Research Article Generalized Robertson-Walker Space-Time ...downloads.hindawi.com/archive/2016/9312525.pdf · Proof. Let x be a vector tangent to a spacelike hypersurface S in .e

Research ArticleGeneralized Robertson-Walker Space-Time Admitting EvolvingNull Horizons Related to a Black Hole Event Horizon

K L Duggal

Department of Mathematics and Statistics University of Windsor Windsor ON Canada N9B 3P4

Correspondence should be addressed to K L Duggal yq8uwindsorca

Received 20 June 2016 Accepted 4 August 2016

Academic Editor Elias C Vagenas

Copyright copy 2016 K L Duggal This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A new technique is used to study a family of time-dependent null horizons called ldquoEvolving Null Horizonsrdquo (ENHs) of generalizedRobertson-Walker (GRW) space-time (119872 119892) such that the metric 119892 satisfies a kinematic conditionThis work is different from ourearly papers on the same issue where we used (1 + 119899)-splitting space-time but only some special subcases of GRW space-time havethis formalism Also in contrast to previous work we have proved that eachmember of ENHs is totally umbilical in (119872 119892) Finallywe show that there exists an ENH which is always a null horizon evolving into a black hole event horizon and suggest some openproblems

1 Introduction

Let (119872 119892) be a (1 + 119899)-dimensional space-time manifoldwhose metric 119892 satisfies the following kinematic condition

nabla119883119880 = 120582 (119883 + 119892 (119883119880)119880) (1)

where 119880 is a timelike unit vector nabla denotes the Levi-Civitaconnection on 119872 120582 is a function and 119883 is an arbitraryvector field on 119872 Consider Gaussian normal coordinates(1199090 = 119905 119909119886) on119872 such that 119880 = 120597120597119905 and 119892 = 119892

119894119895119889119909119894119889119909119895 =

minus1198891199052 + 119892119886119887119889119909119886119889119909119887 where 119894 119895 run over 0 1 119899 and 119886 119887 run

over 1 119899 and 119892119886119887

are functions of all the coordinates 119909119886Let Σ(119905 = a constant) be a spacelike slice orthogonal to119880 and119883 = 120597120597119909

119886 a coordinate vector field tangent to Σ Then (1)reduces tonabla

120597120597119909119886120597120597119905 = 120582(120597120597119909119886)which further implies (with

some computation) that (120597120597119905)119892119886119887= 2120582119892

119886119887 Integrating this

we obtain 119892119886119887= 1198902 int 120582(119905119909

119886

)119889119905120574119886119887 where 120574 is a fixed Riemannian

metric on the initial slice (119905 = 1199050a constant) In this paper we

take 120582 as a function of 119905 only and set 119890int120582(119905)119889119905 = Ω(119905) that is120582 =

ΩΩ where an overdot means derivative with respect to

119905 Then the metric 119892 is of the form

119892 = 119892119894119895119889119909119894119889119909119895= minus119889119905

2+ (Ω)

2

(119905) 120574119886119887119889119909119886119889119909119887 (2)

which is generalized Robertson-Walker (GRW) space-timenotion introduced by Alıas et al [1] It is a warped product

with base of an open interval of a real line with a negativelydefined metric and fibre of a Riemannian manifold andnot of constant sectional curvature in general The readerwill see that the kinematic condition (1) on the space-timemetric plays an important role in the induced geometryof null hypersurfaces of 119872 This is our reason to generateGRW space-time instead of stating it as a definition Thisclass of space-time is inhomogeneous admitting an isotropicradiation whose subcase is classical Robertson-Walker (RW)space-time which includes Einstein-de-Sitter space-timeFriedman cosmologicalmodels and the static Einstein space-time Considerable work is available on GRW space-timeprimarily on spacelike hypersurfaces [1ndash4] and more citedtherein and only recently there has been interest (see Kang[5 6] andNavarro et al [7]) on its null submanifold geometryHowever nothingmuch is available on physical application oftheir null hypersurfaces

It is well-known that the null hypersurfaces have beenused as physical models of black hole horizons in generalrelativity In this paper we show that there exists a physi-cal model of a family of time-dependent null horizons ofGRW space-time which may evolve into a black hole eventhorizon See Sections 3 and 4 for a brief account on eventhorizons recent works on time-dependent null horizons ofDuggal [8ndash10] and Sultana and Dyer [11 12] and more citedtherein We highlight that the physical use of research ontime-dependent null horizons may have connection (though

Hindawi Publishing CorporationInternational Scholarly Research NoticesVolume 2016 Article ID 9312525 6 pageshttpdxdoiorg10115520169312525

2 International Scholarly Research Notices

it is too early to be sure) with the latest LIGO [Laser Inter-ferometer Gravitational-Waves Observatory] experiment (asper Press Release of February 11 2016) which confirms thepresence of gravitational waves produced during the finalfraction of a second of the merger of two black holes into asingle massive spinning black hole Thus this recent LIGOexperiment reconfirms that black hole has a cosmologicalbackground or it is surrounded by a local matter distributionTherefore there is significant difference in the structure andproperties of the surrounding dynamical region of a blackhole and this latest experiment strengthens the ongoingresearch on time-dependent null horizons in particular nearsuch spinning black holes surrounded with gravitationalwaves For information on LIGO via video one may tryhttpmediaassetscaltechedugwave

2 Family of Null Hypersurfaces

Consider GRW space-time (119872 119892) with its metric 119892 given by(2) Let 119901 be a point in119872 and ((119872 119892) 1199091 = a constant) be atimelike hypersurface of119872 passing through 119901 Observe thatthere is a freedom of choice in taking any one of the spacelikecoordinates constant Suppose that n is the future directedtimelike unit vector on119872 induced by 119880 of119872 such that thefollowing holds

nabla119883n = 120582 (119905) (119883 + 119892 (119883n)n) forall119883 isin 119879119872 (3)

where 119892 nabla and 120582(119905) are the induced metric Levi-Civitaconnection and a function on 119872 respectively Let ((S 119902) 1199050= a constant) be a connected spacelike hypersurface in119872

and hence a codimension two-spacelike submanifold of 119872passing through 119901 via the normal exponential map along Sin119872 with metric 119902 induced from 119892 Using (3) and followinga procedure explained in previous section one can show that119872 is also GRW space-time with metric 119892 expressed as

119892 = minus1198891199052+ Ω2(119905) 120574119860119861119889119909

119860119889119909119861

2 le 119860 119861 le 119899 120582 =Ω

Ω

(4)

wherewe take120582 = ΩΩ and 120574119860119861

is a fixedRiemannianmetricAssume that119909 = (119909

2 119909

119899) are the coordinates in (S

119906 119905 = 119906

a constant) centered on 119901 so that the metric 119902119906on S119906is given

by

119902119906= Ω2

119906120574119860119861119889119909119860119889119909119861 2 le 119860 119861 le 119899 (5)

Choose at each point of S119906a null direction perpendicular in

S119906and smoothly depending on the foot-point There are two

such possibilities the ingoing and the outgoing null direc-tionsThen it is easy to see that the union of all geodesics withthe chosen (say outgoing) direction ℓ

119906is a null hypersurface

of 119872 Denote by (119867119906 ℎ119906 ℓ119906) such a null hypersurface of

119872 with degenerate metric ℎ119906and null normal ℓ

119906 Since 119867

119906

includes S119906and its degenerate metric ℎ

119906must be of signature

(0 + +) we assume that the Riemannian metric 119902119906of S119906

coincides with the degenerate metric ℎ119906of119867119906given by

ℎ119906= (ℎ119906)119860119861119889119909119860119889119909119861= Ω2

119867119906

120574119860119861119889119909119860119889119909119861

2 le 119860 119861 le 119899

(6)

where Ω119867119906

is a function on 119867119906induced by the function

Ω119906of 119872 This means that the null hypersurface (119867

119906 ℎ119906)

is conformal to a fixed null hypersurface (119867 120574) that isℎ119906= Ω2119867119906

120574 with the conformal functionΩ119867119906

In this paper we take ℓ

119906in some subset of119872 around119867

119906

This will permit us to well define the space-time covariantderivative nablaℓ

119906 Also in this way we get a foliation of119872 (in

the vicinity of119867119906) by a family (119867

119906) so that ℓ

119906is in the part of

119872 foliated by this family (for details see Carter [13]) such thatat each point in this region ℓ

119906is a null normal to119867

119906for some

value of the parameter 119906 Let (ℎ119906) and (ℓ

119906) be the families of

degenerate metrics and null normal respectively Denote by

119865 = ((119867119906) (ℎ119906) (ℓ119906)) ℎ119906= Ω2

119867119906

120574 isin (ℎ119906) (7)

a family of null hypersurfaces of (119872 119892) conformally related toa fixed null hypersurface (119867 120574 ℓ) and eachΩ

119867119906

is a functionon119867119906for some value of 119906 Also let (S

119906) be the corresponding

family of submanifolds of119872The bending of each 119867

119906of 119865 in 119872 is described by the

Weingarten map

Wℓ119906

119879119901119867119906997888rarr 119879119901119867119906

119883119906997888rarr nabla

119883119906

ℓ119906

(8)

Wℓ119906

associates with each 119883119906of 119867119906the variation of ℓ

119906along

119883119906 with respect to the space-time connection nabla The second

fundamental form say 119861119906 of 119867

119906is the symmetric bilinear

form related to the Weingarten map by

119861119906(119883119906 119884119906) = ℎ119906(Wℓ119906

119883119906 119884119906) = ℎ119906(nabla119883119906

ℓ119906 119884)

119883119906 119884119906isin 119879119867119906

(9)

119861(119883119906 ℓ119906) = 0 for any119883

119906isin 119879119867119906implies that 119861

119906has the same

ℓ119906degeneracy as that of the metric ℎ

119906and it does not depend

on particular choice of ℓ119906

Now we deal with the question of obtaining a normalizedexpression for a null normal ℓ of a null hypersurface 119867 isin

119865 For this purpose we let s isin 119879119872 be a unit spacelikenormal vector to S defined in some open neighborhood ofcorresponding 119867 Since the future directed timelike vectorsn of119872 and s are orthogonal it is immediate that the vectorn + s is null so it is tangent to 119867 Therefore one can takeℓ = n + s as a natural normalization of the null normal to119867 However this normalization can only be defined for asingle fixed hypersurface 119867 whereas we need to normalizethe entire family (ℓ

119906) for any parameter value of 119906 For this

purpose we choose the following normalization of each ℓ119906isin

(ℓ119906)

ℓ119906= (n119906+ s119906)

where s119906sdot s119906= 1 x

119906isin 119879119901S119906lArrrArr s

119906sdot x119906= 0

(10)

International Scholarly Research Notices 3

which implies that each ℓ119906is tangent to each member of

119865 and it has the property of Lie dragging the family ofsubmanifolds (S

119906) Then we define a transversal vector field

k119906to 119879119901119867119906not belonging to 119865 expressed as another suitable

linear combination of n119906and s119906such that it represents the

ingoing direction satisfying

119892 (ℓ119906 k119906) = minus1 k

119906=1

2(n119906minus s119906) (11)

Now we are ready to state and prove the following maintheorem

Theorem 1 Let 119865 = ((119867119906) (ℎ119906) (ℓ119906)) be a family of null

hypersurfaces of the GRW space-time (119872 119892) defined by (2)such that 119865 is given by (7) and each of its members (119867

119906 ℎ119906 ℓ119906)

is conformal to a fixed null hypersurface (119867 120574 ℓ) where ℎ119906=

Ω2119867119906

120574 for every parameter value of 119906 Assume that (119867 120574 ℓ) liesto the future of each (119867

119906 ℎ119906) Then

(a) each (119867119906 ℎ119906 ℓ119906) is totally umbilical in (119872 119892) with its

volume expansion 120579(ℓ119906)= (119899 minus 1)120582

119906where the function

120582119906is as given in (3)

(b) each ℓ119906is conformal Killing vector (m

ℓ119906

ℎ119906= 2120582119906ℎ119906) on

each119867119906

(c) the family 119865 may evolve into a totally geodesic nullhypersurface (119867 120574 ℓ) only if 120582

119906rarr 0 on 119867 and then

120579(ℓ)= 0 on119867

Proof Let x119906be a vector tangent to a spacelike hypersurface

S119906in119872 TheWeingarten formula is given by nablax

119906

n119906= 119860n

119906

x119906

where we denote by 119860n119906

the shape operator of S119906 It follows

from (3) that 119860n119906

x119906= 120582119906x119906 Denote by 119861S

119906

the secondfundamental form of S

119906in 119872 with respect to n

119906 Then for

any x119906 y119906isin 119879119878119906

119861S119906

(x119906 y119906) = 119902119906(nablax119906

n119906 y119906) = 119902119906(119860n

119906

x119906 y119906)

= 120582119906119902119906(x119906 y119906)

(12)

Thus (S119906 119902119906) is totally umbilical in 119872 Then it is easy to

show that any (119867119906 ℎ119906 ℓ119906) is also totally umbilical in 119872

Indeed take 119861119906as the second fundamental form of 119867

119906 As

119861119906(ℓ119906 119883119906) = 0 forall119883

119906isin 119879119867119906 and corresponding S

119906sub 119867119906we

conclude that for every119883119906 119884119906isin 119879119867119906

119861119906(119883119906 119884119906) = 119861119906(x119906 y119906) = 119861S

119906

(x119906 y119906)

forallx119906 y119906isin 119879119878119906(13)

Since the Riemannian metric 119902119906coincides with the corre-

sponding degenerate metric ℎ119906it follows from above relation

and (12) that

119861119906(119883119906 119884119906) = 120582119906ℎ119906(119883 119884) forall119883

119906 119884119906isin 119879119867119906 (14)

which proves that each 119867119906is totally umbilical in 119872 More-

over it follows from (3) that the volume expansion 120579(ℓ119906)=

div(n119906) = (119899 minus 1)120582

119906 where we have used 119861

119906(ℓ119906 119883119906) =

0 forall119883119906isin 119879119867119906 which proves (a) Using the expression

mℓ119906

ℎ119906 (119883 119884) = ℎ119906 (nabla119883ℓ119906 119884) + ℎ119906 (nabla119884ℓ119906 119883) (15)

and 119861119906(119883119906 119884119906) symmetric in the above equation we obtain

119861119906(119883119906 119884119906) =

1

2mℓℎ119906(119883119906 119884119906) forall119883

119906 119884119906isin 119879119867119906 (16)

Since 119867119906

is totally umbilical we use 119861119906(119883119906 119884119906) =

120582119906ℎ119906(119883119906 119884119906) in above relation to get m

ℓ119906

ℎ119906= 2120582119906ℎ119906on 119867119906

Hence each ℓ119906is conformal Killing vector of the metric ℎ

119906

with conformal function 2120582119906which proves (b) For item (c)

since the fixed hypersurface (119872 120574 ℓ) is in the future of itsconformally related family 119865 of hypersurfaces (119867

119906 ℎ119906) rarr

(119867 120574) only if the metric ℎ119906rarr 120574 for that value of 119906 which

further means that Ω119906rarr 1 for that value of 119906 We know

that 120582119906= Ω119906Ω119906 Thus 120582

119906rarr 0 on 119867 for that value of

119906 which implies that 120579(ℓ)

= 0 on 119867 Now we know that anull hypersurface of a semi-Riemannian manifold has zeroexpansion if and only if it is totally geodesic so (c) holdswhichcompletes the proof

Now we quote the following general result for a totallyumbilical submanifold (also holds for totally geodesic case) ofa semi-Riemannian manifold which is needed to address thequestion of howTheorem 1 can be used to show the existenceof null horizons of GRW space-time

Proposition 2 (Perlick [14]) Let119867 be a totally umbilical sub-manifold of a semi-Riemannian manifold119872 Then (a) a nullgeodesic vector field of119872 that starts tangential to 119867 remainswithin 119867 (for some parameter interval around the startingpoint) and (b)119867 is totally geodesic if and only if every geodesicvector field of119872 that starts tangential to119867 remains within119867(for some parameter interval around the starting point)

Above result satisfies a requirement for the existence ofa null horizon in relativity Since we assume that each nullnormal ℓ

119906of the family of hypersurfaces 119865 is null geodesic

using above result of Perlick we state the following corollaryof Theorem 1 (proof is easy)

Corollary 3 Let (119872 119892) be null geodesically complete space-time obeying the null energy condition Ric(119883119883) ge 0 for allnull vectors 119883 such that the hypothesis of Theorem 1 holdsThen each null geodesic vector ℓ

119906of 119865 is contained in its

respective smooth totally umbilical null hypersurface (119867119906 ℓ119906) of

(119872 119892) In particular this property will also hold for the totallygeodesic null hypersurface (119867 120574) of (119872 119892)

3 Physical Interpretation

In this section we present a physical interpretation ofTheorem 1 For this purpose recall that the vorticity-freeRaychaudhuri equation for any member (119867 ℎ ℓ) of thefamily 119865 is given by

119889 (120579(ℓ))

119889119904= minus119877119894119895ℓ119894ℓ119895minus 120590119894119895120590119894119895minus

1205792

119899 minus 1 (17)

where 120590119894119895= nablalarr(119894(ℓ)119895)minus (12)120579

(ℓ)ℎ119894119895is the shear tensor 119904 is a

pseudoarc parameter such that ℓ is null geodesic and 119877119894119895is


the Ricci tensor of119872 We also recall that in two recent papers[8 9] we studied a new class of null horizons of space-timeusing the following definition

Definition 4 A null hypersurface (119867 ℎ ℓ) of the family 119865 ofspace-time (119872 119892) is called an Evolving Null Horizon brieflydenoted by ENH if

(i) 119867 is totally umbilical in119872 and may include a totallygeodesic portion

(ii) all equations of motion hold at 119867 and energy tensor119879119894119895is such that 119879119886

119887ℓ119887 is future-causal for any future

directed null normal ℓ

Comparing this definition withTheorem 1 we notice thatthe first part of its condition (i) is the same as the first partof conclusion (a) of Theorem 1 and energy condition (ii)requires that 119877

119894119895ℓ119894ℓ119895 is nonnegative for any ℓ which implies

(see Hawking and Ellis [15 page 95]) that 120579(ℓ)

monotonicallydecreases in time (whichwe take as119906 parameter for the family119865) along each ℓ that is119872 obeys the null convergence condi-tion Thus in the region where 120579

(ℓ)is nonzero each member

of the family 119865 will be totally umbilical with 119906-parametertime-dependent metric ℎ and may include a totally geodesicportion where 120579

(ℓ)vanishes for a fixed (119867 120574) so condition

(c) will hold It follows from the Raychaudhuri equationthat each ℓ will have nonzero or zero shear accordingly asits expansion function 120579

(ℓ)is nonzero or zero respectively

Moreover it is easy to see that condition (b) of Theorem 1will hold for an ENH This clearly shows that there exists aldquoPhysical Modelrdquo of a class 119865 = ((119867

119906) (ℎ119906) (ℓ119906)) of a family of

totally umbilical null hypersurfaces of the space-time (119872 119892)satisfying the hypothesis and three conclusions ofTheorem 1such that each of its members is an Evolving Null Horizon(ENH) which may evolve into a fixed null hypersurface(119867 120574) with zero expansion 120579

(ℓ)= 0 For up-to-date

information with some examples of ENHs we refer to Duggal[8ndash10 16]

However Theorem 1 is limited by the fact that notevery such totally umbilical null hypersurface evolving intoa totally geodesic null hypersurface can be related to a blackhole horizon Therefore we must show that there exists aprescribed family of ENHs of GRW space-time which isrelated to a black hole horizon To complete our analysis on aphysical interpretation we need the following information ona class of black hole horizons

Event Horizons A boundary of space-time is called a nullevent horizon (briefly denoted by EH) beyond which eventscannot affect the observer An EH is intrinsically a globalconcept as its definition requires the knowledge of the entirespace-time to determine whether null geodesics can reachnull infinity In particular an event horizon is called aKilling horizon if it is represented by a null hypersurfacewhich admits a Killing vector field Most important familyis the Kerr-Newman black holes EHs have played a keyrole and this includes Hawkingrsquos area increasing theoremblack hole thermodynamics black hole perturbation theoryand the topological censorship results Moreover an EH

always exists in black hole asymptotically flat space-timeunder a weak cosmic censorship condition and is representedby a Killing horizon such that the space-time is analyticand the stress tensor obeys the weak energy condition Thenull hypersurface of such space-time admits a nonvanishingKilling vector field say ℓ which may or may not be theKilling vector field of the landing space-time The latter casecorresponds to a rotating asymptotically flat black hole whichwe do not discuss here We refer to Hawkingrsquos paper onldquoevent horizonsrdquo [17] three papers of Hajicekrsquos work [18ndash20] and more cited therein Based on this brief accountwe now recall the following work of Galloway [21] whohas shown that the null hypersurfaces which arise mostnaturally in general relativity such as black hole EHs arein general 1198620 but not 1198621 His approach has its roots inthe well-known geometric maximum principle of E Hopfa powerful analytic tool which is often used in the theoryof minimal or constant mean curvature hypersurfaces Thisprinciple implies that two different minimal hypersurfaces ina Riemannian manifold cannot touch each other from oneside In year 2000 Galloway [21] proved the following resultfor smooth null hypersurfaces restricted to the zero meancurvature case and suitable for asymptotically flat space-time

Theorem 5 Let1198671and119867

2be smooth null hypersurfaces in a

space-time manifold 119872 Suppose that (1) 1198671and 119867

2meet at

119901 isin 119872 and 1198672lies to the future side of 119867

1near 119901 and (2)

the null mean curvatures 1205791of 1198671and 120579

2of 1198672satisfy 120579

2le

0 le 1205791 Then119867

1and119867

2coincide near 119901 and this common null

hypersurface has mean curvature 120579 = 0

Although the above maximum principle theorem is forsmooth null hypersurfaces in reality null hypersurfaces asmodels of black hole event horizons are the null portions ofachronal boundaries as the sets = 120597119868

plusmn(119860) 119860 sub 119872 whichare always 1198620 hypersurfaces and contain nondifferentiablepoints These null geodesics (entirely contained in 1198620 nullhypersurface 119867) are the null geodesic generators of 119867 Forthis reason Galloway also proved his above result for 1198620 nullhypersurface and the following physically meaningful nullsplitting theorem

Theorem6 (see [21]) Let (119872 119892) be null geodesically completespace-time which obeys the null energy condition Ric(119883119883) ge0 for all null vectors 119883 If 119872 admits a null line 120578 then 120578is contained in a smooth closed achronal totally geodesic nullhypersurface

Examples (see [21]) are Minkowski space de-Sitter andanti-de-Sitter spaces Now we state the following theorem asphysical interpretation of Theorem 1

Theorem 7 Let (119872 119892) be null geodesically complete GRWspace-time which obeys the null energy condition Ric(119883119883) ge0 for all null vectors 119883 Suppose F = ((119867

119906) (ℎ119906) (ℓ119906)) is a

family of null hypersurfaces of (119872 119892) defined by (2) and givenby (7) where each of its members (119867

119906 ℎ119906 ℓ119906) is conformally

related to a totally geodesic hypersurface (119867 120574 ℓ) such that the


conformal factor of ℎ119906= Ω2119867119906

120574 for every parameter value of 119906Then conditions (a) and (b) of Theorem 1 will hold Moreoverlet (119867 120574 ℓ) be in the future of each (119867

119906 ℎ119906 ℓ119906) and it admits

a Killing horizon Then (c) the family F evolves into a blackhole event horizon represented by (119867 120574 ℓ) only if 120582

119906rarr 0 at

null infinity

Theproofs of items (a) and (b) follow exactly as the proofsof items (a) and (b) in Theorem 1 For the proof of item (c)we first notice that119867 admits a Killing horizon Secondly theKilling horizon 119867 satisfies a particular case of Corollary 3andGallowayrsquos [21] null splittingTheorem 6This implies thatwhen 120582

119906rarr 0 the Killing horizon119867 will be equivalent to an

event horizon at null infinity which proves (c)We therefore claim that Theorem 1 provides a physical

model of a family of time-dependent Evolving Null Horizons(ENHs) of GRW space-time andTheorem 7 is a step towardsthe ongoing physical use of ENHs of prescribed GRW space-time which admits a null event horizon satisfying Gallowayrsquosnull splitting Theorem 6 whose above listed three examples(which are subcases of GRW space-time) are also examples ofTheorem7 Consequently the twoTheorems 1 and 7 completethe objective of this paper

4 Conclusion

The motivation for this work originated from the need fortime-dependent null horizons which describe the geometryof a null hypersurface of dynamical space-time For thispurpose in our four previous papers [8ndash10 16] we studieda quasilocal family of ldquoEvolving Null Horizonsrdquo (ENHs) of(1 + 119899)-splitting space-time explained the reason for sucha study constructed a variety of examples of ENHs and insome cases established their relation with black hole isolatedhorizons introduced by Ashtekar et al [22] The followingis a brief account on nonexpanding weakly and isolatedhorizons

Definition 8 A null hypersurface (119867 120574) of 4-dimensionalspace-time (119872 119892) is called a nonexpanding horizon (NEH)if (1) 119867 has a topology 119877 times 1198782 (2) any null normal ℓ119886 of119867 has vanishing expansion 120579

(ℓ)= 0 and (3) all equations

of motion hold at 119867 and the stress energy tensor 119879119894119895is such

thatminus119879119894119895ℓ119895 is future-causal for any future directed null normal

ℓ119894

Condition (1) implies that 119867 is ruled by the integralcurves of the null direction field which is normal to itConditions (2) and (3) imply that m

ℓ120574 = 0 on 119867 which

further implies that themetric 120574 is time-independent and119867 istotally geodesic in119872 Also it follows from the Raychaudhuriequation that ℓ is shear-free on119867 In general there does notexist a unique induced connection on 119867 due to degenerate120574 However on an NEH the property m

ℓ120574 = 0 implies that

the space-time connection nabla induces a unique (torsion-free)connection sayD on119867 compatible with 120574 We say that twotypes of null normal ℓ and ℓ belong to the same equivalenceclass [ℓ] if ℓ = 119888ℓ for some positive constant 119888

Definition 9 The pair (119867 [ℓ]) is called a weakly isolatedhorizon (WIH) if it is NEH and each normal ℓ isin [ℓ] satisfies(mℓD119894minus D119894mℓ)ℓ119894 = 0 that is D

119894ℓ119895 is time-independent

MoreoverWIH (119867 120574 [ℓ]) is called an isolated horizon (IH) ifthe full connectionD is time-independent that is if (m

ℓD119894minus

D119894mℓ)119881 = 0 for arbitrary vector fields 119881 tangent to119867

We refer to Ashtekar et al [23] and several other paperslisted therein for information with examples on isolatedhorizons

In this paper for the first time in the literature we havestudied the existence of a family of time-dependent ENHs inGRW space-time which in general is not (1 + 119899)-splittingspace-time Since our early works in which the study onENHs of (1 + 119899)-splitting space-time was explored we haveused a new technique of using the kinematic condition (1) forconstructing the GRW space-time (instead of defining it) andproving Theorems 1 and 7 is an important step towards theongoing research on time-dependent null horizons On theother hand we mention that recently Caballero et al [4] haveproved the following characterization of prescribed space-time which can be globally split as GRW space-time

Theorem 10 A Lorentzian manifold (119872 119892) admits a globaldecomposition as GRW space-time if and only if it has atimelike gradient conformal Killing vector field119870 such that theflow of its normalized vector field 119885 is well defined and ontoa domain 119868 timesL for some interval 119868 le 119877 and some leaf of theorthogonal foliation to 119870

Contrary to this note that our working GRW space-time does not require a conformal Killing vector field Alsoobserve that in this paper we have only related ENHs toblack hole event horizons represented by a Killing horizonwhereas in our previous works [8ndash10 16] we related ENHsto one of the three types of isolated horizons which unlikeevent horizons may not be represented by a Killing horizonNote however that any Killing horizon which is topologically119877times1198782 is an isolated horizon which is the only common result

between this paper andour previous papers on the same issueIt is important to mention that Lewandowski [24] has shownthat Einsteinrsquos equations admit an infinite dimensional familyof solutions with isolated horizons which are not Killinghorizons A subfamily of Robertson-Trautman space-timeprovides explicit examples of space-timewhich admit IHs butdo not admit a Killing vector in any neighborhood of it [25]

Finally we discuss similarity and difference betweenour results in this paper and two papers of Sultana andDyer [11 12] related to common issue of existence of time-dependent null horizons In [11] they considered a conformaltransformation 119892 = Ω2119892 to a stationary asymptoticallyflat space-time (119872 119892) admitting a Killing horizon 119867 Theyhave shown that such a hypersurface 119867 is null geodesiccalled a conformal Killing horizon (CKH) if and only if thetwist of the conformal Killing trajectories on 119872 vanishesMoreover CKHs are time-dependent and they have a linkwith an event horizon In [12] they constructed an exampleof CKH in de-Sitter space-time Consequently although theirresult on existence of time-dependent CKH is similar to the


conclusions ofTheorems 1 and 7 their work is only limited tonull hypersurfaces of asymptotically flat space-time whereasour results in this paper are applicable to a variety of GRWspace-time

5 Future Prospects

(A) Since an event horizon refers to infinity it is moreappropriate (see details in [23]) to use one of the threetypes of isolated horizons which can easily locate ablack hole For this reason one may try differentapproach to show the existence of a family of ENHs ofa prescribed class of GRW space-time which relate toisolated horizons This opens the possibility of usingthe global splitting Theorem 10 and then proving aresult similar to Theorem 1 for existence of ENHsrelated to an isolated horizon

(B) We know that an event horizon always exists inblack hole asymptotically flat space-time and isolatedhorizons approximate event horizons of a black holeat late stages of gravitational collapse and can be easilylocated as stated in our previous papers Howeverthe following question still remains open Does therealways exist an ENH of black hole space-time

(C) Since an ENH comes from a family of null hypersur-faces its existence and uniqueness are questionablewhich is still an open problemWeneed an input frominterested readers for these open problems

Competing Interests

The author declares that there are no competing interestsregarding the publication of this paper

References

[1] L J Alıas A Romero and M Sanchez ldquoUniqueness ofcomplete spacelike hypersurfaces of constantmean curvature ingeneralized Robertson-Walker spacetimesrdquo General Relativityand Gravitation vol 27 no 1 pp 71ndash84 1995

[2] L J Alıas A Romero andM Sanchez ldquoSpacelike hypersurfacesof constant mean curvature and Calabi-Bernstein type prob-lemsrdquo Tohoku Mathematical Journal vol 49 no 3 pp 337ndash3451997

[3] L J Alıas A Romero andM Sanchez ldquoSpacelike hypersurfacesof constant mean curvature in certain spacetimesrdquo NonlinearAnalysis Theory Methods amp Applications vol 30 pp 655ndash6611997

[4] M Caballero A Romero and R M Rubio ldquoConstant meancurvature spacelike hypersurfaces in Lorentzian manifoldswith a timelike gradient conformal vector fieldrdquo Classical andQuantum Gravity vol 28 no 14 Article ID 145009 14 pages2011

[5] T H Kang ldquoOn lightlike hypersurfaces of a GRW space-timerdquoBulletin of the Korean Mathematical Society vol 49 no 4 pp863ndash874 2012

[6] T H Kang ldquoOn lightlike submanifolds of a GRW space-timerdquoCommunications of the KoreanMathematical Society vol 29 no2 pp 295ndash310 2014

[7] M Navarro O Palmas and D A Solis ldquoNull hypersurfaces ingeneralized RobertsonmdashWalker spacetimesrdquo Journal of Geom-etry and Physics vol 106 pp 256ndash267 2016

[8] K L Duggal ldquoFoliations of lightlike hypersurfaces and theirphysical interpretationrdquoCentral European Journal ofMathemat-ics vol 10 no 5 pp 1789ndash1800 2012

[9] K L Duggal ldquoTime-dependent evolving null horizons of adynamical spacetimerdquo ISRN Mathematical Physics vol 2014Article ID 291790 10 pages 2014

[10] K L Duggal ldquoMaximum principle for totally umbilical nullhypersurfaces and time-dependent null horizonsrdquo Journal ofMathematics Research vol 7 no 1 pp 88ndash95 2015

[11] J Sultana and C C Dyer ldquoConformal Killing horizonsrdquo Journalof Mathematical Physics vol 45 no 12 pp 4764ndash4776 2004

[12] J Sultana and C C Dyer ldquoCosmological black holes a blackhole in the Einstein-de Sitter universerdquo General Relativity andGravitation vol 37 no 8 pp 1349ndash1370 2005

[13] B Carter ldquoExtended tensorial curvature analysis for embed-dings and foliationsrdquo Contemporary Mathematics vol 203 pp207ndash219 1997

[14] V Perlick ldquoOn totally umbilic submanifolds of semi-Riemannian manifoldsrdquo Nonlinear Analysis Theory Methodsand Applications vol 63 no 5ndash7 pp e511ndashe518 2005

[15] S W Hawking and G F R Ellis The Large Scale Structure ofSpacetime Cambridge University Press Cambridge UK 1973

[16] K L Duggal ldquoEvolving null horizons near an isolated blackholerdquo Applied Physics Research vol 8 no 3 pp 90ndash93 2016

[17] S W Hawking ldquoThe event horizonsrdquo in Black Holes C DeWittand B DeWitt Eds North-Holland Publishing AmsterdamThe Netherlands 1972

[18] P Hajicek ldquoExactmodels of charged black holes I Geometry oftotally geodesic null hypersurfacerdquo Communications in Mathe-matical Physics vol 34 pp 37ndash52 1973

[19] P Hajıcek ldquoCan outside fields destroy black holesrdquo Journal ofMathematical Physics vol 15 pp 1554ndash1558 1974

[20] PHajıcek ldquoStationary electrovacuum spacetimes with bifurcatehorizonsrdquo Journal of Mathematical Physics vol 16 pp 518ndash5221975

[21] G J Galloway ldquoMaximumprinciples for null hypersurfaces andnull splitting theoremsrdquoAnnales Henri Poincare vol 1 no 3 pp543ndash567 2000

[22] A Ashtekar C Beetle and S Fairhurst ldquoIsolated horizons ageneralization of black holemechanicsrdquoClassical and QuantumGravity vol 16 no 2 pp L1ndashL7 1999

[23] A Ashtekar C Beetle and J Lewandowski ldquoGeometry ofgeneric isolated horizonsrdquo Classical and Quantum Gravity vol19 no 6 pp 1195ndash1225 2002

[24] J Lewandowski ldquoSpacetimes admitting isolated horizonsrdquoClas-sical and Quantum Gravity vol 17 no 4 pp L53ndashL59 2000

[25] P T Chrusciel ldquoOn the global structure of Robinson-Trautmanspace-timesrdquo Proceedings of the Royal Society of A Mathemati-cal Physical andEngineering Sciences vol 436 no 1897 pp 299ndash316 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


it is too early to be sure) with the latest LIGO [Laser Inter-ferometer Gravitational-Waves Observatory] experiment (asper Press Release of February 11 2016) which confirms thepresence of gravitational waves produced during the finalfraction of a second of the merger of two black holes into asingle massive spinning black hole Thus this recent LIGOexperiment reconfirms that black hole has a cosmologicalbackground or it is surrounded by a local matter distributionTherefore there is significant difference in the structure andproperties of the surrounding dynamical region of a blackhole and this latest experiment strengthens the ongoingresearch on time-dependent null horizons in particular nearsuch spinning black holes surrounded with gravitationalwaves For information on LIGO via video one may tryhttpmediaassetscaltechedugwave

2 Family of Null Hypersurfaces

Consider GRW space-time (119872 119892) with its metric 119892 given by(2) Let 119901 be a point in119872 and ((119872 119892) 1199091 = a constant) be atimelike hypersurface of119872 passing through 119901 Observe thatthere is a freedom of choice in taking any one of the spacelikecoordinates constant Suppose that n is the future directedtimelike unit vector on119872 induced by 119880 of119872 such that thefollowing holds

nabla119883n = 120582 (119905) (119883 + 119892 (119883n)n) forall119883 isin 119879119872 (3)

where 119892 nabla and 120582(119905) are the induced metric Levi-Civitaconnection and a function on 119872 respectively Let ((S 119902) 1199050= a constant) be a connected spacelike hypersurface in119872

and hence a codimension two-spacelike submanifold of 119872passing through 119901 via the normal exponential map along Sin119872 with metric 119902 induced from 119892 Using (3) and followinga procedure explained in previous section one can show that119872 is also GRW space-time with metric 119892 expressed as

119892 = minus1198891199052+ Ω2(119905) 120574119860119861119889119909

119860119889119909119861

2 le 119860 119861 le 119899 120582 =Ω

Ω

(4)

wherewe take120582 = ΩΩ and 120574119860119861

is a fixedRiemannianmetricAssume that119909 = (119909

2 119909

119899) are the coordinates in (S

119906 119905 = 119906

a constant) centered on 119901 so that the metric 119902119906on S119906is given

by

119902119906= Ω2

119906120574119860119861119889119909119860119889119909119861 2 le 119860 119861 le 119899 (5)

Choose at each point of S119906a null direction perpendicular in

S119906and smoothly depending on the foot-point There are two

such possibilities the ingoing and the outgoing null direc-tionsThen it is easy to see that the union of all geodesics withthe chosen (say outgoing) direction ℓ

119906is a null hypersurface

of 119872 Denote by (119867119906 ℎ119906 ℓ119906) such a null hypersurface of

119872 with degenerate metric ℎ119906and null normal ℓ

119906 Since 119867

119906

includes S119906and its degenerate metric ℎ

119906must be of signature

(0 + +) we assume that the Riemannian metric 119902119906of S119906

coincides with the degenerate metric ℎ119906of119867119906given by

ℎ119906= (ℎ119906)119860119861119889119909119860119889119909119861= Ω2

119867119906

120574119860119861119889119909119860119889119909119861

2 le 119860 119861 le 119899

(6)

where Ω119867119906

is a function on 119867119906induced by the function

Ω119906of 119872 This means that the null hypersurface (119867

119906 ℎ119906)

is conformal to a fixed null hypersurface (119867 120574) that isℎ119906= Ω2119867119906

120574 with the conformal functionΩ119867119906

In this paper we take ℓ

119906in some subset of119872 around119867

119906

This will permit us to well define the space-time covariantderivative nablaℓ

119906 Also in this way we get a foliation of119872 (in

the vicinity of119867119906) by a family (119867

119906) so that ℓ

119906is in the part of

119872 foliated by this family (for details see Carter [13]) such thatat each point in this region ℓ

119906is a null normal to119867

119906for some

value of the parameter 119906 Let (ℎ119906) and (ℓ

119906) be the families of

degenerate metrics and null normal respectively Denote by

119865 = ((119867119906) (ℎ119906) (ℓ119906)) ℎ119906= Ω2

119867119906

120574 isin (ℎ119906) (7)

a family of null hypersurfaces of (119872 119892) conformally related toa fixed null hypersurface (119867 120574 ℓ) and eachΩ

119867119906

is a functionon119867119906for some value of 119906 Also let (S

119906) be the corresponding

family of submanifolds of119872The bending of each 119867

119906of 119865 in 119872 is described by the

Weingarten map

Wℓ119906

119879119901119867119906997888rarr 119879119901119867119906

119883119906997888rarr nabla

119883119906

ℓ119906

(8)

Wℓ119906

associates with each 119883119906of 119867119906the variation of ℓ

119906along

119883119906 with respect to the space-time connection nabla The second

fundamental form say 119861119906 of 119867

119906is the symmetric bilinear

form related to the Weingarten map by

119861119906(119883119906 119884119906) = ℎ119906(Wℓ119906

119883119906 119884119906) = ℎ119906(nabla119883119906

ℓ119906 119884)

119883119906 119884119906isin 119879119867119906

(9)

119861(119883119906 ℓ119906) = 0 for any119883

119906isin 119879119867119906implies that 119861

119906has the same

ℓ119906degeneracy as that of the metric ℎ

119906and it does not depend

on particular choice of ℓ119906

Now we deal with the question of obtaining a normalizedexpression for a null normal ℓ of a null hypersurface 119867 isin

119865 For this purpose we let s isin 119879119872 be a unit spacelikenormal vector to S defined in some open neighborhood ofcorresponding 119867 Since the future directed timelike vectorsn of119872 and s are orthogonal it is immediate that the vectorn + s is null so it is tangent to 119867 Therefore one can takeℓ = n + s as a natural normalization of the null normal to119867 However this normalization can only be defined for asingle fixed hypersurface 119867 whereas we need to normalizethe entire family (ℓ

119906) for any parameter value of 119906 For this

purpose we choose the following normalization of each ℓ119906isin

(ℓ119906)

ℓ119906= (n119906+ s119906)

where s119906sdot s119906= 1 x

119906isin 119879119901S119906lArrrArr s

119906sdot x119906= 0

(10)








119892 (ℓ119906 k119906) = minus1 k

119906=1

2(n119906minus s119906) (11)




119906 ℎ119906 ℓ119906)


Ω2119867119906


119906 ℎ119906) Then




120582119906is as given in (3)


ℓ119906

ℎ119906= 2120582119906ℎ119906) on

each119867119906



120579(ℓ)= 0 on119867



119906

n119906= 119860n

119906

x119906



from (3) that 119860n119906

x119906= 120582119906x119906 Denote by 119861S

119906



119906 Then for

any x119906 y119906isin 119879119878119906

119861S119906

(x119906 y119906) = 119902119906(nablax119906

n119906 y119906) = 119902119906(119860n

119906

x119906 y119906)

= 120582119906119902119906(x119906 y119906)

(12)




119906 As

119861119906(ℓ119906 119883119906) = 0 forall119883


119906sub 119867119906we


119861119906(119883119906 119884119906) = 119861119906(x119906 y119906) = 119861S

119906

(x119906 y119906)

forallx119906 y119906isin 119879119878119906(13)



and (12) that

119861119906(119883119906 119884119906) = 120582119906ℎ119906(119883 119884) forall119883

119906 119884119906isin 119879119867119906 (14)



div(n119906) = (119899 minus 1)120582


119906(ℓ119906 119883119906) =


mℓ119906



119861119906(119883119906 119884119906) =

1

2mℓℎ119906(119883119906 119884119906) forall119883

119906 119884119906isin 119879119867119906 (16)

Since 119867119906



ℓ119906

ℎ119906= 2120582119906ℎ119906on 119867119906


119906



119906 ℎ119906) rarr



that 120582119906= Ω119906Ω119906 Thus 120582















119889 (120579(ℓ))


1205792

119899 minus 1 (17)





















119906) (ℎ119906) (ℓ119906)) of a family of










2meet at


1near 119901 and (2)


2of 1198672satisfy 120579

2le

0 le 1205791 Then119867

1and119867








119906) (ℎ119906) (ℓ119906)) is a







119906 ℎ119906 ℓ119906) and it admits


119906rarr 0 at

null infinity





4 Conclusion






ℓ119894


ℓ120574 = 0 on 119867 which







ℓD119894minus










5 Future Prospects




Competing Interests


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















119892 (ℓ119906 k119906) = minus1 k

119906=1

2(n119906minus s119906) (11)




119906 ℎ119906 ℓ119906)


Ω2119867119906


119906 ℎ119906) Then




120582119906is as given in (3)


ℓ119906

ℎ119906= 2120582119906ℎ119906) on

each119867119906



120579(ℓ)= 0 on119867



119906

n119906= 119860n

119906

x119906



from (3) that 119860n119906

x119906= 120582119906x119906 Denote by 119861S

119906



119906 Then for

any x119906 y119906isin 119879119878119906

119861S119906

(x119906 y119906) = 119902119906(nablax119906

n119906 y119906) = 119902119906(119860n

119906

x119906 y119906)

= 120582119906119902119906(x119906 y119906)

(12)




119906 As

119861119906(ℓ119906 119883119906) = 0 forall119883


119906sub 119867119906we


119861119906(119883119906 119884119906) = 119861119906(x119906 y119906) = 119861S

119906

(x119906 y119906)

forallx119906 y119906isin 119879119878119906(13)



and (12) that

119861119906(119883119906 119884119906) = 120582119906ℎ119906(119883 119884) forall119883

119906 119884119906isin 119879119867119906 (14)



div(n119906) = (119899 minus 1)120582


119906(ℓ119906 119883119906) =


mℓ119906



119861119906(119883119906 119884119906) =

1

2mℓℎ119906(119883119906 119884119906) forall119883

119906 119884119906isin 119879119867119906 (16)

Since 119867119906



ℓ119906

ℎ119906= 2120582119906ℎ119906on 119867119906


119906



119906 ℎ119906) rarr



that 120582119906= Ω119906Ω119906 Thus 120582















119889 (120579(ℓ))


1205792

119899 minus 1 (17)





















119906) (ℎ119906) (ℓ119906)) of a family of










2meet at


1near 119901 and (2)


2of 1198672satisfy 120579

2le

0 le 1205791 Then119867

1and119867








119906) (ℎ119906) (ℓ119906)) is a







119906 ℎ119906 ℓ119906) and it admits


119906rarr 0 at

null infinity





4 Conclusion






ℓ119894


ℓ120574 = 0 on 119867 which







ℓD119894minus










5 Future Prospects




Competing Interests


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


























119906) (ℎ119906) (ℓ119906)) of a family of










2meet at


1near 119901 and (2)


2of 1198672satisfy 120579

2le

0 le 1205791 Then119867

1and119867








119906) (ℎ119906) (ℓ119906)) is a







119906 ℎ119906 ℓ119906) and it admits


119906rarr 0 at

null infinity





4 Conclusion






ℓ119894


ℓ120574 = 0 on 119867 which







ℓD119894minus










5 Future Prospects




Competing Interests


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119906 ℎ119906 ℓ119906) and it admits


119906rarr 0 at

null infinity





4 Conclusion






ℓ119894


ℓ120574 = 0 on 119867 which







ℓD119894minus










5 Future Prospects




Competing Interests


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











5 Future Prospects




Competing Interests


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Generalized Robertson-Walker Space-Time ...downloads.hindawi.com/archive/2016/9312525.pdf · Proof. Let x be a vector tangent to a spacelike hypersurface S in .e