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Generalized first law of thermodynamics for black holes in spacetimes which are not asymptotically flat
4 April 1994 PHYSICS LETTERS A
ELSEVIER
Physics Letters A 187 (1994) 31-34
Generalized first law of thermodynamics for black holes in spacetimes which are not asymptotically flat Jiliang Jing, Yongjiu Wang, Jiuyun Zhu CCAST (WorldLaboratory), P.O. Box 8730, Beijing 100080, China, Insitute of Physics and Physics Department, Hunan Normal University, Changsha, Hunan 410081, China
Received 17 November 1993; revised manuscript received 1 February 1994; accepted for publication 2 February 1994 Communicated by J.P. Vigier
Abstract
The general formulation of the first law of thermodynamics for static or stationary black holes in spacetimes which are not asymptotically fiat is derived. The only difference between the first law given here and Bardeen, Carter and Hawking's first law (BCH first law) is that the mass M o f t h e black hole in BCH's first law is replaced by the total energy E of the system. This is because the mass and energy of a black hole whose spacetime is not asymptotically fiat are not identical to those of a black hole whose spacetime is asymptotically fiat. The first law given here is also valid for a black hole whose spacetime is asymptotically fiat.
1. I n t r o d u c t i o n
It is well known that the thermodynamics o f static and stationary black holes whose spacetimes are asymptotically flat obeys the four laws proposed by Bardeen, Carter and Hawking [ 1 ]. But for static and stationary black holes whose spacetimes are not asymptotically flat, the important question o f whether the B a r d e e n - C a r t e r - H a w k i n g ( B C H ) laws are valid for those black holes still remains open. It is well known that phase transitions in the early universe can give rise to the formation o f macroscopic topological defects, such as global monopole, cosmic string and texture [ 2 ]. These defects appear as a result o f global symmetry breaking and represent objects which might have been created in the early universe [ 2 ]. If we treat them in general relativity, their nonzero energy mom e n t u m due to the gauge fields will distort the spacetime with respect to their symmetry properties, and these spacetimes would not be asymptotically fiat. The
appearance of the global monopole, cosmic string and texture m a y have important cosmological consequences. They can produce observational effects, and may also affect the galaxy formation [2,3]. The effects of cosmic string, global monopole and texture are diverse physical situations and have been extensively studied in the recent years. Particularly interesting is the inclusion o f a black hole in cosmic string or monopole models [4]. For example, for a Schwarzschild black hole with a cosmic string passing through it, or a global monopole inside, it has been shown that the metrics outside the black holes can be written as [ 5,6 ] d s E = A ( r ) d t 2 - B ( r ) dr 2 _
r E(a 2 d02 + b 2 sin20 d~ 2 ),
withA (r) = B ( r ) -~ = 1 - 2 G M / r , where M i s the mass o f the black holes. The metric describes a Schwarzschild black hole with a global monopole
0375-9601/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0375-9601 ( 94 ) 00105-X
32
J.
Jing et al. /Physics Letters A 187(1994) 31-34
1 - 8 G g r / 2 , where ~/is the scale of symmetry breaking, and a Schwarzschild black hole with a cosmic string passing through it when a 2 = l , b E = ( 1 - / l ) 2, where ~t is the mass per unit length of the cosmic string. Since the metric describes a space with a deficit angle, the spacetimes of these black holes are no longer asymptotically flat. For a black hole with a cosmic string, Aryal, Ford and Vilenkin [ 7 ] have shown that the entropy has to be proportional to onefourth of the horizon area of the combined system. They also discussed the energy, as well as the apparent violation of the Hawking area theorem. In Ref. [ 6 ], we studied the thermodynamics of black holes. We notice that the first law given by Bardeen, Carter and Hawking cannot be used because the relation of the energy E and mass M i s given by E = abMand the area of the event horizon is Ah= 47rabr2+,where r+ is the radial coordinate for the event horizon. In order to obtain the same form as that of classical thermodynamics, we proposed that the first law of thermodynamics of black holes should be d E = 0¢/4n)dA. However, this form of the first law is only suitable for the static black holes given above. The first law of thermodynamics of black holes whose spacetimes are not asymptotically fiat should be given in a general formulation, because black holes, in general, may be static or stationary. In the present Letter, we focus our attention on the problem to extent the BCH first law from black holes whose spacetimes are asymptotically flat to black holes whose spacetimes are not asymptotically flat, since we can easily see that the other three laws coincide with the BCH laws. when a 2 = b 2 =
~(t)u. v{~'~) = {.,)u; v{~',) ,
(2.2)
~[#;v]
,u u
(2.3)
~[~;v]
,u u
<,)~ = - R . ~ c , ) , ~)~ = - R ~ )
,
(2.4)
where a semicolon denotes covariant derivatives, and the square brackets around the indices imply antisymmetrization. Integrating Eq. (2.3) over a hypersurface S of constant time and transferring the volume on the left hand side of the equation to an integral over a twosurface 0S which stretches from the horizon OSh to the spatial boundary 0S~ at infinity, we have R ~(t) dZ'l, S
0Sh
0S~
where dE u and d E ~ are the surface elements of S and 0S respectively. For an asymptotically flat black hole, the last integral on the right hand side of Eq. (2.5) is equal to - 4~M, where M is the mass as measured from infinity [ 8-10 ]. But for the black holes whose spacetimes are not asymptotically flat the meaning of the term should be discussed. For example, for a Schwarzschild black hole with a cosmic string passing through it or a global monopole inside it, let ~'t) = ( 1, 0, 0, 0) be the time-like Killing vector. Using the metric given above, the integral over 0S~ in Eq. (2.5) can be expressed as
f ~'~t);"dZu,,=-4~abM , OS~
2. The general formulation of the first law
Since static spacetime is a special case of the stationary spacetime for which the trajectories of the Killing vector ~u are orthogonal to a family of hypersurfaces, without loss of generality, we will discuss stationary spacetime only. For a stationary axisymmetric black hole, there is a unique time-translational Killing vector ~'t) and a unique rotational Killing vector ~ ) . These Killing vectors obey the equations
which is not equal to - 4=M since ab ~ 1. According to the discussion in Ref. [ 6 ], the integral over 0So~ in Eq. (2.5) for black holes can be regarded as - 4 n E , where E is the total energy of the system. Geroch, Winicour and Wald have proposed have proposed to define the total energy E with the last integral on the right hand side of Eq. (2.5) [8,9]. In the Appendix we will prove that this definition is suitable for stationary spacetime. So, we can write the integral as
f ~(t);~d.Xm,=-4nE. 0S~
(2.6)
J. .ling et aL ~PhysicsLetters A 187 (1994) 31-34 We can introduce a time coordinate t which measures the parameter distance from S along the integral curves of ~t)- The null vector lu= dxU/dt, tangent to the generators of the horizon, can be written as
lu={ft) +I2h{{'~).
33
8 E = ~ ff ( 2 T U - T g ~ )/t~ ( t ) d ~ u S
K
+ 2f2u 8& + ~ BAh.
(2.12)
(2.7)
The coefficient I2h is the angular velocity of the black hole. Using Eqs. (2.6) and (2.7) and Einstein's equations, Eq. (2.5) can be rewritten as
f ( TUp- ½TgU~)~'~t) clZu S
_-½E-&A- ~
P';'aX..,
(2.8)
0Sh
where 1 •/ h = - ~-~ f ~'~);'clZ'u,
(2.9)
OSh
is the angular momentum of the black hole. Introduce another null vector n u orthogonal to dSh, normalized so that nulU=- 1. We can express d27u~ as ltun~l dA, where dA is the surface area element of 0Sh. Thus the last term on the right hand side of Eq. (2.8) can be expressed as
Eq. (2.12) corresponds to Eq. (34) in Ref. [ 1 ], i.e. the first law of thermodynamics of a black hole. The only difference between Eq. (2.12) and Bardeen, Carter and Hawking's first law is that the mass M of the black hole in BCH's first law is replaced by the total energy E of the system. This is because the mass and energy of a black hole whose spacetime is not asymptotically flat is not identical to that of a black hole whose spacetime is asymptotically flat. It is obvious that Eq. (2.12) is also valid for an asymptotically flat black hole. In order to obtain the same form as the first law of general classical thermodynamics, we propose that the general formulation of the first law of thermodynamics for stationary black holes should be described by Eq. (2.12). It is easy to see that the results of Ref. [ 6 ] for a Schwarzschild black hole with a cosmic string passing through it or a global monopole inside it are special cases of the first law described by Eq. (2.12).
Appendix 1 f xdA 8~
(2.10) We first introduce the Euclidean action for the gravitational field [ 10-12 ]
aSh
where x = -lu~nUl~ represents the surface gravity and is constant over the horizon. Thus Eq. (2.8) becomes
S
R7
R d/t- ~ M
# )~(t) v dXu E = ff (2T~u - Tg,,
+ 212hJh + ~xAh,
I, = -
(2.1 1 )
where a h is the area of the event horizon. It should be pointed out that Jh, X and Ah of a black hole whose spacetime is not asymptotically fiat may not be equal to that of a black hole whose spacetime is asymptotically fiat. The differential form of Eq. (2.11 ) can be obtained according to the discussion in Ref. [ 1 ]. It is explicitly given by
[K] d/t,
(A. 1 )
OM
here R is Ricci scalar and Kis the extrinsic curvature of the boundary OM. By the same method as Ref. [ 13 ], the surface term in Eq. (A. 1 ) can be simplified by using an approximation for the metric at large radial distances. After substituting Einstein's equations and using Eq. (2.6), we obtain
Ig = ½fl f T~t) dX u - l flE,
(A.2)
S
where r = 2n/x. It is easily shown that the entropy S of a black hole is given by
J. Jing et al. /Physics Letters A 187 (1994) 31-34
34
S=-
if
~
[K] d , u = ~Ah.
(A.3)
osh S u b s t i t u t i n g Eq. ( A . 3 ) i n t o Eq. (2.11 ), a n d t h e n using Eq. (2.11 ) to e l i m i n a t e T in Eq. (A. 2 ), we o b t a i n
I g = f l E - - S - Q h J h + fl
Tv~(t) d ~ u .
(A.4)
s C o m p a r i n g Eq. ( A . 4 ) with Eq. ( 1 2 ) in Ref. [ 14], or Eq. ( 1.15 ) i n Ref. [ 15 ], we see that the E i n t r o d u c e d i n Eq. ( 2 . 6 ) is the total energy o f the system for a static or s t a t i o n a r y b l a c k hole.
References [ 1 ] J.M. Bardeen, B. Carter and S.W. Hawking, Commun. Math. Phys. 31 (1973) 161. [2] A. Vilenkin, Phys. Rep. 121 (1985) 263; V. Sahni, Phys. Rev. D 43 ( 1991 ) 301. [3] W.H. Press and D.N. Spergel, Phys. Today 42 (1989) 29. [4] F.D. MazziteUi and C.O. Lousto, Phys. Rev. D 43 (1990) 468; V.A. Hiscock, Phys. Rev. Lett. 64 (1990) 344;
A.L. Larsen, Phys. Lett. B 383 (1992) 47; D. Harari and P. Sikivie, Phys. Rev. D 37 (1987) 3438; C.F.P. Lee, Phys. Rev. D 47 (1993) 2260; D 46 (1992) 4169; Jing Jiliang, Yu Hongwei and Wang Yongjiu, Chin. Sci. Bull. 38 (1993) 920; Chin. Phys. Lett. l0 (1993) 445. [5]M. Barriola and A. Vilenkin, Phys. Rev. Lett. 63 (1989) 341; M. Aryal, L.H. Ford and A. Vilenkin, Phys. Rev. D 34 (1986) 2263. [ 6 ] Jing Jiliang, Yu Hongwei and Wang Yongjiu, Chin. Sci. Bull. 39 (1994); Phys. Lett. A 178 (1993) 59. [7] M. Aryal, L.H. Ford and A. Vilenkin, Phys. Rev. D 34 (1986) 2263. [ 8 ] R.P. Geroch and J. Winicour, J. Math. Phys. 22 ( 1981 ) 803. [9] R.M. Wald, General relativity (Univ. of Chicago Press, Chicago, 1984). [ 10] R. Kallosh, T. Ortin and A. Peet, Phys. Rev. D 47 (1993) 5400. [ 11 ] J.W. York, Phys. Rev. Lett. 28 (1992) 1082. [ 12 ] G.W. Gibbons and S.W. Hawking, Phys. Rev. D 15 ( 1977 ) 2752. [13] I. Moss, Phys. Rev. Lett. 69 (1992) 1852. [ 14] C.F.P. Lee, Phys. Rev. D 47 (1993) 2260. [ 15 ] G. Hayward, Phys. Rev. D 43 ( 1991 ) 3861.
ELSEVIER
Physics Letters A 187 (1994) 31-34
Generalized first law of thermodynamics for black holes in spacetimes which are not asymptotically flat Jiliang Jing, Yongjiu Wang, Jiuyun Zhu CCAST (WorldLaboratory), P.O. Box 8730, Beijing 100080, China, Insitute of Physics and Physics Department, Hunan Normal University, Changsha, Hunan 410081, China
Received 17 November 1993; revised manuscript received 1 February 1994; accepted for publication 2 February 1994 Communicated by J.P. Vigier
Abstract
The general formulation of the first law of thermodynamics for static or stationary black holes in spacetimes which are not asymptotically fiat is derived. The only difference between the first law given here and Bardeen, Carter and Hawking's first law (BCH first law) is that the mass M o f t h e black hole in BCH's first law is replaced by the total energy E of the system. This is because the mass and energy of a black hole whose spacetime is not asymptotically fiat are not identical to those of a black hole whose spacetime is asymptotically fiat. The first law given here is also valid for a black hole whose spacetime is asymptotically fiat.
1. I n t r o d u c t i o n
It is well known that the thermodynamics o f static and stationary black holes whose spacetimes are asymptotically flat obeys the four laws proposed by Bardeen, Carter and Hawking [ 1 ]. But for static and stationary black holes whose spacetimes are not asymptotically flat, the important question o f whether the B a r d e e n - C a r t e r - H a w k i n g ( B C H ) laws are valid for those black holes still remains open. It is well known that phase transitions in the early universe can give rise to the formation o f macroscopic topological defects, such as global monopole, cosmic string and texture [ 2 ]. These defects appear as a result o f global symmetry breaking and represent objects which might have been created in the early universe [ 2 ]. If we treat them in general relativity, their nonzero energy mom e n t u m due to the gauge fields will distort the spacetime with respect to their symmetry properties, and these spacetimes would not be asymptotically fiat. The
appearance of the global monopole, cosmic string and texture m a y have important cosmological consequences. They can produce observational effects, and may also affect the galaxy formation [2,3]. The effects of cosmic string, global monopole and texture are diverse physical situations and have been extensively studied in the recent years. Particularly interesting is the inclusion o f a black hole in cosmic string or monopole models [4]. For example, for a Schwarzschild black hole with a cosmic string passing through it, or a global monopole inside, it has been shown that the metrics outside the black holes can be written as [ 5,6 ] d s E = A ( r ) d t 2 - B ( r ) dr 2 _
r E(a 2 d02 + b 2 sin20 d~ 2 ),
withA (r) = B ( r ) -~ = 1 - 2 G M / r , where M i s the mass o f the black holes. The metric describes a Schwarzschild black hole with a global monopole
0375-9601/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0375-9601 ( 94 ) 00105-X
32
J.
Jing et al. /Physics Letters A 187(1994) 31-34
1 - 8 G g r / 2 , where ~/is the scale of symmetry breaking, and a Schwarzschild black hole with a cosmic string passing through it when a 2 = l , b E = ( 1 - / l ) 2, where ~t is the mass per unit length of the cosmic string. Since the metric describes a space with a deficit angle, the spacetimes of these black holes are no longer asymptotically flat. For a black hole with a cosmic string, Aryal, Ford and Vilenkin [ 7 ] have shown that the entropy has to be proportional to onefourth of the horizon area of the combined system. They also discussed the energy, as well as the apparent violation of the Hawking area theorem. In Ref. [ 6 ], we studied the thermodynamics of black holes. We notice that the first law given by Bardeen, Carter and Hawking cannot be used because the relation of the energy E and mass M i s given by E = abMand the area of the event horizon is Ah= 47rabr2+,where r+ is the radial coordinate for the event horizon. In order to obtain the same form as that of classical thermodynamics, we proposed that the first law of thermodynamics of black holes should be d E = 0¢/4n)dA. However, this form of the first law is only suitable for the static black holes given above. The first law of thermodynamics of black holes whose spacetimes are not asymptotically fiat should be given in a general formulation, because black holes, in general, may be static or stationary. In the present Letter, we focus our attention on the problem to extent the BCH first law from black holes whose spacetimes are asymptotically flat to black holes whose spacetimes are not asymptotically flat, since we can easily see that the other three laws coincide with the BCH laws. when a 2 = b 2 =
~(t)u. v{~'~) = {.,)u; v{~',) ,
(2.2)
~[#;v]
,u u
(2.3)
~[~;v]
,u u
<,)~ = - R . ~ c , ) , ~)~ = - R ~ )
,
(2.4)
where a semicolon denotes covariant derivatives, and the square brackets around the indices imply antisymmetrization. Integrating Eq. (2.3) over a hypersurface S of constant time and transferring the volume on the left hand side of the equation to an integral over a twosurface 0S which stretches from the horizon OSh to the spatial boundary 0S~ at infinity, we have R ~(t) dZ'l, S
0Sh
0S~
where dE u and d E ~ are the surface elements of S and 0S respectively. For an asymptotically flat black hole, the last integral on the right hand side of Eq. (2.5) is equal to - 4~M, where M is the mass as measured from infinity [ 8-10 ]. But for the black holes whose spacetimes are not asymptotically flat the meaning of the term should be discussed. For example, for a Schwarzschild black hole with a cosmic string passing through it or a global monopole inside it, let ~'t) = ( 1, 0, 0, 0) be the time-like Killing vector. Using the metric given above, the integral over 0S~ in Eq. (2.5) can be expressed as
f ~'~t);"dZu,,=-4~abM , OS~
2. The general formulation of the first law
Since static spacetime is a special case of the stationary spacetime for which the trajectories of the Killing vector ~u are orthogonal to a family of hypersurfaces, without loss of generality, we will discuss stationary spacetime only. For a stationary axisymmetric black hole, there is a unique time-translational Killing vector ~'t) and a unique rotational Killing vector ~ ) . These Killing vectors obey the equations
which is not equal to - 4=M since ab ~ 1. According to the discussion in Ref. [ 6 ], the integral over 0So~ in Eq. (2.5) for black holes can be regarded as - 4 n E , where E is the total energy of the system. Geroch, Winicour and Wald have proposed have proposed to define the total energy E with the last integral on the right hand side of Eq. (2.5) [8,9]. In the Appendix we will prove that this definition is suitable for stationary spacetime. So, we can write the integral as
f ~(t);~d.Xm,=-4nE. 0S~
(2.6)
J. .ling et aL ~PhysicsLetters A 187 (1994) 31-34 We can introduce a time coordinate t which measures the parameter distance from S along the integral curves of ~t)- The null vector lu= dxU/dt, tangent to the generators of the horizon, can be written as
lu={ft) +I2h{{'~).
33
8 E = ~ ff ( 2 T U - T g ~ )/t~ ( t ) d ~ u S
K
+ 2f2u 8& + ~ BAh.
(2.12)
(2.7)
The coefficient I2h is the angular velocity of the black hole. Using Eqs. (2.6) and (2.7) and Einstein's equations, Eq. (2.5) can be rewritten as
f ( TUp- ½TgU~)~'~t) clZu S
_-½E-&A- ~
P';'aX..,
(2.8)
0Sh
where 1 •/ h = - ~-~ f ~'~);'clZ'u,
(2.9)
OSh
is the angular momentum of the black hole. Introduce another null vector n u orthogonal to dSh, normalized so that nulU=- 1. We can express d27u~ as ltun~l dA, where dA is the surface area element of 0Sh. Thus the last term on the right hand side of Eq. (2.8) can be expressed as
Eq. (2.12) corresponds to Eq. (34) in Ref. [ 1 ], i.e. the first law of thermodynamics of a black hole. The only difference between Eq. (2.12) and Bardeen, Carter and Hawking's first law is that the mass M of the black hole in BCH's first law is replaced by the total energy E of the system. This is because the mass and energy of a black hole whose spacetime is not asymptotically flat is not identical to that of a black hole whose spacetime is asymptotically flat. It is obvious that Eq. (2.12) is also valid for an asymptotically flat black hole. In order to obtain the same form as the first law of general classical thermodynamics, we propose that the general formulation of the first law of thermodynamics for stationary black holes should be described by Eq. (2.12). It is easy to see that the results of Ref. [ 6 ] for a Schwarzschild black hole with a cosmic string passing through it or a global monopole inside it are special cases of the first law described by Eq. (2.12).
Appendix 1 f xdA 8~
(2.10) We first introduce the Euclidean action for the gravitational field [ 10-12 ]
aSh
where x = -lu~nUl~ represents the surface gravity and is constant over the horizon. Thus Eq. (2.8) becomes
S
R7
R d/t- ~ M
# )~(t) v dXu E = ff (2T~u - Tg,,
+ 212hJh + ~xAh,
I, = -
(2.1 1 )
where a h is the area of the event horizon. It should be pointed out that Jh, X and Ah of a black hole whose spacetime is not asymptotically fiat may not be equal to that of a black hole whose spacetime is asymptotically fiat. The differential form of Eq. (2.11 ) can be obtained according to the discussion in Ref. [ 1 ]. It is explicitly given by
[K] d/t,
(A. 1 )
OM
here R is Ricci scalar and Kis the extrinsic curvature of the boundary OM. By the same method as Ref. [ 13 ], the surface term in Eq. (A. 1 ) can be simplified by using an approximation for the metric at large radial distances. After substituting Einstein's equations and using Eq. (2.6), we obtain
Ig = ½fl f T~t) dX u - l flE,
(A.2)
S
where r = 2n/x. It is easily shown that the entropy S of a black hole is given by
J. Jing et al. /Physics Letters A 187 (1994) 31-34
34
S=-
if
~
[K] d , u = ~Ah.
(A.3)
osh S u b s t i t u t i n g Eq. ( A . 3 ) i n t o Eq. (2.11 ), a n d t h e n using Eq. (2.11 ) to e l i m i n a t e T in Eq. (A. 2 ), we o b t a i n
I g = f l E - - S - Q h J h + fl
Tv~(t) d ~ u .
(A.4)
s C o m p a r i n g Eq. ( A . 4 ) with Eq. ( 1 2 ) in Ref. [ 14], or Eq. ( 1.15 ) i n Ref. [ 15 ], we see that the E i n t r o d u c e d i n Eq. ( 2 . 6 ) is the total energy o f the system for a static or s t a t i o n a r y b l a c k hole.
References [ 1 ] J.M. Bardeen, B. Carter and S.W. Hawking, Commun. Math. Phys. 31 (1973) 161. [2] A. Vilenkin, Phys. Rep. 121 (1985) 263; V. Sahni, Phys. Rev. D 43 ( 1991 ) 301. [3] W.H. Press and D.N. Spergel, Phys. Today 42 (1989) 29. [4] F.D. MazziteUi and C.O. Lousto, Phys. Rev. D 43 (1990) 468; V.A. Hiscock, Phys. Rev. Lett. 64 (1990) 344;
A.L. Larsen, Phys. Lett. B 383 (1992) 47; D. Harari and P. Sikivie, Phys. Rev. D 37 (1987) 3438; C.F.P. Lee, Phys. Rev. D 47 (1993) 2260; D 46 (1992) 4169; Jing Jiliang, Yu Hongwei and Wang Yongjiu, Chin. Sci. Bull. 38 (1993) 920; Chin. Phys. Lett. l0 (1993) 445. [5]M. Barriola and A. Vilenkin, Phys. Rev. Lett. 63 (1989) 341; M. Aryal, L.H. Ford and A. Vilenkin, Phys. Rev. D 34 (1986) 2263. [ 6 ] Jing Jiliang, Yu Hongwei and Wang Yongjiu, Chin. Sci. Bull. 39 (1994); Phys. Lett. A 178 (1993) 59. [7] M. Aryal, L.H. Ford and A. Vilenkin, Phys. Rev. D 34 (1986) 2263. [ 8 ] R.P. Geroch and J. Winicour, J. Math. Phys. 22 ( 1981 ) 803. [9] R.M. Wald, General relativity (Univ. of Chicago Press, Chicago, 1984). [ 10] R. Kallosh, T. Ortin and A. Peet, Phys. Rev. D 47 (1993) 5400. [ 11 ] J.W. York, Phys. Rev. Lett. 28 (1992) 1082. [ 12 ] G.W. Gibbons and S.W. Hawking, Phys. Rev. D 15 ( 1977 ) 2752. [13] I. Moss, Phys. Rev. Lett. 69 (1992) 1852. [ 14] C.F.P. Lee, Phys. Rev. D 47 (1993) 2260. [ 15 ] G. Hayward, Phys. Rev. D 43 ( 1991 ) 3861.