PHYSlCAL RF VIEW D VOLUME 14, NUMBER 2 15 JUL Y 1976

Black-hole evaporation in the Klein-Sauter-Heisenberg-Euler formalism

Thib3ut Damour' and Remo RuAini~

Joseph Henry Physical Laboratories, Princeton, ¹wJersey 08540and Physics Department, University of Western Australia, ¹dlands, Australi~

(Received 3 November 1975)

A generalization of the classical approach of barrier penetration introduced by Klein, Sauter, Heisenberg, and

Euler to curved spaces endowed with future horizons is given. This technique allows one to recover most

directly results obtained by Hawking recently. The treatment here presented encompasses, as special cases, the

works of Deruelle and RuAini, of Damour and Ruffini, and of Nakamura and Sato.

one of the most important results obtained inrecent years in black-hole physics has been therealization that the total mais-energy of a blackhole can be separated into three components': theirreducible mass, the Coulomb energy, and therotational energy. That both rotational and Cou-lomb energy could be in principle extractable bya set of classical gedanken experiments has beenknown for some time. ' lt has been only recently,however, that the quantum analog of these pro-cesses occurring in the "effective ergosphere"have been analyzed. ' The use of the Klein-Sauter-Heisenberg-Euler formalism has led to a mostdirect understanding of these processes of vacuumpolarization' and to detailed analyses of possibleastrophysical interest.

Hawking' has suggested, however, that, also,by vacuum polarization processes the irreduciblemass of a black hole could be radiated away. Inthe present paper we show how a generalization ofour previous treatment of barrier penetration4leads to a clear understanding of this phenomenon.

We consider (a) a Kerr-Newman geometry en-dowed with a vacuum future horizon, (b) a massivecharged scalar field @' fulfilling the covariantKlein-Gordon equation in that background geom-etry, and (c) we assume analyticity properties ofthe wave function @ in the complexified manifold.

The result ean be obtained mathematically thanksto the existence of explicit asymptotic expressionsfor the field ~' near the horizon and at spatial in-finity. Physically, it comes from the existence,inside the horizon, of a spacelike Killing vector$„which allows a classical particle as "seen"from infinity to reach a negative-energy state.In the quantum description, this phenomenon al-lows an antipar tie le to reach positive- energystates. These states, classically confined in theblack hole, ean be tunneled out by a wave function"over" the horizon which gives rise to the creationof a pair: one particle (positive energy) going outand one antiparticle (negative energy) falling backtoward the singularity. Note that this approach

dr~/dr = (r'+a')/a,we have when r —r, (r& r, )

(2a)

with (2b)

We shall first treat the case of a Schwarzschildmetric (a = e =0}. The scalar function &' fulfillingthe covariant Klein-Gordon equation in this givenmetric can be separated as

~' =(2vl~lr') '"& (r*, ~)yP(8, 4),the F, being the usual spherical harmonics and

E~ being monochromatic in time. In the followingwe take ~ & 0, that is, a flux of Particles at in-finity, the flux of antiparticles being treated asusual by charge conjugation. It is easy to showthat just outside the horizon H, (r & 2M} two linear-ly independent solutions exist:

~in -iLg(t+t~)»iQJvc ~ —8

and

only requires the existence of a future horizon and

is totally independent of any dynamical details ofthe process leading to the formation of this hori-zon.

As usual we consider the Kerr-Newman metric

ds2 = Z(s ' dr 2+d 8~)

+Z 'sin28[(r'+a')d&f& —adtj'

—Z 'A(dt —a sin'8 dQ)',

with o. =r' —2Mr+a'+e'=(r —r, )(r —r ), wherer, =M +(M' —a' —e2)~' and Z =r'+a' co's,8Mbeing the mass, e the charge, and a the specificangular momentum of the black hole. (Here and

in the following we choose G=c=K =1.) We alsoindicate by H, the future horizon. Introducing thecoordinate &~,

14 332

14 BLACK-HOLE EVAPORATION IN THE KLEIN-SAUTER-. . . 333

gout e- j(tl(t-y'~} e2j(dy'+ +- j(d& (+ 2M)j&NK - j QJv(d I

where we use the usual advanced Eddington-Finkel-stein coordinates, t+&~ = v, &, 8, P in which themetric is well behaved and, in fact, analytic overthe whole coordinate range 0&r&~, —~& L) &~including H, (r =2M, —~ & u & ~).

While Eq. (4a) corresponds to a wave purely in-going on H, and ean be extended inside «2M,Eq. (4b} represents an outgoing wave and has aninfinite number of oscillations as &-2M and there-fore cannot be straightforwardly extended to theregion inside H, . %'e will in the following use andgeneralize to analytic curved spaces the well-knownresult of flat-space relativistic wave theories'8:The wave function @(x)describing a particle state(positive frequencies) can be analytically continuedto complex points of the form z =x+ ip if y lies inthe past cone; similarly, for an antiparticle state(negative frequencies) y has to lie in the futurecone.

Since in Finkelstein coordinates the vectore/sr is everywhere null and past-directed, theprescription r-r —i0 will yield the unique con-tinuation of Eq. (4a) describing an antiParticlestate,

RQRlogs of the Eddington-Finkelstein coox'diDRtesRQd R eox'x'espondlDg gRuge tlansformatlon forthe el.ectromagnetic field, that the normalizedingoing wave 4 is regular at H+ but that 4'"'contains a factor (r -r, )'( 'oui', where x is givenby Eq. (2b) and where

'Ht being the usuRl, azimuthal quRQtum Dumber ofthe particle, & its charge, Rnd 0 Rnd V being

P„=lV 4'„"'(r —2M —i0),

or introducing the Heaviside function 7,P -X [F(r 2M)e "-'(r 2M}-

+s"" y(2itf-r)e:"'{2itf-r)],

where N is a normal. ization factor such that

(5a)

(5b)

(P, P ) = —5(u&, —~s)5» 5, . (5c)

As @~wRs Rlx'eRdy Qornlallzed lt 18 vel y Simpleto obtain N

I& I'=(e'""~-1) '

Now Eq. (5b) describes the splitting of P in

a wave outgoing from the horizon and a wavefalling on the singularity (see Fig. 1). The prob-ability flux carried away by this outgoing waveis simply IN„I'/2s per unit of time [see Eq. (2)]and onl, y a fraction I' of this flux will be trans-mitted to infinity, where I is the transmissioncoefficient of the potential. and centrifugal barrier(Fig. 1). Using Eq. (6) we get at infinity an out-going flux of particles of

(f'/2s)(e"" —1) '

per unit of time and per unit range of frequency,which is Hawking's result. '

If we consider now a scalar field in R Kex'r-Newman geometry, it can be shown, "using the

FIG. I, In usual Eddington-Finkelstein coordinates,a qualitative represelltation is given of the splitting ofthe antiparticle state P~ [see Eq. (51)] into two compo-nents: (aj a particle wave of strength IiV~ It outgoingfrom the horizon and partially transmitted to infinityE'if ~ & p) and partially backscattered into the hole, and

(h) an antiparticle wave of strength IN I

etvs" 1+Iii It=with positive energy flux outgoing in the past from thesingularity. This can always be interpreted as a nega-tive-energy flux of antiparticles IP~ I ingoing in thefuture toward the singularity. P„being normalized,Iiv J /2v yieMs the rate of pair creation per unit fre-quency from H+.

334 THIBAUT DAMOUR AND REMO RUFFINI 14

respectively the angular velocity" and the electricpotential' of the black hole. As in these coordin-ates the vector 8/er is still null and past-directedwe can use the same prescription as before, r-r-i0, to describe an antiparticle, which yieldsthe splitting analogous to Eg. (5b):

P =N [y(r —r )4'„"'(r —r, )

+e' 'n i"F(r -r)C'"'(r —r)] (9)

However, in the present situation of a Kerr-Newman geometry two drastically different situa-tions occur for an energy of the wave +&+0 or(d & QPO.

For ~&coo the norm of 4""' is positive and itsflux is +(2v) ' so that one gets )NJ'=(e'"~—1) '. And as usual only a positive fraction I'of this flux is transmitted to infinity through thecombined potential and centrifugal barriers.

For ~&(do the norm of 4~' is negative as wellas its flux —(2s) ' (antiparticles) so that one getsl&J'=(1 —e"l ~~~") '. But we are preciselyin the conditions of l.evel crossing' betgeeen thehorizon and spatial infinity so that a negativefraction I' of this flux will be transmitted to in-

finity.Therefore in both eases one observes a positive

flux at infinity (particles) of

(1,/2v)(erne(~ ~,)y. 1)-x (10)

per unit of time and per unit range of frequency.The drastic difference between those two re-

gimes appears clearly if we consider the limitwhere the effective temperature «/2v of the blackhole tends to zero. In the Sehwarzschild casethe rate of particle creation goes then to zero.In the Kerr-Newman solution the rate goes alsoto zero if &o&&o„but in the range p &~&~, (where

g is the mass of the particles) the rate of par-ticle creation tends to —1'/2v which is the phe-nomenon studied in Ref. 4.

If we now take into account that the effectivetemperature is given by «/2v=10 'x(MO/M) 'K

we ean conlcude that the only place where thermalcontribution to the vacuum polarization of a blackhole can exist is in the early stages of cosmology(black holes of M&10" g). For macroscopic blackholes, outcoming from stel. lar collapse, only theprocess of vacuum polarization described in Ref.4 can possibly be observed.

~E. S. A. International fellow.)Alfred P. Sloan fellow.D. Christodoulou, Phys. Rev. Lett. 25, 1596 {1970);D. Christodoulou and R. Rudfini, Phys. Rev. D 4, 3552(1971). The result that the irreducible mass of a blackhole can never decrease for classical transformationswas independently shown by S. Hawking, Phys. Rev.Lett. 6, 1344 (1971).

2The extraction of rotational energy has been treated byR. Penrose and R. Floyd, Nature 229, 193 (1971). Theextraction of Coulomb energy has been studied hyG. Denardo and R. Ruffini, Phys. Lett. 45B, 259 (1973),and by G. Denardo, L. Hively, and R. Ruffini, Phys.Lett. 50B, 270 (1974). The region in which energy ex-traction processes can occur has been defined as the'bffective ergosphere" in Ref. 1.

~The pioneering work in this field has been done byYa. B. Zel'dovich, Zh. Eksp. Teor. Fiz. 62, 2076(1972) [Sov. Phys. —JETP 35, 1085 (1972)], and A. A.Starobinsky, ibid. 64, 48 (1973) jibid. 37, 28 (1973)j, forthe rotational energy and by G. W. Gibbons and S. Haw-king, in Gravitational Radiation and GravitationalCollapse, edited by C. DeWitt (Reidel, New York,1974), and W. T. Zaumen, Nature (London) 247, 530(1974), for the Coulomb energy. Then the second-quantized formulations of these phenomena were givenrespectively hy W. Unruh, Phys. Rev. D 10, 3194(1974), and G. Gibbons, Com. mun. Math. Phys. 44, 245(1975).

4The general outline of this approach is in the work byN. Deruelle and R. Ruffini, Phys. Lett. 528, 437 (1974).The analysis of vacuum polarization processes were

given in a Kerr-Newman geometry by T. Damour and

R. Ruffini, Phys. Rev. Lett. 35, 463 (1975); in a Kerrgeometry by N. Deruelle and R. Ruffini, Phys. Lett.57B, 248 (1975); in a Reissner-Nordstrom geometry byT. Nakamura and H. Sato, Phys. Lett. 61B, 371 (1976).

~S. Hawking, Nature (London) 248, 30 (1974); Commun.Math. Phys. 43, 199 (1975).

~Some of the properties of this solution have been recent-ly analyzed by D. G. Boulware, Phys. Rev. D 11, 1404(1974). We disagree with the physical conclusionsreached in that article and we explicitly show how ourprescription of analytic continuation of the wave functionleads to opposite physical conditions.

VSee, e.g. , R. Jost, in The general theory of quantizedfields (American Mathematical Society, Providence,Rhode Island, 1965), p. 73.

R. Penrose, Int. J. Theor. Phys. 1, 61 (1968), especial-ly pp. 77 and 78.

SSee, e.g. , I. M. Guelfand and G. E. Chilov, I.es distri-butions (Dunod, Paris, 1962), p. 43 ff.

~OThe norm is determined as usual by examining broadwave packets.

~'The computation is similar to the one for the simplercase of a Reissner-Nordstrom geometry dealt with byG. Gibbons, Commun. Math. Phys. 44, 245 (1975);see p. 257.

~ See, e.g. , B. Carter, inBlack Holes, edited byB.DeWitt and C. DeWitt (Gordon and Breach, NewYork, 1973).

~~D. Christodoulou and R. Ruffini, in Black Holes (Ref.12).