156
10 Black Holes and Gravitational Collapse
energy dQ = dMc
2 . According to the thermodynamic definition of entropy S the
increase for each such small mass is
dS =
dQ
kT
=
8π G M
c
dM.
(10.35)
(Note that we here use a dimensionless definition of entropy.) The total entropy of
the black hole is thus
S =
4π G
c
M
2
=
1
4
A BH
2
P
, where A BH = 4πr
2
s ,
P =
c 3
G
= 1.6 × 10
−35 m.
(10.36)
It is equal to one fourth of the area of the black hole A BH divided by the square of
the Planck distance P ; the Planck distance may in fact be the smallest physically
meaningful distance as we will discuss in a later section on the Planck scale. The
result (10.36) is called the Bekenstein entropy and was obtained, up to a factor, by
Bekenstein before Hawking obtained his result (Bekenstein 1973). The Bekenstein
formula simply assigns one unit of entropy to each tiny Planck scale area 4
2
P on the
black hole surface.
The black hole entropy has the peculiar feature that it is proportional to a 2dimensional surface area. It is more common that entropy, called an extensive property, is proportional to the volume of a system. Some theorists have made much of
this fact and have introduced the “holographic principle,” that the information on the
surface of the black hole is somehow equivalent to the information one expects for
the volume of the black hole; some have tried to elevate this to a general principle.
This is of course highly speculative.
Hawking’s formula for the temperature is quite remarkable since it predicts a
specific phenomenon that involves both gravity and quantum mechanics. Hawking
radiation has not yet been observed in the real world. For stellar mass black holes
we do not expect to see it since the temperature predicted by (10.34) is only about
6 × 10
−8 K. This is much less than the ambient 2.7 K cosmic background radiation
that permeates the universe, so a black hole at such a low temperature would absorb
more radiation than it emits. For sufficiently small black holes, with temperatures
greater than the background radiation we should be able to detect the radiation; the
relevant mass is roughly 10
−8 M . Indeed, as such a small black hole radiates it will
lose energy and thus become lighter, so its temperature will increase without limit
according to (10.34), and it should end up emitting a very bright flash at the end of
its life. See Exercise 10.17. The absence of observations of such flashes could be due
to the lack of small black holes or the incorrectness of the theory.
The late stages of black hole evaporation likely involve very large energies and
small distances, of order the Planck scale of 10
19 GeV and 10
−35 m. We of course
have no experimental knowledge of such things, and thus no dependable theory, so
the end product of the evaporation is unknown. It is widely believed that distances
10 Black Holes and Gravitational Collapse
energy dQ = dMc
2 . According to the thermodynamic definition of entropy S the
increase for each such small mass is
dS =
dQ
kT
=
8π G M
c
dM.
(10.35)
(Note that we here use a dimensionless definition of entropy.) The total entropy of
the black hole is thus
S =
4π G
c
M
2
=
1
4
A BH
2
P
, where A BH = 4πr
2
s ,
P =
c 3
G
= 1.6 × 10
−35 m.
(10.36)
It is equal to one fourth of the area of the black hole A BH divided by the square of
the Planck distance P ; the Planck distance may in fact be the smallest physically
meaningful distance as we will discuss in a later section on the Planck scale. The
result (10.36) is called the Bekenstein entropy and was obtained, up to a factor, by
Bekenstein before Hawking obtained his result (Bekenstein 1973). The Bekenstein
formula simply assigns one unit of entropy to each tiny Planck scale area 4
2
P on the
black hole surface.
The black hole entropy has the peculiar feature that it is proportional to a 2dimensional surface area. It is more common that entropy, called an extensive property, is proportional to the volume of a system. Some theorists have made much of
this fact and have introduced the “holographic principle,” that the information on the
surface of the black hole is somehow equivalent to the information one expects for
the volume of the black hole; some have tried to elevate this to a general principle.
This is of course highly speculative.
Hawking’s formula for the temperature is quite remarkable since it predicts a
specific phenomenon that involves both gravity and quantum mechanics. Hawking
radiation has not yet been observed in the real world. For stellar mass black holes
we do not expect to see it since the temperature predicted by (10.34) is only about
6 × 10
−8 K. This is much less than the ambient 2.7 K cosmic background radiation
that permeates the universe, so a black hole at such a low temperature would absorb
more radiation than it emits. For sufficiently small black holes, with temperatures
greater than the background radiation we should be able to detect the radiation; the
relevant mass is roughly 10
−8 M . Indeed, as such a small black hole radiates it will
lose energy and thus become lighter, so its temperature will increase without limit
according to (10.34), and it should end up emitting a very bright flash at the end of
its life. See Exercise 10.17. The absence of observations of such flashes could be due
to the lack of small black holes or the incorrectness of the theory.
The late stages of black hole evaporation likely involve very large energies and
small distances, of order the Planck scale of 10
19 GeV and 10
−35 m. We of course
have no experimental knowledge of such things, and thus no dependable theory, so
the end product of the evaporation is unknown. It is widely believed that distances