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Commun. Math. Phys. 87, 577-588 (1983)Communications nMathematical

Physics Springer-Verlag 1983

Thermodynamics of Black Holesin Anti-de Sitter Space

S . W. Hawk in g 1 an d Do n N . P ag e 2

1 University of Cambridge, Department of Applied Mathematics and Theoretical Physics, Silver

Street, Cambridge, England2 Department of Physics, The Pennsylvania State University, UniversityPark, PA 16802, USA

Abs t rac t . The Eins te in equat ions w i th a negative cosmolog ical constan t adm itb lack ho le so lu t ions which are asympto t ic to an t i -de S i t ter space . L ike b lackho les in asympto t ica l ly f la t space , these so lu t ions have thermodynamicproper t ies inc lud ing a character i s t ic temperatu re and an in t r ins ic en t ropyequal to one qu ar ter o f the area o f the even t hor izon in P lanck un its. The re arehow ever som e im por tan t d i fferences f rom the asym pto t ica l ly fla t case. A b lackho le in an ti -de S i t ter space has a m in im um tem pera tu re which occurs when i tssize is of the o rde r o f the ch aracterist ic rad ius of the anti-de S it ter space. Fo rlarger b lack holes the red-shif ted temperature measured at infin i ty is greater.Th is mean s tha t such b lack ho les have posi tive speci fic heat a nd can be instable equil ibriu m with therm al rad iat ion at a f ixed temp erature. I t also impliesthat the canonical ensemble exists for asymptotical ly ant i-de Sit ter space,unlike the case for asymptotical ly f iat space. One can also consider themicro canon ical ensemble. One can avo id the p rob lem that ar ises in asympto t i -cal ly f lat space of having to p ut the system in a bo x with unphys ical perfect lyreflect ing walls because the gr avitat iona l p otential of ant i-de Sit ter space actsas a box of f in ite volume.

1. In t roduct ion

Th e f irs t ind ica t ion tha t b lack ho les have ther mo dyn am ic p roper t ies came wi th thediscove ry that in the classical theor y of general relat iv i ty the are a of the event

hor i zon [ t l (o r equ ivalen t ly, the square o f the i rreducib le mass [2 ] ) neverdecreases . The re i s an obv ious an alogy w i th the second law of therm odyn am icswi th the area o f the even t hor izo n p lay ing the ro le o f en t ropy. There were a lsoanalog ies to the zero th an d f irs t laws o f therm odyn am ics in which the ro te o ftem per atu re was played by a qu anti t y cal led the surface gravity ~c which m eas uredthe stren gth of the gravita t ional f ield at the ev ent horiz on [3] . The se similari tiesled Bekenste in [4 ] to suggest tha t some m ul t ip le o f the area o f the even t hor izon ,

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Black Holes in Anti-de Sitter Space 579

Li k e f l a t s p a c e b u t u n l i k e d e S i t t e r s p a c e , a n t i - d e S i t t e r s p a c e h a s n o n a t u r a lt e m p e r a t u r e a s s o c i a t e d w i t h i t . T h e m o s t s y m m e t r i c " v a c u u m s t a t e " i s t h e r e f o r en o t p e r i o d i c i n t h e i m a g i n a r y t i m e c o o r d i n a t e t h o u g h i t i s p e r i o d i c i n r e a l t i m e .

Th i s is t r u e e v e n i f o n e wo r k s i n t h e c o v e r i n g s p a c e . As i n f la t sp a c e , o n e c a nc o n s t r u c t t h e r m a l s t a t es a t a n y t e m p e r a t u r e T b y i m p o s i n g a p e r i o d i c it y f l = T - 1i n i m a g i n a r y t i m e . T h e g r a v i t a t i o n a l m a s s o f s u c h a t h e r m a l s t a t e i n fl a t s p a c ew o u l d b e i n f in i t e i f i t h a s i n f in i t e v o l u m e a n d t h e r e f o r e t h e s t a t e w o u l d c o l la p s e .E v e n i f o n e r e s t ri c t e d t h e v o l u m e t o b e f in i te b y p u t t i n g i t in s o m e s o r t o f b o x , th es t a t e w o u l d s ti ll b e u n s t a b l e t o t h e f o r m a t i o n o f a b l a c k h o l e , n o m a t t e r h o w l o wt h e t e m p e r a t u r e [ 17 , 1 8] . M o r e o v e r, a l t h o u g h a b l a c k h o l e c an b e i n e q u i l i b r i u mw i t h t h e r m a l r a d i a t i o n a t t h e s a m e t e m p e r a t u r e , t h i s e q u i l ib r i u m i s u n s t a b l e i f t h et e m p e r a t u r e i s h e l d c o n s t a n t : i f t h e b l a c k h o l e w e r e t o g e t a b i t m o r e m a s s , it s

t e m p e r a t u r e w o u l d g o d o w n , t h e r a te o f a b s o r p t i o n w o u l d b e g r ea t e r t h a n t h e r at eo f e m i s s i o n a n d t h e b l a c k h o l e w o u l d c o n t i n u e t o g r o w. T h i s in s t a b i li t y m e a n s t h a tt h e c a n o n i c a l e n s e m b l e c a n n o t b e d e f i n e d in a s y m p t o t i c a l l y f la t sp a c e i f g r a v i-t a t i o n a l e f f ec ts a r e i n c l u d e d . I n s t e a d , o n e h a s t o u s e a m i c r o c a n o n i c a l e n s e m b l e[ 1 7 ] i n w h i c h a c e r t a i n a m o u n t o f e n e rg y is p l a c e d in a n i n s u l a t e d b o x t h o u g h e v e nt h is i s u n p h y s i c a l b e c a u s e o n e c a n n o t c o n s t r u c t a b o x t h a t w i ll p r e v e n t g r a v i to n sf r o m e s c a p in g . I f o n e i g n o r e s t h i s d i f fi c u lt y, o n e f in d s t h a t o n e c a n h a v e a b l a c kh o l e i n s t a b l e e q u i l i b r i u m w i t h t h e r m a l r a d i a t i o n p r o v i d e d t h a t t h e e n e rg yE > ( 2 - 2 1 3 - 1 5 ' * r c - 2 g m ~ V ) l / 5 ,w h e r e V is th e v o l u m e o f t h e b o x a n d g is t h ee ffe c ti v e n u m b e r o f s p i n s t a t es .

I n a n t i - d e S i t te r s p a c e t h e g r a v i t a t i o n a l p o t e n t i a l r e la t i v e t o a n y o r i g i n i n c r e as e sa t l a rg e s p a t ia l d i s t an c e s f r o m t h e o r i g in . T h i s m e a n s t h a t t h e l o c a ll y m e a s u r e dt e m p e r a t u r e o f a t h e r m a l s t a te d e c r e a se s a n d t h a t t h e t o t a l e n e rg y o f t h e t h e r m a lr a d i a t i o n i s f i n it e w i t h o u t a n y n e e d t o p u t i t i n a b o x . In f a c t t h e g r a v i t a t i o n a lp o t e n t i a l c a u s es " c o n f i n e m e n t " o f n o n z e r o r e st m a s s p a r ti c le s a n d p r e v e n t s t h e mf r o m e s c a p i n g t o i n f i n i t y. Ze r o r e s t ma s s p a r t i c l e s c a n e s c a p e t o i n f i n i t y b u t i n at h e r m a l s t a t e t h e i n c o m i n g a n d o u t g o i n g f l u x es a t i n f in i ty a re e q u a l . W e f i n d t h a t i f

t h e t e m p e r a t u r e i s l e s s t h a n To = ~-~g - A )l /z , t h e r m a l r a d i a t i o n i ss t ab le a g a i n s t

c o l l ap s e t o f o r m a b l a c k h o le . A t t e m p e r a t u r e s h i g h e r t h a n TO h e r e a r e t w o v a l u e so f t h e m a s s o f a b l a c k h o l e t h a t c a n b e i n e q u i l i b r iu m w i t h t h e t h e r m a l r a d i a t i o n .T h e e q u i l i b r iu m a t f ix e d t e m p e r a t u r e i s u n s t a b l e f o r th e l o w e r o f t h e s e m a s s e s b u t

T. 1 ( A ~ 1/2is l o c a l ly s t a b le f o r t h e h i g h e r o n e . A t T >~ I = U \ - 3 ] , i fI A [ ~ r n 2a s w e

a s s u m e , t h e c o n f i g u r a t i o n w i t h a b l a c k h o l e a n d t h e r m a l r a d i a t i o n h a s a l o w e r fr eee n e rg y th a n t h e c o n f i g u r a t i o n w i th j u s t t h e r m a l r a d i a t i o n . A t a t e m p e r a t u r e T >Tz

( - m ~ A ) 1/4 ,t h e r e i s n o e q u i l i b r i u m c o n f i g u r a t i o n w i t h o u t a b l a c k h o l e .

O n e c a n a l so c o n s i d e r a m i c r o c a n o n i c a l e n s e m b l e i n w h i c h o n e p u t s a c e rt a i na m o u n t o f e n e rg y i n t o a s y m p t o t i c a l l y a n t i - d e S i t te r s p a ce . O n d o e s n o t n e e d a b o xw i t h u n p h y s i c a I w a l l s b u t o n e h a s t o i m p o s e t h e b o u n d a r y c o n d i t i o n t h a t t h ei n c o m i n g f lu x o f z e r o r e s t m a s s p a r t i c l e s a t i n f i n i ty is e q u a l t o t h e o u t g o i n g f lu x . I ft h e e n e r g y E < E o ~ ( 2 - 2 1 3 - 1 5 4 g r n ~ ) l / 5 ( - A / 3 ) - 3 / l o , t h e d o m i n a n t c o n fi g u ra t io nw i ll b e t h a t o f t h e r m a l r a d i a t i o n . I fE o < E < E ~ I . 3 1 4 E o ,the re wi l l be ac o n f i g u r a t i o n w i t h a b l a c k h o l e a n d t h e r m a l r a d i a t i o n w h i c h is l o c a ll y st a b le b u t is


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580 S.W. Hawking and D. N. Page

less pro bab le th an therm al r adia t ion a lone . I f E I < E E z , the pur e radia t ion conf igura t ion wi ll a lways col lapse .

The re a re a l so cha rged an d ro ta t ing b lack ho le so lu tions in an ti -de S i t t e r spacewhich contr ibute to the grand canonica l ensemble in which the e lec t r ic potent ia land ra te o f ro ta t io n ac t a s chemica l po ten t i a l s fo r cha rge and a ngu la r m om en tumrespec t ive ly. These genera l iza t ions behave m uch as one w ould expec t f rom theasym ptot ica l ly f la t space case , bu t one d i fference is tha t in ant i -de Si t te r space o necan have a ro ta t ing b lack ho le in equ i l ib r ium wi th ro ta t ing rad ia t ion p rov ided tha tthe angu la r m om en tum of the b lack ho le i s su ff ic ien t ly sma ll , whereas inasymptot ica l ly f la t space such an equi l ibr ium is never poss ib le because thero ta t ion a l ve loc ity o f the rad ia t ion would have to exceed tha t o f l igh t a t l a rgedis tances f rom the b lack hole .

Th e p lan of th is pap er i s as fo llows. In Sect. 2 we ad op t the Eucl ideanfo rm ula t ion o f quan tum theory in an t i -de S i t te r space . We ca lcu la te the E uc l ideanact ion of a Schw arzschi ld-ant i -de Si t te r so lu t ion . W e use these resul ts in Sect . 3 tos tudy the canon ica l ensemble. We f ind tha t the b lack ho le has an in tr ins ic en t ro pyequal to a qua r ter of the area of the even t hor izon, as in asymp tot ica l ly f lat space.In Sect . 4 we inves t iga te the microcanonica l ensemble .

2. Euclidean Form ulation

Th e me tr ic of the cove r ing space of ant i -de Si t te r space can be wri t ten in the s ta t icf o r m

ds 2 = - Vd t 2 + V - 1dr2 + r2(dO 2 -t-sin 20d(p2 ), (2.1)

F2V = 1 + b-Z-, (2.2)

b - ( -3 ) 1 / z . (2.3)

Ant i -de Si t te r space can be obta ined f rom th is metr ic by ident i fy ing t per iodica l lywith p er io d 7 = 2rob. A t imel ike geodesic thr ou gh the or ig in r e turns to th e o r ig inaf ter a ha l f per io d y /2. A null geodesic does no t re turn to the or ig in but escapes toin f in i ty. However one can impose the boundary cond i t ion tha t a ze ro re s t masspar t ic le sho uld a lso re turn to the o r ig in af ter a ha l f per iod 7 /2 .

The subs t i tu t ion z = it makes the metric (2.1) Euclidean, i .e . posit ive definite .The mos t na tu ra l and symmet r ic vacuum s ta te fo r quan tum f ie lds on the an t i -deSi t te r back gr ou nd is def ined by a pa th in tegra l over f ie ld conf igu ra t ions wh ich goto zero a t la rge d is tances in the Eucl idean ant i -de Si t te r metr ic . This means tha t

the G ree n fun ct ions wil l be so lu t ions o f e l lip tic equa t ions in the Eucl idea n spacewhich vanish a t la rge d is tances . When analy t ica l ly cont inued to the Lorentz iansect ion of ant i -de Si t te r space , these G ree n func t ions wi ll be per iodic in t wi thper io d 7- On e c an a lso em bed a nt i -de S i t te r space conf orm al ly in to ha l f of thes ta t ic Eins te in universe , tha t is , in to the p ro du ct of ha l f of the spat ia l three- spheresec t ions t imes the t ime ax i s . The an t i -de S i t t e r vacuum s ta te fo r confo rmal lyinvar iant f ie lds i s then the s ta te induced f rom the na tura l vacuum s ta te in the


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Eins te in un ive rse . The reas on tha t the Gre en func t ions a re pe r iod ic is tha t pa r t ic le spass r igh t a round the E ins te in cy l inde r and re tu rn to the i r o r ig ina l pos i t ions inspace af te r a t ime 7 .

One can cons t ruc t the rma l s ta te s in an t i -de S i t t e r space by pe r iod ica l lyiden t ify ing the ima g ina ry t im e coo rd ina te z wi th pe r iod fi = T - 1. These s ta te s wi l lbe in therm al eq ui l ibr ium in the s ta t ic coord ina te sys tem (2 .1) wi th a loca l lym e a s u r e d t e m p e r a t u r e

~oc = fi- 1V - 1/2 (2.4)

The loca l t empera tu re i s r ed -sh i f ted by the g rav i ta t iona l po ten t ia l and dec reasesl ike r- ~ fo r r> b . One wou ld the re fo re expec t the the r ma l ene rgy dens i ty to godow n l ike r -4 fo r ze ro re s t mass pa r t i cle s an d fa s te r fo r pa r t ic le s wi th re s t mass . In

the case o f con fo rma l ly inva r ian t pa r ti c les , one can ve r i fy th is by tak ing a the rm a ls ta te on the E ins te in un ive rse and con fo rm a l ly t r ans fo rming . The re s u l tan t ene rgy-m o m e n t u m t e n s o r i s

T~" = A3~ + f ( T ) V - 2 -~ u -o6, - 4C5o0v), (2.5)

7~2w h e r e f ( T ) = ~-dgT4+ O(b-2 T2) . The f i r s t cons tan t t e rm a r i se s f rom the confo r-

m a l a n o m a ly a n d m a y b e r e g a r d e d a s a r e n o r m a l i z a t i o n o f t h e co s m o lo g i c a lcon s tan t A . The second t e rm has the fo rm o f a pe r fec t fluid wi th P = # and # oc r- 4

for r > b. Thu s the to ta l energ y wil l be f ini te .The Schwarzschi ld-ant i -de Si t te r metr ic has the form (2 .1) , where now

2M r 2V = 1 - m2 + l)~-. (2.6)

Th is has a ho r izon a t r=r+ , w h e r e V ( r + ) = 0 . Th e s u b s t it u t io n z = i t m a k e s t h eme tr ic pos i t ive def in ite for r > r+. Th e a pp are nt s ingular i ty a t r = r+ is jus t l ike thes ingu la r i ty a t the o r ig in o f po la r coo rd ina te s and can be re mo ved i f-c i s r ega rde d a sa n a n g u l a r c o o r d in a t e w i th p e r io d

4rcb2r +fi = b2 + 3r2+ . (2.7)

Thus , a s in a symp to t ica l ly fl a t space, a b lack ho le has a na tu ra l t em pera tu reassoc ia ted w i th i t a l thoug h in th is case the loca lly me asu red t em pera tu re dec reasesinde fin i te ly the fu r the r one i s f rom the b lac k ho le. F ro m the fo rm ula (2.7) one c ansee tha t fi has a m ax im um va lue o f 2 rc3 -1 /2b and the re fo re T has a mi n i m um va lueo f To=(2n)-131/2b-1 w h e n r+=ro=3-1 /Zb. F o r r + > r o , t he t e m p e r a t u r e T

inc reases wi th the mass

(2.8)M=m+ (1+ b2J

One can co mp ute the d ifference be tween the Euc l idean ac t ion o f the b lack ho lemet r ic a nd tha t o f an t i -de S i t t e r space iden t if i ed wi th the same phys ica l pe r iod inima g ina ry t ime . The ca lcu la t ion i s s imi la r to tha t in a sym pto t ic a l ly f l at space [8 ] ,


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bu t in th i s case the con t r ibu t ion o f the su r face te rm i s ze ro . The ac t ion c omes f ro mthe d i fference in fou r-v o lum es o f the tw o m e t r ic s and i s

2?,2 2 2I= nmp +(b -r+) (2.9)

b 2 + 3r2+

Fo r sma l l va lues o f r+ o r M , th i s is the sam e as the f l at space re su lt bu t the ac t ionh a s a m a x im u m w h e n r + = r 0 a n d b e c o m e s n e g a t i v e w h e n r + > b . We s ha llinves t iga te the physica l imp l ica t ions of th is form ula for the ac t ion in the fo l lowingsections.

3 . T h e C a n o n i c a l E n s e m b l e

The canonica l ensemble is def ined by a pa th in tegra l over a l l mat ter f ie lds andmet r ic s wh ich tend a sym pto t ica l ly re spec t ive ly to ze ro and to an t i -de S i t t e r spaceiden t i f i ed pe r iod ica l ly in wi th pe r iod /3 . The dominan t con t r ibu t ion to the pa thin teg ra l i s expec ted to c om e f rom met r ic s wh ich a re nea r c las s ica l so lu t ions to theEins te in e quat ions . P er iodica l ly identi f ied an t i -de Si t te r space is one of these andwe take i t to be the zero o f ac t ion and ene rgy. The p a th in teg ra l ove r the m a t te rf ie ld s and me t r ic f luc tua t ions on the an t i -de S i t t er back gro und can be rega rded a sg iv ing the con t r ib u t ion o f the rm a l rad ia t io n in an t i -de S i t t e r space to the pa r t i t ionfunct ion Z. Fo r a con form al ly invar ian t f ie ld th is wi ll be

T 7~4l o g Z = 3~z2b 3 ! T - 2 ( T ) d T = -90 g(b/fi)3 + O(b/fl). (3.1)

The ene rgy o f the the rma l rad ia t ion wil l be

( E ) = - l o g Z = 3 n z b a f( T ) ~ ~gT~*b 3 . (3.2)

So fa r the g rav i ta t io na l e ffec t o f the rm a l rad ia t io n has been neg lec ted . One can

es t imate th is by so lv ing the Eins te in equat ions with a A term for a perfec t f lu idwi th i= g # . On e f inds tha t so lu t ions ex ist i f the m ass o f the f lu id is less tha n som ecr it i cal va lue M 2 wh ich can be e s t ima ted to be o f o rde rm~b. Th i s w o u ldc o r r e s p o n d t o a t e m p e r a tu r e

T 2 ~ , , - 114~112~-- 112 (3.3)1:1 "~ p ~

Th e r m a l r a d i a t i o n a t a t e m p e r a tu r e g r e a t e r t h a n T2 wou ld no t be ab le to su ppo r ti tse lf aga ins t i ts sel f g rav i ty and wou ld co l lapse to fo rm a b lac k ho le .

The Schwarzsch i ld -an t i -de S i t t e r so lu t ion i s p robab ly the on ly o the r non-s ingular pos i t ive-def in i te so lu t ion of the c lass ica l equat io ns tha t sa t is f ies thepe r iod ic bou nd ary cond i tions . The so lu t ion ex i st s on ly i f /~ TO=(2re)-~31/2b- 1

The Euc l idean ac t ion fo r a b lack ho le so lu t ion g ives a con t r ibu t ion to log Z o f

lo gZ = - I = - m2nr2(bZ - rz+)b2 + 3r2+ (3.4)


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Black Holes in Anti-de Sitter Space 583

The expectat ion value of the energy is

( E ) = - - ~ l ogZ

-grnvr+ 1 + = M .

The en t ropy i s

(3.5)

N=t~ffE) 1 z zo g Z = m s r r +

=m2A, (3.6)

where A is the area of the event horizon. Thus the relat ion between entro py an d

area is the sam e as in asym ptotical ly f lat space. Fo r large M ,

A ~ 4rc(2m~-2b2M)2/3 (3.7)

This mean s th a t the densi ty o f s ta tes N(M) for the black hole grows likeexp [n(2mpb2M)2/3].This is suff iciently slow tha t the in tegral defin ing the p art i t ionfunction

Z = ~ N(M )e-M/r dM (3.s)

converges. This shows tha t the canonica l ensemble in asym ptotical ly ant i-de Sit terspace is well behaved. In asym ptotical ly f lat space the densi ty of b lack hole statesgoes as exp(4rcm22M 2) an d so the canonic al ensemble is pathological .

For temperatu res T< To , the on ly poss ib le equ i l ib r ium is thermal rad ia t ionwi thou t a b lack hole. The free energy is negative and is g iven by

7~4f = - r lo g Z = - 9-0gb3 T4 + O(br2) (3.9)

for conformally invariant fields.

I f T > To, there are two possible b lack ho le masses that can be in equil ibriumwith t herm al radia t ion. T he low er of these has negative specif ic heatOM/OT.It istherefore unstable to decay ei ther in to pure thermal radiat ion or to the largervalue of the b lack ho le mass. The lower value of the mass also has posi t ive freeenergy which means that i t is less probable than pure thermal radiat ion. Thehigher value of the mass has positive specific heat and is therefore at least locallystable. If

r o < T < T1 = (n b )- 1, (3.10)

the free energy of the b lack hole is posi tive so th is co nfigurat ion wo uld reduce i tsfree energy if the b lack ho le evapo rated completely. The tunnelin g probabil i ty forth is to o ccur wil l be o f the fo rm

F = A e - B, (3.11)

where A is some de term inan t and B is the d ifference between the act ions of thelowe r and highe r mass s olution s at the sam e tem pera ture. If T~> T1, the free energy


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586 S.W. Hawking and D. N. Page

Th e + s ign corre spon ds to the h igher mass so lu t ion and the - s ign to the lowermass so lu tion . They wil l thus ma ke a co n t r ibu t ion o f o rde r e - i + o r e - I_ toZ(fl).Th e on e- loop t e rm abo u t the b lack ho le me t r i c s will con t r ibu te a f ac to r o f o rde r

one or z exp respective ly. Th e fac to r of i a r ises in the lower mass case

from the negat ive n onc onf orm al m ode. In the h ig her m ass case, i f E > M 0, thestat ionary phase point in Eq. (4.2) wil l be at

where r+ is the so lu t ion of

4~b2r+fl ~ b2 + 3r2+, (4.7)

= ~ p r ~ _ l + b 2 }. (4.8)

Th e second de r iva t ive o f the loga r i thm of the in teg rand isTZdM/OT>O. Th u s t h epa th of s teepes t descent wil l be para l le l to the im aginar y axis andN(E) will be realand g iven by

N( E) ~ exp(~m2r2+ )

~exp[~(2mvb2E) 2/3] fo r g>>Mo=3-3/22m2b. (4.9)

In the lower mass case the s ta t ionary phase point wi l l be a lso g iven by (4 .7) and(4.8) if g>>go,,,(gm~b3) 1/5 so tha t ther ma l radia t io n mak es a negl ig ib le contr i -but ion . I f g 0 < g ~ M o, the s ta t i ona ry phase poin t wi l l be a t the la rger ro ot of

m~fi ~r4E = M + Erad ~, ~ x + ~ gb3fi-4 (4.10)

where a b lack ho le o f ene rgy M is in equ i l ib r ium wi th the rm al r ad ia t ion o f ene rgyE r a d .

Th e s econd der iva t ive of the loga r i thm of the in tegrand of (4 .2) wi ll be negat ive

at each of these saddle points . Thu s the pa th of s teepes t decent wil l be para l le l tothe rea l axis . This wi ll in t rod uce a f ac tor of i which wi ll cancel the fac tor of i a r is ingf rom the nega t ive nonco nfor ma l mode . T hus N(E) will be rea l and wil l be g iven by

4~ /g b3 \ 1/4 1_ _ 314N(E) , .~exp 4 n m ;Z M 2 + 3 /~3~0) grad] , (4 . t l )

wh ere M an d Era d are the two term s o f (4.10) tha t ad d up to E. If E < E 0, Eq. (4.10)has no solu t ion for a b lack hole in equi l ibr ium with radia t ion , so one obta ins onlythe con tr ib ut io n (4.5) of pure th erm al radia t ion .

We can now es t ima te the p robab le conf igura t ions fo r the mic rocanon ica lensem ble in d i fferent ranges of the en ergy E. I f

- - 2 1 - - 1 4 8 3 1 / 5g < E o ~ ( 2 3 5 gm;b ) , (4.t2)

the only local ly s table conf igura t ion is thermal radia t ion wi thout a b lack hole . I f

E 0 < g < g 1 ~ 1.314E 0 , (4.13)


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Black Holes in Ant i-de Sitter Space 587

there i s a lso a loca l ly s table conf igura t ion wi th a low mass b lack hole inequ i l ib r ium wi th the rmal r ad ia t ion . H ow eve r the pure rad ia t ion s t a te i s mo reprobab le so tha t a l though b lack ho les may fo rm f rom t ime to t ime as a r e su l t o f

f luc tua t ions , they wi ll t end to e vap ora te away by fu r the r f luctua tions. I fEI


12/12

588 S.W . Haw king and D. N. Page

20. Gibbo ns, G.W., Perry, M.J. : Qua ntizing grav itationa l instantons. Nucl. Phys. B 146, 90--108 (1978)21. Page, D.N. : Positive-a ction conjecture. Phys. Rev. D 18, 2733-2738 (1978)22. Perry, M J .: Instabilities in gravity and supergravity. In : Superspace a nd su pergra vity:

Proceed ings of the Nuffield Wo rksho p, Cam bridge, Jun e 16 - July 12, 1980. Hawk ing, S.W., Ro~ek,

M. (eds.). Cam bridg e: Ca mbrid ge Un iversity Press 198123. Hawking, S.W. : Euclidean qua ntum gravity. In: Recent developm ents in gravitation : Carg6se

1978. N A T O Ad van ced Study Institu tes Series, Series B : Physics, Vol. 44. L6vy, M., Deser, S. (eds.).New York: Plenum Press 1979

24. Avis, S.J., Isham , C.J., Storey, D. : Qu an tu m field theo ry in anti -de sitter space-time. Phys. Rev.D 18, 3565-3576 (1978)

25. Page, D.N. : Th erm od yn am ic paradoxes. Physics To da y 30, 11-15 (1977)26. Page, D.N.: Black hole formation in a box. Gen. Rel. Gray. 13, 1117-1126 (1981)

Com munica ted by A. Jaffe

Received August 2, 1982

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