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Dynamical instability of the González-Díaz black hole model
Volume 138, number 3
PHYSICS LETFERS A
19 June 1989
DYNAMICAL INSTABILITY OF THE GONZALEZ-DiAl BLACK HOLE MODEL øyvind GRØN Oslo College ofEngineering, Cort Adelers gt. 30, N-0254 Oslo 2, Norway and Institute of Physics, University ofOslo, P.O. Box 1048 Blindern, N-0316 Oslo 3, Norway
and Harald H. SOLENG Institute of Physics, University of Oslo, P.O. Box 1048 Blindern, N-0316 Oslo 3, Norway Received 19 October 1987; revised manuscript received 17 April 1989; accepted for publication 21 April 1989 Communicated by J.P. Vigier
The black hole model constructed by González-Diaz, with de Sitter metric inside the horizon and Schwarzschild metric outside it, is investigatedby considering a spherical region with false vacuum surrounded by ordinary vacuum. The hypersurface between the two different vacuum regions is a spherical singular layer. A static solution of the equation of motion of the junction surface exists in the limit that the “horizon positions” of the de Sitter and Schwarzschild regions approach each other. This corresponds to the González-Diaz model. A stability analysis shows that this black hole model is dynamically unstable.
1. Introduction Some years ago González-Diazintroduced a model of black holes, with de Sitter metric inside the honzon of the hole and Schwarzschild metric outside it [1]. This model of a black hole seems strange. It is well-known that a region of false vacuum, described by the de Sitter space—time, is a source of gravitational repulsion. Why then does not light escape from this region into the external Schwarzschild space—time? The answer is found by considering radial null-geodesics inside the horizon. It is then found that light takes an infinitely long coordinate time to travel from an arbitrary point in this region to the horizon. It never arrives at the horizon, as seen by an internal observer. The proper separation of test particles in the de Sitter space—time increases exponentially. Thus the event horizon can be understood as an expansion effect [2]. The González-Diaz model was generalized to the charged case by Wenda and Shitong [3,4], and interpreted as a non-singular model of black holes. It was later shown that Israel’s theory of singular sur-
faces in general relativity [5] seems to imply the existence of a surface layer with infinite stress at the horizon of these black hole models [6]. Wenda and Shitong [7] have recently pointed out that Lichnerowicz’s junction conditions are fulfilled by the González-DIaz black hole model, and also that Israel’s theory is not unambiguous as applied to nullhypersurfaces, and consequently that the question of the existence and stability of the González-Diaz model must be reexamined. Israel’s relativistic theory of singular surfaces has recently been generalized so that the resulting theory is applicable to null hypersurfaces [8]. It is possible to describe the González-DIaz model by means of this formalism. However, in the present work we shall follow a different procedure. We shall here perturb the González-DIaz black hole model in such a way that there are no horizons, and that the junction surface between the internal de Sitter space and the external Schwarzschild space is time-like. Then the problems connected with the nulllike nature of the junction surface in this model [7] are taken care of. From the perturbed model we shall
0375-9601 1891$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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obtain the González-Diaz model as a limit, allowing us to investigate the dynamical properties of this space—time by means of Israel’s theory’ of singular surfaces. Calculating the Einstein curvature tensor of the González-DIaz black hole model we find that two of the physical curvature components are singular at the horizon. Einstein’s field equations then imply the existence of a singular surface layer at the horizon. Also the Lanczos energy—momentum tensor is singular in the González-Diazlimit. The equation ofstate ofthe junction surface is not that of a domain wall.
A spherical region of false vacuum surrounded by ordinary vacuum will now be considered. The boundary between the different vacuum states is a non-null spherical singular layer. The equation of motion ofthe boundary wall, as given by Israel’s theory, has been found by Ipser and Sikivie [9] in the case of internal Minkowski space and external Schwarzschild space. The dynamics of false vacuum bubbles has been investigated by Blau, Guendelman and Guth [101. In their work the equation of motion of spherical domain walls is deduced by employing Kruskal—Szekeres coordinates, in order to be able to describe the dynamics of the wall inside as well as outside the horizon. Equations of motion for shells at metric junctions between different combinations of Minkowski, de Sitter, and Schwarzschild space— times have been considered by Sato [11]. A thorough investigation of the dynamics of bubbles in general relativity has recently been given by Berezin, Kuzmin and Thachev [12]. The line-element describing the González-DIaz model of a2—/3~2 black hole maycLQ2, be written dr2—r2 r r =fl~dt 0, (U where 2—X2r2)”2, PD —as(1+i~2-2GJI’1/r)”2, =aD(l+r Ps
90
~deS~tterSchtrsthId
/
Fig. 1. Perturbed González-Diaz model of a black hole.
equal to ±1. This model shall now be perturbed as follows. Let X’=r0+a2e,
2GM=r
0—2e, (3) where a is a real parameter which is determined below and e << r0. Eq. (3) means that the energy density of the internal de Sitter space, and the mass of the external Schwarzschild space are perturbed. Schwarzschild coordinates are well behaved everywhere outside the junction radius r0, provided the perturbed configuration has ~‘> r0 and 2GM< To, thus x ‘>2GM, which means that e>0. The perturbed configuration is depicted in fig. 1. The perturbed space—time has no horizons, because the “horizon position” x’ of the de Sitter space is outside the junction radius r0, and the “horizon position” 2GM of the Schwarzschild space is inside the junction radius r0. The limiting transition e 0÷ means that x’ and 2GM approach r0 from above and below, respectively. ,~
-
—~
3. Gravitational mass of the junction surface Let us calculate the Einstein curvature tensor at the boundary of the perturbed model. For a line-element of the form the by mixed components of the Einstein tensor are(1), given Gt 2=Gr,r2= [r(fl2— 1)]’ 1r
x2=8zGpoI3, r dQ
\
In the static model r=0. o~and a~are sign factors 2. Perturbed González-Dmaz model
0=~’=2GM, 2=d02+sin2OdØ2.
19 June 1989
(2)
In the limit e—+0, we get
,
G’~Qr2=(r2flfl’)’
.
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PHYSICS LETTERS A
~
lim ~
,~
1i j
1
2
G 1r dr=0,
hm ~
1 i
j
~
G
2
3
dr=~r0.
QT
(5) Thus, G~Qhas a d-function singularity at the boundary. This curvature singularity means that the González-DIaz model does not represent a non-singular model of a black hole. From Einstein’s field equations (6) where T~is the energy—momentum (volume) density tensor, and eq. (5) follows that there exists a singular layer at r= r0, with vanishing rest mass and radial pressure, but with finite tangential stress. The gravitational significance of the stress ofjunction surfaces should be noted. Consider a spherically symmetric, static space—time consisting of an external Schwarzschild region and an internal region of finite extent with a non-vanishing energy—momenturn tensor. In this case the Schwarzschild mass M of the external region may be found by integration over the region occupied by a non-vanishing source (T~~ #0), according to the following two formulae [13—171, 2d (7 ~ M —4 T~ 1r r, ‘~ ‘ 0 2 dr. (8) M, =4x J(Tt, Trr~.T~0 T 0)r Using eqs. (6) and inserting eq. (4), we get, respectively
19 June 1989
taken over the complete mass distribution ofthe systern [16]. In order to obtain the total gravitational mass of the system we have to include the contnbution from the junction surface [18]. From eqs. (5) follow that the masses M’11 and ~M2 of the junction surface, as calculated from the integrals (7) and (8) are A ~
_A
A
~
—,
4
~
3
L~JEI2_Jro,.~.J_-?Eporo, where eq. (2) has been used. This gives ~
M,+&f,=M2+i~M2=~irp0r~=M.
(12)
We have shown that the mass MI2 due to the stress of the junction surface is essential in order that the Tolman expression, eq. (8), for gravitational mass shall give the correct result. In other words, it is impossible to avoid the curvature singularity at the junction surface in a consistent description of the González-DIaz black hole model. It may be noted that the González-Diaz model of a black hole has no singularities other than that at the junction. The model is static, so there are no singular points of time (like that of a cosmic big bang). Even at the infinite past (as measured in cosmic time) the de Sitterspace—time has no singularity: The internal space—time has a steady-state character with constant energy-density and constant curvature. Also the external Schwarzschild space—time is singularity free.
—
2GM
2(fl2)’ (9) 1 =r(l ~fl2), 2GM2 =r With fl= fl 5 for the external Schwarzschild metric, we find indeed that M1 = M2 =M is the Schwarzschild mass. Inserting /1=PD for the internal de Sitter metric, the integrated mass out to the junction surface, as given by (9), is (in the limit —p0) .
4. Equation of motion of the junction surface The tensor TM~is related to the geometry ofspace— time. However, the dynamics of a singular layer as described by Israel’s general relativistic theory, is given in terms of Lanczos energy—momentum (surface) density tensor: S~= ([K~]—d~[K]) (13) 8itG where [ ] signifies the discontinuity at the surface and K~is the external curvature tensor, given by K = (14) —
M,=~itp0r~,M2=—~p0r~. (10) The gravitational mass M2 of the internal de Sitter space is negative, in accordance with the well-known result that a positive cosmological constant gives rise to repulsive gravitation. The result that M, and M2 are different reflects the fact that the integrals (7) and (8) give equivalent results only when they are
,
—
°
where n’ is the unit normal vector of the boundary. The tensor S,4,, can be written as 91
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S~= cru~u~ +~
PHYSICS LETTERS A
—
u~u~),
(15)
where ~ is the metric projected into the hypersurface of the wall and u ~= (1, 0, 0, 0) is the four-yelocity of the junction surface. Then we get St 1=a, S~Q=I~. (16) Here a is the energy-density of the surface, and ~ is the surface tension, where ~> 0 means strain and ~< 0 means pressure. The external curvature tensor K’~is most conveniently calculated in Gaussian normal coordinates. In these coordinates 8~g,I, (17) K,~= ~ 19R is a partial derivative with respect to the where Gaussian radial coordinate R. This equation also holds in any coordinate system related to the Gaussian normal coordinates by transformations of the type R=R(r) where r is a radial coordinate. This is the case in the present model where we have 19
19 June 1989
To specify the sign-factors aD and o~,we consider the general expressions for PD and fl~as given in eq. (2). We require that there should be a smooth transition to the trivial [11] Minkowski—Minkowski junction in the limit M—’O and x—~0.By this we mean that the horizon position of the internal de Sitter space—time moves outwards, and that of the external Schwarzschild space—time inwards, towards infinity and origin, respectively. This procedure gives a continuous transition to a Minkowski—Minkowski junction where o~-÷ 0 if and only if a,~ o~= 1. Moreover we apply the weak energy condition a~0.This implies PD~PS. In the case of the static perturbed González-DIaz model with
r=x’—a2e, 2GM=~~—(a2+2)e,
(26)
we get PD =aDa\/~ [1—~a2Xe+O(2)],
Psas~/’~[1+~a2~~+O(2)]
.
(27)
19 = P ~.
(18)
Using eqs. (17) and (18) with the line element of eq. (1), we get K,, = ~,6
~f—, K
1~Q=
— ~
,8
~—.
r
(19)
Hence, with r19r/ôr= ~ and the metric coefficients of eq. (2), we find 2r~) +fl~(P + GM/r2), (20) [K’,] =fl~( P+~ [K~2QJ =Ps/r 0 —PD/re. (21) —
Because of spherical symmetry we have K= K’, + 2K’~.Thus eq. (13) may be written 4xGS’,=47tGcr=—[K’~Q] (22) and 87tGS’2Q=87tG~=—[K’2Q]—[K’,J.
4ltGra=PD—PS,
(24)
81tGr~=rP(fl5’—Pi~)+flD—PS —fli~’x2r2—13i’GMIr.
case. This result has recently been confirmed by Poisson and Israel [191. Since the /3’s contain an undetermined function r(r)geometry describing thenot motion the junction the does fix theof equation ofstatesurface, of this surface. On the other hand, choosing either r( x) or we may calculate the other by help of the equation of motion of the junction surface. 1Q giving The equation ofmotion ofthe junction surface can be deduced from the equations for S’, and S ~
2+flsX2r, (28) aj r $D~PS 2~~~...flDGM/r which is equivalent to eq. (4.29) of ref. [10]. If we allow for a singular equation of state ~/a—~ in the González-DIaz limit, there is a static solution of the above equation. ~(!_
(23)
Inserting from eqs. (20), (21), we get
92
In the limit e—’O the condition PD ~ Ps together with aD=cTs=l implies a>!. Because both P’s are positive, we find by inspection of expressions (25) and (27) that the stress surface density diverges in the limit e—~0,in the static
(25)
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PHYSICS LETTERS A
5. Instability of the Gonzulez-Diaz model
19 June 1989
gives the following metric coefficients at the virtually perturbed position
Wenda and Shitong [7] gave a physical argument to why the González-DIaz model could be stable. They argued that since the junction surface is a null hypersurface, perturbations will only spread on this surface and never reach r= 0, and that this is the reason why stable “non-singular” black holes may exist. This argument is not valid: True enough null hypersurfaces represent the limiting surfaces of physical signals (on a background geometry), but metric perturbations are perturbations of the geometry itself. A general metric perturbation does not leave the light cone structure unaltered, and consequently it cannot be concluded from the null-like nature ofthe unperturbed horizon that the model is stable under general metric perturbations. We now investigate the stability of the GonzalezDiaz black hole model by applying the equation of motion found in the previous section. In ref. [10] it was shown that the equation ofmo-
PD = PD
ö, fi~= fl~ +
—
~—
(5,
(29)
where PD and Ps are given by eqs. (27). Substituting these expressions into eq. (28), calculating to first order in ô, and using that PEC is an equilibrium configuration, gives
((a22)~~
~
aa
3(a2+2) “ (5 + 8a(a— l) 2)X ~
The quotient ~/a for the PEC is ~ a+2 = 8a(a—l ~ —
a> 1. (30)
(3!)
Inserting this into eq. (30), we get 4—2a3—4a2—8a+4 ~ a
Schwarzschild tion ofmotion a domain space—time wall corresponds between de toSitter the and lion of of a particle in a potential (fig.equa6 of
This equation 8(a—l)2a2e showsunder that virtual x~. displacements perturbed models (32) with a <3.627 are stable of
ref. [10]) with only one stationarypoint. This point is a global maximum of the potential, and the equilibrium configuration is unstable. This potential depends on the equation ofstate through the parameter y which is a constant when the junction surface is a domain wall. For other equations of state y is a variable, and consequently the potentials which were plotted in ref. [10] are valid only for domain walls. In view of the latest claims of Wenda and Shitong [7] and the fact that Blau et al. [10] did not explicitly mention the González-DIaz model for which the junction surface does not satisfy the equation of state of a domain wall, we clarify the matter by performing a stability analysis. We have found that the Gonzáiez-DIaz model is in dynamical equilibrium. This holds to any order in Hence there does also exist a perturbed equilibrium configuration, PEC. The significance of the PEC is that it allows a virtual displacement (5 of the boundary surface, maintaming its time-like character, so that the stability of the model may be investigated by means ofthe equation of motion, eq. (28). Putting r=r 0+d, ô’cz<, while leaving M, x and the quotient ~/a unchanged,
the type discussed here. For other values of a the PEC’s are unstable. Letting ö—~~0and —p0, we condude that the González-DIaz model of black holes is dynamically unstable.
~.
6. Conclusion In a recent Letter Wenda and Shitong [7] have examined the González-DIaz black hole model, and concluded: “Our analysis shows that a globally regular space—time metric for a Schwarzschild black hole is indeed singularity free, and that the black hole filled with matter obeying the equation of state p = —p can exist.” The analysis presented above shows that the González-DIaz black hole model has a singular Emstein curvature tensor and a singular Lanczos tensor. Hence, the model is singular. If such a strongly singular layer is accepted, a static solution of the equation of motion does exist. This solution is, however, not stable under general perturbations.
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Acknowledgement We would like to thank a referee for useful comments. References [1] P.F. González-Dfaz, Lett. Nuovo Cimento 32 (1981)161. [2] W. Rindler, Mon. Not. R. Astron. Soc. 116 (1956) 662. [3]S.WendaandZ.Shitong,NuovoCimentoB85 (1985) 142. [4]S.WendaandZ.Shitong,Gen.Rel.Grav. 17 (1985)739. [5] W. Israel, Nuovo Cimento B 44 (1966) 1. [6]0. Grøn, Lett. Nuovo Cimento 44 (1985) 177. [7] 5. Wenda and Z. Shitong, Phys. Lett. A 126 (1988) 229. [8] C.J.S. Clarke and T. Dray, Class. Quantum Gray. 4 (1987) 265.
94
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[91J. Ipser and P. Sikivie, Phys. Rev. D 30 (1984) 712. [10] S.K. Blau, E.I. Guendelman and A.H. Guth, Phys. Rev. D 35 (1987) 1747. [11] H. Sato, Prog. Theor. Phys. 76 (1986)1250. [12] V.A. Berezin, V.A. Kuzmin and 1.1. Thachev, Phys. Rev. D 36 (1987) 2919. [13] R.C. Tolman, Phys. Rev. 35 (1930) 875. [14] L.D. Landau and E.M. Lifshitz, The classical theory offields (Pergamon, Oxford, 1975) chs. 11, 12. [15] Fl. Cooperstock and R.S. Sarracino, J. Phys. A 11(1978) 877. [16] F.I. Cooperstock, R.S. Sarracino and S.S. Bayin, J. Phys. A 14(1981)181. [17]ø.Grøn,Phys.Rev.D3l (1985)2129. [18]R.S. Sarracino, Thesis, University of Victoria (1981). [19]E. Poisson and W. Israel, Class. Quantum Gray. 5 (1988) L201.
PHYSICS LETFERS A
19 June 1989
DYNAMICAL INSTABILITY OF THE GONZALEZ-DiAl BLACK HOLE MODEL øyvind GRØN Oslo College ofEngineering, Cort Adelers gt. 30, N-0254 Oslo 2, Norway and Institute of Physics, University ofOslo, P.O. Box 1048 Blindern, N-0316 Oslo 3, Norway
and Harald H. SOLENG Institute of Physics, University of Oslo, P.O. Box 1048 Blindern, N-0316 Oslo 3, Norway Received 19 October 1987; revised manuscript received 17 April 1989; accepted for publication 21 April 1989 Communicated by J.P. Vigier
The black hole model constructed by González-Diaz, with de Sitter metric inside the horizon and Schwarzschild metric outside it, is investigatedby considering a spherical region with false vacuum surrounded by ordinary vacuum. The hypersurface between the two different vacuum regions is a spherical singular layer. A static solution of the equation of motion of the junction surface exists in the limit that the “horizon positions” of the de Sitter and Schwarzschild regions approach each other. This corresponds to the González-Diaz model. A stability analysis shows that this black hole model is dynamically unstable.
1. Introduction Some years ago González-Diazintroduced a model of black holes, with de Sitter metric inside the honzon of the hole and Schwarzschild metric outside it [1]. This model of a black hole seems strange. It is well-known that a region of false vacuum, described by the de Sitter space—time, is a source of gravitational repulsion. Why then does not light escape from this region into the external Schwarzschild space—time? The answer is found by considering radial null-geodesics inside the horizon. It is then found that light takes an infinitely long coordinate time to travel from an arbitrary point in this region to the horizon. It never arrives at the horizon, as seen by an internal observer. The proper separation of test particles in the de Sitter space—time increases exponentially. Thus the event horizon can be understood as an expansion effect [2]. The González-Diaz model was generalized to the charged case by Wenda and Shitong [3,4], and interpreted as a non-singular model of black holes. It was later shown that Israel’s theory of singular sur-
faces in general relativity [5] seems to imply the existence of a surface layer with infinite stress at the horizon of these black hole models [6]. Wenda and Shitong [7] have recently pointed out that Lichnerowicz’s junction conditions are fulfilled by the González-DIaz black hole model, and also that Israel’s theory is not unambiguous as applied to nullhypersurfaces, and consequently that the question of the existence and stability of the González-Diaz model must be reexamined. Israel’s relativistic theory of singular surfaces has recently been generalized so that the resulting theory is applicable to null hypersurfaces [8]. It is possible to describe the González-DIaz model by means of this formalism. However, in the present work we shall follow a different procedure. We shall here perturb the González-DIaz black hole model in such a way that there are no horizons, and that the junction surface between the internal de Sitter space and the external Schwarzschild space is time-like. Then the problems connected with the nulllike nature of the junction surface in this model [7] are taken care of. From the perturbed model we shall
0375-9601 1891$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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Volume 138, number 3
PHYSICS LETTERS A
obtain the González-Diaz model as a limit, allowing us to investigate the dynamical properties of this space—time by means of Israel’s theory’ of singular surfaces. Calculating the Einstein curvature tensor of the González-DIaz black hole model we find that two of the physical curvature components are singular at the horizon. Einstein’s field equations then imply the existence of a singular surface layer at the horizon. Also the Lanczos energy—momentum tensor is singular in the González-Diazlimit. The equation ofstate ofthe junction surface is not that of a domain wall.
A spherical region of false vacuum surrounded by ordinary vacuum will now be considered. The boundary between the different vacuum states is a non-null spherical singular layer. The equation of motion ofthe boundary wall, as given by Israel’s theory, has been found by Ipser and Sikivie [9] in the case of internal Minkowski space and external Schwarzschild space. The dynamics of false vacuum bubbles has been investigated by Blau, Guendelman and Guth [101. In their work the equation of motion of spherical domain walls is deduced by employing Kruskal—Szekeres coordinates, in order to be able to describe the dynamics of the wall inside as well as outside the horizon. Equations of motion for shells at metric junctions between different combinations of Minkowski, de Sitter, and Schwarzschild space— times have been considered by Sato [11]. A thorough investigation of the dynamics of bubbles in general relativity has recently been given by Berezin, Kuzmin and Thachev [12]. The line-element describing the González-DIaz model of a2—/3~2 black hole maycLQ2, be written dr2—r2 r
90
~deS~tterSchtrsthId
/
Fig. 1. Perturbed González-Diaz model of a black hole.
equal to ±1. This model shall now be perturbed as follows. Let X’=r0+a2e,
2GM=r
0—2e, (3) where a is a real parameter which is determined below and e << r0. Eq. (3) means that the energy density of the internal de Sitter space, and the mass of the external Schwarzschild space are perturbed. Schwarzschild coordinates are well behaved everywhere outside the junction radius r0, provided the perturbed configuration has ~‘> r0 and 2GM< To, thus x ‘>2GM, which means that e>0. The perturbed configuration is depicted in fig. 1. The perturbed space—time has no horizons, because the “horizon position” x’ of the de Sitter space is outside the junction radius r0, and the “horizon position” 2GM of the Schwarzschild space is inside the junction radius r0. The limiting transition e 0÷ means that x’ and 2GM approach r0 from above and below, respectively. ,~
-
—~
3. Gravitational mass of the junction surface Let us calculate the Einstein curvature tensor at the boundary of the perturbed model. For a line-element of the form the by mixed components of the Einstein tensor are(1), given Gt 2=Gr,r2= [r(fl2— 1)]’ 1r
x2=8zGpoI3, r dQ
\
In the static model r=0. o~and a~are sign factors 2. Perturbed González-Dmaz model
0=~’=2GM, 2=d02+sin2OdØ2.
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(2)
In the limit e—+0, we get
,
G’~Qr2=(r2flfl’)’
.
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PHYSICS LETTERS A
~
lim ~
,~
1i j
1
2
G 1r dr=0,
hm ~
1 i
j
~
G
2
3
dr=~r0.
QT
(5) Thus, G~Qhas a d-function singularity at the boundary. This curvature singularity means that the González-DIaz model does not represent a non-singular model of a black hole. From Einstein’s field equations (6) where T~is the energy—momentum (volume) density tensor, and eq. (5) follows that there exists a singular layer at r= r0, with vanishing rest mass and radial pressure, but with finite tangential stress. The gravitational significance of the stress ofjunction surfaces should be noted. Consider a spherically symmetric, static space—time consisting of an external Schwarzschild region and an internal region of finite extent with a non-vanishing energy—momenturn tensor. In this case the Schwarzschild mass M of the external region may be found by integration over the region occupied by a non-vanishing source (T~~ #0), according to the following two formulae [13—171, 2d (7 ~ M —4 T~ 1r r, ‘~ ‘ 0 2 dr. (8) M, =4x J(Tt, Trr~.T~0 T 0)r Using eqs. (6) and inserting eq. (4), we get, respectively
19 June 1989
taken over the complete mass distribution ofthe systern [16]. In order to obtain the total gravitational mass of the system we have to include the contnbution from the junction surface [18]. From eqs. (5) follow that the masses M’11 and ~M2 of the junction surface, as calculated from the integrals (7) and (8) are A ~
_A
A
~
—,
4
~
3
L~JEI2_Jro,.~.J_-?Eporo, where eq. (2) has been used. This gives ~
M,+&f,=M2+i~M2=~irp0r~=M.
(12)
We have shown that the mass MI2 due to the stress of the junction surface is essential in order that the Tolman expression, eq. (8), for gravitational mass shall give the correct result. In other words, it is impossible to avoid the curvature singularity at the junction surface in a consistent description of the González-DIaz black hole model. It may be noted that the González-Diaz model of a black hole has no singularities other than that at the junction. The model is static, so there are no singular points of time (like that of a cosmic big bang). Even at the infinite past (as measured in cosmic time) the de Sitterspace—time has no singularity: The internal space—time has a steady-state character with constant energy-density and constant curvature. Also the external Schwarzschild space—time is singularity free.
—
2GM
2(fl2)’ (9) 1 =r(l ~fl2), 2GM2 =r With fl= fl 5 for the external Schwarzschild metric, we find indeed that M1 = M2 =M is the Schwarzschild mass. Inserting /1=PD for the internal de Sitter metric, the integrated mass out to the junction surface, as given by (9), is (in the limit —p0) .
4. Equation of motion of the junction surface The tensor TM~is related to the geometry ofspace— time. However, the dynamics of a singular layer as described by Israel’s general relativistic theory, is given in terms of Lanczos energy—momentum (surface) density tensor: S~= ([K~]—d~[K]) (13) 8itG where [ ] signifies the discontinuity at the surface and K~is the external curvature tensor, given by K = (14) —
M,=~itp0r~,M2=—~p0r~. (10) The gravitational mass M2 of the internal de Sitter space is negative, in accordance with the well-known result that a positive cosmological constant gives rise to repulsive gravitation. The result that M, and M2 are different reflects the fact that the integrals (7) and (8) give equivalent results only when they are
,
—
°
where n’ is the unit normal vector of the boundary. The tensor S,4,, can be written as 91
Volume 138, number 3
S~= cru~u~ +~
PHYSICS LETTERS A
—
u~u~),
(15)
where ~ is the metric projected into the hypersurface of the wall and u ~= (1, 0, 0, 0) is the four-yelocity of the junction surface. Then we get St 1=a, S~Q=I~. (16) Here a is the energy-density of the surface, and ~ is the surface tension, where ~> 0 means strain and ~< 0 means pressure. The external curvature tensor K’~is most conveniently calculated in Gaussian normal coordinates. In these coordinates 8~g,I, (17) K,~= ~ 19R is a partial derivative with respect to the where Gaussian radial coordinate R. This equation also holds in any coordinate system related to the Gaussian normal coordinates by transformations of the type R=R(r) where r is a radial coordinate. This is the case in the present model where we have 19
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To specify the sign-factors aD and o~,we consider the general expressions for PD and fl~as given in eq. (2). We require that there should be a smooth transition to the trivial [11] Minkowski—Minkowski junction in the limit M—’O and x—~0.By this we mean that the horizon position of the internal de Sitter space—time moves outwards, and that of the external Schwarzschild space—time inwards, towards infinity and origin, respectively. This procedure gives a continuous transition to a Minkowski—Minkowski junction where o~-÷ 0 if and only if a,~ o~= 1. Moreover we apply the weak energy condition a~0.This implies PD~PS. In the case of the static perturbed González-DIaz model with
r=x’—a2e, 2GM=~~—(a2+2)e,
(26)
we get PD =aDa\/~ [1—~a2Xe+O(2)],
Psas~/’~[1+~a2~~+O(2)]
.
(27)
19 = P ~.
(18)
Using eqs. (17) and (18) with the line element of eq. (1), we get K,, = ~,6
~f—, K
1~Q=
— ~
,8
~—.
r
(19)
Hence, with r19r/ôr= ~ and the metric coefficients of eq. (2), we find 2r~) +fl~(P + GM/r2), (20) [K’,] =fl~( P+~ [K~2QJ =Ps/r 0 —PD/re. (21) —
Because of spherical symmetry we have K= K’, + 2K’~.Thus eq. (13) may be written 4xGS’,=47tGcr=—[K’~Q] (22) and 87tGS’2Q=87tG~=—[K’2Q]—[K’,J.
4ltGra=PD—PS,
(24)
81tGr~=rP(fl5’—Pi~)+flD—PS —fli~’x2r2—13i’GMIr.
case. This result has recently been confirmed by Poisson and Israel [191. Since the /3’s contain an undetermined function r(r)geometry describing thenot motion the junction the does fix theof equation ofstatesurface, of this surface. On the other hand, choosing either r( x) or we may calculate the other by help of the equation of motion of the junction surface. 1Q giving The equation ofmotion ofthe junction surface can be deduced from the equations for S’, and S ~
2+flsX2r, (28) aj r $D~PS 2~~~...flDGM/r which is equivalent to eq. (4.29) of ref. [10]. If we allow for a singular equation of state ~/a—~ in the González-DIaz limit, there is a static solution of the above equation. ~(!_
(23)
Inserting from eqs. (20), (21), we get
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In the limit e—’O the condition PD ~ Ps together with aD=cTs=l implies a>!. Because both P’s are positive, we find by inspection of expressions (25) and (27) that the stress surface density diverges in the limit e—~0,in the static
(25)
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5. Instability of the Gonzulez-Diaz model
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gives the following metric coefficients at the virtually perturbed position
Wenda and Shitong [7] gave a physical argument to why the González-DIaz model could be stable. They argued that since the junction surface is a null hypersurface, perturbations will only spread on this surface and never reach r= 0, and that this is the reason why stable “non-singular” black holes may exist. This argument is not valid: True enough null hypersurfaces represent the limiting surfaces of physical signals (on a background geometry), but metric perturbations are perturbations of the geometry itself. A general metric perturbation does not leave the light cone structure unaltered, and consequently it cannot be concluded from the null-like nature ofthe unperturbed horizon that the model is stable under general metric perturbations. We now investigate the stability of the GonzalezDiaz black hole model by applying the equation of motion found in the previous section. In ref. [10] it was shown that the equation ofmo-
PD = PD
ö, fi~= fl~ +
—
~—
(5,
(29)
where PD and Ps are given by eqs. (27). Substituting these expressions into eq. (28), calculating to first order in ô, and using that PEC is an equilibrium configuration, gives
((a22)~~
~
aa
3(a2+2) “ (5 + 8a(a— l) 2)X ~
The quotient ~/a for the PEC is ~ a+2 = 8a(a—l ~ —
a> 1. (30)
(3!)
Inserting this into eq. (30), we get 4—2a3—4a2—8a+4 ~ a
Schwarzschild tion ofmotion a domain space—time wall corresponds between de toSitter the and lion of of a particle in a potential (fig.equa6 of
This equation 8(a—l)2a2e showsunder that virtual x~. displacements perturbed models (32) with a <3.627 are stable of
ref. [10]) with only one stationarypoint. This point is a global maximum of the potential, and the equilibrium configuration is unstable. This potential depends on the equation ofstate through the parameter y which is a constant when the junction surface is a domain wall. For other equations of state y is a variable, and consequently the potentials which were plotted in ref. [10] are valid only for domain walls. In view of the latest claims of Wenda and Shitong [7] and the fact that Blau et al. [10] did not explicitly mention the González-DIaz model for which the junction surface does not satisfy the equation of state of a domain wall, we clarify the matter by performing a stability analysis. We have found that the Gonzáiez-DIaz model is in dynamical equilibrium. This holds to any order in Hence there does also exist a perturbed equilibrium configuration, PEC. The significance of the PEC is that it allows a virtual displacement (5 of the boundary surface, maintaming its time-like character, so that the stability of the model may be investigated by means ofthe equation of motion, eq. (28). Putting r=r 0+d, ô’cz<, while leaving M, x and the quotient ~/a unchanged,
the type discussed here. For other values of a the PEC’s are unstable. Letting ö—~~0and —p0, we condude that the González-DIaz model of black holes is dynamically unstable.
~.
6. Conclusion In a recent Letter Wenda and Shitong [7] have examined the González-DIaz black hole model, and concluded: “Our analysis shows that a globally regular space—time metric for a Schwarzschild black hole is indeed singularity free, and that the black hole filled with matter obeying the equation of state p = —p can exist.” The analysis presented above shows that the González-DIaz black hole model has a singular Emstein curvature tensor and a singular Lanczos tensor. Hence, the model is singular. If such a strongly singular layer is accepted, a static solution of the equation of motion does exist. This solution is, however, not stable under general perturbations.
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Acknowledgement We would like to thank a referee for useful comments. References [1] P.F. González-Dfaz, Lett. Nuovo Cimento 32 (1981)161. [2] W. Rindler, Mon. Not. R. Astron. Soc. 116 (1956) 662. [3]S.WendaandZ.Shitong,NuovoCimentoB85 (1985) 142. [4]S.WendaandZ.Shitong,Gen.Rel.Grav. 17 (1985)739. [5] W. Israel, Nuovo Cimento B 44 (1966) 1. [6]0. Grøn, Lett. Nuovo Cimento 44 (1985) 177. [7] 5. Wenda and Z. Shitong, Phys. Lett. A 126 (1988) 229. [8] C.J.S. Clarke and T. Dray, Class. Quantum Gray. 4 (1987) 265.
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[91J. Ipser and P. Sikivie, Phys. Rev. D 30 (1984) 712. [10] S.K. Blau, E.I. Guendelman and A.H. Guth, Phys. Rev. D 35 (1987) 1747. [11] H. Sato, Prog. Theor. Phys. 76 (1986)1250. [12] V.A. Berezin, V.A. Kuzmin and 1.1. Thachev, Phys. Rev. D 36 (1987) 2919. [13] R.C. Tolman, Phys. Rev. 35 (1930) 875. [14] L.D. Landau and E.M. Lifshitz, The classical theory offields (Pergamon, Oxford, 1975) chs. 11, 12. [15] Fl. Cooperstock and R.S. Sarracino, J. Phys. A 11(1978) 877. [16] F.I. Cooperstock, R.S. Sarracino and S.S. Bayin, J. Phys. A 14(1981)181. [17]ø.Grøn,Phys.Rev.D3l (1985)2129. [18]R.S. Sarracino, Thesis, University of Victoria (1981). [19]E. Poisson and W. Israel, Class. Quantum Gray. 5 (1988) L201.