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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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