PHYSICAL REVIEW D, VOLUME 62, 107502
Higher dimensional radiation collapse and cosmic censorship S. G. Ghosh* Department of Mathematics, Science College, Congress Nagar, Nagpur-440 012, India
R. V. Saraykar† Department of Mathematics, Nagpur University, Nagpur-440 010, India 共Received 11 May 2000; published 16 October 2000兲 We study the occurrence of naked singularities in the spherically symmetric collapse of radiation shells in a higher dimensional spacetime. The necessary conditions for the formation of a naked singularity or a black hole are obtained. The naked singularities are found to be strong in the Tipler sense and thus violate the cosmic censorship conjecture. PACS number共s兲: 04.20.Dw
The cosmic censorship conjecture 共CCC兲 put forward some three decades ago by Penrose 关1兴 says that in a generic situation all singularities arising from regular initial data are clothed by an event horizon. Since the CCC remains an unresolved issue in classical general relativity, examples which appear to violate this conjecture remain important and may be valuable when one attempts to formulate the notion of the CCC in a concrete mathematical form. The Vaidya solution 关2兴 is most commonly used as a testing ground for various forms of the CCC. In particular, Papapetrou 关3兴 first showed that this solution can give rise to the formation of naked singularities and thus provided one of the earlier counterexamples to the CCC. Since then, the solution has been extensively studied in gravitational collapse with reference to the CCC 共see, e.g., 关4兴 and references therein兲. Lately, there has been interest in studying gravitational collapse in higher dimensions 关5兴. This Brief Report searches for the occurrence of naked singularities in higher dimensional Vaidya spacetime and, if they exist, to investigate whether the dimensionality of spacetime has any role in the nature of singularities. We find that higher dimensional Vaidya spacetime admits strong-curvature naked singularities in the Tipler 关6兴 sense. The idea that spacetime should be extended from four to higher dimensions was introduced by Kaluza and Klein 关7兴 to unify gravity and electromagnetism. Five dimensional 共5D兲 spacetime is particularly more relevant because both 10D and 11D supergravity theories yield solutions where a 5D spacetime results after dimensional reduction 关8兴. Hence, we shall confine ourselves to a 5D case. It should be noted that higher dimensional spacetimes where all dimensions are in the equal foot, like the ones being considered here, are not so realistic, as we are living in effectively 4-dimensional spacetime. So in principle one might expect that by dimensional reduction the higher dimensional spacetime should reduce to our 4-dimensional world. In this sense, the models considered in this paper are ideal. The metric of collapsing Vaidya models in the 5D case is 关9兴 *Corresponding author. Email address:
[email protected] †
Email address:
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0556-2821/2000/62共10兲/107502共4兲/$15.00
冉
ds 2 ⫽⫺ 1⫺
m共 v 兲 r2
冊
d v 2 ⫹2d v dr⫹r 2 d⍀ 2 ,
共1兲
where v is a null coordinate which represents advanced Eddington time with ⫺⬁⬍ v ⬍⬁, r is a radial coordinate with 0⭐r⬍⬁, d⍀ 2 ⫽d 21 ⫹sin21(d22⫹sin22d23) is a metric of 3 sphere, and the arbitrary function m( v ) 共which is restricted only by the energy conditions兲 represents the mass at advanced time v . The energy-momentum tensor can be written in the form T ab ⫽
3 2r 3
˙ 共 v 兲k ak b , m
共2兲
with the null vector k a satisfying k a ⫽⫺ ␦ va and k a k a ⫽0. We have used the units which fix the speed of light and gravitational constant via 8 G⫽c⫽1. Clearly, for the weak energy ˙ ( v ) be noncondition to be satisfied we require that m negative, where an overdot represents a derivative with respect to v . Thus the mass function is a non-negative increasing function of v and that radiation is imploding. The physical situation here is that of a radial influx of null fluid in an initially empty region of the higher dimensional Minkowskian spacetime. The first shell arrives at r⫽0 at time v ⫽0 and the final at v ⫽T. A central singularity of growing mass is developed at r⫽0. For v ⬍0 we have m( v )⫽0, i.e., higher dimensional Minkowskian spacetime, ˙ ( v )⫽0, m( v ) is positive definite. The metand for v ⬎T, m ric for v ⫽0 to v ⫽T is higher dimensional Vaidya, and for v ⬎T we have the higher dimensional Schwarzschild solution. In order to get an analytical solution for our higher dimensional case, we choose m( v )⬀ v 2 . To be specific we take
m共 v 兲⫽
再
v ⬍0,
0, v 共 ⬎0 兲 ,
0⭐ v ⭐T,
m 0 共 ⬎0 兲 ,
v ⬎T,
2
共3兲
and with this choice of m( v ) the spacetime is self-similar 关10兴, admitting a homothetic Killing vector a given by
62 107502-1
©2000 The American Physical Society
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PHYSICAL REVIEW D 62 107502
a ⫽r
⫹v , r v
共4兲
which satisfies the condition
X 0⫽
lim X⫽ r→0 v →0
v . r→0 v →0 r
lim
共14兲
Using Eq. 共13兲 and L’Hoˆpital’s rule we get
Ł g ab ⫽ a;b ⫹ b;a ⫽2g ab ,
共5兲 X 0⫽
where Ł denotes the Lie derivative. It follows that a k a is constant along radial null geodesics and hence a constant of motion:
lim X⫽ r→0 v →0
dv 2 v , ⫽ lim ⫽ 1⫺X 20 r→0 v →0 r r→0 v →0 dr lim
共15兲
which implies
a K a ⫽rK r ⫹ v K v ⫽C.
共6兲
Let K a ⫽dx a /dk be the tangent vector to the null geodesics, where k is an affine parameter. Geodesic equations, on using the null condition K a K a ⫽0, take the simple form dK v m 共 v 兲 v 2 ⫹ 3 共 K 兲 ⫽0, dk r
共7兲
˙ 共v兲 dK r m ⫹ 共 K v 兲 2 ⫽0. dk 2r 2
共8兲
Following 关11,12兴, we introduce K v⫽
P r
共9兲
and, from the null condition, we obtain
冉
K r ⫽ 1⫺
m共 v 兲 r
2
冊
P . 2r
共10兲
The function P( v ,r) obeys the differential equation
冉
冊
3m 共 v 兲 P 2 dP ⫺ 1⫺ ⫽0. dk r2 2r 2
共11兲
Radial null geodesics of the metric 共1兲, by virtue of Eqs. 共9兲 and 共10兲, satisfy
冋
册
m共 v 兲 dr 1 ⫽ 1⫺ 2 . dv 2 r
共12兲
Clearly, the above differential equation has a singularity at r⫽0, v ⫽0. The nature 共a naked singularity or a black hole兲 of the collapsing solutions can be characterized by the existence of radial null geodesics coming out from the singularity. Equation 共12兲, upon using Eq. 共3兲, turns out to be dr 1 ⫽ 关 1⫺X 2 兴 , dv 2
共13兲
X 30 ⫺X 0 ⫹2⫽0.
共16兲
This algebraic equation governs the behavior of the tangent vector near the singular point. Thus by studying the solution of this algebraic equation, the nature of the singularity can be determined. The central shell focusing singularity is at least locally naked 共for brevity we have addressed it as naked throughout this paper兲, if Eq. 共16兲 admits one or more positive real roots. When there are no positive real roots to Eq. 共16兲, the central singularity is not naked because in that case there are no outgoing future directed null geodesics from the singularity. Hence in the absence of positive real roots, the collapse will always lead to a black hole. Thus the occurrence of positive real roots implies that the strong CCC is violated, though not necessarily the weak CCC. We now examine the condition for the occurrence of a naked singularity. Equation 共16兲 has two positive roots if ⭐1/27, e.g., the two positive roots of Eq. 共16兲 X 0 ⫽2.21833 and 5.69593, correspond to ⫽1/50. Thus referring to the above discussion, gravitational collapse of null fluid in higher dimensions leads to a naked singularity if ⭐1/27 and to formation of a black hole otherwise. The degree of inhomogeneity of collapse is defined as ⬅1/ 共see 关13兴兲, we see that for a collapse sufficiently inhomogeneous, naked singularities develop. Comparison with the analogous 4D case shows that naked singularity occurs for a slightly larger value of the inhomogeneity factor in 5D. The analysis of geodesics escaping to far away observers emanating from singularity is similar to that discussed in 关3,11兴. To see this, we first note that the apparent horizon in our case is given by X⫽1/冑. We write the equation of geodesics in the form r⫽r(X). Using Eqs. 共3兲 and 共13兲 we obtain r
dX X 3 ⫺X⫹2 . ⫽ dr 1⫺X 2
共17兲
Integration of the above equation yields the equation of geodesics 共integral curves兲. Here, we briefly describe constraints when geodesics can escape to a far away observer. Basically this follows from analysis of Eq. 共17兲 and its solution. We consider separately the following two cases. Case (a) ⬍1/27. The solution of Eq. 共17兲 is
where X⬅ v /r is the tangent to a possible outgoing geodesic. In order to determine the nature of the limiting value of X at r⫽0, v ⫽0 on a singular geodesic, we let 107502-2
r⫽
共 X⫺a 1 兲 C 1 共 X⫺a 2 兲 C 2 共 X⫹a 3 兲 C 3
,
共18兲
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PHYSICAL REVIEW D 62 107502
where a 1 and a 2 are two positive roots with a 1 ⬎a 2 and ⫺a 3 is the negative root of Eq. 共16兲. C 1 , C 2 , and C 3 are given by C 1⫽
a 2a 3 , 共 a 1 ⫺a 2 兲共 a 1 ⫹a 3 兲
C 2⫽
dX 共 X⫺3 兲 2 共 X⫹6 兲 ⫽ , dr 27⫺X 2
共20兲
which has a trivial solution X⫽3. In the variables (t,r), it takes the form t⫽2r which is a special null geodesic. In the analogous 4D case one gets t⫽3r 关3兴. For X⫽3, Eq. 共17兲 can be integrated to exp r⫽
冉 冊 2 3⫺X
共 3⫺X 兲 8/9共 X⫹6 兲 1/9
共21兲
.
As in the 4D case 关3兴, we can conclude that there is an infinite number of geodesics escaping to a far away observer. The strength of the singularity is an important issue because there have been attempts to relate it to stability 关14兴. A singularity is termed gravitationally strong or simply strong if it destroys by crushing or stretching any object which falls into it. A sufficient condition 关15兴 for a strong singularity as defined by Tipler 关6兴 is that for at least one non-space-like geodesic with affine parameter k, in a limiting approach to singularity, we must have lim k 2 ⫽ lim k 2 R ab K a K b ⬎0, k→0
共22兲
k→0
where R ab is the Ricci tensor. Equation 共22兲, with the help of Eqs. 共2兲, 共3兲, and 共9兲, can be expressed as lim k ⫽ lim 3X 2
k→0
k→0
冉 冊 kP r2
2
.
2⫺X⫹X 3
共24兲
,
dX 1 v X r ⫽ K ⫺ K , dk r r
共19兲
For X⭓1/冑, all geodesics are ingoing and no geodesics escape. For X⬍1/冑, geodesic families can meet the singularity with tangent X⫽a 1 and r⫽⬁ can be realized in the future along the same geodesic at X⫽a 2 . Thus the singularity will be globally naked and an infinity of geodesics would escape from singularity to reach a far away observer. It is also seen that for a very small value of , say O(10⫺17), X⭓1/冑 and hence singularity is only locally naked. Case (b) ⫽1/27. In this case, the Eq. 共17兲 takes the form r
2C
which is a solution of Eq. 共11兲 and geodesics are completely determined. Further, we note that
a 1a 3 , 共 a 1 ⫺a 2 兲共 a 2 ⫹a 3 兲
a 1a 2 . C 3⫽ 共 a 1 ⫹a 3 兲共 a 2 ⫹a 3 兲
P⫽
共23兲
Our purpose here is to investigate the above condition along future directed null geodesics coming out from the singularity. For this, solution of Eq. 共11兲 is required. Equation 共6兲, because of Eqs. 共3兲, 共9兲 and 共10兲, yields
共25兲
which, on inserting the expressions for K v and K r , becomes P C dX ⫽ 共 2⫺X⫺X 3 兲 2 ⫽ 2 . dk 2r r
共26兲
Using the fact that as singularity is approached, k→0, r →0, and X→a ⫹ 关a root of Eq. 共16兲兴 and using L’Hoˆpital’s rule, we observe kP
lim
r
k→0
2
2
⫽
1⫹X 20
共27兲
and hence Eq. 共23兲 gives lim k 2 ⫽ k→0
12X 0 共 1⫹X 20 兲 2
⬎0.
共28兲
Thus along radial null geodesics the strong curvature condition is satisfied. Recently, Nolan 关16兴 gave an alternative approach to check the nature of singularities without having to integrate the geodesic equations. It was shown in 关16兴 that a radial null geodesic which runs into r⫽0 terminates in a gravitationally weak singularity if and only if r˙ is finite in the limit as the singularity is approached 共this occurs at k⫽0); the overdot here indicates differentiation along the geodesics. So assuming a weak singularity, we have r˙ ⬃d 0 ,
r⬃d 0 k.
共29兲
Using the asymptotic relationship above and the form of m( v ), the geodesic equations yield d 2v dk 2
⬃ ␣ k ⫺1 ,
共30兲
where ␣ ⫽⫺X 40 d 0 ⫽a nonzero const, which is inconsistent with v˙ ⬃d 0 X 0 , which is finite. Since, the coefficient ␣ of k ⫺1 is nonzero, the singularity is gravitationally strong. Having seen that naked singularity is a strong curvature singularity, we now turn our attention to checking it for scalar polynomial singularity. We therefore examine the behavior of the Kretschmann scalar (K⫽R abcd R abcd , R abcd is the Riemann tensor兲. The Kretschmann scalar with the help of Eq. 共3兲 takes the form
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K⫽
72 r4
X 4,
共31兲
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PHYSICAL REVIEW D 62 107502
which diverges at the naked singularity and hence the singularity is a scalar polynomial singularity. The Weyl scalar (C⫽C abcd C abcd , C abcd is the Weyl tensor兲 has the same expression as the Kretschmann scalar and thus the Weyl scalar also diverges at the naked singularity and so the singularity is physically significant 关17兴. The Vaidya metric in the 4D case has been extensively used to study the formation of naked singularity in spherical gravitational collapse 关4兴. We have extended this study to the higher dimensional Vaidya metric, and found that strong curvature naked singularities do arise for a slightly higher value
of the inhomogeneity parameter. We have checked for naked singularities to be gravitationally strong, by the method in 关15兴 and by the alternative approach proposed by Nolan 关16兴 as well, and both seem to be in agreement. It is straightforward to extend the above analysis to nonradial causal curves. Furthermore, the Kretschmann and Weyl scalars blow up as singularity is approached. In conclusion, this offers a counterexample to CCC. The authors would like to thank IUCAA, Pune 共INDIA兲 for the hospitality. Thanks are also due to Brien C. Nolan for helpful correspondence.
关1兴 R. Penrose, Riv. Nuovo Cimento 1, 252 共1969兲. 关2兴 P.C. Vaidya, Proc.-Indian Acad. Sci., Sect. A 33, 264 共1951兲; reprinted in Gen. Relativ. Gravit. 31, 119 共1999兲. 关3兴 A. Papapetrou, in A Random Walk in Relativity and Cosmology, edited by N. Dadhich, J.K. Rao, J.V. Narlikar, and C.V. Vishveshwara 共Wiley, New York, 1985兲. 关4兴 P.S. Joshi, Global Aspects in Gravitation and Cosmology 共Clarendon, Oxford, 1993兲; K. Rajagopal and K. Lake, Phys. Rev. D 35, 1531 共1987兲; K. Lake ibid. 43, 1416 共1991兲. 关5兴 A. llha and J.P.S. Lemos, Phys. Rev. D 55, 1788 共1997兲; A. llha, A. Kleber, and J.P.S. Lemos, J. Math. Phys. 40, 3509 共1999兲; J.F.V. Rocha and A. Wang, gr-qc/9910109. 关6兴 F.J. Tipler, C.J.S. Clarke, and G.F.R. Ellis, in General Relativity and Gravitation, edited by A. Held 共Plenum, New York, 1980兲. 关7兴 T. Kaluza, Sitzungsber. K. Preuss. Akad. Wiss. D33, 966
共1921兲; O. Klein, Z. Phys. 37, 895 共1926兲. 关8兴 J.J. Schwarz, Nucl. Phys. B226, 269 共1983兲. 关9兴 B.R. Iyer and C.V. Vishveshwara, Pramana 32, 749 共1989兲. 关10兴 A spherical symmetric space-time is self-similar if g tt (ct,cr) ⫽g tt (t,r) and g rr (ct,cr)⫽g rr (t,r) for every c⬎0. 关11兴 I.H. Dwivedi and P.S. Joshi, Class. Quantum Grav. 6, 1599 共1989兲. 关12兴 S.G. Ghosh and A. Beesham, Phys. Rev. D 61, 067502 共2000兲. 关13兴 J.P.S. Lemos, Phys. Rev. Lett. 68, 1447 共1992兲; Phys. Rev. D 59, 044020 共1992兲. 关14兴 S.S. Deshingkar, P.S. Joshi, and I.H. Dwivedi, Phys. Rev. D 59, 044018 共1999兲. 关15兴 C.J.S. Clarke and A. Kro´lak, J. Geom. Phys. 2, 127 共1986兲. 关16兴 B.C. Nolan, Phys. Rev. D 60, 024014 共1999兲. 关17兴 S. Barve and T.P. Singh, Mod. Phys. Lett. A 12, 2415 共1997兲.
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