research article time-dependent evolving null horizons of

Research ArticleTime-Dependent Evolving Null Horizons ofa Dynamical Spacetime

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 November 2013 Accepted 19 December 2013 Published 22 January 2014

Academic Editors U Kulshreshtha and W-H Steeb

Copyright copy 2014 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

Totally geodesic null hypersurfaces have been widely used in the study of isolated black holes In this paper we introduce a newquasilocal notion of a family of totally umbilical null hypersurfaces called evolving null horizons (ENH) of a dynamical spacetimesatisfied under an appropriate energy condition We focus on a variety of examples of ENHs and in some cases establish theirrelation with event and isolated horizonsWe also present two specific physical models of an ENH in a black hole spacetime Besidethe examples for further study we propose two open problems on possible general existence of an ENH in a black hole spacetimeand its canonical or unique existenceThe results of this paper have ample scope of working on totally umbilical null hypersurfacesof Lorentzian and in general semiRiemannian manifolds

1 Introduction

It is well-known that null hypersurfaces play an importantroll in the study of black hole horizons A black hole is aregion of spacetime which contains a huge amount of masscompacted into an extremely small volume Shortly afterEinsteinrsquos first version of the theory of gravitation that waspublished in 1915 in 1916 Karl Schwarzschild computed thegravitational fields of stars using Einsteinrsquos field equations Heassumed that the star is spherical gravitationally collapsedand nonrotating His solution is called a Schwarzschildsolution which is an exact solution of static vacuum fieldsof the point-mass Since then considerable work has beendone on black hole physics of asymptotically flat and time-independent spacetimes Such isolated black holes deal withthe following concepts of event and isolated horizons

11 Event Horizons A boundary of a spacetime is called anevent horizon briefly denoted by EH beyond which eventscannot affect the observer Note that an event horizon isintrinsically a global concept since its definition requires theknowledge of the whole spacetime to determine whether nullgeodesics can reach null infinity EHs have played a key roleand this includes Hawkingrsquos area increase theorem blackhole thermodynamics black hole perturbation theory and

the topological censorship resultsThemost important familyis the Kerr-Newman black holes Moreover an EH alwaysexists in black hole asymptotically flat spacetime under aweak cosmic censorship conditionWe refer Hawkingrsquos paperon ldquoevent horizonrdquo [1] three papers of Hajıcekrsquos work [2ndash4]on ldquoperfect horizonsrdquo (later called ldquononexpanding horizonsrdquoby Ashtekar et al [5]) and a paper by Galloway [6] in whichhe has shown that the null hypersurfaces which arise mostnaturally in spacetime geometry and general relativity suchas black hole event horizons are in general 1198620 but not 1198621Also Chrusciel and his collaboratorsrsquo papers include key useof EHs (see his review paper [7] on ldquoRecent results in mathe-matical relativityrdquo and several latest papers in particular ldquoNoHairrdquo Theorems) However an event horizon is too global tobe useful in a number of physical situations ranging fromquantum gravity to numerical relativity and to astrophysicsIn particular since it refers to infinity it cannot be used inspecially compact spacetime Moreover to actually locate ablack hole one needs to know the full spacetime metric upto the infinite future Even if one locates the event horizonusing it to calculate the physical parameters is extremelydifficult See Ashtekar-Krishnan [8] for more information onwhy the notion of an event horizon is inappropriate for avariety of physical situations Therefore attempts were made

Hindawi Publishing CorporationISRN Mathematical PhysicsVolume 2014 Article ID 291790 10 pageshttpdxdoiorg1011552014291790

2 ISRNMathematical Physics

to find a quasilocal concept of a horizon which requiresonly minimum number of conditions to detect a black holeand study its properties To achieve this objective in a 1999paper [5] Asktekar et al introduced the following concept ofisolated horizons

12 Isolated Horizons Before giving the definition of isolatedhorizons we recall some features of the intrinsic geometryof 3-dimensional null hypersurface say H of a spacetime(119872 119892) where themetric 119892

119894119895has signature (minus + + +) Denote

by 119902119894119895=119892119894119895

larr997888the intrinsic degenerate induced metric on H

which is the pull back of 119892119894119895 where an under arrow denotes

the pullback toH Degenerate 119902119894119895has signature (0 + +) and

does not have an inverse in the standard sense but in theweaker sense it admits an inverse 119902119894119895 if it satisfies 119902

119894119896119902119895119898119902119896119898

=

119902119894119895 Using this the expansion 120579

(ℓ)is defined by

120579(ℓ)= 119902119894119895nabla119894ℓ119895 (1)

where ℓ119894 is a future-directed null normal to H and nabla is theLevi-Civita connection on 119872 The vorticity-free Raychaud-huri equation is given by

119889 (120579(ℓ))

119889119904= minus119877119894119895ℓ119894ℓ119895minus 120590119894119895120590119894119895minus1205792

2 (2)

where 120590119894119895= nablalarr997888(119894ℓ119895)minus(12)120579

(ℓ)119902119894119895is the shear tensor 119904 is a

pseudoarc parameter such that ℓ is null geodesic and 119877119894119895is

the Ricci tensor of119872 We say that two null normals ℓ119894 and ℓ119894

belong to the same equivalence class [ℓ] if ℓ119894 = 119888ℓ119894 for some

positive constant 119888 The following are three notions of iso-lated horizons namely nonexpanding weakly and strongerisolated horizons respectively

Definition 1 A null hypersurface (H 119902) of a 4-dimensionalspacetime (119872 119892) is called a nonexpanding horizon (NEH) if

(1) H has a topology 119877 times 1198782(2) any null normal ℓ of H has vanishing expansion

120579(ℓ)= 0

(3) all equations of motion hold at H and stress energytensor 119879

119894119895is such that minus119879119894

119895ℓ119895 is future-causal for any

future-directed null normal ℓ119894

Condition (1) is a restriction on topology of H whichguarantee that marginally trapped surfaces are related to ablack hole spacetime Condition (2) and the energy condition

of (3) imply from the Raychaudhuri equation (2) that119879119894119895ℓ119895

larr997888= 0

and = nablalarr997888(119894ℓ119895)equiv poundℓ119902119894119895= 0 on H which further implies that

the metric 119902119894119895is time-independent Note that pound

ℓ119902119894119895= 0 onH

does not necessarily imply that ℓ is a Killing vector of the fullmetric 119892

119894119895 In general there does not exist a unique induced

connection onH due to degenerate 119902119894119895 However on anNEH

the property poundℓ119902119894119895= 0 implies that the spacetime connection

nabla induces a unique (torsion-free) connection say D on Hwhich is compatible with 119902

119894119895

Definition 2 The pair (H [ℓ]) is called a weakly isolatedhorizon (WIH) if H is an NEH and each normal ℓ isin [ℓ]

satisfies

(poundℓD119894minusD119894poundℓ) ℓ119894= 0 (3)

Condition (3) implies that in addition to the metric 119902119894119895

the connection componentD119894ℓ119895 is also time-independent for

a WIH Given a NEH one can always have an equivalenceclass [ℓ] (which is not unique) of null normals such that(H [ℓ]) is aWIH Ashtekar et al [9] have discussed the issueof ldquoFreedom of the choice of ℓrdquo in satisfying condition (3)

Definition 3 A WIH (H [ℓ]) is called an isolated horizon(IH) if the full connectionD is time-independent that is if

(poundℓD119894minusD119894poundℓ) 119881119895= 0 (4)

for arbitrary vector fields 119881 tangent toH

An IH is stronger notion of isolation as its condition(4) cannot always be satisfied by a choice of null normalsIsolated horizons are quasilocal and do not require theknowledge of the whole spacetime The class of spacetimeshaving isolated horizons is quite big Any Killing horizonwhich is topologically 119877times1198782 is a trivial example of an isolatedhorizon See Ashtekar and Krishnan [8] Ashtekar et al [10]Lewandowski [11] and Gourgoulhon and Jaramillo [12] forexamples and their physical use

On the other hand we know that the isolated horizonsmodel specifically quasilocal equilibrium regimes of blackhole spacetimes However in nature black holes are rarely inequilibrium This led to research on a quasilocal frameworkto describe the geometry of the surface of the dynamicalblack hole not just at its equilibrium state First attempt inthis direction was made by Hayward [13] in 1994 usingthe framework of (2 + 2)-formalism based on the notion oftrapped surfaces He proposed the following notion of futureouter and trapping horizons (FOTH)

Definition 4 A future outer and trapped horizon (FOTH)is a three-manifold Σ foliated by family of closed 2-surfacessuch that (i) one of its future-directed null normal say ℓ haszero expansion 120579

(ℓ)= 0 (ii) the other null normal k has

negative expansion 120579(k) lt 0 and (iii) the directional deriva-

tive of 120579(ℓ)

along k is negative poundn120579(ℓ) lt 0

Σ is either spacelike or null for which 120579(k) = 0 and poundk120579(ℓ) =

0 Hayward [13] derived the following general laws of blackhole dynamics

(a) Zeroth Law The total trapping gravity of a compactouter marginal surface has an upper bound attainedif and only if the trapping gravity is constant

(b) First Law The variation of the area form along anouter trapping horizon is determined by the trappinggravity and an energy flux

After this Ashktekar and Krishnan [14 15] observedthat in dynamical situations Haywardrsquos condition (iii) is not

ISRNMathematical Physics 3

required for most of the key physical results For this reasonthey introduced in [14] the following quasilocal concept ofldquodynamical horizonsrdquo denoted by DH which model theevolving black holes and their asymptotic states are isolatedhorizons

Definition 5 A smooth 3-dimensional spacelike submani-fold (possibly with boundary) 119867 of a spacetime is said to bea dynamical horizon (DH) if it can be foliated by a family ofclosed 2-manifolds such that

(1) on each leaf 119878 its future-directed null normal ℓ haszero expansion 120579

(ℓ)= 0

(2) and the other null normal k has negative expansion120579(k) lt 0

They first required that 119867 be spacelike everywhere andthen studied the case in which portions ofmarginally trappedsurfaces lie on a spacelike horizon and the remainder ona null horizon In the null case 119867 reaches equilibrium forwhich the shear and the matter flux vanish and this portion isrepresented by a weakly isolated horizonThe Vaidya metricsare explicit examples of dynamical horizons with their equi-librium states of the isolated horizonsThe horizon geometryof DHs is time-dependent Compared to Haywardrsquos (2 + 2)-formalism the DH framework is based on the standard (1 +3)-formulism and has the advantage that it only refers to theintrinsic structure of 119867 without any conditions on the evo-lution of fields in transverse directions to119867 In [15] Ashtekarand Krishnan have obtained the following main results

(i) Expressions of fluxes of energy and angular momen-tum carried by gravitational waves across these hori-zons were obtained

(ii) A detailed area balance law relating to the change inthe area of119867 to the flux of energy across it provided

(iii) The cross sections S of119867 have the topology 1198782 if thecosmological constant (of Einsteinrsquos filed equations)is positive and of 1198782 or a 2-torus The 2-torus case isdegenerate as the matter and the gravitational energyflux vanish the intrinsic metric on each S is flat andthe shear of the (expansion free) null normal vanishes

(iv) A generalization of the first and the second laws ofmechanics was obtained

(v) A relation between DHs and IH was established

DH has provided a new perspective covering all areas ofblack holes that is quantum gravity mathematical physicsnumerical relativity and gravitational wave phenomenologyleading to the underlying unity of the subject

As explained in [15] the definition of DH rules out thepossibility of 119867 null except when 119867 reaches equilibriumThus we do not have any null horizon which is time-dependent and quasilocal and is always a null geodesichypersurface Such a null horizon is desired for informationon the geometry andphysics of the null surface of a dynamicalspacetime along with DHs which are models of evolvingblack holes Moreover there is a need to know the null

version of the known results provided by FOTH and the DHframeworks For these reasons in this paper we study a newclass of null horizons as explained below

Observe that all types of null horizons (defined above)have a common condition that their future null normalhas vanishing expansion This means that their underlyingnull hypersurface is totally geodesic in the correspondingspacetime and these horizons are time-independent In thispaper we use totally umbilical geometry to show the existenceof a class of null hypersurfaces called ldquoEvolving Null Horizons(ENH)rdquo (see Definition 8) (this notion is slightly general thanrecently introduced in [16] fromwhere we take somematerialto present their improved version) which is time-dependentquasilocal and suitable for the null geometry of the surface ofa dynamical spacetime We focus on a variety of examples ofENHs some of them having a totally geodesic null portionwhich represents an event or an isolated horizon at theequilibrium state of a black hole spacetime This paper hasbeen written with a twofold objective inmind firstly initiate anewway of research on time-dependent null hypersurfaces ofdynamical spacetimes by using the totally umbilical geome-try Secondly themathematical theory (using intrinsic geom-etry) on the foliations of null hypersurfaces presented in thispaper (see also [16] for more general details) is expected togenerate interest in further research on differential geometryof totally umbilical null hypersurfaces of Lorentzian and ingeneral semiRiemannian manifolds

2 Totally Umbilical Null Hypersurfaces

Recall that a hypersurface (H ℎ) of a 4-dimensional space-time (119872119892) is null if there exists a nonvanishing null vectorfield ℓ in 119879(H) which is orthogonal (with respect to ℎ) to allvector fields in 119879(H) that is

ℎ (ℓ 119883) = 0 forall119883 isin 119879 (H) (5)

where ℎ is the degenerate metric of H In this paper weassume that the null normal ℓ is future-directed and itis not entirely in H but is defined in some open subsetof 119872 around H This will permit us to well define thespacetime covariant derivative nablaℓ and is suitable for theintrinsic geometry For general extrinsic geometry of nullhypersurfaces of semiRiemannian manifolds (where nullnormal is taken entirely in the hypersurface) we refer Duggaland Bejancu [17 Chapter 4] and Duggal and Jin [18 Chapter7] A simple way to take this extended ℓ is to consider afoliation of119872 (in the vicinity ofH) by a family (H

119906) so that

ℓ is in the part of119872 foliated by this family such that at eachpoint in this region ℓ is a null normal toH

119906for some value of

119906 Denote by (ℎ119906) the respective family of degenerate metrics

Although the family (H119906) is not unique for our purpose we

can manage (with some reasonable condition(s)) to involveonly those quantities which are independent of the choice ofthe foliation (H

119906) once evaluated atH

119906=const For simplicityin this paper we consider (H ℎ) amember of the family (H

119906)

and its respectivemetricℎ for some value of119906 with the under-standing that the results are the same for any other member


The ldquobendingrdquo ofH in119872 is described by the Weingartenmap

Wℓ 119879119901H 997904rArr 119879

119901H

119883 997904rArr nabla119883ℓ

(6)

Wℓassociates each 119883 of H the variation of ℓ along 119883

with respect to the spacetime connection nabla The secondfundamental form say119861 ofH is the symmetric bilinear formand is related to the Weingarten map by

119861 (119883 119884) = ℎ (Wℓ119883119884) = ℎ (nabla

119883ℓ 119884) (7)

119861(119883 ℓ) = 0 for any null normal ℓ and for any 119883 isin

119879(H) imply that 119861 has the same ℓ degeneracy as that ofthe induced metric ℎ Hence it is natural to study a class ofnull hypersurfaces such that 119861 is conformally equivalent tothe metric ℎ Geometrically this means that (H ℎ) is totallyumbilical in119872 if and only if there is a smooth function 119891 onH such that

119861 (119883 119884) = 119891ℎ (119883 119884) forall119883 119884 isin 119879 (H) (8)

The above definition does not depend on particular choiceof ℓ H is called proper totally umbilical if and only if 119891 isnonzero on entire H In particular a portion of H is calledtotally geodesic if and only if 119861 vanishes that is if and onlyif 119891 vanishes on that portion of H From (7) 119861(119883 ℓ) = 0

for any null normal ℓ and (8) we conclude that H is totallyumbilical in 119872 if and only if on each neighborhood U theconformal function 119891 satisfies

Wℓ119883 = 119891119883 forall non-null 119883 isin Γ (119879H) (9)

The following result is important in the study of null hyper-surfaces

Proposition 6 Let (H119906) be a family of null hypersurfaces of

a Lorentzian manifold Then each member (H ℎ) of (H119906) is

totally umbilical if and only if its null normal ℓ is a conformalKilling vector of the degenerate metric ℎ

Proof Consider a member (H ℎ) of (H119906) Using the expres-

sion poundℓℎ(119883 119884) = ℎ(nabla

119883ℓ 119884) + ℎ(nabla

119884ℓ 119883) and 119861(119883 119884)

symmetric in the above equation we obtain

119861 (119883 119884) =1

2poundℓℎ (119883 119884) forall119883 119884 isin 119879 (H) (10)

which is well defined up to conformal rescaling (related to thechoice of ℓ) SupposeH is totally umbilical that is (8) holdsUsing this in (10) we have pound

ℓℎ = 2119891ℎ on H Therefore ℓ is

conformal Killing vector of the metric ℎ Conversely assumepoundℓℎ = 2119891ℎ onH Then

poundℓℎ (119883 119884) = ℎ (nabla

119883ℓ 119884) + ℎ (nabla

119884ℓ 119883) = 2ℎ (nabla

119883ℓ 119884)

= 2ℎ (Wℓ119883119884) = 2119891ℎ (119883 119884)

(11)

which implies that (9) holds soH is totally umbilical

21 Normalization of ℓ and Projector onto (Hℎ) Due todegenerate metric ℎ there is no canonical way from the nullstructure alone of H to define a projector mapping II

119879119901119872 rarr 119879

119901(H) Thus to obtain normalized expressions

for ℓ there is a need for some extrastructure on 119872 For thispurpose we consider a (1+3)-spacetime (119872 119892)This assumesa thin sandwich of119872 evolved from a spacelike hypersurfaceΣ119905at a coordinate time 119905 to another spacelike hypersurface

Σ119905+119889119905

at coordinate time 119905 + 119889119905 whose metric 119892 is given by

119892119894119895119889119909119894119889119909119895= (minus120582

2+ |119880|2) 1198891199052+ 2119892119886119887119880119886119889119909119887119889119905 + 119892

119886119887119889119909119886119889119909119887

(12)

where 1199090 = 119905 and 119909119886 (119886 = 1 2 3) are spatial coordinates ofΣ119905with 119892

119886119887its 3-metric induced from 119892 120582 = 120582(119905 119909

1 1199092 1199093)

is the lapse function and 119880 is a spacelike shift vector Thechoice of (1+3)-spacetime comes from the works of Ashtekarand Krishnan [14 15] and Gourgoulhon and Jaramillo [12] ondynamical and isolated horizons respectivelyThe coordinatetime vector t = 120597120597119905 is such that 119892(119889119905 t) = 1 For the (1 + 3)-spacetime119872 one can write

t = 120582n + 119880 with n sdot 119880 = 0 (13)

where n is the future timelike unit vector field The questionis how to normalize ℓ In general each spacelike hypersurfaceΣ119905intersects the null hypersurfaceH on some 2-dimensional

submanifold S119905 that is S

119905= H cap Σ

119905 Consider a familyF =

(H119906) in the vicinity ofH defined by

S119905119906= H119906cap Σ119905 (14)

where S119905119906=1

is an element of this family (S119905119906) Let s isin Σ

119905be a

unit vector normal to S119905defined in some open neighborhood

of H Taking (S119905) a foliation of H the coordinate 119905 can

be used as a nonaffine parameter along each null geodesicgenerating H We normalize ℓ of H such that it is tangentvector associated with this parameterization of the nullgenerators that is

ℓ119894=119889119909119894

119889119905 (15)

This means that ℓ is a vector field ldquodualrdquo to the 1-form 119889119905Equivalently the function 119905 can be regarded as a coordinatecompatible with ℓ that is

119892 (119889119905 ℓ) = nablaℓ119905 = 1 (16)

Based on this it is easy to see that ℓ has the followingnormalization

ℓ = 120582 (n + s) where s sdot 119904 = 1 V isin 119879119901(S119905) lArrrArr s sdot V = 0

(17)

which implies that ℓ is tangent toH with the property of Liedragging the family of surfaces (S

119905119906)

As anyH is defined by 119906 = a constant the gradient 119889119906 isits normal that is

ℎ (119889119906119883) = 0 lArrrArr forall119883 isin 119879 (H) (18)


Thus the 1-form ℓ associated with the null normal ℓ iscollinear to 119889119906 that is

ℓ = 119890120588119889119906 (19)

where 120588 isin H is a scalar field Now the question is how to findsome direction transverse toH We see from the normalizedequation (17) that there are timelike and spacelike transversedirections n and s (as they both do not belong toH) respec-tively which are normal to the 2-dimensional spacelike sub-manifold S

119905 Since we already have the outgoing null normal ℓ

tangent toH we define a transversal vector fieldk of119879119901119872not

belonging toH expressed as another suitable linear combina-tion of n and s such that it represents the light rays emitted inthe opposite direction called the ingoing direction satisfying

119892 (ℓ k) = minus1 119892 (k k) = 0 (20)

Using the normalization (17) of ℓ and (20) we get the follow-ing normalized expression of the null transversal vector k

k = 1

2120582(n minus s) (21)

Since two null normals ℓ and ℓ of H belong to the sameequivalence class [ℓ] if ℓ = 119888ℓ for some positive constant 119888 itfollows from (20) that with respect to change of ℓ to ℓ thereis another k = (1119888) k satisfying (20) Now we define theprojector ontoH along k by

II 119879119901119872 997888rarr 119879

119901H

119883 997888rarr 119883 + 119892 (ℓ119883) k(22)

The above mapping is well defined that is its image is in119879119901H Indeed

forall119883 isin 119879119901119872 119892 (ℓ II (119883)) = 119892 (ℓ 119883) 119892 (ℓ k) = 0 (23)

Observe that II leaves any vector in 119879119901H invariant and

II(k) = 0Moreover the definition of the projector IIdoes notdepend on the normalization of ℓ and k as long as they satisfythe relation (20) In other words II is determined only by thefoliation of the family (S

119905119906) ofH

119906and not by any rescaling of

ℓConsider a spacelike orthonormal frames field 119864 =

1198901 1198902 on S

119905of H Since 119861(ℓ ℓ) = 0 the expansion scalar

field (also called null mean curvature) 120579(ℓ)

ofH with respectto ℓ can be defined by

120579(ℓ)= 119861 (119890

1 1198901) + 119861 (119890

2 1198902) (24)

which is equivalent to trace (119861) and therefore it does notdepend on the frame 119864 For a totally umbilicalH this meansthat

120579(ℓ)= 119861 (119890

1 1198901) + 119861 (119890

2 1198902) = 119891 ℎ (119890

1 1198901) + ℎ (119890

2 1198902) = 2119891

(25)

Then H is totally geodesic (also called minimal) if 120579(ℓ)

vanishes that is

119861 (1198901 1198901) + 119861 (119890

2 1198902) = 0 (26)

22 Induced Extrinsic Structure Equations Recall that Dug-gal and Bejancu [17 Chapter 4] used a screen distribution toobtain induced extrinsic objects of a lightlike hypersurfaceAlthough we are not using any screen for H we do have avector bundle 119879119878 of the 2-dimensional submanifolds of 119872In order to use the extrinsic structure equations given in [17Chapter 4] we replace the role of screen by the vector bundle119879119878 of 119872 which has the added advantage that it is obviouslyintegrable With this understanding from (20) we have thefollowing decomposition of 119879119872

|H

119879119872|H = 119879Hoplusorth tr (119879H) (27)

where tr(119879H) = k denotes a null transversal vector bundleof rank 1 over H Using the above decomposition and thesecond fundamental form119861 we obtain the following extrinsicGauss and Weingarten formulas [17 Chapter 4]

nabla119883119884 = D

119883119884 + 119861 (119883 119884) k (28)

nabla119883k = minus119860k119883 + 120591 (119883) k forall119883 119884 isin Γ (119879H) (29)

where119860k is the shape operator on119879119901H in119872 120591 is a 1-formonH andD is the induced linear connection on a pair (H [ℓ])By setting 119884 = ℓ in (28) the Weingarten mapW

ℓ defined by

(6) will satisfy

D119883ℓ = W

ℓ119883 forall119883 isin 119879H (30)

with respect to the induced linear connection D on a pair(H [ℓ]) In generalD is not a Levi-Civita connection and itsatisfies

(D119883ℎ) (119884 119885) = 119861 (119883 119884) 120578 (119885) + 119861 (119883 119885) 120578 (119884)

forall119883 119884 119885 isin Γ (119879H|U)

(31)

where 120578(119883) = 119892(119883 k) forall119883 isin Γ(119879H|U) Let 119877 andR denote

the curvature tensors of the Levi-Civita connection nabla on119872and the induced linear connectionD onH respectivelyTheGauss-Codazzi equations are

119892 (119877 (119883 119884)119885 119881) = ℎ (R (119883 119884)119885 119881) forall119881 isin 119879119878

119892 (119877 (119883 119884)119885 ℓ) = ℎ (R (119883 119884)119885 ℓ)

= (nabla119883119861) (119884 119885) minus (nabla

119884119861) (119883 119885)

+ 119861 (119884 119885) 120591 (119883) minus 119861 (119883 119885) 120591 (119884)

119892 (119877 (119883 119884)119885 k) = ℎ (R (119883 119884)119885 k)

(32)

The induced Ricci tensor of (H ℎ) is given by the followingformula

R (119883 119884) = trace 119885 997888rarr R (119883 119885) 119884 forall 119883 119884 isin Γ (119879H)

(33)

Since D on H is not a Levi-Civita connection in generalRicci tensor is not symmetric Indeed let 119865 = ℓ k 119890

1 1198902

be a quasiorthonormal frame on119872 Then we obtain

R (119883 119884) = ℎ (R (1198901 119883) 119884 119890

1)

+ ℎ (R (1198902 119883) 119884 119890

2) + ℎ (R (ℓ 119883) 119884 k)

(34)


Using Gauss-Codazzi equations and the first Bianchi identitywe get

R (119883 119884) minusR (119884119883) = 2119889120591 (119883 119884) (35)

Also the 1-form 120591 in (29) depends on the choice of the normalℓTherefore wemust require thatR is symmetric (otherwiseit has no geometric or physical meaning) and 120591 vanishes sothat119860k is independent of the choice of the foliation (H119906) Tofix this problem we assume the following known result

Proposition 7 (see [17 page 99]) Let (H ℎ) be a nullhypersurface of a semiRiemannian manifold (119872 119892) If theinduced Ricci tensor ofH is symmetric then there exists a localnull pair ℓ k 119892(ℓ k) = minus1 such that the corresponding 1-form 120591 from (29) vanishes

For a Ricci symmetric totally umbilical H (28)ndash(30)reduce to

nabla119883119884 = D

119883119884 + 119891119892 (119883 119884) k

nabla119883k = minus119860k119883

(D119883ℎ) (119884 119885) = 119891 119892 (119883 119884) 120578 (119885) + 119892 (119883 119885) 120578 (119885)

forall119883 119884 119885 isin Γ (119879H|U)

(36)

3 Evolving Null Horizons

Here we state the notion of an ldquoEvolving Null Horizonrdquodenoted by ENH explain the implications of its conditionsconstruct some examples of ENHs establish their relationwith event and isolated horizons and construct two physicalmodels of an ENH of a black hole spacetime

Definition 8 A null hypersurface (H ℎ) of a 4-dimensionalspacetime (119872 119892) is called an evolving null horizon (ENH) if

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

(ii) all equations of motion hold at H and stress energytensor 119879




For condition (i) it follows from Proposition 6 that poundℓℎ =

2119891ℎ on H that is ℓ is a conformal Killing vector (CKV) ofthemetric ℎ IfH includes a totally geodesic portion then onits portion 119891 vanishes and ℓ reduces to a Killing vector Thisdoes not necessarily imply that ℓ is a CKV (or Killing vector)of the full metric 119892 The shear tensor 120590 is given by

120590 = poundℓℎ minus

1

2120579(ℓ)ℎ = (2119891 minus

1

2120579(ℓ)) ℎ (37)

The energy condition of (ii) requires that 119877119894119895ℓ119894ℓ119895 is nonnega-

tive for any ℓ which implies (see Hawking and Ellis [19 page95]) that 120579

(ℓ)monotonically decreases in time along ℓ that is

119872 obeys the null convergence condition Condition (i) alsoimplies that we have two classes of ENHs namely genericENH (119891 does not vanish on H) and nongeneric ENH for

which 119891may vanish on a possible totally geodesic portion ofH Themetric on a generic ENHwill be time-dependent and120579(ℓ)

will not vanish onH In the case of a non-generic ENH120579(ℓ)

may eventually vanish on its totally geodesic portion forwhich its transformed metric will be time-independent Wegive examples of both classes

31 Mathematical Model Let (H ℎ) be a member of thefamily (H

119906) of null hypersurfaces of a spacetime (119872 119892)

whose metric 119892 is given by (12) With respect to each S119905 the

shift vector 119880 can be expressed as

119880 = 120572s minus 119881 120572 = s sdot 119880 119881 isin 119879119901(S119905) (38)

Using (13) (17) and (38) we obtain

ℓ = t + 119881 + (120582 minus 120572) s (39)

To relate above decomposition of ℓ with the conformalfunction 119891 of the totally umbilical condition (8) we choose120582 minus 120572 = 119891 on H which means that a portion of H may betotally geodesic if and only if 120582 = 120572 on that portion Thus

ℓH= t + 119881 + 119891s (40)

Consider a coordinate system (119909119860) = (119905 119909

2 1199093) on (H ℎ)

defined by 1199091 = constant Then the degenerate metric ℎis

ℎ = ℎ119860119861119889119909119860119889119909119861= ℎ1199051199051198891199052+ 2ℎ119905119896119889119905 119889119909119896+ ℎ119896119898119889119909119896119889119909119898

(41)

where 2 le 119896119898 le 3 and

ℎ119905119905= 119881119896119881119896+ 1198912 ℎ

119905119896= 119880119896= 120572s119896minus 119881119896= minus119881119896 (42)

Observe that one can also take 1199092 or 1199093 constant Assumethat (H ℎ) is proper totally umbilical in (119872 119892) and119872 obeysthe null convergence condition with respect to each future-directed null normal of this family (H

119906) Then a member

(H ℎ 1199091= constant) of this family is a model of a generic

ENH whose degenerate metric is given by (41)

Nongeneric ENH Consider a null hypersurface H = Σ cup Δ

such that Σ and Δ are null proper totally umbilical and totallygeodesic portions of H and 119872 obeys the null convergencecondition Thus as per Definition 8 H is an ENH whichincludes a totally geodesic portion (Δ 119902) where 119902 denotesits metric Since 120579

(ℓ)monotonically decreases in time along

ℓ and for this case 119891 can vanish on Δ a state may reach at atime when 120579

(ℓ)vanishes on the portionΔ ofH and by putting

119891 = 0 in the metric (41) we recover the metric 119902 of Δ given by

119902 = 119902119860119861119889119909119860119889119909119861= 119902119896119898(119889119909119896minus 119881119896119889119905) (119889119909

119898minus 119881119898119889119905)

(43)

where poundℓ119902 = 0 and

120572(H119902)= 120582 is constant on (H 119902)

ℓ(H119902)= t + 119881

(44)


For this case the transformed coordinate system (119909119860) is

stationary with respect to the hypersurface (H 119902) as itsmetric (43) is time-independent In other words the locationof S119905is fixed as 119905 varies Moreover

((119909119860) stationary wrt (H 119902)) lArrrArr

120597119906

120597119905= 0

lArrrArr t isin 119879 (H)

(45)

At this state of transition Raychaudhuri equation (2) impliesthat shear also vanishes Consequently the two conditionsof the definition of a totally geodesic evolving null horizon(H 119902) reduce to conditions (2) and (3) of a nonexpandinghorizon (NEH) In case (H 119902) is a null horizon of some blackhole then the vector 119881 isin 119879

119901(S119905) is called the surface velocity

of the black hole (see Damour [20])We present the followingmathematical model

Monge Null Hypersurfaces Consider a smooth function 119865

Ω rarr R where Ω is an open set of R3 Then a hypersurfaceH of R4

1is called a Monge hypersurface [17 page 129] given

by the equation

119905 = 119865 (119909 119910 119911) (46)

where (119905 119909 119910 119911) are the standard Minkowskian coordinateswith origin 0 The scalar 119906 generates a family of Mongehypersurfaces (H

119906) as the level sets of 119906 and is given by

119906 (119905 119909 119910 119911) = 119865 minus 119905 + 119887 (47)

where 119887 is a constant and we takeH119887= H a member of the

family (H119906) nabla119894119906 = (minus1 119865

1015840

119909 1198651015840

119910 1198651015840

119911)H is null if and only if 119865

is a solution of the following partial differential equation

(1198651015840

119909)2

+ (1198651015840

119910)2

+ (1198651015840

119911)2

= 1 (48)

From (19) the null normal ℓ ofH is

ℓ119894= 119890120588(1 1198651015840

119909 1198651015840

119910 1198651015840

119911) ℓ

119894= 119890minus120588(minus1 119865

1015840

119909 1198651015840

119910 1198651015840

119911) (49)

The components of the transversal vector field k can be takenas

k119894 = 1

2119890120588(minus1 minus119865

1015840

119909 minus1198651015840

119910 minus1198651015840

119911)

k119894=1

2119890minus120588(1 minus119865

1015840

119909 minus1198651015840

119910 minus1198651015840

119911)

(50)

so that ℎ(ℓ k) = minus1 The corresponding base vectors say11988211198822 of St are

1198821= 1198651015840

119911

120597

120597119909minus 1198651015840

119909

120597

120597119911 1198822= 1198651015840

119911

120597

120597119910minus 1198651015840

119910

120597

120597119911 (51)

where we take 1198651015840

119911= 0 on H Then the 1-form 120591 in (29)

vanishes Indeed 120591(119883) = 119892(nabla119883k ℓ) = minus(12)119892(nabla

119883ℓ ℓ) = 0

Therefore it is quite straightforward (see [17 page 121]) thatthe Ricci tensor of the linear connection D on this Mongehypersurface H is symmetric Consequently our extrinsic

objects on null Monge hypersurfaces are independent ofthe choice of a family (H

119906) To find the expansion 120579

(ℓ)of

H we use the base vectors (51) to construct the followingorthonormal basis 119890

1 1198902 of St

1198901=

1

100381710038171003817100381711988211003817100381710038171003817

1198821

1198902=

1

10038161003816100381610038161198651015840

119911

1003816100381610038161003816

100381710038171003817100381711988211003817100381710038171003817

10038171003817100381710038171198821

1003817100381710038171003817

21198822minus 119892 (119882

11198822)1198821

(52)

Then by direct calculations we obtain

119861 (1198901 1198901) + 119861 (119890

2 1198902)

=1

(1198651015840

119911)2ℎ11

times ((1198651015840

119911)2

+ (ℎ12)2) 11986111minus 2ℎ11ℎ1211986112+ (ℎ11)211986122

(53)

where we put ℎ120572120573= ℎ(119882

120572119882120573) and 119861

120572120573= 119861(119882

120572119882120573) 120572 120573 isin

1 2 Taking into account that

ℎ11= 1 minus (119865

1015840

119910)2

ℎ12= 1198651015840

1199091198651015840

119910 ℎ

22= 1 minus (119865

1015840

119909)2

(54)

we finally (see pages 118ndash132 in [17] for some missing details)obtain

120579(ℓ)= 119861 (119890

1 1198901) + 119861 (119890

2 1198902) = minus

1

(1198651015840

119911)2ℎ11

(11986510158401015840

119909119909+ 11986510158401015840

119910119910+ 11986510158401015840

119911119911)

(55)

Consider a family (H119906) of null Monge hypersurfaces of

R41whose each member (H ℎ) obeys the null convergence

condition that is 120579(ℓ)

monotonically decreases in time alongℓ Thus it follows from (26) and (55) that (H ℎ) may have atotally geodesic portion say Δ if and only if a state reacheswhen 120579

(ℓ)vanishes and at that state 119865 satisfies the Laplace

equation

11986510158401015840

119909119909+ 11986510158401015840

119910119910+ 11986510158401015840

119911119911= 0 (56)

which means that 119865 is harmonic on Δ As explained inSection 2 take a family (H

119906) of null Monge hypersurfaces of

R41whose eachmember (H ℎ) is totally umbilical (119861 = 2119891ℎ)

Then as per (24) and (55) we have

120579(ℓ)= minus

1

(1198651015840

119911)211989211

(11986510158401015840

119909119909+ 11986510158401015840

119910119910+ 11986510158401015840

119911119911) = 2119891 (57)

which implies that 119865 is harmonic on Δ if and only if 119891vanishes on Δ Thus there exists a model of a class of Mongenull hypersurfaces which is union of a family of generic ENHsand its totally geodesic null portions

Generic ENHs Now we show by the following example thatthere are some Monge ENHs which are generic that is theydo not evolve into a totally geodesic ENH


Example 9 (see [16]) Let (Λ30)119906be a family of Monge null

cones in R41given by

119905 = 119865 (119909 119910 119911) = 119903 = radic1199092+ 1199102+ 1199112 (58)

where (119905 119909 119910 119911) are the Minkowskian coordinates withorigin 0 Exclude 0 to keep each null cone smooth LetΛ3

0be

a member at the level 119906 = 119887 = 0 Then the scalar 119906 generatesa family of Monge null cones ((Λ3

0)119906) given by

119906 (119905 119909 119910 119911) = 119903 minus 119905 + 119887 with 119905 = radic1199092 + 1199102 + 1199112 (59)

nabla119894119906 = (minus1 119909119905 119910119905 119911119905) Thus the components of the null

normal to Λ30are

ℓ119894= 119890120588(1

119909

119905119910

119905119911

119905) ℓ

119894= 119890minus120588(minus1

119909

119905119910

119905119911

119905) (60)

Since ℓ is the position vector field for the above constructionof the set of null cones it follows from the Gauss equation(28) that

nabla119883ℓ = D

119883ℓ = 119906119883 forall119883 isin (Λ

3

0)119906 (61)

where 119906 = 0 as 0 is excludedThen using theWeingartenmapequation (6) we obtain

Wℓ119883 = 119906119883 = 119891119883 forall non-null 119883 isin (Λ

3

0)119906 (62)

Hence by (9) we conclude that any member of the set ofnull cones ((Λ3

0)119906 = 0

) is proper totally umbilical with theconformal function119891 = 119906 = 0Theunit timelike and spacelikenormals n and s in the normalized expression of ℓ are

n119894 = (1 0 0 0) s119894 = (0 119909119905119910

119905119911

119905) (63)

Therefore it follows from (19) that120582 = 119891+120572 = 119890120588 To calculatethe expansion 120579

(ℓ)= 2119891 we use (57) for 119865 = 119903 and obtain

120579(ℓ)= minus

1

(1199031015840

119911)2ℎ11

(11990310158401015840

119909119909+ 11990310158401015840

119910119910+ 11990310158401015840

119911119911)

= minus21199053

1199112(1199052minus 1199102)= minus

21199053

1199112(1199092+ 1199112)= 2119891 = 0

(64)

Since119891 does not vanish onH subject to the null convergencecondition the above is an example of an ENHwhich does notevolve into an event or isolated horizon

4 Two Physical Models Having an ENH ofa Black Hole Spacetime

Nongeneric Model Let (119872 119892) be a spacetime which admits atotally geodesic null hypersurface (H 119902 ℓ) where its degen-erate metric 119902 is the pull back of 119892 and ℓ is its future-directednull normal defined in some open subset of 119872 around H(see details in Section 1) Then we know that pound

ℓ119902 = 0

that is ℓ is Killing with respect to the metric 119902 which is notnecessarily Killing with respect to full metric 119892 Consider ametric conformal transformation defined by

119866 = Ω2119892 (65)

whereΩ is a scalar function on119872 and (119872119866) is its conformalspacetime manifold Since the causal structure and nullgeodesics are invariant under a conformal transformation asdiscussed in Section 2 we have a family of null hypersurfaces(H119906) in (119872119866) with its corresponding family of induced

metrics (ℎ119906) such that each one is conformally related to 119902

by the following transformation

ℎ119906= Ω2

119906119902 (66)

for some value of the parameter 119906 of scalar functions (Ω119906) on

H and ℓ is in the part of (119872119866) foliated by this family suchthat at each point in this region ℓ is a future null normal toH119906for some value of 119906 Set (H ℎ Ω) as a member of (H

119906)

for some 119906 It is easy to show that poundℓℎ = 2ℓ(ln(Ω))ℎ that is ℓ

is conformal Killing with respect to the metric ℎ which is theimage of the corresponding Killing null vector with respectto the metric 119902 Thus as per Proposition 6 ((H

119906) (ℎ119906)) is a

family of totally umbilical null hypersurfaces of (119872119866) suchthat the second fundamental form 119861 of its each member(H ℎ Ω) satisfies

119861 (119883 119884) = ℓ (ln (Ω)) ℎ (119883 119884) forall119883 119884 isin 119879 (H) (67)

where the conformal function 119891 = ℓ(ln(Ω)) Thus subject tocondition (ii) ofDefinition 8 eachmember of (H

119906) is an ENH

of (119872119866) for some value of 119906 Suppose the null hypersurface(H 119902) of (119872 119892) is an event or an isolated horizon such that(65) and (66) hold Then the family ((H

119906) (ℎ119906)) of ENHs

evolves into the corresponding event or isolated horizon(H 119902) when 119866 rarr 119892 that is Ω rarr 1 The following is anexample of such an ENH of a BH spacetime which evolvesinto a weakly isolated horizon

Example 10 Let (119872 119892) be Einstein static universe withmetric 119892 given by

119892 = minus1198891199052+ 1198891199032+ 1199032(1198891205792+ sin21205791198891206012) (68)

where (119905 119903 120579 120601) is a spherical coordinate systemThis metricis singular at 119903 = 0 and sin 120579 = 0 We therefore choose theranges 0 lt 119903 lt infin 0 lt 120579 lt 120587 and 0 lt 120601 lt 2120587 for which itis a regular metric Take two null coordinates V and 119908 withrespect to a pseudo-orthonormal basis such that V = 119905+119903 and119908 = 119905 minus 119903 (V ge 119908) We get the following transformed metric

119892 = minus119889V119889119908 + (V minus 1199082

)

2

(1198891205792+ sin21205791198891206012) (69)

whereminusinfin lt V119908 lt infinThe absence of the terms119889V2 and1198891199082in (69) implies that the hypersurfaces 119908 = constant and V =constant are null future and past-directed hypersurfacesrespectively since 119908

119886119908119887120578119886119887

= 0 = V119886V119887120578119886119887 Consider a

future-directed null hypersurface (H 119902 119908 = constant) of


(119872 119892) It is easy to see that the degenerate metric 119902 of thisH is given by

119902 = (V minus 1199082

)

2

(1198891205792+ sin2120579 1198891206012) 119908 = constant (70)

which is time-independent and has topology 119877 times 1198782 There-fore pound

ℓ119902 = 0 that is H 119902 119908 = constant is totally geodesic

in (119872 119892) which further implies that its future-directed nullnormal ℓ has vanishing expansion 120579

(ℓ)= 0 Thus subject to

the energy condition (3) of Definition 1 this H is an NEHwhich (see Ashtekar-Fairhust-Krishnan [9]) can be a weaklyisolated horizon

Consider (119872119866) the de-Sitter spacetime [19] which is ofconstant positive curvature and is topologically 1198771 times 1198783 withits metric 119866 given by

119866 = minus1198891199052+ 1198862cosh2 ( 119905

119886) 1198891199032+ 1199032(1198891205792+ sin2120579 1198891206012)

(71)

where 119886 is nonzero constant Its spacial slices (119905 = constant)are 3-spheres Introducing a new timelike coordinate 120591 =

2 arctan(exp(1199052)) minus (1205872) we get

119866 = 1198862cosh2 (120591

119886) 119892 (minus

120587

2lt 120591 lt

120587

2) (72)

where the metric 119892 is given by (69)Thus the de-Sitter space-time (119872119866) is locally conformal to the Einstein static uni-verse (119872 119892) with the conformal function Ω = 119886 cosh(120591119886)As discussed above we have a family of null hypersurfaces((H119906) (ℎ119906)) in (119872119866) whose metrics (ℎ

119906) are conformally

related to the metric 119902 by the transformation

ℎ119906= 1198862cosh2 (120591

119886) 119902 = 119886

2cosh2 (120591119886)

times (V minus 1199082

)

2

(1198891205792+ sin21205791198891206012)

(73)

for some value of the parameter 119906 = 120591 of scalar functions(Ω120591) on H where 119908 = constant and 119902 is the metric of the

null hypersurface (H 119902) of (119872 119892) Set (H ℎ ℓ) as a memberof this family ((H

119906) (ℎ119906)) Then the following holds

poundℓℎ = 2119886ℓ (ln(cosh (120591

119886)) ℎ)

119861 (119883 119884) = 119891ℎ (119883 119884) = 119886ℓ (ln(cosh (120591119886)) ℎ (119883 119884))

forall119883 119884 isin 119879 (H)

(74)

Therefore we have a family of totally umbilical time-dependent null hypersurfaces ((H

119906) (ℎ119906)) of (119872119866) Subject

to the condition (ii) of Definition 8 this is a family of ENHs ofthe de-Sitter spacetime (119872119866) which can evolve into a WIHwhen 119866 rarr 119892 at the equilibrium state of119872

Generic Model Consider a spacetime (119872 119892) withSchwarzschild metric

119892 = minus119860(119903)21198891199032+ 119860(119903)

minus21198891199032+ 1199032(sin2120579 1198891206012 + 1198891205792)

(75)

where 119860(119903)2 = 1 + 2119898119903 and 119898 and 119903 are the mass and theradius of a spherical body Construct a 2-dimensional sub-manifold S of119872 as an intersection of a timelike hypersurface119903 = constant with a spacelike hypersurface 119905 = constantChoose at each point of S a null direction perpendicular inS smoothly depending on the foot-point There are two suchpossibilities the ingoing and the outgoing radical directionsLet H be the union of all geodesics with the chosen (sayoutgoing) initial direction This is a submanifold near SMoreover the symmetry of the situation guarantees that His a submanifold everywhere except at points where it meetsthe centre of symmetry It is very easy to verify that H is atotally umbilical null hypersurface of 119872 Since H does notinclude the points where it meets the centre of symmetry it isproper totally umbilical in119872 that isH does not evolve intoa totally geodesic portionTherefore subject to condition (ii)of Definition 8 and considering H a member of the family(H119906) we have a physical model of generic ENHs of the black

hole Schwarzschild spacetime (119872 119892)

5 Discussion

In this paper we introduced a quasilocal definition of a familyof null hypersurfaces called ldquoevolving null horizons (ENH)rdquoof a spacetime manifold This definition uses some basicresults taken from the differential geometry of totally umbil-ical hypersurfaces as opposed to present day use of totallygeodesic geometry for event and isolated horizons The firstpart of this paper includes enough background informationon the event three types of isolated and dynamical horizonswith a brief on their respective use in the study of black holespacetimes and need for introducing evolving null horizons

The rest of this paper is focused on a variety of examples tojustify the existence of ENHs and in some cases relate themwith event and isolated horizons of a black hole spacetimeThus we have shown that an ENH describes the geometry ofthe null surface of a dynamical spacetime in particular a BHspacetime that also can be evolved into an event or isolatedhorizon at the equilibrium state of a landing spacetimeThereis ample scope of further study on geometricphysical prop-erties of ENHs An input of the interested readers on what ispresented so far is desired before one starts working on prop-erties of ENHs in particular reference to the null version ofsome known results on dynamical horizons As is the case ofintroducing any new concept there may be several questionsonwhat we have presented in this paper At this point in timethe following two fundamental issues need to be addressed

We know that an event horizon always exists in blackhole asymptotically flat spacetimes under a weak cosmiccensorship condition and isolated horizons are preciselymeant to model specifically quasilocal equilibrium regimesby offering a precise geometric approximation Since so farwe only have two specific models of an ENH in a black holespacetime one may ask Does there always exist an ENH in ablack hole spacetime

Secondly since an ENHH comes with an assigned folia-tion of null hypersurfaces its unique existence is questionableTherefore one may ask Is there a canonical or unique choiceof an evolving null horizon


Interested readers are invited for an input on these twoand any other issues

Finally it is well-known that the rich geometry of totallyumbilical submanifolds has a variety of uses in the worldof mathematics and physics Besides the needed study onthe properties of evolving null horizons in black hole theoryfurther study on the mathematical theory of this paper alonehas ample scope to a new way of research on the geometry oftotally umbilical null hypersurfaces in a general semiRieman-nian and for physical applications a spacetime manifold

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] S W Hawking ldquoThe event horizonsrdquo in Black Holes CDeWitt and B DeWitt Eds Norht Holland Amsterdam TheNetherlands 1972

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

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

[4] P Hajıcek ldquoStationary electrovacuum spacetimes with bifurcatehorizonsrdquo Journal of Mathematical Physics vol 16 pp 518ndash5221975

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

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

[7] P T Chrusciel ldquoRecent results in mathematical relativityrdquoClassical Quantum Gravity pp 1ndash17 2005

[8] A Ashtekar and B Krishnan ldquoIsolated and dynamical horizonsand their applicationsrdquo Living Reviews in Relativity vol 7 pp1ndash91 2004

[9] A Ashtekar S Fairhurst and B Krishnan ldquoIsolated horizonshamiltonian evolution and the first lawrdquo Physical Review D vol62 no 10 Article ID 104025 29 pages 2000

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

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

[12] E Gourgoulhon and J L Jaramillo ldquoA 3 + 1 perspective on nullhypersurfaces and isolated horizonsrdquo Physics Reports vol 423no 4-5 pp 159ndash294 2006

[13] S A Hayward ldquoGeneral laws of black-hole dynamicsrdquo PhysicalReview D vol 49 no 12 pp 6467ndash6474 1994

[14] A Ashtekar and B Krishnan ldquoDynamical horizons energyangular momentum fluxes and balance lawsrdquo Physical ReviewLetters vol 89 no 26 Article ID 261101 4 pages 2002

[15] A Ashtekar and B Krishnan ldquoDynamical horizons and theirpropertiesrdquo Physical ReviewD vol 68 no 10 Article ID 10403025 pages 2003

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

[17] K L Duggal and A Bejancu Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications vol 364 of Math-ematics and Its Applications Kluwer Academic PublishersDordrecht The Netherlands 1996

[18] K L Duggal and D H Jin Null Curves and Hypersurfaces ofSemi-Riemannian Manifolds World Scientific Hackensack NJUSA 2007

[19] S W Hawking and G F R Ellis The Large Scale Structure ofSpace-Time Cambridge Monographs onMathematical Physicsno 1 Cambridge University Press London UK 1973

[20] T Damour ldquoBlack-hole eddy currentsrdquo Physical Review D vol18 no 10 pp 3598ndash3604 1978

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Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


to find a quasilocal concept of a horizon which requiresonly minimum number of conditions to detect a black holeand study its properties To achieve this objective in a 1999paper [5] Asktekar et al introduced the following concept ofisolated horizons

12 Isolated Horizons Before giving the definition of isolatedhorizons we recall some features of the intrinsic geometryof 3-dimensional null hypersurface say H of a spacetime(119872 119892) where themetric 119892

119894119895has signature (minus + + +) Denote

by 119902119894119895=119892119894119895

larr997888the intrinsic degenerate induced metric on H

which is the pull back of 119892119894119895 where an under arrow denotes

the pullback toH Degenerate 119902119894119895has signature (0 + +) and

does not have an inverse in the standard sense but in theweaker sense it admits an inverse 119902119894119895 if it satisfies 119902

119894119896119902119895119898119902119896119898

=

119902119894119895 Using this the expansion 120579

(ℓ)is defined by

120579(ℓ)= 119902119894119895nabla119894ℓ119895 (1)

where ℓ119894 is a future-directed null normal to H and nabla is theLevi-Civita connection on 119872 The vorticity-free Raychaud-huri equation is given by

119889 (120579(ℓ))

119889119904= minus119877119894119895ℓ119894ℓ119895minus 120590119894119895120590119894119895minus1205792

2 (2)

where 120590119894119895= nablalarr997888(119894ℓ119895)minus(12)120579

(ℓ)119902119894119895is the shear tensor 119904 is a

pseudoarc parameter such that ℓ is null geodesic and 119877119894119895is

the Ricci tensor of119872 We say that two null normals ℓ119894 and ℓ119894

belong to the same equivalence class [ℓ] if ℓ119894 = 119888ℓ119894 for some

positive constant 119888 The following are three notions of iso-lated horizons namely nonexpanding weakly and strongerisolated horizons respectively

Definition 1 A null hypersurface (H 119902) of a 4-dimensionalspacetime (119872 119892) is called a nonexpanding horizon (NEH) if

(1) H has a topology 119877 times 1198782(2) any null normal ℓ of H has vanishing expansion

120579(ℓ)= 0

(3) all equations of motion hold at H and stress energytensor 119879




Condition (1) is a restriction on topology of H whichguarantee that marginally trapped surfaces are related to ablack hole spacetime Condition (2) and the energy condition

of (3) imply from the Raychaudhuri equation (2) that119879119894119895ℓ119895

larr997888= 0

and = nablalarr997888(119894ℓ119895)equiv poundℓ119902119894119895= 0 on H which further implies that

the metric 119902119894119895is time-independent Note that pound

ℓ119902119894119895= 0 onH

does not necessarily imply that ℓ is a Killing vector of the fullmetric 119892

119894119895 In general there does not exist a unique induced

connection onH due to degenerate 119902119894119895 However on anNEH

the property poundℓ119902119894119895= 0 implies that the spacetime connection

nabla induces a unique (torsion-free) connection say D on Hwhich is compatible with 119902

119894119895

Definition 2 The pair (H [ℓ]) is called a weakly isolatedhorizon (WIH) if H is an NEH and each normal ℓ isin [ℓ]

satisfies

(poundℓD119894minusD119894poundℓ) ℓ119894= 0 (3)

Condition (3) implies that in addition to the metric 119902119894119895

the connection componentD119894ℓ119895 is also time-independent for

a WIH Given a NEH one can always have an equivalenceclass [ℓ] (which is not unique) of null normals such that(H [ℓ]) is aWIH Ashtekar et al [9] have discussed the issueof ldquoFreedom of the choice of ℓrdquo in satisfying condition (3)

Definition 3 A WIH (H [ℓ]) is called an isolated horizon(IH) if the full connectionD is time-independent that is if

(poundℓD119894minusD119894poundℓ) 119881119895= 0 (4)

for arbitrary vector fields 119881 tangent toH

An IH is stronger notion of isolation as its condition(4) cannot always be satisfied by a choice of null normalsIsolated horizons are quasilocal and do not require theknowledge of the whole spacetime The class of spacetimeshaving isolated horizons is quite big Any Killing horizonwhich is topologically 119877times1198782 is a trivial example of an isolatedhorizon See Ashtekar and Krishnan [8] Ashtekar et al [10]Lewandowski [11] and Gourgoulhon and Jaramillo [12] forexamples and their physical use

On the other hand we know that the isolated horizonsmodel specifically quasilocal equilibrium regimes of blackhole spacetimes However in nature black holes are rarely inequilibrium This led to research on a quasilocal frameworkto describe the geometry of the surface of the dynamicalblack hole not just at its equilibrium state First attempt inthis direction was made by Hayward [13] in 1994 usingthe framework of (2 + 2)-formalism based on the notion oftrapped surfaces He proposed the following notion of futureouter and trapping horizons (FOTH)

Definition 4 A future outer and trapped horizon (FOTH)is a three-manifold Σ foliated by family of closed 2-surfacessuch that (i) one of its future-directed null normal say ℓ haszero expansion 120579

(ℓ)= 0 (ii) the other null normal k has

negative expansion 120579(k) lt 0 and (iii) the directional deriva-

tive of 120579(ℓ)

along k is negative poundn120579(ℓ) lt 0

Σ is either spacelike or null for which 120579(k) = 0 and poundk120579(ℓ) =

0 Hayward [13] derived the following general laws of blackhole dynamics

(a) Zeroth Law The total trapping gravity of a compactouter marginal surface has an upper bound attainedif and only if the trapping gravity is constant

(b) First Law The variation of the area form along anouter trapping horizon is determined by the trappinggravity and an energy flux

After this Ashktekar and Krishnan [14 15] observedthat in dynamical situations Haywardrsquos condition (iii) is not





(ℓ)= 0














ℎ (ℓ 119883) = 0 forall119883 isin 119879 (H) (5)


119906) so that








119906)




Wℓ 119879119901H 997904rArr 119879

119901H

119883 997904rArr nabla119883ℓ

(6)



119861 (119883 119884) = ℎ (Wℓ119883119884) = ℎ (nabla

119883ℓ 119884) (7)



119861 (119883 119884) = 119891ℎ (119883 119884) forall119883 119884 isin 119879 (H) (8)










119883ℓ 119884) + ℎ(nabla

119884ℓ 119883) and 119861(119883 119884)


119861 (119883 119884) =1





poundℓℎ (119883 119884) = ℎ (nabla

119883ℓ 119884) + ℎ (nabla

119884ℓ 119883) = 2ℎ (nabla

119883ℓ 119884)

= 2ℎ (Wℓ119883119884) = 2119891ℎ (119883 119884)

(11)



119879119901119872 rarr 119879



Σ119905+119889119905


119892119894119895119889119909119894119889119909119895= (minus120582

2+ |119880|2) 1198891199052+ 2119892119886119887119880119886119889119909119887119889119905 + 119892

119886119887119889119909119886119889119909119887

(12)



1 1199092 1199093)


t = 120582n + 119880 with n sdot 119880 = 0 (13)



119905= H cap Σ



S119905119906= H119906cap Σ119905 (14)

where S119905119906=1


119905be a




ℓ119894=119889119909119894

119889119905 (15)


119892 (119889119905 ℓ) = nablaℓ119905 = 1 (16)



(17)


119905119906)





ℓ = 119890120588119889119906 (19)





119892 (ℓ k) = minus1 119892 (k k) = 0 (20)


k = 1

2120582(n minus s) (21)


II 119879119901119872 997888rarr 119879

119901H

119883 997888rarr 119883 + 119892 (ℓ119883) k(22)





119905119906) ofH



1198901 1198902 on S




120579(ℓ)= 119861 (119890

1 1198901) + 119861 (119890

2 1198902) (24)


120579(ℓ)= 119861 (119890

1 1198901) + 119861 (119890

2 1198902) = 119891 ℎ (119890

1 1198901) + ℎ (119890

2 1198902) = 2119891

(25)


vanishes that is

119861 (1198901 1198901) + 119861 (119890

2 1198902) = 0 (26)


|H

119879119872|H = 119879Hoplusorth tr (119879H) (27)


nabla119883119884 = D

119883119884 + 119861 (119883 119884) k (28)



ℓ defined by

(6) will satisfy

D119883ℓ = W

ℓ119883 forall119883 isin 119879H (30)


(D119883ℎ) (119884 119885) = 119861 (119883 119884) 120578 (119885) + 119861 (119883 119885) 120578 (119884)

forall119883 119884 119885 isin Γ (119879H|U)

(31)



119892 (119877 (119883 119884)119885 119881) = ℎ (R (119883 119884)119885 119881) forall119881 isin 119879119878

119892 (119877 (119883 119884)119885 ℓ) = ℎ (R (119883 119884)119885 ℓ)


119884119861) (119883 119885)

+ 119861 (119884 119885) 120591 (119883) minus 119861 (119883 119885) 120591 (119884)

119892 (119877 (119883 119884)119885 k) = ℎ (R (119883 119884)119885 k)

(32)



(33)


1 1198902


R (119883 119884) = ℎ (R (1198901 119883) 119884 119890

1)

+ ℎ (R (1198902 119883) 119884 119890

2) + ℎ (R (ℓ 119883) 119884 k)

(34)



R (119883 119884) minusR (119884119883) = 2119889120591 (119883 119884) (35)




nabla119883119884 = D

119883119884 + 119891119892 (119883 119884) k

nabla119883k = minus119860k119883

(D119883ℎ) (119884 119885) = 119891 119892 (119883 119884) 120578 (119885) + 119892 (119883 119885) 120578 (119885)

forall119883 119884 119885 isin Γ (119879H|U)

(36)












1

2120579(ℓ)ℎ = (2119891 minus

1

2120579(ℓ)) ℎ (37)














ℓ = t + 119881 + (120582 minus 120572) s (39)


ℓH= t + 119881 + 119891s (40)


2 1199093) on (H ℎ)


ℎ = ℎ119860119861119889119909119860119889119909119861= ℎ1199051199051198891199052+ 2ℎ119905119896119889119905 119889119909119896+ ℎ119896119898119889119909119896119889119909119898

(41)

where 2 le 119896119898 le 3 and

ℎ119905119905= 119881119896119881119896+ 1198912 ℎ

119905119896= 119880119896= 120572s119896minus 119881119896= minus119881119896 (42)











119902 = 119902119860119861119889119909119860119889119909119861= 119902119896119898(119889119909119896minus 119881119896119889119905) (119889119909

119898minus 119881119898119889119905)

(43)


120572(H119902)= 120582 is constant on (H 119902)

ℓ(H119902)= t + 119881

(44)





120597119906

120597119905= 0


(45)







by the equation

119905 = 119865 (119909 119910 119911) (46)



119906 (119905 119909 119910 119911) = 119865 minus 119905 + 119887 (47)



1015840

119909 1198651015840

119910 1198651015840



(1198651015840

119909)2

+ (1198651015840

119910)2

+ (1198651015840

119911)2

= 1 (48)


ℓ119894= 119890120588(1 1198651015840

119909 1198651015840

119910 1198651015840

119911) ℓ

119894= 119890minus120588(minus1 119865

1015840

119909 1198651015840

119910 1198651015840

119911) (49)


k119894 = 1

2119890120588(minus1 minus119865

1015840

119909 minus1198651015840

119910 minus1198651015840

119911)

k119894=1

2119890minus120588(1 minus119865

1015840

119909 minus1198651015840

119910 minus1198651015840

119911)

(50)


1198821= 1198651015840

119911

120597

120597119909minus 1198651015840

119909

120597

120597119911 1198822= 1198651015840

119911

120597

120597119910minus 1198651015840

119910

120597

120597119911 (51)




119883ℓ ℓ) = 0




(ℓ)of


1 1198902 of St

1198901=

1

100381710038171003817100381711988211003817100381710038171003817

1198821

1198902=

1

10038161003816100381610038161198651015840

119911

1003816100381610038161003816

100381710038171003817100381711988211003817100381710038171003817

10038171003817100381710038171198821

1003817100381710038171003817

21198822minus 119892 (119882

11198822)1198821

(52)


119861 (1198901 1198901) + 119861 (119890

2 1198902)

=1

(1198651015840

119911)2ℎ11

times ((1198651015840

119911)2

+ (ℎ12)2) 11986111minus 2ℎ11ℎ1211986112+ (ℎ11)211986122

(53)

where we put ℎ120572120573= ℎ(119882

120572119882120573) and 119861

120572120573= 119861(119882

120572119882120573) 120572 120573 isin


ℎ11= 1 minus (119865

1015840

119910)2

ℎ12= 1198651015840

1199091198651015840

119910 ℎ

22= 1 minus (119865

1015840

119909)2

(54)


120579(ℓ)= 119861 (119890

1 1198901) + 119861 (119890

2 1198902) = minus

1

(1198651015840

119911)2ℎ11

(11986510158401015840

119909119909+ 11986510158401015840

119910119910+ 11986510158401015840

119911119911)

(55)






equation

11986510158401015840

119909119909+ 11986510158401015840

119910119910+ 11986510158401015840

119911119911= 0 (56)





120579(ℓ)= minus

1

(1198651015840

119911)211989211

(11986510158401015840

119909119909+ 11986510158401015840

119910119910+ 11986510158401015840

119911119911) = 2119891 (57)






119905 = 119865 (119909 119910 119911) = 119903 = radic1199092+ 1199102+ 1199112 (58)


0be


0)119906) given by

119906 (119905 119909 119910 119911) = 119903 minus 119905 + 119887 with 119905 = radic1199092 + 1199102 + 1199112 (59)


normal to Λ30are

ℓ119894= 119890120588(1

119909

119905119910

119905119911

119905) ℓ

119894= 119890minus120588(minus1

119909

119905119910

119905119911

119905) (60)


nabla119883ℓ = D

119883ℓ = 119906119883 forall119883 isin (Λ

3

0)119906 (61)



3

0)119906 (62)


0)119906 = 0


n119894 = (1 0 0 0) s119894 = (0 119909119905119910

119905119911

119905) (63)



120579(ℓ)= minus

1

(1199031015840

119911)2ℎ11

(11990310158401015840

119909119909+ 11990310158401015840

119910119910+ 11990310158401015840

119911119911)

= minus21199053

1199112(1199052minus 1199102)= minus

21199053

1199112(1199092+ 1199112)= 2119891 = 0

(64)




ℓ119902 = 0


119866 = Ω2119892 (65)




ℎ119906= Ω2

119906119902 (66)



119906)



119906) (ℎ119906)) is a




119906) is an ENH


119906) (ℎ119906)) of ENHs



119892 = minus1198891199052+ 1198891199032+ 1199032(1198891205792+ sin21205791198891206012) (68)


119892 = minus119889V119889119908 + (V minus 1199082

)

2

(1198891205792+ sin21205791198891206012) (69)


119886119908119887120578119886119887

= 0 = V119886V119887120578119886119887 Consider a




119902 = (V minus 1199082

)

2

(1198891205792+ sin2120579 1198891206012) 119908 = constant (70)







119866 = minus1198891199052+ 1198862cosh2 ( 119905

119886) 1198891199032+ 1199032(1198891205792+ sin2120579 1198891206012)

(71)



119866 = 1198862cosh2 (120591

119886) 119892 (minus

120587

2lt 120591 lt

120587

2) (72)




ℎ119906= 1198862cosh2 (120591

119886) 119902 = 119886

2cosh2 (120591119886)


)

2

(1198891205792+ sin21205791198891206012)

(73)





119886)) ℎ)

119861 (119883 119884) = 119891ℎ (119883 119884) = 119886ℓ (ln(cosh (120591119886)) ℎ (119883 119884))

forall119883 119884 isin 119879 (H)

(74)


119906) (ℎ119906)) of (119872119866) Subject



119892 = minus119860(119903)21198891199032+ 119860(119903)

minus21198891199032+ 1199032(sin2120579 1198891206012 + 1198891205792)

(75)



5 Discussion










References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













(ℓ)= 0














ℎ (ℓ 119883) = 0 forall119883 isin 119879 (H) (5)


119906) so that








119906)




Wℓ 119879119901H 997904rArr 119879

119901H

119883 997904rArr nabla119883ℓ

(6)



119861 (119883 119884) = ℎ (Wℓ119883119884) = ℎ (nabla

119883ℓ 119884) (7)



119861 (119883 119884) = 119891ℎ (119883 119884) forall119883 119884 isin 119879 (H) (8)










119883ℓ 119884) + ℎ(nabla

119884ℓ 119883) and 119861(119883 119884)


119861 (119883 119884) =1





poundℓℎ (119883 119884) = ℎ (nabla

119883ℓ 119884) + ℎ (nabla

119884ℓ 119883) = 2ℎ (nabla

119883ℓ 119884)

= 2ℎ (Wℓ119883119884) = 2119891ℎ (119883 119884)

(11)



119879119901119872 rarr 119879



Σ119905+119889119905


119892119894119895119889119909119894119889119909119895= (minus120582

2+ |119880|2) 1198891199052+ 2119892119886119887119880119886119889119909119887119889119905 + 119892

119886119887119889119909119886119889119909119887

(12)



1 1199092 1199093)


t = 120582n + 119880 with n sdot 119880 = 0 (13)



119905= H cap Σ



S119905119906= H119906cap Σ119905 (14)

where S119905119906=1


119905be a




ℓ119894=119889119909119894

119889119905 (15)


119892 (119889119905 ℓ) = nablaℓ119905 = 1 (16)



(17)


119905119906)





ℓ = 119890120588119889119906 (19)





119892 (ℓ k) = minus1 119892 (k k) = 0 (20)


k = 1

2120582(n minus s) (21)


II 119879119901119872 997888rarr 119879

119901H

119883 997888rarr 119883 + 119892 (ℓ119883) k(22)





119905119906) ofH



1198901 1198902 on S




120579(ℓ)= 119861 (119890

1 1198901) + 119861 (119890

2 1198902) (24)


120579(ℓ)= 119861 (119890

1 1198901) + 119861 (119890

2 1198902) = 119891 ℎ (119890

1 1198901) + ℎ (119890

2 1198902) = 2119891

(25)


vanishes that is

119861 (1198901 1198901) + 119861 (119890

2 1198902) = 0 (26)


|H

119879119872|H = 119879Hoplusorth tr (119879H) (27)


nabla119883119884 = D

119883119884 + 119861 (119883 119884) k (28)



ℓ defined by

(6) will satisfy

D119883ℓ = W

ℓ119883 forall119883 isin 119879H (30)


(D119883ℎ) (119884 119885) = 119861 (119883 119884) 120578 (119885) + 119861 (119883 119885) 120578 (119884)

forall119883 119884 119885 isin Γ (119879H|U)

(31)



119892 (119877 (119883 119884)119885 119881) = ℎ (R (119883 119884)119885 119881) forall119881 isin 119879119878

119892 (119877 (119883 119884)119885 ℓ) = ℎ (R (119883 119884)119885 ℓ)


119884119861) (119883 119885)

+ 119861 (119884 119885) 120591 (119883) minus 119861 (119883 119885) 120591 (119884)

119892 (119877 (119883 119884)119885 k) = ℎ (R (119883 119884)119885 k)

(32)



(33)


1 1198902


R (119883 119884) = ℎ (R (1198901 119883) 119884 119890

1)

+ ℎ (R (1198902 119883) 119884 119890

2) + ℎ (R (ℓ 119883) 119884 k)

(34)



R (119883 119884) minusR (119884119883) = 2119889120591 (119883 119884) (35)




nabla119883119884 = D

119883119884 + 119891119892 (119883 119884) k

nabla119883k = minus119860k119883

(D119883ℎ) (119884 119885) = 119891 119892 (119883 119884) 120578 (119885) + 119892 (119883 119885) 120578 (119885)

forall119883 119884 119885 isin Γ (119879H|U)

(36)












1

2120579(ℓ)ℎ = (2119891 minus

1

2120579(ℓ)) ℎ (37)














ℓ = t + 119881 + (120582 minus 120572) s (39)


ℓH= t + 119881 + 119891s (40)


2 1199093) on (H ℎ)


ℎ = ℎ119860119861119889119909119860119889119909119861= ℎ1199051199051198891199052+ 2ℎ119905119896119889119905 119889119909119896+ ℎ119896119898119889119909119896119889119909119898

(41)

where 2 le 119896119898 le 3 and

ℎ119905119905= 119881119896119881119896+ 1198912 ℎ

119905119896= 119880119896= 120572s119896minus 119881119896= minus119881119896 (42)











119902 = 119902119860119861119889119909119860119889119909119861= 119902119896119898(119889119909119896minus 119881119896119889119905) (119889119909

119898minus 119881119898119889119905)

(43)


120572(H119902)= 120582 is constant on (H 119902)

ℓ(H119902)= t + 119881

(44)





120597119906

120597119905= 0


(45)







by the equation

119905 = 119865 (119909 119910 119911) (46)



119906 (119905 119909 119910 119911) = 119865 minus 119905 + 119887 (47)



1015840

119909 1198651015840

119910 1198651015840



(1198651015840

119909)2

+ (1198651015840

119910)2

+ (1198651015840

119911)2

= 1 (48)


ℓ119894= 119890120588(1 1198651015840

119909 1198651015840

119910 1198651015840

119911) ℓ

119894= 119890minus120588(minus1 119865

1015840

119909 1198651015840

119910 1198651015840

119911) (49)


k119894 = 1

2119890120588(minus1 minus119865

1015840

119909 minus1198651015840

119910 minus1198651015840

119911)

k119894=1

2119890minus120588(1 minus119865

1015840

119909 minus1198651015840

119910 minus1198651015840

119911)

(50)


1198821= 1198651015840

119911

120597

120597119909minus 1198651015840

119909

120597

120597119911 1198822= 1198651015840

119911

120597

120597119910minus 1198651015840

119910

120597

120597119911 (51)




119883ℓ ℓ) = 0




(ℓ)of


1 1198902 of St

1198901=

1

100381710038171003817100381711988211003817100381710038171003817

1198821

1198902=

1

10038161003816100381610038161198651015840

119911

1003816100381610038161003816

100381710038171003817100381711988211003817100381710038171003817

10038171003817100381710038171198821

1003817100381710038171003817

21198822minus 119892 (119882

11198822)1198821

(52)


119861 (1198901 1198901) + 119861 (119890

2 1198902)

=1

(1198651015840

119911)2ℎ11

times ((1198651015840

119911)2

+ (ℎ12)2) 11986111minus 2ℎ11ℎ1211986112+ (ℎ11)211986122

(53)

where we put ℎ120572120573= ℎ(119882

120572119882120573) and 119861

120572120573= 119861(119882

120572119882120573) 120572 120573 isin


ℎ11= 1 minus (119865

1015840

119910)2

ℎ12= 1198651015840

1199091198651015840

119910 ℎ

22= 1 minus (119865

1015840

119909)2

(54)


120579(ℓ)= 119861 (119890

1 1198901) + 119861 (119890

2 1198902) = minus

1

(1198651015840

119911)2ℎ11

(11986510158401015840

119909119909+ 11986510158401015840

119910119910+ 11986510158401015840

119911119911)

(55)






equation

11986510158401015840

119909119909+ 11986510158401015840

119910119910+ 11986510158401015840

119911119911= 0 (56)





120579(ℓ)= minus

1

(1198651015840

119911)211989211

(11986510158401015840

119909119909+ 11986510158401015840

119910119910+ 11986510158401015840

119911119911) = 2119891 (57)






119905 = 119865 (119909 119910 119911) = 119903 = radic1199092+ 1199102+ 1199112 (58)


0be


0)119906) given by

119906 (119905 119909 119910 119911) = 119903 minus 119905 + 119887 with 119905 = radic1199092 + 1199102 + 1199112 (59)


normal to Λ30are

ℓ119894= 119890120588(1

119909

119905119910

119905119911

119905) ℓ

119894= 119890minus120588(minus1

119909

119905119910

119905119911

119905) (60)


nabla119883ℓ = D

119883ℓ = 119906119883 forall119883 isin (Λ

3

0)119906 (61)



3

0)119906 (62)


0)119906 = 0


n119894 = (1 0 0 0) s119894 = (0 119909119905119910

119905119911

119905) (63)



120579(ℓ)= minus

1

(1199031015840

119911)2ℎ11

(11990310158401015840

119909119909+ 11990310158401015840

119910119910+ 11990310158401015840

119911119911)

= minus21199053

1199112(1199052minus 1199102)= minus

21199053

1199112(1199092+ 1199112)= 2119891 = 0

(64)




ℓ119902 = 0


119866 = Ω2119892 (65)




ℎ119906= Ω2

119906119902 (66)



119906)



119906) (ℎ119906)) is a




119906) is an ENH


119906) (ℎ119906)) of ENHs



119892 = minus1198891199052+ 1198891199032+ 1199032(1198891205792+ sin21205791198891206012) (68)


119892 = minus119889V119889119908 + (V minus 1199082

)

2

(1198891205792+ sin21205791198891206012) (69)


119886119908119887120578119886119887

= 0 = V119886V119887120578119886119887 Consider a




119902 = (V minus 1199082

)

2

(1198891205792+ sin2120579 1198891206012) 119908 = constant (70)







119866 = minus1198891199052+ 1198862cosh2 ( 119905

119886) 1198891199032+ 1199032(1198891205792+ sin2120579 1198891206012)

(71)



119866 = 1198862cosh2 (120591

119886) 119892 (minus

120587

2lt 120591 lt

120587

2) (72)




ℎ119906= 1198862cosh2 (120591

119886) 119902 = 119886

2cosh2 (120591119886)


)

2

(1198891205792+ sin21205791198891206012)

(73)





119886)) ℎ)

119861 (119883 119884) = 119891ℎ (119883 119884) = 119886ℓ (ln(cosh (120591119886)) ℎ (119883 119884))

forall119883 119884 isin 119879 (H)

(74)


119906) (ℎ119906)) of (119872119866) Subject



119892 = minus119860(119903)21198891199032+ 119860(119903)

minus21198891199032+ 1199032(sin2120579 1198891206012 + 1198891205792)

(75)



5 Discussion










References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Wℓ 119879119901H 997904rArr 119879

119901H

119883 997904rArr nabla119883ℓ

(6)



119861 (119883 119884) = ℎ (Wℓ119883119884) = ℎ (nabla

119883ℓ 119884) (7)



119861 (119883 119884) = 119891ℎ (119883 119884) forall119883 119884 isin 119879 (H) (8)










119883ℓ 119884) + ℎ(nabla

119884ℓ 119883) and 119861(119883 119884)


119861 (119883 119884) =1





poundℓℎ (119883 119884) = ℎ (nabla

119883ℓ 119884) + ℎ (nabla

119884ℓ 119883) = 2ℎ (nabla

119883ℓ 119884)

= 2ℎ (Wℓ119883119884) = 2119891ℎ (119883 119884)

(11)



119879119901119872 rarr 119879



Σ119905+119889119905


119892119894119895119889119909119894119889119909119895= (minus120582

2+ |119880|2) 1198891199052+ 2119892119886119887119880119886119889119909119887119889119905 + 119892

119886119887119889119909119886119889119909119887

(12)



1 1199092 1199093)


t = 120582n + 119880 with n sdot 119880 = 0 (13)



119905= H cap Σ



S119905119906= H119906cap Σ119905 (14)

where S119905119906=1


119905be a




ℓ119894=119889119909119894

119889119905 (15)


119892 (119889119905 ℓ) = nablaℓ119905 = 1 (16)



(17)


119905119906)





ℓ = 119890120588119889119906 (19)





119892 (ℓ k) = minus1 119892 (k k) = 0 (20)


k = 1

2120582(n minus s) (21)


II 119879119901119872 997888rarr 119879

119901H

119883 997888rarr 119883 + 119892 (ℓ119883) k(22)





119905119906) ofH



1198901 1198902 on S




120579(ℓ)= 119861 (119890

1 1198901) + 119861 (119890

2 1198902) (24)


120579(ℓ)= 119861 (119890

1 1198901) + 119861 (119890

2 1198902) = 119891 ℎ (119890

1 1198901) + ℎ (119890

2 1198902) = 2119891

(25)


vanishes that is

119861 (1198901 1198901) + 119861 (119890

2 1198902) = 0 (26)


|H

119879119872|H = 119879Hoplusorth tr (119879H) (27)


nabla119883119884 = D

119883119884 + 119861 (119883 119884) k (28)



ℓ defined by

(6) will satisfy

D119883ℓ = W

ℓ119883 forall119883 isin 119879H (30)


(D119883ℎ) (119884 119885) = 119861 (119883 119884) 120578 (119885) + 119861 (119883 119885) 120578 (119884)

forall119883 119884 119885 isin Γ (119879H|U)

(31)



119892 (119877 (119883 119884)119885 119881) = ℎ (R (119883 119884)119885 119881) forall119881 isin 119879119878

119892 (119877 (119883 119884)119885 ℓ) = ℎ (R (119883 119884)119885 ℓ)


119884119861) (119883 119885)

+ 119861 (119884 119885) 120591 (119883) minus 119861 (119883 119885) 120591 (119884)

119892 (119877 (119883 119884)119885 k) = ℎ (R (119883 119884)119885 k)

(32)



(33)


1 1198902


R (119883 119884) = ℎ (R (1198901 119883) 119884 119890

1)

+ ℎ (R (1198902 119883) 119884 119890

2) + ℎ (R (ℓ 119883) 119884 k)

(34)



R (119883 119884) minusR (119884119883) = 2119889120591 (119883 119884) (35)




nabla119883119884 = D

119883119884 + 119891119892 (119883 119884) k

nabla119883k = minus119860k119883

(D119883ℎ) (119884 119885) = 119891 119892 (119883 119884) 120578 (119885) + 119892 (119883 119885) 120578 (119885)

forall119883 119884 119885 isin Γ (119879H|U)

(36)












1

2120579(ℓ)ℎ = (2119891 minus

1

2120579(ℓ)) ℎ (37)














ℓ = t + 119881 + (120582 minus 120572) s (39)


ℓH= t + 119881 + 119891s (40)


2 1199093) on (H ℎ)


ℎ = ℎ119860119861119889119909119860119889119909119861= ℎ1199051199051198891199052+ 2ℎ119905119896119889119905 119889119909119896+ ℎ119896119898119889119909119896119889119909119898

(41)

where 2 le 119896119898 le 3 and

ℎ119905119905= 119881119896119881119896+ 1198912 ℎ

119905119896= 119880119896= 120572s119896minus 119881119896= minus119881119896 (42)











119902 = 119902119860119861119889119909119860119889119909119861= 119902119896119898(119889119909119896minus 119881119896119889119905) (119889119909

119898minus 119881119898119889119905)

(43)


120572(H119902)= 120582 is constant on (H 119902)

ℓ(H119902)= t + 119881

(44)





120597119906

120597119905= 0


(45)







by the equation

119905 = 119865 (119909 119910 119911) (46)



119906 (119905 119909 119910 119911) = 119865 minus 119905 + 119887 (47)



1015840

119909 1198651015840

119910 1198651015840



(1198651015840

119909)2

+ (1198651015840

119910)2

+ (1198651015840

119911)2

= 1 (48)


ℓ119894= 119890120588(1 1198651015840

119909 1198651015840

119910 1198651015840

119911) ℓ

119894= 119890minus120588(minus1 119865

1015840

119909 1198651015840

119910 1198651015840

119911) (49)


k119894 = 1

2119890120588(minus1 minus119865

1015840

119909 minus1198651015840

119910 minus1198651015840

119911)

k119894=1

2119890minus120588(1 minus119865

1015840

119909 minus1198651015840

119910 minus1198651015840

119911)

(50)


1198821= 1198651015840

119911

120597

120597119909minus 1198651015840

119909

120597

120597119911 1198822= 1198651015840

119911

120597

120597119910minus 1198651015840

119910

120597

120597119911 (51)




119883ℓ ℓ) = 0




(ℓ)of


1 1198902 of St

1198901=

1

100381710038171003817100381711988211003817100381710038171003817

1198821

1198902=

1

10038161003816100381610038161198651015840

119911

1003816100381610038161003816

100381710038171003817100381711988211003817100381710038171003817

10038171003817100381710038171198821

1003817100381710038171003817

21198822minus 119892 (119882

11198822)1198821

(52)


119861 (1198901 1198901) + 119861 (119890

2 1198902)

=1

(1198651015840

119911)2ℎ11

times ((1198651015840

119911)2

+ (ℎ12)2) 11986111minus 2ℎ11ℎ1211986112+ (ℎ11)211986122

(53)

where we put ℎ120572120573= ℎ(119882

120572119882120573) and 119861

120572120573= 119861(119882

120572119882120573) 120572 120573 isin


ℎ11= 1 minus (119865

1015840

119910)2

ℎ12= 1198651015840

1199091198651015840

119910 ℎ

22= 1 minus (119865

1015840

119909)2

(54)


120579(ℓ)= 119861 (119890

1 1198901) + 119861 (119890

2 1198902) = minus

1

(1198651015840

119911)2ℎ11

(11986510158401015840

119909119909+ 11986510158401015840

119910119910+ 11986510158401015840

119911119911)

(55)






equation

11986510158401015840

119909119909+ 11986510158401015840

119910119910+ 11986510158401015840

119911119911= 0 (56)





120579(ℓ)= minus

1

(1198651015840

119911)211989211

(11986510158401015840

119909119909+ 11986510158401015840

119910119910+ 11986510158401015840

119911119911) = 2119891 (57)






119905 = 119865 (119909 119910 119911) = 119903 = radic1199092+ 1199102+ 1199112 (58)


0be


0)119906) given by

119906 (119905 119909 119910 119911) = 119903 minus 119905 + 119887 with 119905 = radic1199092 + 1199102 + 1199112 (59)


normal to Λ30are

ℓ119894= 119890120588(1

119909

119905119910

119905119911

119905) ℓ

119894= 119890minus120588(minus1

119909

119905119910

119905119911

119905) (60)


nabla119883ℓ = D

119883ℓ = 119906119883 forall119883 isin (Λ

3

0)119906 (61)



3

0)119906 (62)


0)119906 = 0


n119894 = (1 0 0 0) s119894 = (0 119909119905119910

119905119911

119905) (63)



120579(ℓ)= minus

1

(1199031015840

119911)2ℎ11

(11990310158401015840

119909119909+ 11990310158401015840

119910119910+ 11990310158401015840

119911119911)

= minus21199053

1199112(1199052minus 1199102)= minus

21199053

1199112(1199092+ 1199112)= 2119891 = 0

(64)




ℓ119902 = 0


119866 = Ω2119892 (65)




ℎ119906= Ω2

119906119902 (66)



119906)



119906) (ℎ119906)) is a




119906) is an ENH


119906) (ℎ119906)) of ENHs



119892 = minus1198891199052+ 1198891199032+ 1199032(1198891205792+ sin21205791198891206012) (68)


119892 = minus119889V119889119908 + (V minus 1199082

)

2

(1198891205792+ sin21205791198891206012) (69)


119886119908119887120578119886119887

= 0 = V119886V119887120578119886119887 Consider a




119902 = (V minus 1199082

)

2

(1198891205792+ sin2120579 1198891206012) 119908 = constant (70)







119866 = minus1198891199052+ 1198862cosh2 ( 119905

119886) 1198891199032+ 1199032(1198891205792+ sin2120579 1198891206012)

(71)



119866 = 1198862cosh2 (120591

119886) 119892 (minus

120587

2lt 120591 lt

120587

2) (72)




ℎ119906= 1198862cosh2 (120591

119886) 119902 = 119886

2cosh2 (120591119886)


)

2

(1198891205792+ sin21205791198891206012)

(73)





119886)) ℎ)

119861 (119883 119884) = 119891ℎ (119883 119884) = 119886ℓ (ln(cosh (120591119886)) ℎ (119883 119884))

forall119883 119884 isin 119879 (H)

(74)


119906) (ℎ119906)) of (119872119866) Subject



119892 = minus119860(119903)21198891199032+ 119860(119903)

minus21198891199032+ 1199032(sin2120579 1198891206012 + 1198891205792)

(75)



5 Discussion










References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











ℓ = 119890120588119889119906 (19)





119892 (ℓ k) = minus1 119892 (k k) = 0 (20)


k = 1

2120582(n minus s) (21)


II 119879119901119872 997888rarr 119879

119901H

119883 997888rarr 119883 + 119892 (ℓ119883) k(22)





119905119906) ofH



1198901 1198902 on S




120579(ℓ)= 119861 (119890

1 1198901) + 119861 (119890

2 1198902) (24)


120579(ℓ)= 119861 (119890

1 1198901) + 119861 (119890

2 1198902) = 119891 ℎ (119890

1 1198901) + ℎ (119890

2 1198902) = 2119891

(25)


vanishes that is

119861 (1198901 1198901) + 119861 (119890

2 1198902) = 0 (26)


|H

119879119872|H = 119879Hoplusorth tr (119879H) (27)


nabla119883119884 = D

119883119884 + 119861 (119883 119884) k (28)



ℓ defined by

(6) will satisfy

D119883ℓ = W

ℓ119883 forall119883 isin 119879H (30)


(D119883ℎ) (119884 119885) = 119861 (119883 119884) 120578 (119885) + 119861 (119883 119885) 120578 (119884)

forall119883 119884 119885 isin Γ (119879H|U)

(31)



119892 (119877 (119883 119884)119885 119881) = ℎ (R (119883 119884)119885 119881) forall119881 isin 119879119878

119892 (119877 (119883 119884)119885 ℓ) = ℎ (R (119883 119884)119885 ℓ)


119884119861) (119883 119885)

+ 119861 (119884 119885) 120591 (119883) minus 119861 (119883 119885) 120591 (119884)

119892 (119877 (119883 119884)119885 k) = ℎ (R (119883 119884)119885 k)

(32)



(33)


1 1198902


R (119883 119884) = ℎ (R (1198901 119883) 119884 119890

1)

+ ℎ (R (1198902 119883) 119884 119890

2) + ℎ (R (ℓ 119883) 119884 k)

(34)



R (119883 119884) minusR (119884119883) = 2119889120591 (119883 119884) (35)




nabla119883119884 = D

119883119884 + 119891119892 (119883 119884) k

nabla119883k = minus119860k119883

(D119883ℎ) (119884 119885) = 119891 119892 (119883 119884) 120578 (119885) + 119892 (119883 119885) 120578 (119885)

forall119883 119884 119885 isin Γ (119879H|U)

(36)












1

2120579(ℓ)ℎ = (2119891 minus

1

2120579(ℓ)) ℎ (37)














ℓ = t + 119881 + (120582 minus 120572) s (39)


ℓH= t + 119881 + 119891s (40)


2 1199093) on (H ℎ)


ℎ = ℎ119860119861119889119909119860119889119909119861= ℎ1199051199051198891199052+ 2ℎ119905119896119889119905 119889119909119896+ ℎ119896119898119889119909119896119889119909119898

(41)

where 2 le 119896119898 le 3 and

ℎ119905119905= 119881119896119881119896+ 1198912 ℎ

119905119896= 119880119896= 120572s119896minus 119881119896= minus119881119896 (42)











119902 = 119902119860119861119889119909119860119889119909119861= 119902119896119898(119889119909119896minus 119881119896119889119905) (119889119909

119898minus 119881119898119889119905)

(43)


120572(H119902)= 120582 is constant on (H 119902)

ℓ(H119902)= t + 119881

(44)





120597119906

120597119905= 0


(45)







by the equation

119905 = 119865 (119909 119910 119911) (46)



119906 (119905 119909 119910 119911) = 119865 minus 119905 + 119887 (47)



1015840

119909 1198651015840

119910 1198651015840



(1198651015840

119909)2

+ (1198651015840

119910)2

+ (1198651015840

119911)2

= 1 (48)


ℓ119894= 119890120588(1 1198651015840

119909 1198651015840

119910 1198651015840

119911) ℓ

119894= 119890minus120588(minus1 119865

1015840

119909 1198651015840

119910 1198651015840

119911) (49)


k119894 = 1

2119890120588(minus1 minus119865

1015840

119909 minus1198651015840

119910 minus1198651015840

119911)

k119894=1

2119890minus120588(1 minus119865

1015840

119909 minus1198651015840

119910 minus1198651015840

119911)

(50)


1198821= 1198651015840

119911

120597

120597119909minus 1198651015840

119909

120597

120597119911 1198822= 1198651015840

119911

120597

120597119910minus 1198651015840

119910

120597

120597119911 (51)




119883ℓ ℓ) = 0




(ℓ)of


1 1198902 of St

1198901=

1

100381710038171003817100381711988211003817100381710038171003817

1198821

1198902=

1

10038161003816100381610038161198651015840

119911

1003816100381610038161003816

100381710038171003817100381711988211003817100381710038171003817

10038171003817100381710038171198821

1003817100381710038171003817

21198822minus 119892 (119882

11198822)1198821

(52)


119861 (1198901 1198901) + 119861 (119890

2 1198902)

=1

(1198651015840

119911)2ℎ11

times ((1198651015840

119911)2

+ (ℎ12)2) 11986111minus 2ℎ11ℎ1211986112+ (ℎ11)211986122

(53)

where we put ℎ120572120573= ℎ(119882

120572119882120573) and 119861

120572120573= 119861(119882

120572119882120573) 120572 120573 isin


ℎ11= 1 minus (119865

1015840

119910)2

ℎ12= 1198651015840

1199091198651015840

119910 ℎ

22= 1 minus (119865

1015840

119909)2

(54)


120579(ℓ)= 119861 (119890

1 1198901) + 119861 (119890

2 1198902) = minus

1

(1198651015840

119911)2ℎ11

(11986510158401015840

119909119909+ 11986510158401015840

119910119910+ 11986510158401015840

119911119911)

(55)






equation

11986510158401015840

119909119909+ 11986510158401015840

119910119910+ 11986510158401015840

119911119911= 0 (56)





120579(ℓ)= minus

1

(1198651015840

119911)211989211

(11986510158401015840

119909119909+ 11986510158401015840

119910119910+ 11986510158401015840

119911119911) = 2119891 (57)






119905 = 119865 (119909 119910 119911) = 119903 = radic1199092+ 1199102+ 1199112 (58)


0be


0)119906) given by

119906 (119905 119909 119910 119911) = 119903 minus 119905 + 119887 with 119905 = radic1199092 + 1199102 + 1199112 (59)


normal to Λ30are

ℓ119894= 119890120588(1

119909

119905119910

119905119911

119905) ℓ

119894= 119890minus120588(minus1

119909

119905119910

119905119911

119905) (60)


nabla119883ℓ = D

119883ℓ = 119906119883 forall119883 isin (Λ

3

0)119906 (61)



3

0)119906 (62)


0)119906 = 0


n119894 = (1 0 0 0) s119894 = (0 119909119905119910

119905119911

119905) (63)



120579(ℓ)= minus

1

(1199031015840

119911)2ℎ11

(11990310158401015840

119909119909+ 11990310158401015840

119910119910+ 11990310158401015840

119911119911)

= minus21199053

1199112(1199052minus 1199102)= minus

21199053

1199112(1199092+ 1199112)= 2119891 = 0

(64)




ℓ119902 = 0


119866 = Ω2119892 (65)




ℎ119906= Ω2

119906119902 (66)



119906)



119906) (ℎ119906)) is a




119906) is an ENH


119906) (ℎ119906)) of ENHs



119892 = minus1198891199052+ 1198891199032+ 1199032(1198891205792+ sin21205791198891206012) (68)


119892 = minus119889V119889119908 + (V minus 1199082

)

2

(1198891205792+ sin21205791198891206012) (69)


119886119908119887120578119886119887

= 0 = V119886V119887120578119886119887 Consider a




119902 = (V minus 1199082

)

2

(1198891205792+ sin2120579 1198891206012) 119908 = constant (70)







119866 = minus1198891199052+ 1198862cosh2 ( 119905

119886) 1198891199032+ 1199032(1198891205792+ sin2120579 1198891206012)

(71)



119866 = 1198862cosh2 (120591

119886) 119892 (minus

120587

2lt 120591 lt

120587

2) (72)




ℎ119906= 1198862cosh2 (120591

119886) 119902 = 119886

2cosh2 (120591119886)


)

2

(1198891205792+ sin21205791198891206012)

(73)





119886)) ℎ)

119861 (119883 119884) = 119891ℎ (119883 119884) = 119886ℓ (ln(cosh (120591119886)) ℎ (119883 119884))

forall119883 119884 isin 119879 (H)

(74)


119906) (ℎ119906)) of (119872119866) Subject



119892 = minus119860(119903)21198891199032+ 119860(119903)

minus21198891199032+ 1199032(sin2120579 1198891206012 + 1198891205792)

(75)



5 Discussion










References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











R (119883 119884) minusR (119884119883) = 2119889120591 (119883 119884) (35)




nabla119883119884 = D

119883119884 + 119891119892 (119883 119884) k

nabla119883k = minus119860k119883

(D119883ℎ) (119884 119885) = 119891 119892 (119883 119884) 120578 (119885) + 119892 (119883 119885) 120578 (119885)

forall119883 119884 119885 isin Γ (119879H|U)

(36)












1

2120579(ℓ)ℎ = (2119891 minus

1

2120579(ℓ)) ℎ (37)














ℓ = t + 119881 + (120582 minus 120572) s (39)


ℓH= t + 119881 + 119891s (40)


2 1199093) on (H ℎ)


ℎ = ℎ119860119861119889119909119860119889119909119861= ℎ1199051199051198891199052+ 2ℎ119905119896119889119905 119889119909119896+ ℎ119896119898119889119909119896119889119909119898

(41)

where 2 le 119896119898 le 3 and

ℎ119905119905= 119881119896119881119896+ 1198912 ℎ

119905119896= 119880119896= 120572s119896minus 119881119896= minus119881119896 (42)











119902 = 119902119860119861119889119909119860119889119909119861= 119902119896119898(119889119909119896minus 119881119896119889119905) (119889119909

119898minus 119881119898119889119905)

(43)


120572(H119902)= 120582 is constant on (H 119902)

ℓ(H119902)= t + 119881

(44)





120597119906

120597119905= 0


(45)







by the equation

119905 = 119865 (119909 119910 119911) (46)



119906 (119905 119909 119910 119911) = 119865 minus 119905 + 119887 (47)



1015840

119909 1198651015840

119910 1198651015840



(1198651015840

119909)2

+ (1198651015840

119910)2

+ (1198651015840

119911)2

= 1 (48)


ℓ119894= 119890120588(1 1198651015840

119909 1198651015840

119910 1198651015840

119911) ℓ

119894= 119890minus120588(minus1 119865

1015840

119909 1198651015840

119910 1198651015840

119911) (49)


k119894 = 1

2119890120588(minus1 minus119865

1015840

119909 minus1198651015840

119910 minus1198651015840

119911)

k119894=1

2119890minus120588(1 minus119865

1015840

119909 minus1198651015840

119910 minus1198651015840

119911)

(50)


1198821= 1198651015840

119911

120597

120597119909minus 1198651015840

119909

120597

120597119911 1198822= 1198651015840

119911

120597

120597119910minus 1198651015840

119910

120597

120597119911 (51)




119883ℓ ℓ) = 0




(ℓ)of


1 1198902 of St

1198901=

1

100381710038171003817100381711988211003817100381710038171003817

1198821

1198902=

1

10038161003816100381610038161198651015840

119911

1003816100381610038161003816

100381710038171003817100381711988211003817100381710038171003817

10038171003817100381710038171198821

1003817100381710038171003817

21198822minus 119892 (119882

11198822)1198821

(52)


119861 (1198901 1198901) + 119861 (119890

2 1198902)

=1

(1198651015840

119911)2ℎ11

times ((1198651015840

119911)2

+ (ℎ12)2) 11986111minus 2ℎ11ℎ1211986112+ (ℎ11)211986122

(53)

where we put ℎ120572120573= ℎ(119882

120572119882120573) and 119861

120572120573= 119861(119882

120572119882120573) 120572 120573 isin


ℎ11= 1 minus (119865

1015840

119910)2

ℎ12= 1198651015840

1199091198651015840

119910 ℎ

22= 1 minus (119865

1015840

119909)2

(54)


120579(ℓ)= 119861 (119890

1 1198901) + 119861 (119890

2 1198902) = minus

1

(1198651015840

119911)2ℎ11

(11986510158401015840

119909119909+ 11986510158401015840

119910119910+ 11986510158401015840

119911119911)

(55)






equation

11986510158401015840

119909119909+ 11986510158401015840

119910119910+ 11986510158401015840

119911119911= 0 (56)





120579(ℓ)= minus

1

(1198651015840

119911)211989211

(11986510158401015840

119909119909+ 11986510158401015840

119910119910+ 11986510158401015840

119911119911) = 2119891 (57)






119905 = 119865 (119909 119910 119911) = 119903 = radic1199092+ 1199102+ 1199112 (58)


0be


0)119906) given by

119906 (119905 119909 119910 119911) = 119903 minus 119905 + 119887 with 119905 = radic1199092 + 1199102 + 1199112 (59)


normal to Λ30are

ℓ119894= 119890120588(1

119909

119905119910

119905119911

119905) ℓ

119894= 119890minus120588(minus1

119909

119905119910

119905119911

119905) (60)


nabla119883ℓ = D

119883ℓ = 119906119883 forall119883 isin (Λ

3

0)119906 (61)



3

0)119906 (62)


0)119906 = 0


n119894 = (1 0 0 0) s119894 = (0 119909119905119910

119905119911

119905) (63)



120579(ℓ)= minus

1

(1199031015840

119911)2ℎ11

(11990310158401015840

119909119909+ 11990310158401015840

119910119910+ 11990310158401015840

119911119911)

= minus21199053

1199112(1199052minus 1199102)= minus

21199053

1199112(1199092+ 1199112)= 2119891 = 0

(64)




ℓ119902 = 0


119866 = Ω2119892 (65)




ℎ119906= Ω2

119906119902 (66)



119906)



119906) (ℎ119906)) is a




119906) is an ENH


119906) (ℎ119906)) of ENHs



119892 = minus1198891199052+ 1198891199032+ 1199032(1198891205792+ sin21205791198891206012) (68)


119892 = minus119889V119889119908 + (V minus 1199082

)

2

(1198891205792+ sin21205791198891206012) (69)


119886119908119887120578119886119887

= 0 = V119886V119887120578119886119887 Consider a




119902 = (V minus 1199082

)

2

(1198891205792+ sin2120579 1198891206012) 119908 = constant (70)







119866 = minus1198891199052+ 1198862cosh2 ( 119905

119886) 1198891199032+ 1199032(1198891205792+ sin2120579 1198891206012)

(71)



119866 = 1198862cosh2 (120591

119886) 119892 (minus

120587

2lt 120591 lt

120587

2) (72)




ℎ119906= 1198862cosh2 (120591

119886) 119902 = 119886

2cosh2 (120591119886)


)

2

(1198891205792+ sin21205791198891206012)

(73)





119886)) ℎ)

119861 (119883 119884) = 119891ℎ (119883 119884) = 119886ℓ (ln(cosh (120591119886)) ℎ (119883 119884))

forall119883 119884 isin 119879 (H)

(74)


119906) (ℎ119906)) of (119872119866) Subject



119892 = minus119860(119903)21198891199032+ 119860(119903)

minus21198891199032+ 1199032(sin2120579 1198891206012 + 1198891205792)

(75)



5 Discussion










References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













120597119906

120597119905= 0


(45)







by the equation

119905 = 119865 (119909 119910 119911) (46)



119906 (119905 119909 119910 119911) = 119865 minus 119905 + 119887 (47)



1015840

119909 1198651015840

119910 1198651015840



(1198651015840

119909)2

+ (1198651015840

119910)2

+ (1198651015840

119911)2

= 1 (48)


ℓ119894= 119890120588(1 1198651015840

119909 1198651015840

119910 1198651015840

119911) ℓ

119894= 119890minus120588(minus1 119865

1015840

119909 1198651015840

119910 1198651015840

119911) (49)


k119894 = 1

2119890120588(minus1 minus119865

1015840

119909 minus1198651015840

119910 minus1198651015840

119911)

k119894=1

2119890minus120588(1 minus119865

1015840

119909 minus1198651015840

119910 minus1198651015840

119911)

(50)


1198821= 1198651015840

119911

120597

120597119909minus 1198651015840

119909

120597

120597119911 1198822= 1198651015840

119911

120597

120597119910minus 1198651015840

119910

120597

120597119911 (51)




119883ℓ ℓ) = 0




(ℓ)of


1 1198902 of St

1198901=

1

100381710038171003817100381711988211003817100381710038171003817

1198821

1198902=

1

10038161003816100381610038161198651015840

119911

1003816100381610038161003816

100381710038171003817100381711988211003817100381710038171003817

10038171003817100381710038171198821

1003817100381710038171003817

21198822minus 119892 (119882

11198822)1198821

(52)


119861 (1198901 1198901) + 119861 (119890

2 1198902)

=1

(1198651015840

119911)2ℎ11

times ((1198651015840

119911)2

+ (ℎ12)2) 11986111minus 2ℎ11ℎ1211986112+ (ℎ11)211986122

(53)

where we put ℎ120572120573= ℎ(119882

120572119882120573) and 119861

120572120573= 119861(119882

120572119882120573) 120572 120573 isin


ℎ11= 1 minus (119865

1015840

119910)2

ℎ12= 1198651015840

1199091198651015840

119910 ℎ

22= 1 minus (119865

1015840

119909)2

(54)


120579(ℓ)= 119861 (119890

1 1198901) + 119861 (119890

2 1198902) = minus

1

(1198651015840

119911)2ℎ11

(11986510158401015840

119909119909+ 11986510158401015840

119910119910+ 11986510158401015840

119911119911)

(55)






equation

11986510158401015840

119909119909+ 11986510158401015840

119910119910+ 11986510158401015840

119911119911= 0 (56)





120579(ℓ)= minus

1

(1198651015840

119911)211989211

(11986510158401015840

119909119909+ 11986510158401015840

119910119910+ 11986510158401015840

119911119911) = 2119891 (57)






119905 = 119865 (119909 119910 119911) = 119903 = radic1199092+ 1199102+ 1199112 (58)


0be


0)119906) given by

119906 (119905 119909 119910 119911) = 119903 minus 119905 + 119887 with 119905 = radic1199092 + 1199102 + 1199112 (59)


normal to Λ30are

ℓ119894= 119890120588(1

119909

119905119910

119905119911

119905) ℓ

119894= 119890minus120588(minus1

119909

119905119910

119905119911

119905) (60)


nabla119883ℓ = D

119883ℓ = 119906119883 forall119883 isin (Λ

3

0)119906 (61)



3

0)119906 (62)


0)119906 = 0


n119894 = (1 0 0 0) s119894 = (0 119909119905119910

119905119911

119905) (63)



120579(ℓ)= minus

1

(1199031015840

119911)2ℎ11

(11990310158401015840

119909119909+ 11990310158401015840

119910119910+ 11990310158401015840

119911119911)

= minus21199053

1199112(1199052minus 1199102)= minus

21199053

1199112(1199092+ 1199112)= 2119891 = 0

(64)




ℓ119902 = 0


119866 = Ω2119892 (65)




ℎ119906= Ω2

119906119902 (66)



119906)



119906) (ℎ119906)) is a




119906) is an ENH


119906) (ℎ119906)) of ENHs



119892 = minus1198891199052+ 1198891199032+ 1199032(1198891205792+ sin21205791198891206012) (68)


119892 = minus119889V119889119908 + (V minus 1199082

)

2

(1198891205792+ sin21205791198891206012) (69)


119886119908119887120578119886119887

= 0 = V119886V119887120578119886119887 Consider a




119902 = (V minus 1199082

)

2

(1198891205792+ sin2120579 1198891206012) 119908 = constant (70)







119866 = minus1198891199052+ 1198862cosh2 ( 119905

119886) 1198891199032+ 1199032(1198891205792+ sin2120579 1198891206012)

(71)



119866 = 1198862cosh2 (120591

119886) 119892 (minus

120587

2lt 120591 lt

120587

2) (72)




ℎ119906= 1198862cosh2 (120591

119886) 119902 = 119886

2cosh2 (120591119886)


)

2

(1198891205792+ sin21205791198891206012)

(73)





119886)) ℎ)

119861 (119883 119884) = 119891ℎ (119883 119884) = 119886ℓ (ln(cosh (120591119886)) ℎ (119883 119884))

forall119883 119884 isin 119879 (H)

(74)


119906) (ℎ119906)) of (119872119866) Subject



119892 = minus119860(119903)21198891199032+ 119860(119903)

minus21198891199032+ 1199032(sin2120579 1198891206012 + 1198891205792)

(75)



5 Discussion










References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119905 = 119865 (119909 119910 119911) = 119903 = radic1199092+ 1199102+ 1199112 (58)


0be


0)119906) given by

119906 (119905 119909 119910 119911) = 119903 minus 119905 + 119887 with 119905 = radic1199092 + 1199102 + 1199112 (59)


normal to Λ30are

ℓ119894= 119890120588(1

119909

119905119910

119905119911

119905) ℓ

119894= 119890minus120588(minus1

119909

119905119910

119905119911

119905) (60)


nabla119883ℓ = D

119883ℓ = 119906119883 forall119883 isin (Λ

3

0)119906 (61)



3

0)119906 (62)


0)119906 = 0


n119894 = (1 0 0 0) s119894 = (0 119909119905119910

119905119911

119905) (63)



120579(ℓ)= minus

1

(1199031015840

119911)2ℎ11

(11990310158401015840

119909119909+ 11990310158401015840

119910119910+ 11990310158401015840

119911119911)

= minus21199053

1199112(1199052minus 1199102)= minus

21199053

1199112(1199092+ 1199112)= 2119891 = 0

(64)




ℓ119902 = 0


119866 = Ω2119892 (65)




ℎ119906= Ω2

119906119902 (66)



119906)



119906) (ℎ119906)) is a




119906) is an ENH


119906) (ℎ119906)) of ENHs



119892 = minus1198891199052+ 1198891199032+ 1199032(1198891205792+ sin21205791198891206012) (68)


119892 = minus119889V119889119908 + (V minus 1199082

)

2

(1198891205792+ sin21205791198891206012) (69)


119886119908119887120578119886119887

= 0 = V119886V119887120578119886119887 Consider a




119902 = (V minus 1199082

)

2

(1198891205792+ sin2120579 1198891206012) 119908 = constant (70)







119866 = minus1198891199052+ 1198862cosh2 ( 119905

119886) 1198891199032+ 1199032(1198891205792+ sin2120579 1198891206012)

(71)



119866 = 1198862cosh2 (120591

119886) 119892 (minus

120587

2lt 120591 lt

120587

2) (72)




ℎ119906= 1198862cosh2 (120591

119886) 119902 = 119886

2cosh2 (120591119886)


)

2

(1198891205792+ sin21205791198891206012)

(73)





119886)) ℎ)

119861 (119883 119884) = 119891ℎ (119883 119884) = 119886ℓ (ln(cosh (120591119886)) ℎ (119883 119884))

forall119883 119884 isin 119879 (H)

(74)


119906) (ℎ119906)) of (119872119866) Subject



119892 = minus119860(119903)21198891199032+ 119860(119903)

minus21198891199032+ 1199032(sin2120579 1198891206012 + 1198891205792)

(75)



5 Discussion










References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119902 = (V minus 1199082

)

2

(1198891205792+ sin2120579 1198891206012) 119908 = constant (70)







119866 = minus1198891199052+ 1198862cosh2 ( 119905

119886) 1198891199032+ 1199032(1198891205792+ sin2120579 1198891206012)

(71)



119866 = 1198862cosh2 (120591

119886) 119892 (minus

120587

2lt 120591 lt

120587

2) (72)




ℎ119906= 1198862cosh2 (120591

119886) 119902 = 119886

2cosh2 (120591119886)


)

2

(1198891205792+ sin21205791198891206012)

(73)





119886)) ℎ)

119861 (119883 119884) = 119891ℎ (119883 119884) = 119886ℓ (ln(cosh (120591119886)) ℎ (119883 119884))

forall119883 119884 isin 119879 (H)

(74)


119906) (ℎ119906)) of (119872119866) Subject



119892 = minus119860(119903)21198891199032+ 119860(119903)

minus21198891199032+ 1199032(sin2120579 1198891206012 + 1198891205792)

(75)



5 Discussion










References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article time-dependent evolving null horizons of

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