May 15, 1988 - Correction to "Acceleration radiation and the generalized second law of thermodynamics" ... Department ofPhysics, University ofBritish Columbia,Vancouver, British Columbia, Canada .... 1988 The American Physical Society ...
PHYSICAL REVIEW D
VOLUME 37, NUMBER 10
15 MAY 1988
Correction to "Acceleration radiation and the generalized second law of thermodynamics" Marek Department
J. Radzikowski'
and W.
G. Unruh
Cosmology Program, Canadian Institute for Advanced Research, and of Physics, University of British Columbia, Vancouver, British Columbia, Canada V6T2A6
(Received 18 November
1987)
We correct the calculation for the "inertial viewpoint" in Unruh and Wald [Phys. Rev. D 25, 942 (1982)], without altering the conclusions of that paper.
Unruh and Wald' analyzed two gedanken experiments for attempting to violate the proposed "generalized with the use of the second law of thermodynamics" Hawking radiation from a black hole. They gave explanations for the failure of these attempts in terms of existing quantum phenomena. The second gedanken experiment consisted of 6lling a box with thermal radiation, adiabatically lowering it by a rope toward a Schwarzchild black hole of mass M, dropping its contents into the hole, and then raising the empty box. In Ref. 1 (hereafter UW), this situation was analyzed from the points of view of accelerated and inertial observers. It was claimed that the tension in the rope at infinity was the same in both viewpoints. However, Myhrvold pointed out that, in the inertialviewpoint analysis, the expression UW (4. 16),
5E=
1
12' (ab
f
(g&da&b
X, d
)a/— X, ,
a=- dgdl 1
37
I1
12m
~g
—R
da
—R 12m
d dl
X
da dl
t 1
dg
X di
da dl
&
2
where R is the proper length of the box and we have used the fact that a =da /dl =0 at infinity. Thus,
a5E=
—R
da dl
12m
—R 1d(Xa 24m
1
21 dX
2
X dl
)
dl
X
1
1&&
2
24sr X
(3)
Also, the tension in the rope at the box should be
F =(Eo 5E)a —1 h(— XT„;)—
(4)
[instead of UW (4.23)], because forces must also be redshifted to the same point (say the center of the box) before they are summed. It is now easy to see that using Eq. (3) and following the same steps leading to UW (4.25), we have 1 F=E a —— 6 7— T 6
—(XP)
R =E a+ —
(2)
2M/r)'~ are red-shift factors. Similar where the X =(1 — corrections must also be made in UW (4. 14) and (4. 15). Ignoring subscripts, and using the identity
—R
12m'
—a, ),
for the total energy Bow out of the box during the lowering process, is incorrect. The differentials da/12m are local energies dE [by UW (4. 13)] and need to be transported to the same point before they are summed. In his reanalysis Myhrvold adds up the energies out of the top and bottom faces separately, and only at the end of the lowering process does he refer the totals to the same point and sum them, getting zero net emitted energy. Physically, his procedure assumes that the energy emitted from each of the faces remains concentrated there, and the energies are mixed only at the end when the box is emptied, but this is clearly not the situation Unruh and Wald wanted to analyze. Instead, the energy should be distributed throughout the box in a continual manner. Thus, the energies emitted during a short time interval from the two surfaces must be red-shifted to the center of the box (say), added together, and then integrated over the total process. Thus the net emitted energy at any distance I from the horizon should be
5E =
Eq. (2) simplifies to
(5)
The second term in Eq. (5) corresponds exactly to the "extra" force F2 of UW (2. 15) needed to oppose the acceleration-radiation pressure. [Note that in UW (2. 15), the force F2 is the tension in the rope at inanity caused by the bouyant force, while Eq. (5) gives the tension in 3059
1988
The American Physical Society
BRIEF REPORTS
3060
the rope at the box as in UW (4.25). The tensions at infinity and at the box differ by a red-shift factor, so Eq. in order to (5) and UW (4.25) must be multiplied by compare them directly with UW (2. 15).] Thus the tension is indeed the same in both viewpoints. This work was done under the support of Natural Sci-
J
Current address: Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544. W. G. Unruh and R. M. Wald, Phys. Rev. D 25, 942 (1982).
37
ences and Engineering Research Council Grant No. 580441 and of the Canadian Institute for Advanced Research. One of use (M.J.R.) would like to thank the Challenge 87 Program of the government of British Columbia for partial support.
N. P. Myhrvold, Ph. D. thesis, Princeton University, available from U. Microfilms, Ann Arbor, Michigan.
1983,
