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Rate of black hole formation in a thermal box
Volume 90A, number 1,2
PHYSICS LETTERS
21June1982
RATE OF BLACK HOLE FORMATION IN A THERMAL BOX
Tsvi PIRAN Racah Institute of Physics, Hebrew Universityof Jerusalem, Jerusalem, Israel 91904 and Institute for Advanced Study, Rinceton, NJ 08540, USA
and Robert M. WALD Enrico Fermi Institute, Universityof Chicago, Chicago, IL 60637, USA Received 18 February 1982
We derive an order of magnitude estimate for the rate at which ordinary thermal fluctuations in a box filled with radiation will produce black holes. Our results are compared with a recent calculation (using euclidean quantum gravity) of Gross, Perry and Yaffe for the rate of formation of black holes at the equilibrium massMeq = 1/8nT,-,.
One of the most striking developments in the theory of black holes is the analogy between the laws satisfied by black holes and the ordinary laws of thermodynamics (see, e.g. ref. [l] and the references cited therein). Many insights into black hole thermodynamics and the nature of quantum gravity have been obtained by considering gedanken experiments involving a box filled with thermal matter in which black holes form and evaporate. The purpose of this letter is to derive an order of magnitude estimate for the rate at which black holes of a given mass should form in such a box as a result of ordinary, thermal fluctuations. We calculate this rate as follows. Consider a “large” box of volume PO containing energy E. in the form of radiation, i.e. free, massless fields. Our starting point is the assumption that the probability of any configuration of this radiation is proportional to exp(s), where S is the entropy of that configuration. We focus attention on a volume I/ and calculate the probability that an energy E is found in this volume, where we use the standard formulae for the entropy of black body radation and ignore the self-gravitation of the radiation when calculating this probability. We then assume that if E - V113(in Planck units C = c
20
= h = k =l), self-gravitation will result in collapse to a black hole. (For spherically distributed radiation, the condition E = 1. 2R expresses the condition that the radiation be just within its own Schwarzschild radius, while E * 0.25R is the point at which stable equilibrium configurations of self-gravitating radation no longer can exist [2], i.e. the point at which Jeans’ instability sets in.) Thus, we obtain the probability that as a result of ordinary thermal fluctuations the box of radiation will be found in a configuration which will result in the formation of a black hole of energy E.This probability then is converted to a rate of black hole formation by dynamical arguments. For free radiation in its maximum entropy, thermal state, the energy, E, and entropy, S, are given in terms of the temperature T by,
E=apV,
(1)
S=&T3V,
(2)
where we have neglected boundary effects resulting from finite box size. Here Q = &(7?/30)c+g~ wheregi is the spin degeneracy factor, c+ takes the value 1 for bosons and 7/8 for fermions and we employ Planck umts in all our calculations. By eliminating T, we
0 031-9163/82/0000-0000/$02.75 0 1982 North-Holland
Volume 90A, number 1,2
PHYSICS LETTERS
fmd that the maximum entropy, S of radiation having total energy E in a volume V is given by, S = &l/4 Vl’4E3’4 .
(3)
Consider, now, a box of volume V. containing radiation of total energy E. . In its maximum entropy state, this radiation would be thermally distributed at temperature To = (Eo/cz Vo)1’4. Although this is the most probable distribution of the radiation, other ditrrbutions can occur with probability proportional to the exponential of their entropy. We focus attention on a particular volume V contained within the box. If an energy E is found in V, the energy in the remaining volume V. - V will be E. - E. Thus, the probability that the volume V will contain an energy between E and E + dE is given by, P&Y)
dE.a exp(Stot) dE = exp[&1’4 V114E314
t ~c&~(V,
- V)1’4(Eo Y!?)~“]
dE.
(4)
P&l?) dE a exp&1’4 Vo1’4E03’4 - ⁢ V - E/To t ;cY~‘~ V1’4E3’4) dE t $c$‘~ V1’4E3’4)
dE ,
(5)
where in the last line we have dropped the terms independent of E from our un-normalized probability distribution. Thus, we obtain the familiar result that the energy distribution of a small component of a large system is given by the canonical ensemble, and the probability of any configuration of the small system is proportional to exp(-F/To) where F 3 E ToS is the free energy of the small system. Defining 7 by y=aT,+
(6)
NY) = &V
dr exp(-x + 45+x3/4)
0
u,
Inthelimitr
.
(7 s 1) .
00)
Eq. (7) yields the probability that if we were to look in at the volume V “at random” then we would find an energy between E and E t dE contained within it. As stated above, we assume that a black hole of energy E will form if an energy E ii found within a volume V = W3 with X N 1. (It is useful to introduce the factor Xhere in order to keep track of where the terms in our fmal expression arise from.) The number of such volumes in our box is Vo/V, so the probability per unit volume that a black hole of energy E is about to form is obtained by dividing by Vand setting V= AE3 in eq. (7): rs(E = v-lp”(‘%~
= hE3
.
(11)
re-examine this volume again at a time much less than the light travel time, N3, across this volwne, the configuration will not have had an opportunity to change, and the probability of finding that a black hole is about to form will be much smaller than our above “random” probability. On the other hand, if we reexamine the volume at a time greater then the light travel time across this volume, the configuration will have changed completely, and our probability formula (7) should be applicable. Hence, the rate per unit volume of black hole formation should be obtained by dividing 3Q?) by the light travel time V1’3 = @E3)1’3. Thus, our final formula - expressed in Planck units - for the rate per unit volume for formation of black holes of energy between E and E + dE is, exp(-E/To
(7)
where N(r) = J
exp(7/3)
+ $Y~‘~A~‘~E~‘~)
R(E)dE=
t &x1” Vl’4E3’4) T@‘(Y)
(9)
In the opposite limit 7 % 1, we can approximate the probability distribution as a gaussian peaked at the equilibrium value EA = czT$ V, and we find
we obtain the normalized probability distribution, exp(-E/To P&!!q dE =-
(^I
Finally, the rate of black hole formation at energy E is determined as follows. If we find that a black hole is not about to form in the volume V, and then we
Assuming that V Q V. and E Q Eo, we obtain,
a exb(-E/To
N-1
21June1982
(f-0
dE, W3)4’3
(12)
T0.N~)
where N(r) is given by eq. (8) and V should now be set equal to AE3 in our definition, eq. (6), of 7. Our expression (12) for R(E) diverges at low energies E -+ 0. However, quantum gravitational effects certainly should invalidate our analysis for black hole energies below the Planck mass mp = (ll~/G)~‘~ w 1O-5 gm, thus providing a natural low energy cut-off for the 21
of our formula. On the other hand, the validity of our derivation at high energies is limited by the condition E Q E. and the requirement that xE$ < V. so that the large box is not within its own Schwarzschild radius. This yields the upper limit E 4 1/@I2 h1i2 TG). (Note that together with our lower limit, this implies the restriction To 4 T,, = k- 1 (hc5 / IqW - 1032 K.) Hence, within the range of applicability of our formula, the “entropy term” $1/4X1/4 X E3i2 in the exponential is always small compared with the “energy term” E/.To. Within the range of validity of our formula, the maximum rate of black hole formulation occurs at the Planck energy, E = EP. In this case, since To 4 TP we have y 4 1 and hence N(r) w 1. Assuming a! = 1 and setting X = 1, we obtain the rate per unit volume in Planck units, applicability
R(ER) = (1 /TO) exp(1 /To) .
(13)
i.e., in ordinary cgs units, R(E P )~(l/kTo)exp(-TP/To)i;3t-1
(14)
where 45 1 = (Gfi/~~)~/~ - 1O-33 cm, “,d’t, = (Gfi/~~)1~~ - los. Thus, at room temperature, To = 300 K, we find that the timescale, t, for formation of a black hole within AE - E, of the Planck energy EP in one cm3 by ordinary thermal fluctuations is roughly t - 1IR(E
- exp(1029) s .
(15)
Of particular interest is the rate of formation of black holes at energy Eq = 1/&TO, since such black holes would be in (unstable) thermal equilibrium with the radiation. (Black holes with E
= exp[-l/&T:
x (8?r)4T,3/x4’3N(~o) )
t ~a1/4h1/4(1/8aTo)3/2] (16)
where 7. = ~X(877)-~.Recently, Gross et al. [3] have calculated this rate by a quantum mechanical treatment. The dominant factor, exp(-1/16nTi), in their expression is very similar to the dominant factor, exp(-1/8nTi) in our formula. However, the other factors do not appear to be related. The factor of two discrepancy in the argument of the exponential of the dominant factor is important; the rate predicted by the Gross-Perry-Yaffe formula is much higher than ours. It is not difficult to trace the origin of this dis22
21 June 1982
PHYSICSLETTERS
Volume90A, number 1,2
crepancy. The dominant term in both formulae is just exp(-F/To) where F is the free energy, but we use F=E-
ToSmd =E=
1/8~T~,
whereas in effect Gross-Perry-Yaffe F = E - To&,
(17)
use,
= E - To(4?rE2) = l/14aTo
.
(18)
The disagreement between our formula and that of Gross-Perry-Yaffe does not imply that either calculation is in error. Our calculation treated the radiation in an entirely classical manner except, of course, for the quantum discreteness which gives the radiation a well defined, finite entropy. Correspondingly, we considered only ordinary thermal fluctuations which result in a classically describable gravitational collapse to a black hole. On the other hand, the Gross-Perry-Yaffe formula includes the process of quantum tunneling directly into a black hole configuration. Furthermore, our formula gives the rate of black hole formation by direct collapse to a black hole within A,!?of E,, whereas the Gross-Perry-Yaffe formula presumably includes the contributions from formation of smaller black holes which then grow to this size. (Most black holes with E
PHYSICS LETTERS
21June1982
RATE OF BLACK HOLE FORMATION IN A THERMAL BOX
Tsvi PIRAN Racah Institute of Physics, Hebrew Universityof Jerusalem, Jerusalem, Israel 91904 and Institute for Advanced Study, Rinceton, NJ 08540, USA
and Robert M. WALD Enrico Fermi Institute, Universityof Chicago, Chicago, IL 60637, USA Received 18 February 1982
We derive an order of magnitude estimate for the rate at which ordinary thermal fluctuations in a box filled with radiation will produce black holes. Our results are compared with a recent calculation (using euclidean quantum gravity) of Gross, Perry and Yaffe for the rate of formation of black holes at the equilibrium massMeq = 1/8nT,-,.
One of the most striking developments in the theory of black holes is the analogy between the laws satisfied by black holes and the ordinary laws of thermodynamics (see, e.g. ref. [l] and the references cited therein). Many insights into black hole thermodynamics and the nature of quantum gravity have been obtained by considering gedanken experiments involving a box filled with thermal matter in which black holes form and evaporate. The purpose of this letter is to derive an order of magnitude estimate for the rate at which black holes of a given mass should form in such a box as a result of ordinary, thermal fluctuations. We calculate this rate as follows. Consider a “large” box of volume PO containing energy E. in the form of radiation, i.e. free, massless fields. Our starting point is the assumption that the probability of any configuration of this radiation is proportional to exp(s), where S is the entropy of that configuration. We focus attention on a volume I/ and calculate the probability that an energy E is found in this volume, where we use the standard formulae for the entropy of black body radation and ignore the self-gravitation of the radiation when calculating this probability. We then assume that if E - V113(in Planck units C = c
20
= h = k =l), self-gravitation will result in collapse to a black hole. (For spherically distributed radiation, the condition E = 1. 2R expresses the condition that the radiation be just within its own Schwarzschild radius, while E * 0.25R is the point at which stable equilibrium configurations of self-gravitating radation no longer can exist [2], i.e. the point at which Jeans’ instability sets in.) Thus, we obtain the probability that as a result of ordinary thermal fluctuations the box of radiation will be found in a configuration which will result in the formation of a black hole of energy E.This probability then is converted to a rate of black hole formation by dynamical arguments. For free radiation in its maximum entropy, thermal state, the energy, E, and entropy, S, are given in terms of the temperature T by,
E=apV,
(1)
S=&T3V,
(2)
where we have neglected boundary effects resulting from finite box size. Here Q = &(7?/30)c+g~ wheregi is the spin degeneracy factor, c+ takes the value 1 for bosons and 7/8 for fermions and we employ Planck umts in all our calculations. By eliminating T, we
0 031-9163/82/0000-0000/$02.75 0 1982 North-Holland
Volume 90A, number 1,2
PHYSICS LETTERS
fmd that the maximum entropy, S of radiation having total energy E in a volume V is given by, S = &l/4 Vl’4E3’4 .
(3)
Consider, now, a box of volume V. containing radiation of total energy E. . In its maximum entropy state, this radiation would be thermally distributed at temperature To = (Eo/cz Vo)1’4. Although this is the most probable distribution of the radiation, other ditrrbutions can occur with probability proportional to the exponential of their entropy. We focus attention on a particular volume V contained within the box. If an energy E is found in V, the energy in the remaining volume V. - V will be E. - E. Thus, the probability that the volume V will contain an energy between E and E + dE is given by, P&Y)
dE.a exp(Stot) dE = exp[&1’4 V114E314
t ~c&~(V,
- V)1’4(Eo Y!?)~“]
dE.
(4)
P&l?) dE a exp&1’4 Vo1’4E03’4 - &IT; V - E/To t ;cY~‘~ V1’4E3’4) dE t $c$‘~ V1’4E3’4)
dE ,
(5)
where in the last line we have dropped the terms independent of E from our un-normalized probability distribution. Thus, we obtain the familiar result that the energy distribution of a small component of a large system is given by the canonical ensemble, and the probability of any configuration of the small system is proportional to exp(-F/To) where F 3 E ToS is the free energy of the small system. Defining 7 by y=aT,+
(6)
NY) = &V
dr exp(-x + 45+x3/4)
0
u,
Inthelimitr
.
(7 s 1) .
00)
Eq. (7) yields the probability that if we were to look in at the volume V “at random” then we would find an energy between E and E t dE contained within it. As stated above, we assume that a black hole of energy E will form if an energy E ii found within a volume V = W3 with X N 1. (It is useful to introduce the factor Xhere in order to keep track of where the terms in our fmal expression arise from.) The number of such volumes in our box is Vo/V, so the probability per unit volume that a black hole of energy E is about to form is obtained by dividing by Vand setting V= AE3 in eq. (7): rs(E = v-lp”(‘%~
= hE3
.
(11)
re-examine this volume again at a time much less than the light travel time, N3, across this volwne, the configuration will not have had an opportunity to change, and the probability of finding that a black hole is about to form will be much smaller than our above “random” probability. On the other hand, if we reexamine the volume at a time greater then the light travel time across this volume, the configuration will have changed completely, and our probability formula (7) should be applicable. Hence, the rate per unit volume of black hole formation should be obtained by dividing 3Q?) by the light travel time V1’3 = @E3)1’3. Thus, our final formula - expressed in Planck units - for the rate per unit volume for formation of black holes of energy between E and E + dE is, exp(-E/To
(7)
where N(r) = J
exp(7/3)
+ $Y~‘~A~‘~E~‘~)
R(E)dE=
t &x1” Vl’4E3’4) T@‘(Y)
(9)
In the opposite limit 7 % 1, we can approximate the probability distribution as a gaussian peaked at the equilibrium value EA = czT$ V, and we find
we obtain the normalized probability distribution, exp(-E/To P&!!q dE =-
(^I
Finally, the rate of black hole formation at energy E is determined as follows. If we find that a black hole is not about to form in the volume V, and then we
Assuming that V Q V. and E Q Eo, we obtain,
a exb(-E/To
N-1
21June1982
(f-0
dE, W3)4’3
(12)
T0.N~)
where N(r) is given by eq. (8) and V should now be set equal to AE3 in our definition, eq. (6), of 7. Our expression (12) for R(E) diverges at low energies E -+ 0. However, quantum gravitational effects certainly should invalidate our analysis for black hole energies below the Planck mass mp = (ll~/G)~‘~ w 1O-5 gm, thus providing a natural low energy cut-off for the 21
of our formula. On the other hand, the validity of our derivation at high energies is limited by the condition E Q E. and the requirement that xE$ < V. so that the large box is not within its own Schwarzschild radius. This yields the upper limit E 4 1/@I2 h1i2 TG). (Note that together with our lower limit, this implies the restriction To 4 T,, = k- 1 (hc5 / IqW - 1032 K.) Hence, within the range of applicability of our formula, the “entropy term” $1/4X1/4 X E3i2 in the exponential is always small compared with the “energy term” E/.To. Within the range of validity of our formula, the maximum rate of black hole formulation occurs at the Planck energy, E = EP. In this case, since To 4 TP we have y 4 1 and hence N(r) w 1. Assuming a! = 1 and setting X = 1, we obtain the rate per unit volume in Planck units, applicability
R(ER) = (1 /TO) exp(1 /To) .
(13)
i.e., in ordinary cgs units, R(E P )~(l/kTo)exp(-TP/To)i;3t-1
(14)
where 45 1 = (Gfi/~~)~/~ - 1O-33 cm, “,d’t, = (Gfi/~~)1~~ - los. Thus, at room temperature, To = 300 K, we find that the timescale, t, for formation of a black hole within AE - E, of the Planck energy EP in one cm3 by ordinary thermal fluctuations is roughly t - 1IR(E
- exp(1029) s .
(15)
Of particular interest is the rate of formation of black holes at energy Eq = 1/&TO, since such black holes would be in (unstable) thermal equilibrium with the radiation. (Black holes with E
= exp[-l/&T:
x (8?r)4T,3/x4’3N(~o) )
t ~a1/4h1/4(1/8aTo)3/2] (16)
where 7. = ~X(877)-~.Recently, Gross et al. [3] have calculated this rate by a quantum mechanical treatment. The dominant factor, exp(-1/16nTi), in their expression is very similar to the dominant factor, exp(-1/8nTi) in our formula. However, the other factors do not appear to be related. The factor of two discrepancy in the argument of the exponential of the dominant factor is important; the rate predicted by the Gross-Perry-Yaffe formula is much higher than ours. It is not difficult to trace the origin of this dis22
21 June 1982
PHYSICSLETTERS
Volume90A, number 1,2
crepancy. The dominant term in both formulae is just exp(-F/To) where F is the free energy, but we use F=E-
ToSmd =E=
1/8~T~,
whereas in effect Gross-Perry-Yaffe F = E - To&,
(17)
use,
= E - To(4?rE2) = l/14aTo
.
(18)
The disagreement between our formula and that of Gross-Perry-Yaffe does not imply that either calculation is in error. Our calculation treated the radiation in an entirely classical manner except, of course, for the quantum discreteness which gives the radiation a well defined, finite entropy. Correspondingly, we considered only ordinary thermal fluctuations which result in a classically describable gravitational collapse to a black hole. On the other hand, the Gross-Perry-Yaffe formula includes the process of quantum tunneling directly into a black hole configuration. Furthermore, our formula gives the rate of black hole formation by direct collapse to a black hole within A,!?of E,, whereas the Gross-Perry-Yaffe formula presumably includes the contributions from formation of smaller black holes which then grow to this size. (Most black holes with E