The Physical Basis of The Direction of Time (The Frontiers ...

5.1 Thermodynamics of Black Holes 147
1983), while their entangled partners with positive energy may then propagate
towards infinity. (Again, all energy values refer to an asymptotic frame
of reference.) The probabilities for these processes lead precisely to a black
body radiation with temperature
T bh = κ
2π , (5.14)
with κ in units of /ck B , and therefore to the black hole entropy 1
S bh = A 4 . (5.15)
The mean wavelength of the emitted radiation is of order √ A.
A black hole not coupled to any quantum fields (α = ∞) would possess
zero temperature and infinite entropy, corresponding to an ideal absorber in
the sense of Sect. 2.2. This result would also be obtained for classical black body
radiation, that is, for classical electromagnetic waves in thermal equilibrium
– reflecting the historically important infrared catastrophe for classical fields
(Gould 1987).
According to (5.14), a black hole of solar mass would possess a temperature
of no more than T bh ≈ 10 −6 K. In the presence of a cosmic background radiation
of 2.7 K, it would therefore absorb far more energy than it emits (even in
the complete absence of interstellar dust). Only black holes with mass below
3 × 10 −7 solar masses could effectively lose mass under the present conditions
of the Universe (Hawking 1976). Black holes that have formed by gravitational
collapse (that is, with a mass above 1.4 solar masses) require a further expansion
and cooling of the Universe by a factor of almost 10 7 or more in order
to be able to disappear by radiation. ‘Black-and-white holes’ in equilibrium
with a heat bath would not possess any horizon, but according to classical
general relativity require a spatial singularity at r = 0, which corresponds to
a negative singular mass – signalling the need for quantum gravity (Zurek and
Page 1984).
In vacuo (at T = 0), a black hole would eventually completely decay into
thermal radiation. The resulting entropy can be estimated to be somewhat
larger than that of the black hole (Zurek 1982b). Since the future horizon and
the singularity would thereby also disappear, this process seems to represent
a genuine global indeterminism – known as the ‘information loss paradox’.
It is remarkable, though, that no conservation laws would be violated in a
Schwarzschild foliation. The diverging time dilation close to the horizon prevents
all matter from ever reaching the horizon on these simultaneities, which
define a global history that covers the complete external world.
1 It is important to realize that this result is quite independent of the nature of
existing fields. Therefore, it cannot be used to support any specific theory, such
as M-theory, by its explicit confirmation.