Introduction to Cosmology

102 Cosmological Models

B 1

B 2

Infalling

stars

B 3

Figure 5.4The merging of two black holes with event horizons퐵 1 and퐵 2 , respectively. The
black hole 2 grows initially by swallowing infalling stars. The final black hole has an event
horizon with event horizon퐵 3 ⩾퐵 1 +퐵 2.

in quantum theory the radius푟and the time푡are conjugate quantities which must
obey Heisenberg’s uncertainty relation, so the conditions for Hawking radiation are
uncertain.
The timescale of complete evaporation is

푡≈10 Gyr

(

푀

1012 kg

) 3

. (5.82)

Thus small black holes evaporate fast, whereas heavy ones may have lifetimes exceed-
ing the age of the Universe. Paradoxically, in some black holes the infall of matter may
cause the black hole to evaporate faster than the horizon has time to form, as seen
by a distant observer. This is a consequence of the fact that during gravitational col-
lapse time dilation can increase without bound, sufficiently retarding implosion so
as to prevent the formation of an event horizon and the potential loss of matter and
information from the observable universe.
The analogy with entropy can be used even further. A system in thermal equi-
librium is characterized by a unique temperature푇 throughout. When Hawking
applied quantum theory to black holes, he found that the radiation emitted from
particle–antiparticle creation at the event horizon is exactly thermal. The rate of parti-
cle emission is as if the hole were a black body with a unique temperature proportional
to the gravitational field on the horizon, theHawking temperature:

푘푇H= 푐

(^3) ℎ
8 휋GM

= 6. 15 × 10 −^8

푀⊙

푀

K. (5.83)