

I shal l now show that us ing the normal ordered energy momentum opera tor ,

the average energy flux across a surface of consta nt r between retar ded times Ul

and u2

(u 2 - u 1)-1 j'~ (0 _ IT~bt0_ ) K d S b (4. 5)

is directed outwards and is equal to the energy flux for the thermal emission from

a hot body. Because the {y .} contain n o negative frequencies on J - , they will

not make any contrib ution to the expectation value (4.5) of the normal ordere d

energy-momentum opera tor . Let

x j , = I ~ ( j , o ' f ~ ' + ~ m o , f o , ,) d c o ' .

(4.6)

N e a r J +

Xj ,~ = (Fjn ) -~ p jn . (4.7)

Thus

(4.5) = ( u 2 - Ul) -1 Re {Y4,, E, ",," 5~ f:~ coc0"r),p,,~,,~,,

(F~ , ,. ,, t ~ , , . , , ~ 2 , , . .~ , , -F~ , ,~ , , p j , ,~ , , 2 , , . , , o , , )dco 'du}

(4.8)

wh ere co an d co" are the frequ encie s of the w ave -p ack ets pj~ and p~,,,,, respectively

In the limit u 2 - u l tends to infinity, the seco nd te rm in the integran d in (4.8) will

integrate out and the first term will con tri but e only for (j", n") = (j, n). By argu-

ments similar to those used in Section 2,

f;o i~a,~,,12dco,__ {exp(27zco~c- 1)_ 1} - 1 (4.9)

Therefore

(4.5) = S~ Fo, co{exp(2~zco~:- 1)_ 1} - ldco (4.10)

where Fo,= lim F: n is the fraction of wave-pac ket of frequency that woul d be

n-- oo J

absorbed by the black hole. The energy flux (4.10) corresponds exactly to the rate

of thermal emiss ion ca lcula ted in Sect ion 2 . Any renormalised energy mo me ntu m

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Particle Creation by Black Holes 219

rigin fpolar i d~ +

coordinatesventorizSing?\arity

---4D

C [ [ a p s i n g ~~ - b o d y

Fig. 5. The Penrose diagram for a gravitational collapse followed by the slow evaporation and even-

tual disappearance of the black hole, leaving empty space with no singularity at the origin

operator which agrees with the normal ordered operator near J+, which obeys

the conservation equations, and which is stationary in the final quasi-stationary

region will give the same energy flux over any surface of constant r. Thus it will

give positive energy flux out across the event horizon or, equivalently, a negative

energy flux in across the event horizon.

This negative energy flux will cause the area of the event horizon to decrease

and so the black hole will not, in fact, be in a stationary state. However, as long

as the mass of the black hole is large compared to the Planck mass 10-5 g, the

rate of evolution of the black hole will be very slow compared to the characteristic

time for light to cross the Schwarzchild radius. Thus it is a reasonable approxima-

tion to describe the black hole by a sequence of stationary solutions and to cal-

culate the rate of particle emission in each solution. Eventually, when the mass

of the black hole is reduced to 10-5 g, the quasi-stationary approximation will

break down. At this point, one cannot continue to use the concept of a classical

metric. However, the total mass or energy remaining in the system is very small.

Thus, provided the black hole does not evolve into a negative mass naked sin-

gularity there is not much it can do except disappear altogether. The baryons or

leptons that formed the original collapsing body cannot reappear because all

their rest mass energy has been carried away by the thermal radiation. It is

tempting to speculate that this might be the reason why the universe now contains

so few baryons compared to photons: the universe might have started out with

baryons only, and no radiation. Most of the baryons might have fallen into small

black holes which then evaporated giving back the rest mass energy of baryons

in the form of radiation, but not the baryons themselves.

The Penrose diagram of a black hole which evaporates and leaves only empty

space is shown in Fig. 5. The horizontal line marked singularity is really a

region where the radius of curvature is of the order the Planck length. The matter

that runs into this region might reemerge in another universe or it might even

reemerge in our universe through the upper vertical line thus creating a naked

singularity of negative mass.

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220 S .W. Hawking

e f erences

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Com munica ted by J . Ehle rs

S. W. Hawking

Cal i forn ia Ins t i tu te o f Technology

W. K. Kellogg Radiat ion Lab. 106-38

Pasadena, California 91125, USA