- Email: [email protected]
Mass inflation in 2d dilaton gravity
15 August 1994 PHYSICS LETTERS A
EL-SEWER
PhysicsLettersA
191 (1994) 211-215
Mass inflation in 2d dilaton gravity S. Droz Theoretical Physics Institute, University of Alberta, Edmonton, Alberta, Canada T6G 2JI Received
16 May 1994; accepted for publication Communicated by J.P. Vigier
29 June 1994
Recently two-dimensional dilaton gravity has received great attention. We investigate two-dimensional charged black holes. These solutions possess two horizons, an event horizon and an inner, so-called Cauchy horizon. It has been argued on general grounds that the inner horizon must be unstable. We show that a minimally coupled scalar field indeed causes a mass inflation singularity to be formed at the Cauchy horizon, in striking similarity to the four-dimensionalcase.
1. Introduction
Usually one associates with a black hole a region of space-time that is concealed from the rest of the universe by an event horizon. Traditionally the interior structure of black holes did not get much attention, as it is removed from any experimental investigation. There is however no reason to doubt the validity of our theories of gravity behind the event horizon, which is a perfectly regular part of space-time. An electrically neutral (in four dimensions also spherically symmetric) black hole possesses a fairly simple inner structure. Once something crosses the event horizon, it is doomed to end up in a curvature singularity. This simple picture changes radically as soon as one adds the slightest amount of charge (or angular momentum in four dimensions). It now seems that one can travel through the black hole and emerge out of a white hole into another universe. During this journey one crosses the inner or Cauchy horizon of the black hole. After this, one would face bizarre, unpredictable things: A direct view onto the r = 0 singularity makes any Cauchy problem ill defined. Al-
ready 25 years ago Penrose [ 1] noted that the Cauchy horizon is unstable. The slightest radiation streaming along the Cauchy horizon gets infinitely blueshifted. This only produces a so-called whimpersingularity: all curvature scalars stay finite, but a free falling observer crossing the Cauchy horizon measures an infinite energy density [7]. Generally whimpers are unstable [2] and once some outflux is added, a true curvature singularity will form [ 81. In this Letter we would like to demonstrate that this behavior, well known in four-dimensional Einstein gravity, is reproduced by its smaller brother, 2d string gravity. The paper is organized as follows: In the first part we derive the general equations for 2d string gravity with a minimally coupled scalar field. We then derive a solution which describes a black hole, with matter streaming in along the Cauchy horizon. In the last section we discuss the effects caused by considering inflowing as well as outflowing radiation.
037%9601/94/$07.00 @ 1994 Elsevier Science B.V. All rights reserved XSDZO375-9601(94)00544-3
212
S. Droz /Physics Letters A 191(1994) 21 I-215
2. Basic equations
3. Vaidya solution
Let us consider the following string theory inspired action [ 3 1,
Before we can solve Eqs. (6)-(g) we have to choose our coordinates. The most convenient choice for our purpose are ingoing Eddington-Finkelstein coordinates v, r. The metric is then written as
s=
dx2~{e-2’[R+4(Va,)2+4A-2FPyF”“] J - ; (vf)2}.
(1)
R is the Ricci scalar, a, the dilaton field and F,, is the the Maxwell field tensor. f denotes a minimally coupled scalar field and A is a cosmological constant. Note that the Maxwell contribution has been introduced by hand. It does not arise as a low-energy limit of string theory. Varying ( 1) with respect to the metric elements g,B gives the energy momentum tensor 7’,‘,~.For the gravitational part we find = e-2v{2gap [ (09)
Ts -
- (VP)’
‘- +F,,VFpv + 412]
2V,V,3q -2FppFap),
ds2 = -B(v,r)dw2-2ddrdv.
(9)
Note that (9) represents indeed the most general case, as every two-dimensional manifold is conformally flat, i.e. determined by one free function. Let us now concentrate on the Maxwell equation. In 1 + 1 dimensions the Maxwell field tensor can be written as Ffi” = J-gF
(v, rk,w,
(10)
where c/1”is the totally antisymmetric sor. Then (8) reads
Levi-Civita
(ePZQF),@ = 0,
ten-
(11)
(2) which is obviously
satisfied by
and for the matter part we get Tu”s = $]V&‘,S
-
We can use the equation field, R - am
F = Qe2,.
~g&‘f)21. of motion
for the dilaton
+ 4A - 2FpvFpV + 4 0 p = 0,
(4)
to simplify the expression for the gravitational energy momentum tensor. One finally obtains for the total energy momentum tensor Tap = e-2’(-igapR
- 2V,Vbv,
+ 4FgpFap)
+ T$. (5)
The “Einstein T,, =
equations”
0,
(12)
(3)
in two dimensions
then read
(6)
Q is a constant and denotes the “total charge” of the system. The wave equation takes the form of
and is identically
satisfied by the ansatz (14)
f = f(v).
This is clearly not the general solution, but it is sufficient for our purposes, and we will restrict ourselves to this particular form for the rest of this paper. Let us now focus on the Einstein equations. The simplest is the rr-equation, which now reads
as the Einstein tensor vanishes identically. Finally, the equation of motion for the scalar field is just the homogeneous wave equation
Therefore y, becomes
qp = 0.
v, = a(v)r
(7)
The electromagnetic equations (e2’ Fpv);,
= 0.
field is described by the Maxwell
(8)
(13)
= 2&, + Br1r + BfS, = 0
T,, = 0 = -2 e-2cp v,~~.
+ b(v).
To solve the remaining ing ansatz for B, B = Bo + Bl(w)e2’
(15)
(16) equations
+Q2e4p,
we make the follow-
(17)
213
S. Droz /Physics LettersA 191 (I 994) 21 l-21 5
where Bs is a constant and Bi = B1(v ) is a function of w only. To get Be and BI we have to work a little bit harder. First, combining T,, and T,,, gives
B
BT,, + Trv
This metric represents, as its four-dimensional analogue, a charged black hole, with charge Q and massfunction M (w) . It possesses two horizons at
=e -*‘(-260,vv-2Bq,,,+B,l~,,-B.v~,,)+~(~”)* = 0.
1
2M(v)
e-2A’
+Ee-4Ar
/I
M,, =
#I*
y
;A(&)* =: L(v).
(25)
(18)
Inserting ( 17 ) in ( 18 ) and using the integrability condition u(v)B,” - B,r~,v = e*‘a(w)B~,~
(19)
gives (P,,,~+ Ba,” + f ezr Bl,va = $(_&)2e2P.
(20)
We now differentiate this equation with respect to r to get a simple expression for a (21)) u,~~ + e29(r,“)[BI (a* ),v + BI,~u* - f (XV)*a] + e4r(r,v) 2Q2 (u*),~ = 0.
(21)
This equation has to hold for all values of r, but the first term is independent of r. Thus a,,,,, = 0. Similarly it then follows that a,” = 0, i.e. a(v) = a = const and
Substituting this result into (20) now yields b,V, = 0 =+ b=h+c. Using T$ = 0 then gives a*=1
=
=*a=*1
(23)
and BO = A* - 2ah.
(24)
It is now convenient to perform a coordinate transformation
r +
A(rr-
6w -c),
Bl(v)
---+ -2M(w)A,
v -+
‘fw/A,
so that we finally obtain the following result, ds* = -Bdv*
+ 2drdv,
r* = &log
r(l%
d-)
(
>
and a curvature singularity at r = -XL For &, = 0, i.e in the static case this solution agrees with the one given in Ref. [ 41, Ch. 3. The scalar field f produces an influx with luminosity L (w ), which forces the massfunction to grow.
4. Mass inflation
In this section we want to find a solution to our equations, with boundary conditions, which are inspired by a four-dimensional analysis of a gravitational collapse. During a collapse a star will emit gravitational waves, which will be backscattered by the gravitational potential outside the event horizon [ 5 1. This produces a decaying tail of infalling radiation, which gives rise to the following energy momentum tensor, T,, = f&&
= L(v)l,Ju,
(26)
where 1, = &v. This radiation gets enormously blueshifted near the inner horizon, and thus causes an infalling observer to measure an infinite energy density [ 71. Once the influx crosses the event horizon, one expects these waves to be scattered again, thus giving raise to a tail of outflowing radiation [ $61. This outflow then forces the Cauchy horizon to contract slowly. The massfunction starts growing exponentially with advanced time, and a curvature singularity evolves. This phenomenon is called ma.ss inflation [ 8 1. In general it is not possible to solve the equations for the crossflow region. However, Ori considered a simple model [ 91 that still catches the essential physics. He replaced the outflux by a thin lightlike shell Z of free falling matter. The space-time on either side of the shell is then described by solutions of the type (25 ) (see Fig. 1) . This system
214
S. Droz / Physics Letters A 191 (1994) 211-215
Let us now solve these equations: Price’s analysis [5,6] suggests L- (v) = alvpP where p = 41 + 4, and I denotes the multipole moment of the ingoing radiation. a is constant of dimension [length(P+2)]. Thus, integrating (27) for M- gives M_(v)
= Ml)-
-v
1-P
.
P-l
Along the shell we have -B_dr
+ 2dv = 0,
which can be solved for an asymptotic for r. One finds r(v)
= r0 + v
x
-cp-l)a(p
- 1M a’Ic0
1 +p~+o(l/v*)
still cannot be solved in closed form, but one can perform an asymptotic analysis that reveals the nature of the singularity. Let us denote quantities behind the shell with the subscript plus, and before the shell with the subscript minus. We then have to solve the following equations,
aM* = L*(v). dV*
-
Continuity demands
(27)
of the line element and
r
across the shell
B+ dv+ = B- dv-, and a continuous enforced by
(28) crossflow of radiation
through X is
8V
B+ = XL_(V).
ro = r_ is the value of r at the inner horizon, LY= exp(2lrc)l = Ma - ,/m, and 1~0= 2i(Mo ,)/a is the surface gravity of the inner horizon. Substituting this expression for r into (30) gives for M+ alog(M+(u))
=/co-t
ih
M+(v)
+0(1/v*)
*
= M;eKo”v-PO(l/v).
(32)
Thus the massfunction diverges exponentially near the Cauchy horizon. It is clear that this is not just a coordinate singularity, as the curvature scalar R goes like R 0; e’@”vep. Note however, that in suitable coordinates is finite,
the metric
du : = e+22r dr + 2M+ (v ) dv+ = e+2ir dr + 2 M+ (v) eKKoVdv. / . KU-P
where no = (l/B, i). Combining Eqs. (27)-(29) gives the equation for M+ along C (from now on we drop the subscript minus in v_, i.e. v_ = v):
aM+(v)
(31)
. >
(
Fig. 1. The conformal diagram of a collapsing two-dimensional star. The double line denotes the star’s surface. The global horizons are represented by dashed lines, if they do not coincide with the apparent horizons.
series in l/v
(30)
In terms of u and v, the line element near the Cauchy horizon reads d.s* = 2 eeZLr du dv, which is perfectly well behaved near the Cauchy horizon.
S. Droz/PhysicsLettersA
191(1994)211-215
5. Conclusion
Acknowledgement
The derivation presented in this paper shows that mass inflation not only occurs within the framework of spherically symmetric Einstein gravity, but also in two-dimensional string gravity. This is a strong indication that the mass inflation scenario is a generic phenomenon, and does not depend on the particular details of the theory. So far we have not touched the problem of boundary conditions, which seems crucial to us. Until now all the work done [ 8,9] assumed the existence of a portion of the Cauchy horizon, by tuming the outflux on behind the event horizon. It is not clear if a Cauchy horizon forms otherwise. We are currently investigating this assumptions [ lo]. Maybe the simple two-dimensional models help understand the essential basic physics of the much more complex four-dimensional black holes.
I would like to thank Warren Anderson Israel for some helpful discussions.
215
and Werner
References [ 1] R. Penrose, in: Battele rencontres, eds. C.M. Dewitt and J.A. Wheeler (Benjamin, New York, 1968) p. 222. [2] S.T.C. Siklos, J. Gen. Relat. Gravit. 10 (1979) 1031. [3] C.G. Callan, S.B. Giddings, J.A. Harvey and A. Strominger, Phys. Rev. D 45 (1992) R1005. [4] V.P. Frolov, Phys. Rev. D 46 (1992) 5383. [5] R.H. Price, Phys. Rev. D 5 (1972) 2419. [6] C. Gundlach, R.H. Price and J. Pullin, Phys. Rev. D 49 (1994) 883. [7] W.A. Hiscock, Phys. L&t. A 83 (1981) 110. [S] E. Poisson and W. Israel, Phys. Rev. D 41 (1990) 1796. [9] A. Ori, Phys. Rev. Lett. 67 (1991) 789. [lo] A. Bonanno, S. Droz, W. Israel and S. Morsink, in preparation.
EL-SEWER
PhysicsLettersA
191 (1994) 211-215
Mass inflation in 2d dilaton gravity S. Droz Theoretical Physics Institute, University of Alberta, Edmonton, Alberta, Canada T6G 2JI Received
16 May 1994; accepted for publication Communicated by J.P. Vigier
29 June 1994
Recently two-dimensional dilaton gravity has received great attention. We investigate two-dimensional charged black holes. These solutions possess two horizons, an event horizon and an inner, so-called Cauchy horizon. It has been argued on general grounds that the inner horizon must be unstable. We show that a minimally coupled scalar field indeed causes a mass inflation singularity to be formed at the Cauchy horizon, in striking similarity to the four-dimensionalcase.
1. Introduction
Usually one associates with a black hole a region of space-time that is concealed from the rest of the universe by an event horizon. Traditionally the interior structure of black holes did not get much attention, as it is removed from any experimental investigation. There is however no reason to doubt the validity of our theories of gravity behind the event horizon, which is a perfectly regular part of space-time. An electrically neutral (in four dimensions also spherically symmetric) black hole possesses a fairly simple inner structure. Once something crosses the event horizon, it is doomed to end up in a curvature singularity. This simple picture changes radically as soon as one adds the slightest amount of charge (or angular momentum in four dimensions). It now seems that one can travel through the black hole and emerge out of a white hole into another universe. During this journey one crosses the inner or Cauchy horizon of the black hole. After this, one would face bizarre, unpredictable things: A direct view onto the r = 0 singularity makes any Cauchy problem ill defined. Al-
ready 25 years ago Penrose [ 1] noted that the Cauchy horizon is unstable. The slightest radiation streaming along the Cauchy horizon gets infinitely blueshifted. This only produces a so-called whimpersingularity: all curvature scalars stay finite, but a free falling observer crossing the Cauchy horizon measures an infinite energy density [7]. Generally whimpers are unstable [2] and once some outflux is added, a true curvature singularity will form [ 81. In this Letter we would like to demonstrate that this behavior, well known in four-dimensional Einstein gravity, is reproduced by its smaller brother, 2d string gravity. The paper is organized as follows: In the first part we derive the general equations for 2d string gravity with a minimally coupled scalar field. We then derive a solution which describes a black hole, with matter streaming in along the Cauchy horizon. In the last section we discuss the effects caused by considering inflowing as well as outflowing radiation.
037%9601/94/$07.00 @ 1994 Elsevier Science B.V. All rights reserved XSDZO375-9601(94)00544-3
212
S. Droz /Physics Letters A 191(1994) 21 I-215
2. Basic equations
3. Vaidya solution
Let us consider the following string theory inspired action [ 3 1,
Before we can solve Eqs. (6)-(g) we have to choose our coordinates. The most convenient choice for our purpose are ingoing Eddington-Finkelstein coordinates v, r. The metric is then written as
s=
dx2~{e-2’[R+4(Va,)2+4A-2FPyF”“] J - ; (vf)2}.
(1)
R is the Ricci scalar, a, the dilaton field and F,, is the the Maxwell field tensor. f denotes a minimally coupled scalar field and A is a cosmological constant. Note that the Maxwell contribution has been introduced by hand. It does not arise as a low-energy limit of string theory. Varying ( 1) with respect to the metric elements g,B gives the energy momentum tensor 7’,‘,~.For the gravitational part we find = e-2v{2gap [ (09)
Ts -
- (VP)’
‘- +F,,VFpv + 412]
2V,V,3q -2FppFap),
ds2 = -B(v,r)dw2-2ddrdv.
(9)
Note that (9) represents indeed the most general case, as every two-dimensional manifold is conformally flat, i.e. determined by one free function. Let us now concentrate on the Maxwell equation. In 1 + 1 dimensions the Maxwell field tensor can be written as Ffi” = J-gF
(v, rk,w,
(10)
where c/1”is the totally antisymmetric sor. Then (8) reads
Levi-Civita
(ePZQF),@ = 0,
ten-
(11)
(2) which is obviously
satisfied by
and for the matter part we get Tu”s = $]V&‘,S
-
We can use the equation field, R - am
F = Qe2,.
~g&‘f)21. of motion
for the dilaton
+ 4A - 2FpvFpV + 4 0 p = 0,
(4)
to simplify the expression for the gravitational energy momentum tensor. One finally obtains for the total energy momentum tensor Tap = e-2’(-igapR
- 2V,Vbv,
+ 4FgpFap)
+ T$. (5)
The “Einstein T,, =
equations”
0,
(12)
(3)
in two dimensions
then read
(6)
Q is a constant and denotes the “total charge” of the system. The wave equation takes the form of
and is identically
satisfied by the ansatz (14)
f = f(v).
This is clearly not the general solution, but it is sufficient for our purposes, and we will restrict ourselves to this particular form for the rest of this paper. Let us now focus on the Einstein equations. The simplest is the rr-equation, which now reads
as the Einstein tensor vanishes identically. Finally, the equation of motion for the scalar field is just the homogeneous wave equation
Therefore y, becomes
qp = 0.
v, = a(v)r
(7)
The electromagnetic equations (e2’ Fpv);,
= 0.
field is described by the Maxwell
(8)
(13)
= 2&, + Br1r + BfS, = 0
T,, = 0 = -2 e-2cp v,~~.
+ b(v).
To solve the remaining ing ansatz for B, B = Bo + Bl(w)e2’
(15)
(16) equations
+Q2e4p,
we make the follow-
(17)
213
S. Droz /Physics LettersA 191 (I 994) 21 l-21 5
where Bs is a constant and Bi = B1(v ) is a function of w only. To get Be and BI we have to work a little bit harder. First, combining T,, and T,,, gives
B
BT,, + Trv
This metric represents, as its four-dimensional analogue, a charged black hole, with charge Q and massfunction M (w) . It possesses two horizons at
=e -*‘(-260,vv-2Bq,,,+B,l~,,-B.v~,,)+~(~”)* = 0.
1
2M(v)
e-2A’
+Ee-4Ar
/I
M,, =
#I*
y
;A(&)* =: L(v).
(25)
(18)
Inserting ( 17 ) in ( 18 ) and using the integrability condition u(v)B,” - B,r~,v = e*‘a(w)B~,~
(19)
gives (P,,,~+ Ba,” + f ezr Bl,va = $(_&)2e2P.
(20)
We now differentiate this equation with respect to r to get a simple expression for a (21)) u,~~ + e29(r,“)[BI (a* ),v + BI,~u* - f (XV)*a] + e4r(r,v) 2Q2 (u*),~ = 0.
(21)
This equation has to hold for all values of r, but the first term is independent of r. Thus a,,,,, = 0. Similarly it then follows that a,” = 0, i.e. a(v) = a = const and
Substituting this result into (20) now yields b,V, = 0 =+ b=h+c. Using T$ = 0 then gives a*=1
=
=*a=*1
(23)
and BO = A* - 2ah.
(24)
It is now convenient to perform a coordinate transformation
r +
A(rr-
6w -c),
Bl(v)
---+ -2M(w)A,
v -+
‘fw/A,
so that we finally obtain the following result, ds* = -Bdv*
+ 2drdv,
r* = &log
r(l%
d-)
(
>
and a curvature singularity at r = -XL For &, = 0, i.e in the static case this solution agrees with the one given in Ref. [ 41, Ch. 3. The scalar field f produces an influx with luminosity L (w ), which forces the massfunction to grow.
4. Mass inflation
In this section we want to find a solution to our equations, with boundary conditions, which are inspired by a four-dimensional analysis of a gravitational collapse. During a collapse a star will emit gravitational waves, which will be backscattered by the gravitational potential outside the event horizon [ 5 1. This produces a decaying tail of infalling radiation, which gives rise to the following energy momentum tensor, T,, = f&&
= L(v)l,Ju,
(26)
where 1, = &v. This radiation gets enormously blueshifted near the inner horizon, and thus causes an infalling observer to measure an infinite energy density [ 71. Once the influx crosses the event horizon, one expects these waves to be scattered again, thus giving raise to a tail of outflowing radiation [ $61. This outflow then forces the Cauchy horizon to contract slowly. The massfunction starts growing exponentially with advanced time, and a curvature singularity evolves. This phenomenon is called ma.ss inflation [ 8 1. In general it is not possible to solve the equations for the crossflow region. However, Ori considered a simple model [ 91 that still catches the essential physics. He replaced the outflux by a thin lightlike shell Z of free falling matter. The space-time on either side of the shell is then described by solutions of the type (25 ) (see Fig. 1) . This system
214
S. Droz / Physics Letters A 191 (1994) 211-215
Let us now solve these equations: Price’s analysis [5,6] suggests L- (v) = alvpP where p = 41 + 4, and I denotes the multipole moment of the ingoing radiation. a is constant of dimension [length(P+2)]. Thus, integrating (27) for M- gives M_(v)
= Ml)-
-v
1-P
.
P-l
Along the shell we have -B_dr
+ 2dv = 0,
which can be solved for an asymptotic for r. One finds r(v)
= r0 + v
x
-cp-l)a(p
- 1M a’Ic0
1 +p~+o(l/v*)
still cannot be solved in closed form, but one can perform an asymptotic analysis that reveals the nature of the singularity. Let us denote quantities behind the shell with the subscript plus, and before the shell with the subscript minus. We then have to solve the following equations,
aM* = L*(v). dV*
-
Continuity demands
(27)
of the line element and
r
across the shell
B+ dv+ = B- dv-, and a continuous enforced by
(28) crossflow of radiation
through X is
8V
B+ = XL_(V).
ro = r_ is the value of r at the inner horizon, LY= exp(2lrc)l = Ma - ,/m, and 1~0= 2i(Mo ,)/a is the surface gravity of the inner horizon. Substituting this expression for r into (30) gives for M+ alog(M+(u))
=/co-t
ih
M+(v)
+0(1/v*)
*
= M;eKo”v-PO(l/v).
(32)
Thus the massfunction diverges exponentially near the Cauchy horizon. It is clear that this is not just a coordinate singularity, as the curvature scalar R goes like R 0; e’@”vep. Note however, that in suitable coordinates is finite,
the metric
du : = e+22r dr + 2M+ (v ) dv+ = e+2ir dr + 2 M+ (v) eKKoVdv. / . KU-P
where no = (l/B, i). Combining Eqs. (27)-(29) gives the equation for M+ along C (from now on we drop the subscript minus in v_, i.e. v_ = v):
aM+(v)
(31)
. >
(
Fig. 1. The conformal diagram of a collapsing two-dimensional star. The double line denotes the star’s surface. The global horizons are represented by dashed lines, if they do not coincide with the apparent horizons.
series in l/v
(30)
In terms of u and v, the line element near the Cauchy horizon reads d.s* = 2 eeZLr du dv, which is perfectly well behaved near the Cauchy horizon.
S. Droz/PhysicsLettersA
191(1994)211-215
5. Conclusion
Acknowledgement
The derivation presented in this paper shows that mass inflation not only occurs within the framework of spherically symmetric Einstein gravity, but also in two-dimensional string gravity. This is a strong indication that the mass inflation scenario is a generic phenomenon, and does not depend on the particular details of the theory. So far we have not touched the problem of boundary conditions, which seems crucial to us. Until now all the work done [ 8,9] assumed the existence of a portion of the Cauchy horizon, by tuming the outflux on behind the event horizon. It is not clear if a Cauchy horizon forms otherwise. We are currently investigating this assumptions [ lo]. Maybe the simple two-dimensional models help understand the essential basic physics of the much more complex four-dimensional black holes.
I would like to thank Warren Anderson Israel for some helpful discussions.
215
and Werner
References [ 1] R. Penrose, in: Battele rencontres, eds. C.M. Dewitt and J.A. Wheeler (Benjamin, New York, 1968) p. 222. [2] S.T.C. Siklos, J. Gen. Relat. Gravit. 10 (1979) 1031. [3] C.G. Callan, S.B. Giddings, J.A. Harvey and A. Strominger, Phys. Rev. D 45 (1992) R1005. [4] V.P. Frolov, Phys. Rev. D 46 (1992) 5383. [5] R.H. Price, Phys. Rev. D 5 (1972) 2419. [6] C. Gundlach, R.H. Price and J. Pullin, Phys. Rev. D 49 (1994) 883. [7] W.A. Hiscock, Phys. L&t. A 83 (1981) 110. [S] E. Poisson and W. Israel, Phys. Rev. D 41 (1990) 1796. [9] A. Ori, Phys. Rev. Lett. 67 (1991) 789. [lo] A. Bonanno, S. Droz, W. Israel and S. Morsink, in preparation.