300
19 Inflation and Some Questions
be left behind The GUP may prevent total evaporation in exactly the same way that
the uncertainty principle prevents a hydrogen atom from total collapse: the complete
decay of a black hole is prevented, not by symmetry, but by dynamics, as a minimum
size and mass are approached (Adler 2001). See Exercise 19.11.
We may use the GUP to derive a modified black hole temperature exactly as we
derived the Hawking temperature in Chap. 10. The basic idea is the same; a virtual
pair of charged particles and a photon are formed near the black hole surface, the
particles are absorbed by the black hole, and the photon is emitted as black body
thermal radiation as shown in Fig. 10.8. From (19.27) we solve for the emitted
photon momentum in terms of the distance uncertainty, which we take to be the
Schwarzschild radius x = 2G M/c
2 , and obtain
p =
x
2L
2
P
⎡
⎣ 1 ±
1 − 4
L
2
P
x 2
⎤
⎦ = Mc
⎡
⎣ 1 −
1 −
M
2
P
M 2
⎤
⎦ .
(19.48)
We have chosen the negative sign to agree with the results of Sect. 10.7. The energy
of the photon is of course E = pc. Thus we estimate the temperature of the black
hole to be kT = E with
kT ≈ E = Mc
2
⎡
⎣ 1 −
1 −
M
2
P
M 2
⎤
⎦ .
(19.49)
It is easy to check that this agrees with the previous results of Chap. 10 for masses
large compared to the Planck mass: we expand (19.49) and find
kT ≈
M
2
P c
2
2M
=
c
3
2G M
,
(19.50)
which is roughly the same as our estimate (10.33) and the Hawking temperature
(10.34). Notice also that the temperature (19.49) is well-behaved as the mass of the
black hole approaches the Planck mass, whereas in the standard Hawking result it is
infinite.
With the modified temperature (19.49) it is straightforward to calculate the entropy
of a black hole in terms of its mass, and also its lifetime and the rate of energy
radiated; all are well-behaved as the mass approaches the Planck mass and the black
hole becomes a remnant. Figure 19.8 shows the mass as a function of time and
compares the Hawking result to what we obtained with the GUP—corrected to agree
with Hawking at t = 0.
In summary the picture that follows from the above calculation is that a small
black hole, with temperature greater than the ambient temperature, should radiate
photons, as well as other particles, until it approaches the Planck mass and size.
At the Planck mass it ceases to radiate and its entropy reaches zero, even though
19 Inflation and Some Questions
be left behind The GUP may prevent total evaporation in exactly the same way that
the uncertainty principle prevents a hydrogen atom from total collapse: the complete
decay of a black hole is prevented, not by symmetry, but by dynamics, as a minimum
size and mass are approached (Adler 2001). See Exercise 19.11.
We may use the GUP to derive a modified black hole temperature exactly as we
derived the Hawking temperature in Chap. 10. The basic idea is the same; a virtual
pair of charged particles and a photon are formed near the black hole surface, the
particles are absorbed by the black hole, and the photon is emitted as black body
thermal radiation as shown in Fig. 10.8. From (19.27) we solve for the emitted
photon momentum in terms of the distance uncertainty, which we take to be the
Schwarzschild radius x = 2G M/c
2 , and obtain
p =
x
2L
2
P
⎡
⎣ 1 ±
1 − 4
L
2
P
x 2
⎤
⎦ = Mc
⎡
⎣ 1 −
1 −
M
2
P
M 2
⎤
⎦ .
(19.48)
We have chosen the negative sign to agree with the results of Sect. 10.7. The energy
of the photon is of course E = pc. Thus we estimate the temperature of the black
hole to be kT = E with
kT ≈ E = Mc
2
⎡
⎣ 1 −
1 −
M
2
P
M 2
⎤
⎦ .
(19.49)
It is easy to check that this agrees with the previous results of Chap. 10 for masses
large compared to the Planck mass: we expand (19.49) and find
kT ≈
M
2
P c
2
2M
=
c
3
2G M
,
(19.50)
which is roughly the same as our estimate (10.33) and the Hawking temperature
(10.34). Notice also that the temperature (19.49) is well-behaved as the mass of the
black hole approaches the Planck mass, whereas in the standard Hawking result it is
infinite.
With the modified temperature (19.49) it is straightforward to calculate the entropy
of a black hole in terms of its mass, and also its lifetime and the rate of energy
radiated; all are well-behaved as the mass approaches the Planck mass and the black
hole becomes a remnant. Figure 19.8 shows the mass as a function of time and
compares the Hawking result to what we obtained with the GUP—corrected to agree
with Hawking at t = 0.
In summary the picture that follows from the above calculation is that a small
black hole, with temperature greater than the ambient temperature, should radiate
photons, as well as other particles, until it approaches the Planck mass and size.
At the Planck mass it ceases to radiate and its entropy reaches zero, even though