let ( m,g ) be a ( 1 + n)-dimensional space - time manifold whose metric g satisfies the following kinematic condition:(1)xu=x+gx , uu , where u is a timelike unit vector ,  denotes the levi - civita connection on m ,  is a function , and x is an arbitrary vector field on m. consider gaussian normal coordinates ( x = t , x ) on m such that u = /t and g=gijdxidxj =- dt2+gabdxadxb , where i , j run over 0,1 ,  , n and a , b run over 1 ,  , n and gab are functions of all the coordinates x. let (t = a constant ) be a spacelike slice orthogonal to u and x = /x a coordinate vector field tangent to . then ( 1 ) reduces to /xa/t=(/xa ) which further implies ( with some computation ) that ( /t)gab=2gab . integrating this we obtain gab = e2(t , xa)dtab , where  is a fixed riemannian metric on the initial slice ( t = t0 a constant ) . in this paper , we take  as a function of t only and set e(t)dt=t ; that is , =/ , where an overdot means derivative with respect to t. then the metric g is of the form(2)g=gijdxidxj=dt2+2tabdxadxb , which is generalized robertson - walker ( grw ) space - time notion introduced by alas et al . . it is a warped product , with base of an open interval of a real line with a negatively defined metric and fibre of a riemannian manifold and not of constant sectional curvature , in general . the reader will see that the kinematic condition ( 1 ) on the space - time metric plays an important role in the induced geometry of null hypersurfaces of m. this is our reason to generate grw space - time instead of stating it as a definition . this class of space - time is inhomogeneous admitting an isotropic radiation whose subcase is classical robertson - walker ( rw ) space - time which includes einstein - de - sitter space - time , friedman cosmological models , and the static einstein space - time . considerable work is available on grw space - time , primarily on spacelike hypersurfaces [ 14 ] and more cited therein , and only recently there has been interest ( see kang [ 5 , 6 ] and navarro et al . ) on its null submanifold geometry . it is well - known that the null hypersurfaces have been used as physical models of black hole horizons in general relativity . in this paper we show that there exists a physical model of a family of time - dependent null horizons of grw space - time which may evolve into a black hole event horizon . see sections 3 and 4 for a brief account on event horizons , recent works on time - dependent null horizons of duggal [ 810 ] and sultana and dyer [ 11 , 12 ] , and more cited therein . we highlight that the physical use of research on time - dependent null horizons may have connection ( though it is too early to be sure ) with the latest ligo [ laser interferometer gravitational - waves observatory ] experiment ( as per press release of february 11 , 2016 ) which confirms the presence of gravitational waves produced during the final fraction of a second of the merger of two black holes into a single massive spinning black hole . thus , this recent ligo experiment reconfirms that black hole has a cosmological background or it is surrounded by a local matter distribution . therefore , there is significant difference in the structure and properties of the surrounding dynamical region of a black hole and this latest experiment strengthens the ongoing research on time - dependent null horizons , in particular , near such spinning black holes surrounded with gravitational waves . for information on ligo via video one may try http://mediaassets.caltech.edu / gwave/. consider grw space - time ( m,g ) with its metric g given by ( 2 ) . let p be a point in m and ( ( m , g ) : x = a constant ) be a timelike hypersurface of m passing through p. observe that there is a freedom of choice in taking any one of the spacelike coordinates constant . suppose that n is the future directed timelike unit vector on m induced by u of m such that the following holds:(3)xn=tx+gx , nn,xtm , where g ,  , and (t ) are the induced metric , levi - civita connection , and a function on m , respectively . let ( ( s , q ) : t0 = a constant ) be a connected spacelike hypersurface in m and hence a codimension two - spacelike submanifold of m passing through p via the normal exponential map along s in m with metric q induced from g. using ( 3 ) and following a procedure explained in previous section one can show that m is also grw space - time with metric g expressed as ( 4)g=dt2+2tabdxadxb,2a , bn ,  =,where we take =/ and ab is a fixed riemannian metric . assume that x = ( x2 ,  , xn ) are the coordinates in ( su , t = u a constant ) centered on p so that the metric qu on su is given by ( 5)qu=u2abdxadxb,2a , bn.choose at each point of su a null direction perpendicular in su and smoothly depending on the foot - point . then , it is easy to see that the union of all geodesics with the chosen ( say outgoing ) direction u is a null hypersurface of m. denote by ( hu , hu , u ) such a null hypersurface of m with degenerate metric hu and null normal u . since hu includes su and its degenerate metric hu must be of signature ( 0 , + ,  , + ) , we assume that the riemannian metric qu of su coincides with the degenerate metric hu of hu given by(6)hu = huabdxadxb=hu2abdxadxb,2a , bn , where hu is a function on hu induced by the function u of m. this means that the null hypersurface ( hu , hu ) is conformal to a fixed null hypersurface ( h ,  ) ; that is , hu = hu , with the conformal function hu . in this paper , we take u in some subset of m around hu . also , in this way we get a foliation of m ( in the vicinity of hu ) by a family ( hu ) so that u is in the part of m foliated by this family ( for details see carter ) such that at each point in this region u is a null normal to hu for some value of the parameter u. let ( hu ) and ( u ) be the families of degenerate metrics and null normal , respectively . denote by(7)f = hu , hu , lu : hu=hu2hua family of null hypersurfaces of ( m,g ) conformally related to a fixed null hypersurface ( h ,  ,  ) and each hu is a function on hu for some value of u. also , let ( su ) be the corresponding family of submanifolds of m. the bending of each hu of f in m is described by the weingarten map : ( 8)wlu : tphutphu , xuxulu.u associates with each xu of hu the variation of u along xu , with respect to the space - time connection . the second fundamental form , say bu , of hu is the symmetric bilinear form related to the weingarten map by(9)buxu , yu = huwluxu , yu = huxulu , y , xu , yuthu.b(xu , u ) = 0 for any xu  thu implies that bu has the same u degeneracy as that of the metric hu and it does not depend on particular choice of u . now we deal with the question of obtaining a normalized expression for a null normal  of a null hypersurface h  f. for this purpose , we let stm be a unit spacelike normal vector to s defined in some open neighborhood of corresponding h. since the future directed timelike vectors n of m and s are orthogonal , it is immediate that the vector n + s is null so it is tangent to h. therefore , one can take  = n + s as a natural normalization of the null normal to h. however , this normalization can only be defined for a single fixed hypersurface h , whereas we need to normalize the entire family ( u ) for any parameter value of u. for this purpose , we choose the following normalization of each u  ( u):(10)lu = nu+su , where  susu=1 ,  xutpsusuxu=0,which implies that each u is tangent to each member of f and it has the property of lie dragging the family of submanifolds ( su ) . then , we define a transversal vector field ku to tphu not belonging to f expressed as another suitable linear combination of nu and su such that it represents the ingoing direction satisfying(11)glu , ku=1,ku=12nusu.now we are ready to state and prove the following main theorem . let f = ( ( hu ) , ( hu ) , ( u ) ) be a family of null hypersurfaces of the grw space - time ( m,g ) defined by ( 2 ) such that f is given by ( 7 ) and each of its members ( hu , hu , u ) is conformal to a fixed null hypersurface ( h ,  ,  ) , where hu = hu for every parameter value of u. assume that ( h ,  ,  ) lies to the future of each ( hu , hu ) . then ( a)each ( hu , hu , u ) is totally umbilical in ( m,g ) , with its volume expansion (u ) = ( n  1)u where the function u is as given in ( 3);(b)each u is conformal killing vector (  uhu = 2uhu ) on each hu;(c)the family f may evolve into a totally geodesic null hypersurface ( h ,  ,  ) only if u  0 on h and then ( ) = 0 on h. 
  let f = ( ( hu ) , ( hu ) , ( u ) ) be a family of null hypersurfaces of the grw space - time ( m,g ) defined by ( 2 ) such that f is given by ( 7 ) and each of its members ( hu , hu , u ) is conformal to a fixed null hypersurface ( h ,  ,  ) , where hu = hu for every parameter value of u. assume that ( h ,  ,  ) lies to the future of each ( hu , hu ) . then ( a)each ( hu , hu , u ) is totally umbilical in ( m,g ) , with its volume expansion (u ) = ( n  1)u where the function u is as given in ( 3);(b)each u is conformal killing vector (  uhu = 2uhu ) on each hu;(c)the family f may evolve into a totally geodesic null hypersurface ( h ,  ,  ) only if u  0 on h and then ( ) = 0 on h. each ( hu , hu , u ) is totally umbilical in ( m,g ) , with its volume expansion (u ) = ( n  1)u where the function u is as given in ( 3 ) ; each u is conformal killing vector (  uhu = 2uhu ) on each hu ; the family f may evolve into a totally geodesic null hypersurface ( h ,  ,  ) only if u  0 on h and then ( ) = 0 on h. 
 prooflet xu be a vector tangent to a spacelike hypersurface su in m. the weingarten formula is given by xunu = anuxu , where we denote by anu the shape operator of su . denote by bsu the second fundamental form of su in m with respect to nu . then , for any xu , yu  tsu(12)bsuxu , yuquxunu , yu = quanuxu , yu=uquxu , yu.thus , ( su , qu ) is totally umbilical in m. then , it is easy to show that any ( hu , hu , u ) is also totally umbilical in m. indeed , take bu as the second fundamental form of hu . as bu(u , xu ) = 0 , xu  thu , and corresponding su  hu we conclude that for every xu , yu  thu(13)buxu , yu = buxu , yu = bsuxu , yu,xu , yutsu.since the riemannian metric qu coincides with the corresponding degenerate metric hu it follows from above relation and ( 12 ) that ( 14)buxu , yu=uhux , y,xu , yuthu , which proves that each hu is totally umbilical in m. moreover , it follows from ( 3 ) that the volume expansion (u ) = div(nu ) = ( n  1)u , where we have used bu(u , xu ) = 0 , xu  thu , which proves ( a ) . using the expression ( 15)luhux , y = huxlu , y+huylu , xand bu(xu , yu ) symmetric in the above equation we obtain ( 16)buxu , yu=12lhuxu , yu,xu , yuthu.since hu is totally umbilical , we use bu(xu , yu ) = uhu(xu , yu ) in above relation to get  uhu = 2uhu on hu . hence , each u is conformal killing vector of the metric hu with conformal function 2u which proves ( b ) . for item ( c ) , since the fixed hypersurface ( m ,  ,  ) is in the future of its conformally related family f of hypersurfaces , ( hu , hu)(h ,  ) only if the metric hu   for that value of u , which further means that u  1 for that value of u. we know that u=u/u . thus , u  0 on h for that value of u which implies that ( ) = 0 on h. now we know that a null hypersurface of a semi - riemannian manifold has zero expansion if and only if it is totally geodesic so ( c ) holds which completes the proof . let xu be a vector tangent to a spacelike hypersurface su in m. the weingarten formula is given by xunu = anuxu , where we denote by anu the shape operator of su . denote by bsu the second fundamental form of su in m with respect to nu . then , for any xu , yu  tsu(12)bsuxu , yuquxunu , yu = quanuxu , yu=uquxu , yu.thus , ( su , qu ) is totally umbilical in m. then , it is easy to show that any ( hu , hu , u ) is also totally umbilical in m. indeed , take bu as the second fundamental form of hu . as bu(u , xu ) = 0 , xu  thu , and corresponding su hu we conclude that for every xu , yu  thu(13)buxu , yu = buxu , yu = bsuxu , yu,xu , yutsu.since the riemannian metric qu coincides with the corresponding degenerate metric hu it follows from above relation and ( 12 ) that ( 14)buxu , yu=uhux , y,xu , yuthu , which proves that each hu is totally umbilical in m. moreover , it follows from ( 3 ) that the volume expansion (u ) = div(nu ) = ( n  1)u , where we have used bu(u , xu ) = 0 , xu  thu , which proves ( a ) . using the expression ( 15)luhux , y = huxlu , y+huylu , xand bu(xu , yu ) symmetric in the above equation we obtain ( 16)buxu , yu=12lhuxu , yu,xu , yuthu.since hu is totally umbilical , we use bu(xu , yu ) = uhu(xu , yu ) in above relation to get  uhu = 2uhu on hu . hence , each u is conformal killing vector of the metric hu with conformal function 2u which proves ( b ) . for item ( c ) , since the fixed hypersurface ( m ,  ,  ) is in the future of its conformally related family f of hypersurfaces , ( hu , hu)(h ,  ) only if the metric hu   for that value of u , which further means that u  1 for that value of u. we know that u=u/u . thus , u  0 on h for that value of u which implies that ( ) = 0 on h. now we know that a null hypersurface of a semi - riemannian manifold has zero expansion if and only if it is totally geodesic so ( c ) holds which completes the proof . now we quote the following general result for a totally umbilical submanifold ( also holds for totally geodesic case ) of a semi - riemannian manifold which is needed to address the question of how theorem 1 can be used to show the existence of null horizons of grw space - time . let h be a totally umbilical submanifold of a semi - riemannian manifold m. then , ( a ) a null geodesic vector field of m that starts tangential to h remains within h ( for some parameter interval around the starting point ) and ( b ) h is totally geodesic if and only if every geodesic vector field of m that starts tangential to h remains within h ( for some parameter interval around the starting point ) . let h be a totally umbilical submanifold of a semi - riemannian manifold m. then , ( a ) a null geodesic vector field of m that starts tangential to h remains within h ( for some parameter interval around the starting point ) and ( b ) h is totally geodesic if and only if every geodesic vector field of m that starts tangential to h remains within h ( for some parameter interval around the starting point ) . above result satisfies a requirement for the existence of a null horizon in relativity . since we assume that each null normal u of the family of hypersurfaces f is null geodesic , using above result of perlick , we state the following corollary of theorem 1 ( proof is easy ) . let ( m,g ) be null geodesically complete space - time obeying the null energy condition ric(x , x)0 for all null vectors then , each null geodesic vector u of f is contained in its respective smooth totally umbilical null hypersurface ( hu , u ) of ( m,g ) . in particular , this property will also hold for the totally geodesic null hypersurface ( h ,  ) of ( m,g ) . let ( m,g ) be null geodesically complete space - time obeying the null energy condition ric(x , x)0 for all null vectors x such that the hypothesis of theorem 1 holds . then , each null geodesic vector u of f is contained in its respective smooth totally umbilical null hypersurface ( hu , u ) of ( m,g ) . in particular , this property will also hold for the totally geodesic null hypersurface ( h ,  ) of ( m,g ) . in this section we present a physical interpretation of theorem 1 . for this purpose , recall that the vorticity - free raychaudhuri equation for any member ( h , h ,  ) of the family f is given by(17)dlds=rijliljijij2n1,where ij=-  ( i()j)-(1/2)hij is the shear tensor , s is a pseudoarc parameter such that  is null geodesic , and rij is the ricci tensor of m. we also recall that in two recent papers [ 8 , 9 ] we studied a new class of null horizons of space - time using the following definition . definition 4 . a null hypersurface ( h , h ,  ) of the family f of space - time ( m,g ) is called an evolving null horizon , briefly denoted by enh , if ( i)h is totally umbilical in m and may include a totally geodesic portion;(ii)all equations of motion hold at h and energy tensor tij is such that tb is future - causal for any future directed null normal . 
  a null hypersurface ( h , h ,  ) of the family f of space - time ( m,g ) is called an evolving null horizon , briefly denoted by enh , if ( i)h is totally umbilical in m and may include a totally geodesic portion;(ii)all equations of motion hold at h and energy tensor tij is such that tb is future - causal for any future directed null normal . h is totally umbilical in m and may include a totally geodesic portion ; all equations of motion hold at h and energy tensor tij is such that tb is future - causal for any future directed null normal . comparing this definition with theorem 1 , we notice that the first part of its condition ( i ) is the same as the first part of conclusion ( a ) of theorem 1 and energy condition ( ii ) requires that rijij is nonnegative for any  , which implies ( see hawking and ellis [ 15 , page 95 ] ) that ( ) monotonically decreases in time ( which we take as u parameter for the family f ) along each  ; that is , m obeys the null convergence condition . thus , in the region where ( ) is nonzero each member of the family f will be totally umbilical with u - parameter time - dependent metric h and may include a totally geodesic portion where ( ) vanishes for a fixed ( h ,  ) , so condition ( c ) will hold . it follows from the raychaudhuri equation that each  will have nonzero or zero shear accordingly as its expansion function ( ) is nonzero or zero , respectively . moreover , it is easy to see that condition ( b ) of theorem 1 will hold for an enh . this clearly shows that there exists a  physical model  of a class f = ( ( hu ) , ( hu ) , ( u ) ) of a family of totally umbilical null hypersurfaces of the space - time ( m,g ) , satisfying the hypothesis and three conclusions of theorem 1 , such that each of its members is an evolving null horizon ( enh ) which may evolve into a fixed null hypersurface ( h ,  ) with zero expansion , ( ) = 0 . for up - to - date information with some examples of enhs we refer to duggal [ 810 , 16 ] . however , theorem 1 is limited by the fact that not every such totally umbilical null hypersurface evolving into a totally geodesic null hypersurface can be related to a black hole horizon . therefore , we must show that there exists a prescribed family of enhs of grw space - time which is related to a black hole horizon . to complete our analysis on a physical interpretation we need the following information on a class of black hole horizons . 
 event horizons . a boundary of space - time is called a null event horizon ( briefly denoted by eh ) beyond which events can not affect the observer . an eh is intrinsically a global concept as its definition requires the knowledge of the entire space - time to determine whether null geodesics can reach null infinity . in particular , an event horizon is called a killing horizon if it is represented by a null hypersurface which admits a killing vector field . ehs have played a key role and this includes hawking 's area increasing theorem , black hole thermodynamics , black hole perturbation theory , and the topological censorship results . moreover , an eh always exists in black hole asymptotically flat space - time under a weak cosmic censorship condition and is represented by a killing horizon such that the space - time is analytic and the stress tensor obeys the weak energy condition . the null hypersurface of such space - time admits a nonvanishing killing vector field , say  , which may or may not be the killing vector field of the landing space - time . the latter case corresponds to a rotating asymptotically flat black hole which we do not discuss here . we refer to hawking 's paper on  event horizons  , three papers of hjiek 's work [ 1820 ] , and more cited therein . based on this brief account , we now recall the following work of galloway   who has shown that the null hypersurfaces which arise most naturally in general relativity , such as black hole ehs , are in general c but not c. his approach has its roots in the well - known geometric maximum principle of e. hopf , a powerful analytic tool which is often used in the theory of minimal or constant mean curvature hypersurfaces . this principle implies that two different minimal hypersurfaces in a riemannian manifold can not touch each other from one side . in year 2000 , galloway   proved the following result for smooth null hypersurfaces restricted to the zero mean curvature case and suitable for asymptotically flat space - time . let h1 and h2 be smooth null hypersurfaces in a space - time manifold m. suppose that ( 1 )  h1 and h2 meet at pm and h2 lies to the future side of h1 near p and ( 2 ) the null mean curvatures 1 of h1 and 2 of h2 satisfy 2  0  1 . then h1 and h2 coincide near p and this common null hypersurface has mean curvature  = 0 . let h1 and h2 be smooth null hypersurfaces in a space - time manifold m. suppose that ( 1 )  h1 and h2 meet at pm and h2 lies to the future side of h1 near p and ( 2 ) the null mean curvatures 1 of h1 and 2 of h2 satisfy 2  0  1 . then h1 and h2 coincide near p and this common null hypersurface has mean curvature  = 0 . although the above maximum principle theorem is for smooth null hypersurfaces , in reality null hypersurfaces as models of black hole event horizons are the null portions of achronal boundaries as the sets = i(a ) ,  am , which are always c hypersurfaces and contain nondifferentiable points . these null geodesics ( entirely contained in c null hypersurface h ) are the null geodesic generators of h. for this reason , galloway also proved his above result for c null hypersurface and the following physically meaningful null splitting theorem . let ( m,g ) be null geodesically complete space - time which obeys the null energy condition , ric(x , x)0 for all null vectors x. if m admits a null line  , then  is contained in a smooth closed achronal totally geodesic null hypersurface . let ( m,g ) be null geodesically complete space - time which obeys the null energy condition , ric(x , x)0 for all null vectors x. if m admits a null line  , then  is contained in a smooth closed achronal totally geodesic null hypersurface . examples ( see ) are minkowski space , de - sitter , and anti - de - sitter spaces . let ( m,g ) be null geodesically complete grw space - time which obeys the null energy condition , ric(x , x)0 for all null vectors x. suppose  = ( ( hu ) , ( hu ) , ( u ) ) is a family of null hypersurfaces of ( m,g ) defined by ( 2 ) and given by ( 7 ) where each of its members ( hu , hu , u ) is conformally related to a totally geodesic hypersurface ( h ,  ,  ) such that the conformal factor of hu = hu for every parameter value of u. then , conditions ( a ) and ( b ) of theorem 1 will hold . moreover , let ( h ,  ,  ) be in the future of each ( hu , hu , u ) and it admits a killing horizon . then , ( c ) the family  evolves into a black hole event horizon represented by ( h ,  ,  ) only if u  0 at null infinity . let ( m,g ) be null geodesically complete grw space - time which obeys the null energy condition , ric(x , x)0 for all null vectors x. suppose  = ( ( hu ) , ( hu ) , ( u ) ) is a family of null hypersurfaces of ( m,g ) defined by ( 2 ) and given by ( 7 ) where each of its members ( hu , hu , u ) is conformally related to a totally geodesic hypersurface ( h ,  ,  ) such that the conformal factor of hu = hu for every parameter value of u. then , conditions ( a ) and ( b ) of theorem 1 will hold . moreover , let ( h ,  ,  ) be in the future of each ( hu , hu , u ) and it admits a killing horizon . then , ( c ) the family  evolves into a black hole event horizon represented by ( h ,  ,  ) only if u  0 at null infinity . the proofs of items ( a ) and ( b ) follow exactly as the proofs of items ( a ) and ( b ) in theorem 1 . for the proof of item ( c ) we first notice that h admits a killing horizon . secondly , the killing horizon h satisfies a particular case of corollary 3 and galloway 's   null splitting theorem 6 . this implies that when u  0 the killing horizon h will be equivalent to an event horizon at null infinity , which proves ( c ) . we , therefore , claim that theorem 1 provides a physical model of a family of time - dependent evolving null horizons ( enhs ) of grw space - time and theorem 7 is a step towards the ongoing physical use of enhs of prescribed grw space - time which admits a null event horizon satisfying galloway 's null splitting theorem 6 whose above listed three examples ( which are subcases of grw space - time ) are also examples of theorem 7 . the motivation for this work originated from the need for time - dependent null horizons which describe the geometry of a null hypersurface of dynamical space - time . for this purpose , in our four previous papers [ 810 , 16 ] we studied a quasilocal family of  evolving null horizons  ( enhs ) of ( 1 + n)-splitting space - time , explained the reason for such a study , constructed a variety of examples of enhs , and in some cases established their relation with black hole isolated horizons introduced by ashtekar et al . . the following is a brief account on nonexpanding , weakly , and isolated horizons . definition 8 . a null hypersurface ( h ,  ) of 4-dimensional space - time ( m,g ) is called a nonexpanding horizon ( neh ) if ( 1 )  h has a topology r  s , ( 2 ) any null normal  of h has vanishing expansion , ( ) = 0 , and ( 3 ) all equations of motion hold at h and the stress energy tensor tij is such that tj is future - causal for any future directed null normal . 
  a null hypersurface ( h ,  ) of 4-dimensional space - time ( m,g ) is called a nonexpanding horizon ( neh ) if ( 1 )  h has a topology r  s , ( 2 ) any null normal  of h has vanishing expansion , ( ) = 0 , and ( 3 ) all equations of motion hold at h and the stress energy tensor tij is such that tj is future - causal for any future directed null normal . condition ( 1 ) implies that h is ruled by the integral curves of the null direction field which is normal to it . conditions ( 2 ) and ( 3 ) imply that   = 0 on h , which further implies that the metric  is time - independent and h is totally geodesic in m. also , it follows from the raychaudhuri equation that  is shear - free on h. in general , there does not exist a unique induced connection on h due to degenerate . however , on an neh , the property   = 0 implies that the space - time connection  induces a unique ( torsion - free ) connection , say  , on h compatible with . we say that two types of null normal ,  and  , belong to the same equivalence class [  ] if =c for some positive constant c. 
 definition 9 . the pair ( h , [  ] ) is called a weakly isolated horizon ( wih ) if it is neh and each normal   [  ] satisfies (  i  i) = 0 ; that is , i is time - independent . moreover , wih ( h ,  , [  ] ) is called an isolated horizon ( ih ) if the full connection  is time - independent , that is , if (  i  i)v = 0 for arbitrary vector fields v tangent to h. 
  the pair ( h , [  ] ) is called a weakly isolated horizon ( wih ) if it is neh and each normal   [  ] satisfies (  i  i) = 0 ; that is , i is time - independent . moreover , wih ( h ,  , [  ] ) is called an isolated horizon ( ih ) if the full connection  is time - independent , that is , if (  i  i)v = 0 for arbitrary vector fields v tangent to h. we refer to ashtekar et al .   and several other papers listed therein for information with examples on isolated horizons . in this paper , for the first time in the literature we have studied the existence of a family of time - dependent enhs in grw space - time which , in general , is not ( 1 + n)-splitting space - time . since our early works in which the study on enhs of ( 1 + n)-splitting space - time was explored , we have used a new technique of using the kinematic condition ( 1 ) for constructing the grw space - time ( instead of defining it ) and proving theorems 1 and 7 is an important step towards the ongoing research on time - dependent null horizons . on the other hand we mention that recently caballero et al . have proved the following characterization of prescribed space - time which can be globally split as grw space - time . manifold ( m,g ) admits a global decomposition as grw space - time if and only if it has a timelike gradient conformal killing vector field k , such that the flow of its normalized vector field , z , is well defined and onto a domain i   for some interval i  r and some leaf of the orthogonal foliation to k. 
  a lorentzian manifold ( m,g ) admits a global decomposition as grw space - time if and only if it has a timelike gradient conformal killing vector field k , such that the flow of its normalized vector field , z , is well defined and onto a domain i   for some interval i  r and some leaf of the orthogonal foliation to k. contrary to this , note that our working grw space - time does not require a conformal killing vector field . also , observe that in this paper we have only related enhs to black hole event horizons represented by a killing horizon , whereas in our previous works [ 810 , 16 ] we related enhs to one of the three types of isolated horizons which , unlike event horizons , may not be represented by a killing horizon . note however that any killing horizon which is topologically r  s is an isolated horizon which is the only common result between this paper and our previous papers on the same issue . it is important to mention that lewandowski   has shown that einstein 's equations admit an infinite dimensional family of solutions with isolated horizons which are not killing horizons . a subfamily of robertson - trautman space - time provides explicit examples of space - time which admit ihs but do not admit a killing vector in any neighborhood of it . finally , we discuss similarity and difference between our results in this paper and two papers of sultana and dyer [ 11 , 12 ] related to common issue of existence of time - dependent null horizons . in , they considered a conformal transformation g=2 g to a stationary , asymptotically flat space - time ( m , g ) admitting a killing horizon h. they have shown that such a hypersurface h is null geodesic , called a conformal killing horizon ( ckh ) if and only if the twist of the conformal killing trajectories on m vanishes . moreover , ckhs are time - dependent and they have a link with an event horizon . in   they constructed an example of ckh in de - sitter space - time . consequently , although their result on existence of time - dependent ckh is similar to the conclusions of theorems 1 and 7 , their work is only limited to null hypersurfaces of asymptotically flat space - time whereas our results in this paper are applicable to a variety of grw space - time . since an event horizon refers to infinity , it is more appropriate ( see details in ) to use one of the three types of isolated horizons which can easily locate a black hole . for this reason , one may try different approach to show the existence of a family of enhs of a prescribed class of grw space - time which relate to isolated horizons . this opens the possibility of using the global splitting theorem 10 and then proving a result similar to theorem 1 for existence of enhs related to an isolated horizon.we know that an event horizon always exists in black hole asymptotically flat space - time and isolated horizons approximate event horizons of a black hole at late stages of gravitational collapse and can be easily located , as stated in our previous papers . however , the following question still remains open : does there always exist an enh of black hole space - time?since an enh comes from a family of null hypersurfaces , its existence and uniqueness are questionable , which is still an open problem . since an event horizon refers to infinity , it is more appropriate ( see details in ) to use one of the three types of isolated horizons which can easily locate a black hole . for this reason , one may try different approach to show the existence of a family of enhs of a prescribed class of grw space - time which relate to isolated horizons . this opens the possibility of using the global splitting theorem 10 and then proving a result similar to theorem 1 for existence of enhs related to an isolated horizon . we know that an event horizon always exists in black hole asymptotically flat space - time and isolated horizons approximate event horizons of a black hole at late stages of gravitational collapse and can be easily located , as stated in our previous papers . however , the following question still remains open : does there always exist an enh of black hole space - time ? since an enh comes from a family of null hypersurfaces , its existence and uniqueness are questionable , which is still an open problem .
a new technique is used to study a family of time - dependent null horizons , called  evolving null horizons  ( enhs ) , of generalized robertson - walker ( grw ) space - time ( m,g ) such that the metric g satisfies a kinematic condition . 
 this work is different from our early papers on the same issue where we used ( 1 + n)-splitting space - time but only some special subcases of grw space - time have this formalism . 
 also , in contrast to previous work , we have proved that each member of enhs is totally umbilical in ( m,g ) . 
 finally , we show that there exists an enh which is always a null horizon evolving into a black hole event horizon and suggest some open problems .