What, no black hole evaporation?

Volume 80A, number 1

PHYSICS LETTERS

10 November 1980

WHAT, NO BLACK HOLE EVAPORATION? P. HAJICEK’ and W. ISRAEL2

Institute for Theoretical Physics, University ofBerne, CH-3012 Berne, Switzerland Received 29 September 1980

Tipler has claimed that the inward flux of negative energy across the horizon which (according to the semi-classical approximation) accompanies the evaporation of a black hole would cause a solar mass black hole to evaporate in less than a second. It is shown that this claim is in error.

In the semi-classical description of black hole evaporation, which treats gravity as a classical background field, the Hawking radiation flux to infinity is accompanied by an inward flux of negative energy across the horizon [1] which causes the black hole to shrink. A solar mass black hole is conventionally expected to last about ~ years. However, in a paper entitled “Do black holes really evaporate thermally ?“ Tipler [2] has claimed that the horizon would shrink to zero radius in less than a second. We shall show that

this claim is fallacious. The renormalized stress-energy tensor which describes the back-reaction of a quantized zero mass field propagating on a Schwarzschild backgroundis a fairly object [I], complete has notcomplicated yet been obtained [3].whose For our present form purposes it is sufficient to adopt a simplified picture which ignores vacuum polarization and approximates T, 2~near the horizon as a radial influx of negative energy which balances the outward Hawking flux at infinity. We believe this yields a tolerable description of the evaporation until the fInal 1 0~ seconds, when both the quasi-static and semi-classical approximations break down. In any case, this model is quite sufficient to serve as a counterexample to Tipler’s argument, which is based on Raychaudhuri’s equation, 2

Work supported partially by Schweizerischer Nationalfonds. On leave of absence from Theoretical Physics Institute, University of Alberta, Edmonton, Canada. Work partially supported by National Science and Engineering Council of Canada.

and for which the energy flux at the horizon is the only relevant part of Trn,. Near the horizon the metric then has the Vaidya form [4] .,

ds =2drdv— [1 —2m(v)/r]dv +r d~ with the associated energy tensor I

=

2

[m(v)/4~’rr](~u)(~v), J2

V

for v 0
with m and v measured in Planck units. This is the standard result, according to which the evaporation time for a black hole of astronomical mass is enormous. We now examine the evolution of the event horizon. The interior of the evaporating black hole is rea~ sonably defined as the set of events which distant external observers cannot become aware of if the terminal point 0 of the evaporation (fig. 1) is a barrier for causal signals, i.e. as M J_(9+), with event 0 ex—

cluded from the space—time manifold M. The event horizon HO is the boundary of this region, and is an 9

Volume 80A, number 1

PHYSICS LETTERS

10 November 1980

is given by r

o

-

_____________________________

4rn/[1 + (1

—

16th)h12]

and lies to the right of the event horizon, so the horizon has 7 <0 everywhere. Tipler’s incorrect result stems from his assumption that (3) P=2cP_4ir(T~)l0l13r=0, initially; this would then really imply 7 >0 afterwards. To check eq. (3), we set our values

/

apparent horizon

-,

=

~,nf1ex,on points

e = m/2r~, in eq.

(T~)l011l= ,iz/4irr2,

(3) and obtain

One sees from (2) that P ~—2(—v)~/~.

H

Fig. 1. History of spherical evaporating black hole, showing outgoing radial light rays, the event horizon HO, the apparent horizon, and the locus of inflexion points.

This is negative, but of course very small. Nevertheless, it suffices to invalidate Tipler’s argument: One sees from fig. 1 that the light rays which start from the inflexion point curve with 7 0 reach the apparent horizon very quickly (may be, form m®, in less than a second) and expand to 9~.For v ~ —1, the event horizon, which satisfies V <0 safely along its whole (1064 years) history, lies very close to the inflexion points curve. One has, therefore, to be very ‘—

outgoing radial light-like surface r = r(v), satisfying dr/dy = ~(l

—

2m(v)/r),

with r(0) = 0. It is easy to verify the asymptotic forms of the solution: r(v)

~

(—v)213

2m(v)

—

~(—v)~/3

(v

—~

0),

(v —~ _oo).

References (2)

We note from fig. 1 that the event horizon lies 117 side the apparent horizon r = 2m(v), in contrast to what happens in the more conventional case of positive energy influx [5]. The essential feature is that the event horizon remains very close to the apparent horizon for most of the lifetime of the black hole. The locus of inflexion points of solutions of eq. (1)

10

careful in making approximations like eq. (3).

[I] S.M. Christensen and S.A. Fulling, Phys. Rev. D15 (1977) 2088. [2] F.J. Tipler, Phys. Rev. Lett., to be published; see also: Contribution to the 9th Intern. Congress on General reiativity and gravitation (Jena, July 1980). [3] Rev. D21 2185. [4] P. SeeCandelas, e.g. R.W.Phys. Lindquist, R.A.(1980) Schwartz and C.W. Misner, Phys. Rev. 137 (1965) B1364. [5]

S.W. Hawking and G.F.R. Ellis, The large scale structure of space—time (Cambridge UP., Cambridge, l973)p.320.