10.7 Hawking Radiation from a Black Hole
155
The brief violation of energy conservation is allowed according to the energytime version of the uncertainty principle; this relates the lifetime of a state to the
uncertainty in its energy by
Et ≈ .
(10.29)
Thus the particles in the left Fig. 10.8 with energy E equal to the combined rest mass
or greater can only exist for a time t. However, near a black hole we can imagine
that the electron and positron fall toward the black hole rather than recombining with
the photon, while the photon escapes outward as in the right of Fig. 10.8. Rather
than being forced to recombine by energy conservation the photon becomes real and
escapes with energy provided by the black hole.
Consider the escaping particle and the uncertainty principle in its usual form
px ≈ /2.
(10.30)
This lets us estimate the momentum and energy the escaping photon can have. The
only scale in the system is the Schwarzschild radius of the black hole r s = 2m, so
we take that to be the uncertainty x and have for the photon momentum
p ≈ p ≈ /2x ≈ /4m,
(10.31)
(See Exercise 10.13 for a stronger motivation for this choice). This gives for the
photon energy
E = pc ≈
c
4m
=
c
3
4G M
.
(10.32)
If we now assume that the spectrum of such escaping photons is thermal then the
temperature and energy are related by
kT ≈ E ≈
c
3
4G M
.
(10.33)
This is about the same relation for the temperature obtained by Hawking using
quantum field theory, except that his result contains 8π rather than 4 in the
denominator,
kT H =
c
3
8π G M
, Hawking temperature.
(10.34)
The result (10.33) can also be obtained with a thermodynamic argument, as outlined
in Exercise 10.15 (Ohanian 1994).
Having obtained the temperature of a black hole we can also calculate an entropy.
We imagine building the black hole by assembling small masses, each with rest
155
The brief violation of energy conservation is allowed according to the energytime version of the uncertainty principle; this relates the lifetime of a state to the
uncertainty in its energy by
Et ≈ .
(10.29)
Thus the particles in the left Fig. 10.8 with energy E equal to the combined rest mass
or greater can only exist for a time t. However, near a black hole we can imagine
that the electron and positron fall toward the black hole rather than recombining with
the photon, while the photon escapes outward as in the right of Fig. 10.8. Rather
than being forced to recombine by energy conservation the photon becomes real and
escapes with energy provided by the black hole.
Consider the escaping particle and the uncertainty principle in its usual form
px ≈ /2.
(10.30)
This lets us estimate the momentum and energy the escaping photon can have. The
only scale in the system is the Schwarzschild radius of the black hole r s = 2m, so
we take that to be the uncertainty x and have for the photon momentum
p ≈ p ≈ /2x ≈ /4m,
(10.31)
(See Exercise 10.13 for a stronger motivation for this choice). This gives for the
photon energy
E = pc ≈
c
4m
=
c
3
4G M
.
(10.32)
If we now assume that the spectrum of such escaping photons is thermal then the
temperature and energy are related by
kT ≈ E ≈
c
3
4G M
.
(10.33)
This is about the same relation for the temperature obtained by Hawking using
quantum field theory, except that his result contains 8π rather than 4 in the
denominator,
kT H =
c
3
8π G M
, Hawking temperature.
(10.34)
The result (10.33) can also be obtained with a thermodynamic argument, as outlined
in Exercise 10.15 (Ohanian 1994).
Having obtained the temperature of a black hole we can also calculate an entropy.
We imagine building the black hole by assembling small masses, each with rest