On the question of singularity in higher-dimensional space–time A. Banerjee, D. Panigrahi, and S. Chatterjee Citation: Journal of Mathematical Physics 36, 331 (1995); View online: https://doi.org/10.1063/1.531307 View Table of Contents: http://aip.scitation.org/toc/jmp/36/1 Published by the American Institute of Physics On the question space-time of singularity in higher-dimensional A. Banerjee, D. Panigrahi, and S. Chatterjeea) Relativity and CosmologyResearchCentre, Department of Physics, Jadavpur University Calcutta 700032, India (Received 8 June 1994; accepted for publication 20 September 1994) The problem of singularity is discussed in a (d+4)-dimensional space-time with 00. Some important relations are obtained, which may determine the conditions of occurrence of singularities in this more general situation. 0 1995 Americun Institute of Physics. I. INTRODUCTION The possibility that space-time has more than four dimension has recently received much attention in its attempts to unify gravity with other gauge types of interactions. The observational evidence that we apparently live in a four-dimensional world is sought to be explained by the fact that solutions of Einstein’s equations in higher-dimensional space-time exist where as the fourdimensional (4-D) space expands the extra dimensions shrink with time (for more details see Chatterjee’). In most cases in this (d-t-4)-dimensional gravity theory attention has been primarily focused so far on dealing with exact solutions in the domain of cosmology,2 compact objects,3 and gravitational waves4 with the pious hope that some of its observational consequences, in principle at least, may provide us with “windows” to the primordial relics of the higher-dimensional phase. While the issues concerning the singularity structure of many of these solutions have been discussed, no attempt has been made so far to discuss the question of singularity from a general framework. In 4-D space-time the relations like the Raychudhuri equation5 and also other focusing theorems6 exist which prove the inevitability of the space-time singularity if one starts with a regular energy condition. In view of the above discussions we have thought it fit, if not imperative, to seek a deeper basis of the question of singularity in multidimensional space-time and to look for singularity theorems in these space-times. II. RAYCHAUDHURI EQUATION IN MULTIDIMENSIONAL COSMOLOGY It is worthwhile to generalize the well-known 4-D Raychaudhuri equation in the KaluzaKlien-type space-time with more than four dimensions and reexamine the question of singularity. For any vector uP the definition of the Riemann Christoffel curvature tensor enables one to write (~~;~u~);~-u~;~~~;~=R~~~~u~. (2.1) (Here p runs from zero to d+3.) The shear tensor, the vorticity tensor, and the expansion scalar are now defined for the velocity vector vP as usual with the necessary modifications demanded by the addition of extradimensions in the following manner: ~pv=qI*;v)-~(pb) -W(3+41vQ,,h,,, ~pY=q/L;P]-~[pvY] and $=v?+. (2.2) ‘)Permanent address: Department of Physics, New Alipore College, Calcutta 700053, India. J. Math. Phys. 36 8 1995 American Institute of Physics 331 332 Banerjee, Panigrahi, and Chatterjee: Question of singularity in higher-D space-time In the expression given in Eq. (2.2) “d” stands for the number of extra dimensions. With these definitions and following the standard procedure as in four dimensions we obtain the Raychaudhuri equation5 in higher dimensional space in the following form: (2.3) If the vorticity vanishes and the motion is geodesic, Eq. (2.3) reduces to B,,v”= -2u2-[ 1/(3+d)]82+RV,uVvn. (2.4) Thus 6’in this restricted situation will monotonically decrease along the timelike geodesic if Rvav “v “SO. In view of Einstein’s field equation the last term on the right-hand side of Eq. (2.4) may be replaced by -87rG[T,-(1/(2+d))T], where G is here the (4+d)-dimensional gravitational constant. One has little idea of the behavior of matter under extreme conditions of density and pressure particularly in the presence of extra dimensions. It may be questionable whether Hawking’s energy conditions proposed as physically reasonable conditions in the usual 4-D space will still remain valid in higher-dimensional space-time. In fact Eq. (2.4) when expressed in terms of matter density and pressures finally gives the following relation for the dynamics of the universe: 1 (3+d) e= -22- 8TG e2- (2+d) (2.5) ((1 +d)p+3p+dp,l. The time behavior of the expansion scalar 8 is therefore somewhat uncertain because of the absence of exact knowledge regarding the behavior of the pressure p t corresponding to the extra dimensions. Here, of course, it is assumed for simplicity that the three-dimensional pressure p and the extra dimensional pressure p r do not show any anisotropy separately in them. The conclusion, however, is valid for a more general situation as well. In the next step one can proceed to verify the validity of Eq. (2.5) for a typical five-dimensional inhomogeneous cosmological model where the three-space is maximally symmetric and is of zero spatial curvature, whereas the extra fifth dimension has the corresponding metric component depending on space as well as time coordinates.7 The equations of state p=3p and p # p, are satisfied and the following metric holds: ds2=dt2-eX(dr2+r2 d02+r2 sin2 6’d+2)-eP dy2, (2.6) where A=A(t) and CL=& r,t). The energy-momentum tensor components in comoving coordinates are c=p, TI = T$= T:= -p, e= -p, and the appropriate solutions of Einstein’s field equations in this case are ,A= p/4 p p=5= e~=[~c-br2)t3/8- (26/85)b+(9/32)(c-br2)t-5’4 (c-br2)t3’4-(192/85)bt2 $&‘3/8]2, (2.7) ’ p,=(9/32)P. It is now a straightforward task to check that the solutions given above are consistent with the Raychaudhuri-type equation (2.5) valid for higher-dimensional space-time. This is done in the following way. In our case, d = 1, the expansion scalar 8= ($A + f/;) and the shear is given by 2a2= &(b ->;)2. Using the solutions given in Eq. (2.7) it is not difficult to verify that each side of Eq. (2.5) is given by --t 3 -*_ {(9/8)(c-br2)t-“2 8 -(372/85)b(c-br2)t3’4+(87552/7225)b2t2} {(c-br2)t3’4-(192/85)bt2}2 J. Math. Phys., Vol. 36, No. 1, January 1995 Banerjee, Panigrahi, and Chatterjee: Question of singularity in higher-l) space-time 333 III. THE NULL CONVERGENCE CONDITION AND THE FOCUSING THEOREM The null convergence condition may be stated from the Raychaudhuri equation, which is valid for null geodesics. The derivation is similar with no wide divergence from what has been done above. Here for a bundle of null geodesics the propagation vector of the wave front characterized by an equation of the type f(xa) =const is given by K,= (Jfl%~) a [here the suffix LYruns from 0 to (3+d)]. For a null geodesic with the propagation vector being orthogonal to the constant phase hypersurface, KaLpKp=O and K,.p= Kp;, . So that the vorticity tensor weP corresponding to this null congruence is zero. 5*6For the (4+d)-dimensional space-time the shear tensor of null geodesics is defined as a,p=K~,;p)-[1/(2+d)lK~;~~,p. (3.1) Here hap stands for the projection operator onto the set of elementary (2+d) spaces orthogonal to the velocity vector u, and also to the null vector K,, so that’ 2K( au /3, + h,p=gap- ~,v” Kc&p W,U~“)* ’ (3.2) Here also oaP is symmetric and tracefree and satisfies the relations aapKa= u,~u~= 0 yielding 2a2=a,p~ap=K,;pKa;B-[ 1/(2+d)](K”,,)*. (3.3) Following the procedure adopted in the case of timelike geodesics the higher-dimensional analog of the Raychaudhuri equation for null geodesic congruence can be written explicitly in the form In the field equations written so far the gravitational constant G is actually the product of the four-dimensional gravitational constant and V-the volume of the extra space. In this case 0 will monotonically decrease along the null geodesic if R,JCpK’GO for any null vector Kp, that is, it implies that matter has a converging (or more strictly nondiverging) effect on the congruences of null geodesics. IV. THE FOCUSING THEOREM As stated earlier the Raychaudhuri equation plays a crucial role in understanding the question of singularity in general relativity. In what follows we derive the so-called focusing theorem as applied to our higher-dimensional space-time. The focusing of the bundle of null geodesics may also be looked at as a change in the cross-sectional area “S” of the bundle around a central ray viewed by the observer proceeding along the ray. In the context of (4+d)-dimensional space-time one should define “area” with some caution. In the 4-D space-time the three-dimensional cone casts a two-dimensional shadow in its path. At this stage we argue that the (3+d)-dimensional null geodesic cone gives rise to a (2+d)-dimensional shadow. If A be an affine parameter chosen along the null trajectory one can write dldX = Ka(dldxa) so that from the general hydrodynamical conservation law we get Ka-$ = (Ka;,)S. J. Math. Phys., Vol. 36, No. 1, January 1995 (4.1) 334 Banerjee, Panigrahi, and Chatterjee: Question of singularity in higher-D space-time Elementary calculations show that in this case one can use Eq. (4.1) to obtain the following relation: d*s”(*+d) dX* Sll(2fd) =- Of4 1 WdpKP+ c2+dj (K”;,>* 1 . (4.2) The relation (3.4) when applied to Eq. (4.2) finally yields for a pencil of light rays in a gravitational field d2s’/(*+d) s-“(2fd) dX* =&[-2 CT*- ~ITGT,~K”K~]. (4.3) The above equation is already known6 in four dimensions where d=O. This equation shows that for a regular matter field the right-hand side is negative, so that the cross-sectional area for the pencil of rays once contracting, would continue to accelerate the process of contraction and in a finite value of the affine parameter would focus the congruence of rays to points which are, in fact, the nodes of the null geodesics. As in Eq. (3.4) the shear term helps focusing and the second term on the right-hand side, in fact, represents the attraction due to the matter field. In conclusion the question, however, remains-whether the relation TaBKnK p>O still remains valid even in the presence of extra dimensions, where the stress components corresponding to the extra dimensions may behave differently from the usual three-dimensional quantities? Lastly a few comments on the applicability of general singularity theorems in this context may be in order before conclusion. While the focusing theorems considered above are sufficient to prove the existence of singularities in high symmetry cases, there are a series of causal properties that are required to establish the complete theorems. The holes left behind by the removal of singularities should be detectable by the fact that there will be geodesics which have finite afline length or in other words incomplete geodesics. We could thus define a space-time to be singular if it possesses at least one incomplete geodesic. According to the singularity theorem of Hawking and Penrose (see Ref. 6) the space-time must contain at least one incomplete timelike or null geodesic provided it satisfies certain physically reasonable conditions which in turn point towards the existence of singularity in our Universe. It is quite plausible that the analogous results will hold in higher dimensions too-indeed many of the standard proofs may go over almost unchanged. The detailed proof is, however, quite involved and is not attempted here. ACKNOWLEDGMENT Financial support from the DST, India is acknowledged. ‘P. Cl. 0. Freund, Nucl. Phys. B 209, 146 (1982); M. J. Duff, B. E. W. Nilsson, and C. N. Pope, Phys. Rep. 130, 1 (1986); A. Salam and E. Sezgin, Supergravities in Diverse Dimensions (World Scientific, Singapore, 1989), Vol. 2, p. 1251. ‘A. Beloborodov, M. Demianski, P. Ivanov, and A. Cl. Polnarev, Phys. Rev. D 48, 503 (1993); S. Chatterjee and A. Banerjee, Class. Quantum Gravit. 10, Ll (1993). 3H. Liu and P S. Wesson, J. Math. Phys. 33.3888 (1992) (and references therein); D. J. Gross and M. J. Perry, Nucl Phys. B 226, 29 (1983); S. Chatterjee, Astron. Astrophys. 179, 1 (1987). 4L. M. Sokolowski, Class. Quantum Gravit. 6, L257 (1989). ‘A. K. Raychaudhuri, S. Banerjee, and A. Banerjee, General Relativity, Astmphysics, and Cosmology (Springer-Verlag, Berlin, 1992). p. 228. ‘S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time (Cambridge University, Cambridge, 1973). ‘A. Banerjee, D. Pan&&i, and S. Chatterjee, Class. Quantum Gravit. 11, 1405 (1994). *R. K. Sachs, Proc. R. Sot. A 264, 309 (1961). J. Math. Phys., Vol. 36, No. 1, January 1995

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