Patil, Kishor D

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In d ia n  J. Phys. 77B (3). 345-350 (2003)$ UP >The final fate of spherical inhomogeneous dust collapse in higher dimensional spacetimeKishorDPatilB D College of Engineenng, Sewagram,Wardha-442 001, Maharashtra, IndiaE-mail kishordpaiil@yahoo.com Received H May 2002, accepted 24 January 200JAbstract ’ The work given in this paper searches for the occurrence of naked singularities in higher dimensional Tolman type spherically sMumotne spacetime and, if they exist, to investigate whether the dimensionality of spacetime has any role in the nature of singulonties. We show that (limeiisionality of spacetime does not essentially change the basic nature of singularity of an inhomogeneous dust collapse. We examine here the nature ol ihc central singularity forming in the sphencally symmetric collapse of dust cloud and it is shown that this is always a strong curvature singularity where giavitational tidal force diverge powerfully.Kt‘vwurd.s ’ Higher dimensional .spacetime, naked singularity, cosmic censorship hypothesis, black hole, gravitational collapse l'A( S No. 04.20.DwI- IntroductionAn outstanding problem in gravitation theory and relativistic asirc)[)hysics today is to understand the final outcome of an cn(j|css gravitational collapse. Such a continual collapse would liike place when stars more massive than few limes the mass of the sun collapse under their own gravity on exhausting their nuclear fuel. According to the general theory of relativity, this results either in a black hole or a naked singularity which can eniiimumcate with faraway observers in the universe. Various niotlcls of spherical collapse have been studied over the last lew years, and these show that both black holes and naked ''ttigulaniies arise during gravitational collapse.l^ hc models studied so far includes collapse of dust [1], radiation [2], perfect fluids [3] and imperfect fluids [4]. In each of ‘he.se ca.ses, the formation of covered as well as naked ""‘Jigularities has been observed.Nowadays there has been a growing interest in studying t’raviiational collapse in higher dimensions [5-10). Many workshigher dimensional solutions have appeared recently in *“eraurc because of their implications in astrophysics, ^^ s^mology, string theory and particle physics [6].The results of gravitational collapse in higher dimensions are of interest in the view of current possibilities being explored for higher dimensional gravity. An interesting problem that arises is the effect that higher dimensions can have on the formation of naked singularities. Sil and Chatterjee [8] studied dust collapse in five dimensional spacetime. By considering a self-similar Tolman type model in higher dimensional spacetime, they showed the occurrence of a naked shell focusing singularity which may develop into a strong curvature singularity.In this paper, we consider nature and structure of singularities in both marginally as well as non-marginally bound dust collapse in 5-D. We show that the central singularity of collapse may indeed be a (strong or weak curvature) naked one depending on the conditions on initial density distribution.The focus of our investigation will be the singularity that may possibly form during the collapse, at the origin r = 0 of the spherical coordinates. This is the so-called central shell- focussing singularity. Hence, we will specify initial conditions only in a small neighbourhood of the center and investigate the nature of collapse in that region, without considering the evolution in the other regions of the spherical object. We will also assume that the initial conditions are such that shell­crossing singularities do not form during evolution.©2003 lACS346 Kishor D PatilThe organization of the paper is as below.In Sction 2, the basic parameter of the Tolman-Bondi models in 5D describing the inhomogeneous dust collapse arc specified. The existence and nature of the naked singularity is analyzed in Section 3. Strength of this singularity is examined in Section 4. We end the paper by giving concluding remarks in Section 5.2. Tolman type model in higher dimensional spacetimeThe metric of a spherically symmetric inhomogeneous dust cloud in five dimensional spacetime |8J is given by^  -dr^ l + / ( r )■^R^^dO^ -Hsin  ^0, dOl +sin^ 0, sin^ 02^ 0 3 ), (1)where f ( r )  is an arbitrary function of comoving coordinate r, satisfying f  > -\ .R{ t, r)  is the physical radius at a time t of the shell labeled by r, in the sense that 47cR^(rj)  is the proper area of the shell at time t. A prime denotes the partial derivative with respect to r.The energy momentum tensor is given by(2)where e{r j )  is the energy density of a cloud of radius r and is given by£ (r,0  =3F 'IR^R'The function R{rt) is the solution of(3)where G(y) is a strictly real positive function and is given hyG { y ) = ' ^ * ^  y * 0 .= 1/2, ^ = 0. (6iand r^(r) is a constant of integration which can be fixed by the choice of scaling on the initial surface ( r = 0). Using this scalm|z freedom, if we choose R so that /?(0, r) = r thenn)gives the time at which the physical radius (R) ot the shell labeled by r becomes zero.Thus, the range of coordinates is given by '0 < r < r ^ ,  -oo< r< f^ (r ) ,  (S)where r = denotes the boundary of the dust cloud.It has been shown [11] that shell crossing singuhintics (characterized by R* >0  and /? > 0) are gravitationally weak and hence such singularities need not be considered seriouslyWe therefore consider only the shell focusing singularity We thus assume in the following discussion.From eqs. (3) - (7), we can write R '  =^ ( R - P ) X  + X^G(PX'^)j/:X^(4)where a dot denotes the partial derivative with respect to t. the function/(r), F(r) are arbitrary, and the results from integration of the field equations.As wc are only concerned with gravitational collapse of dust, we requireF ( / ,r )< 0 .We have used units which fix the speed of light and the gravitational constant via %jtG = =1.For physical reasons, one assumes that the energy density C(r,f) is everywhere non-negative.Eq. (4) after integration yields^ f R ^(5)where we have used the following notationsD rF' r f 'U = r \  X = - ,  = p{r) = - ^ .p ( r ) = — , A = -----, P = —^F u F0  =1.2(a-l) G(p)H ( X , r )  = - ( r i - P ) X( 11)( 12)Since we are considering collapsing case, we requireThe final fa te  o f  spherical inhomogeneous dust collapse in higher dimensional spacetime 347The function P(r) is defined to be zero when f ( r )  is zero (inarginally bound case).Tlic parameter a  (which satisfies a >  I ) is introduced here (oi examining the structure of the central singularity at r = 0.One can write the energy density as£ =3t]AV^2R^H(X,r) (13)Since F' = T \ ^ r ^  ' it follows from above that everywhereH(X ,r)S0 and riA^ ^ 0  as a consequence of the weakenergy condition.Krelschmann scalar (K = metric (1) is1:1 veil byR^R'^144FF' 288F^■+-----T— 0 4 )hor radial null geodesies, these arc dk RK' _ dr _ K ' y [ u f  _  P y [ i T Jdie R ' R R '(15)(16)"'here the faction F(f.r) satisfies the differential equationdP ni  —  + P^ dkR' R R'R/• + / = 0. (17)Let u = r“ ( a S l ) .  then(18)L'rom eq. (1) we see that for outgoing radial null geodesic.*drR'(19)- + / •Using eqs. (19) and (20), eq. (18) becomes(20)duR'a r a-iI - - RT + /V iT / (21)W(X,M)a1 - = U(X,u).(22)R'’R' R"In Tolman-Bondi spacetimes, singularities occur, as one can sL‘c Irnm cqs. (3) and (14), at points where R = 0, which are calledshell loc'using singularities.3. Existence of naked singularityIn Older to check whether the singularity is naked, we examine ihe null geodesic equations for the tangent vector K" /dk  ’ 'vhcre k is an affine parameter along theijcodcsics.The function H(X,r) in the above equation is strictly positive and nonzero for all r > 0. If the null geodesics terminate in the past at the singularity with a definite tangent, then at the singularity, the tangent to the geodesic dR/du is positive and must have a finite value. From eq. (21), we note that dR/du is positive if R^ > F. Thus boundary of the trapped surface i.e. apparent horizon is given byR = 4 f . (23)Using above equation, we find from eq. (5) thattah(r) = l d r ) - y l F G ( f ) ,  (24)where r^^(r) denotes the time at whieh apparent horizon forms.It can be easily seen from the above equation that all the points on the singularity curve /^(r), other than the central point (r = 0) are covered by the apparent horizon. This is because, since both the functions F(r) and G(r) are strictly positive for r > 0, with F(r) = 0 at r= 0 , therefore for all r>0t , ( r )> r ^ J r )  and f,(0) = r„;,(0).Thus only central singularity could be naked while non­central singularities are covered.After simplifying differential eq. (22), we see that dR/du is of the form 0/0 in the limit approach to the singularity (R = 0, u = 0). The point M = r ® = 0 ,/?  = 0 i s  thus a singularity of the differential equation (22).Hence we study the detailed behaviour of the characteristic curves in the vicinity of the singularity. Defining quantityXn = lim —  = hm — = lim —r->0/t->0 U R’^ Odur-^0(25)348 Kishor D Patilwhich gives the tangent to characteristics which terminate in the past at the singularity, it can be shown [ 12] that if the equationV{x) = U { X f } ) - X  = \ -  -X = 0 ,(2 6 )«  /l + /oadmits a real positive root, then the central singularity at /* = 0, /? = 0 is naked.In other cases a black hole will be formed as the end product of collapse. We extend the earlier examples given in Ref. 112) to SD case.A. Marginally hound collapse :Let us consider first the marginally bound spacetime (/ = 0) characteri/ed by the function F(r) and f[r) asf ( r ) = 0 ,  F{r)= /i> 2 . (27)Here, F^ is a constant. Thus,«„ = «. p  = 0. p = 0 ,  P = Q. 0  = (28)For 0  to be finite we choose a  = l , hence we gett.e.AlsoandnX  ^ 4 - n  nAp ( 4 - w) / \q ^  4 4X 4 4 X ‘ (31)In ihc case n>2  where y\o = 0 , the above equation becomes( n - 4 ) ( X ^ - l )  = 0. ,12,But Since ^  4 , we can write X = ± \ .Thus eq. (31) has only one positive root X =  1 w h u h  satisfies the equation V(X) = 0 for all /? > 2, thus establishing the existence of naked singularity for all these spacetimesIn the case n = 2, A^  ^ = .JFq , hence eq. (3 1 );bccomcsx V ^ X ^ - X + 7 ^  = 0.\Numerical calculations show that the above equation has real and positive root if-n + 5 > /5i.e. 0.09017. ((•I)0 = 1 - - .4f f /  1/ VA ( n  = ~  = ^  r -  .Limiting value of A(r) in the neighbourhood of r = 0 is given by^(0) = for /I = 2- 0  for 2^ >2. (30)With the help of above functions, eq. (26) becomesFor example, for F^  ^= 0.08, there are two positive mnis X =  0.5984 and 0.3759. Hence for all such values given by q  (34), the singularity is naked. On the other hand, il thi ^ . r- -11+ 5^5  ,  ^ - ,inequality is reversed, i.e. fj, < ------------ , no naked singulanivoccurs and gravitational collapse of dust results in a black hole In the analogous 4-D case, one gets quartic equation ami the shell focusing singularity is naked if and only if Fj^<() iS09[12].B. Non-margitially hound collapse :Next, consider non-marginally bound spacetimes characterized by the functions F{r) and f(r) as/ ( r )  =  / o r ^ ( l + / , r h .F { r ) = F o r \-{^ = P o > - l .where, F ,^ and /j are constants. Hence, we get(33)Po=2, rj(0 = 4. p(r)=po(l + / / ). « = 2,®o = / iV' + Fo- 2 G ( p o )  M r ) ^ ^The fin al fa te  o f  spherical inhomogeneous dust collapse in higher dimensional spacetime 349H{X,0) = X + (36)H e n c e ,  eq. (26) becomesX i= , elsewhere. (42)Hence, we get[2oX 'o+A o(4a-no)]"x ' + J F oX ' - - & „ X + 0 0 ^ = 0 . (3T)Numerical calculations show that eq. (37) has positive realnmls 11(-K- ii+ sV s(38)4. Curvature strengthWc now discuss the strength of the singularity by considering ihc curvature growth near it.f ollowing Clarke and Krolak [ 13], a sufficient condition for a singularity to be a strong in the sense of Tipler [14] is that at IcLisi along one null geodesic (with affine parameter ^ ), we should have m ihc limit of approach to the singularityhm k ‘ R.,1, K" k '’ > 0 . (39)s\hcic A'*' IS the tangent to null geodesies.Koi ific Tolman type model in five dimension, the condition (^ 91 lor radial null geodesics becomes3F' tk-,0 2R^R' (K' )3 ^ , 4  kP2aX^ (40)^^ 'hc^c(41)andl i m f ^ V^2a j2aXl2oc( X q +2A q) — AqTJqif 0^ “■ linn P = 0 , » ;A-40if lim P = = 0 , o o ;8oY52 ’ elsewhere. (43)K ' . t  k ’ ^ U L LR ' /?' ^^ satisfies the differential eq. (17).I'or radial null geodesics, using L-Hospitals' rule and cqs.  ^ H9) and (15)-( 17) and the fact that at the singularity r —>0, X find thatindicating that the naked singularity is strong curvature one.5. ConclusionThe Tolman-Bondi metric in the 4-D case has been extensively used to study the formation of naked singularities in spherical gravitational collapse, We have extended this study to higher dimensional Tolrnan-bondi metric and found (hat strong curvature naked singularities do arise in these spacetimes. In this work we have shown that the dimensionality of spacetime does not essentially change the basic nature of the singularity of an inhomogeneous dust collapse.Referencesf i )  D M  E a r d l e y  a n d  L S m a r r  Phyx. Rev  D19 2 2 3 9  ( 1 9 7 9 )  . D  C h r i s l o d o l o u  Commun. Phyx  93 171 ( 1 9 8 4 )  , B  W a u g h  a n d  K L a k e  Phys, Rev D38 131.S ( 1 9 8 8 )  ; P  S  J u s h i  a n d  T  P  S in g h  Phys Rev D51 6 7 7 8  ( I9 9 .S )  ; T  P  S in g h  a n d  P  S  J o s h i  Class. Quantum  Grav  13 5 5 9  ( 1 9 9 6 )  . S a n ja y  J h in g a n ,  P  S  J o s h i  a n d  T  P  S in g h  Class Quantum (h a v  13 3 0 5 7  ( 1 9 9 6 )  . I H D w ie d i  a n d  P  S J o s h i  Class Quantum G rav  14 1 2 2 3  ( 1 9 9 7 )  , H I g u c h i ,  K  N a k a o  a n d  T  H a r a d a  Phys Rev D72 6 2  ( 1 9 9 8 )  . S  B u r v e .  T  P  S in g h ,  C  Vo7 a n d  L W i t l e n  C lass Q uan tum  G rav  16 1 7 2 7  ( 1 9 9 9 )  ; R V S a r a y k a r  a n d  S  H G h a l e  C lass Q uan tum  G rav  16 2 8 1 - 2 8 9  ( 1 9 9 9 )  ; S  H G h a te ,  R V  S a r a y k a r  a n d  K D  P a u l  Pramana  53 2 5 3  ( 1 9 9 9 )  , S  S  r ^ e s h in g k a r  a n d  P  S  J o s h i .  g r -q c / (X )1 0 0 1 5  ( 2 0 0 0 )  , S S  D e .s h m g k a r , I H D w iv e d i  a n d  P  S  J o s h i  Phy Rev D59 0 4 4 0 1 8( 1 9 9 9 )  ; F  C  M e n a .  R  T a v a k o l ,  P  S  J o s h i  Phy Rev D62 0 4 4 0 0 1( 2000 )f 2 J  W  A  H is c o k ,  L G  W i l l i a m s  a n d  D  M  E r a d le y  Phys. Rev D26 7 5 1  ( 1 9 8 2 )  . Y  K u r o d a  Prof> Theor. Phys. 72 6 3  ( 1 9 8 4 )  ; K  R a ja g o p a l  a n d  K  L a k e  P hy  Rev. D35 1 5 3 1  ( 1 9 8 7 )  ; I H  D w i v e d i  a n d  P  S  J o s h i  Class Quantum Grav. 6  1 5 9 9  ( 1 9 8 9 )  ; P  S J o s h i  a n d  I H  D w iv e d i  Gen Rel. Grav. 24 1 2 9  ( 1 9 9 2 )  ; J L e m o s  Phys Rev Lett. 6 8  1 4 4 7  ( 1 9 9 2 )  ; K  D  P a u l ,  R  V  S a r a y k a r  a n d  S  H  G h a te ,  P ra im m i 52 5 5 3  ( 1 9 9 9 )  ; S G  G h o s h  a n d  R V  S a r a y k a r  Phys Rev. D62 1 0 7 5 0 2  ( 2 0 0 0 )[ 3 ]  A  O r i  a n d  T  P ir a n  Phys. Rev D42 1 0 6 8  ( 1 9 9 0 )  ; P  S  J o s h i  a n d  1 H  D w iv e d i  Comm. Math. Phys 146 3 3 3  ( 1 9 9 2 )  . T  H a r a d a  Phys Rev. D58 1 0 4 0 1 5  ( 1 9 9 8 )  . F  I C o o p e r s to c k ,  S  J h in g a n .  P  S  J o s h i  a n d  T  P  S in g h  Class Quantum Grav. 14 2 1 9 5  ( 1 9 9 7 )[ 4 ]  P  S z e k e r e s  a n d  V  I y e r  Phys. Rev. D47 4 3 6 2  ( 1 9 9 3 )  , S  B a r v e ,  T  P  S in g h ,  a n d  L  W i t le n  Gen. Rel. G rav  Vol 32 N o . 4  6 9 7  ( 2 0 0 0 )[ 5 ]  J  F  V  R o c h a  a n d  A  W a n g ,  g r - q c / 9 9 1 0 1 0 9  ( 1 9 9 9 )350 Kishor D Patil[61 SEP Bergliaffa Phys U t t  A15, 531 (2(XX)) and reference therein[7] S G Ghosh and R V Saraykar Phys. Rev D62 107502 ( 2000)|H| A Sil and S Chatterjec Gen Ret Grav 26 Vol 999 (1994)[9J K D Parjl, S H Ghatc and R V Suraykar Pratnana J Ph\s. 56 503 ( 2001)[101 K D Patil, S H Ghate and R V Soraykor In d ia n  J  Pure Anni 33 p379 (2002)[11] R P A C Newman Ciass Quanium Grav. 3 527 (1986)[12] PS Joshi and I H Dwivedi Phys. Rev. D47 5357 (1993)[13] C J S Clarke and A Krolok J. Geo. Phys 2 127 (1986)[14] F J Tipler Phys. Lett. A64 8 (1977)

The Final Fate of Spherical Inhomogeneous Dust Collapse in Higher Dimensional Spacetime

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The Final Fate of Spherical Inhomogeneous Dust Collapse in Higher Dimensional Spacetime

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