Black Holes 101information contained in the book is lost forever. This is a paradox because quantum
theory states that the information of the initial state should never disappear. In order
to resolve this paradox it is necessary to construct microscopic states of the black hole
and to give a statistical-mechanical explanation for the black hole entropy, which is
difficult within general relativity because of the no-hair theorem. So far the paradox
still remains since a complete description of an evaporating black hole has not yet
been established.
In 1973J. Bekensteinnoted [7] that there are certain similarities between the size of
the event horizon of a black hole and entropy. When a star has collapsed to the size of
its Schwarzschild radius, its event horizon will never change (to an outside observer)
although the collapse continues. Thus entropy푠could be defined as the surface area
퐴of the event horizon times some proportionality factor,
푠=
Ak푐^3
4 퐺ℏ, (5.79)
theBekenstein–Hawking formula. For a spherically symmetric black hole of mass푀
the surface area is given by
퐴= 16 휋푀^2 퐺^2 ∕푐^4. (5.80)퐴can increase only if the black hole devours more mass from the outside, but
퐴can never decrease because no mass will leave the horizon. In Hawking’s par-
lance [6], however, this is true for the classical event horizon but not for the apparent
horizon.
Inserting퐴into Equation (5.79), entropy comes out (in units of erg/K) proportional
to푀^2 :
푠=푀^24 휋kG∕푐ℏ. (5.81)Thus two black holes coalesced into one possess more entropy than they both had
individually. This is illustrated in Figure 5.4.
Hawking Radiation. Stephen Hawking has shown [8,9] that although no light can
escape from black holes, they can nevertheless radiate if one takes quantum mechan-
ics into account. It is a property of thevacuumthat particle–antiparticle pairs such as
e−e+are continuously created out of nothing, to disappear in the next moment byanni-
hilation, which is the inverse process. Since energy cannot be created or destroyed,
one of the particles must have positive energy and the other one an equal amount of
negative energy. They form avirtual pair, neither one is real in the sense that it could
escape to infinity or be observed by us.
In a strong electromagnetic field the electron e−and the positron e+may become
separated by a Compton wavelength휆of the order of the Schwarzschild radius푟c.
Hawking has previously shown that there is a small but finite probability for one of
them to ‘tunnel’ through the barrier of the quantum vacuum and escape the black
hole horizon as a real particle with positive energy, leaving the negative-energy par-
ticle inside the horizon of the hole. Since energy must be conserved, the hole loses
mass in this process, a phenomenon calledHawking radiation.But,asnotedbefore,