Symmetries of Associated Standard Static Spacetimes and Applications

Home , Spacetime symmetries, Static spacetime

ARTICLE IN PRESS JID: JOEMS [m5G; July 22, 2017;16:58 ]

Journal of the Egyptian Mathematical Society 0 0 0 (2017) 1–5

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Original Article

Symmetries of f −associated standard static spacetimes and applications

∗ H.K. El-Sayied a, Sameh Shenawy b, , Noha Syied b a Mathematics Department, Faculty of Science, Tanata University, Tanta, Egypt b Modern Academy for engineering and Technology, Maadi, Egypt

a r t i c l e i n f o a b s t r a c t

Article history: The purpose of this note is to study and explore some collineation vector ﬁelds on standard static space-

Received 25 March 2017 times I × M (also called f − associated SSST). Conformal vector ﬁelds, Ricci and matter collineations are f Revised 30 June 2017 studied. Many implications for the existence of these collineations on f −associated SSSTs are obtained. Accepted 10 July 2017 Moreover, Ricci soliton structures on f − associated SSSTs admitting a potential conformal vector ﬁeld are Available online xxx considered.

MSC: © 2017 Egyptian Mathematical Society. Production and hosting by Elsevier B.V. Primary 53C21 This is an open access article under the CC BY-NC-ND license. Secondary 53C25 53C50 ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ) Keywords: Conformal vector ﬁelds Ricci collineations Standard static spacetime Ricci soliton

1. An introduction those preserving metric, curvature and Ricci curvature by virtue of their essential role in general relativity. An extensive work has Warped product manifolds are extensively studied as a method been done in the last two decades studying collineations and their to generate general exact solutions to Einstein’s field equation generalizations on classical 4 −dimensional spacetimes. [1–3] . A standard static spacetime (also called f −associated SSST) The main purpose of this work is to study and explore some is often pictured in the form of a Lorentzian warped product man- collineation vector fields on f −associated standard static space- × − ifold If M [4,5]. An f associated standard static spacetime is, to times. Many answers are given to the following questions: Un- some extent, a generalization of some well-known classical space- der what condition(s) is a vector field on an f −associated stan- times such as the Einstein static universe and Minkowski space- dard static spacetime a certain collineation or a conformal vec- time [1,6] . tor field? What does the base factor submanifold M inherit − × The study of spacetime symmetries is essential for solving Ein- from an f associated standard static spacetime If M admitting stein field equation and for providing further insight into conser- a collineation or a conformal vector field? Ricci soliton structures vative laws of dynamical systems (see [7] an important reference on f − associated standard static spacetimes admitting a potential for symmetries of classical spacetimes). Collineation vector fields, conformal vector field are considered. in general, enable physicists to portray the geometry of a space- The distribution of this article is as follows. In Section 2 , the time [8–10] . The presence of a non-trivial collineation vector field basic definitions and related formulas of both f −associated stan- on a spacetime is sufficient to guarantee some kind of symme- dard static spacetimes and collineation vector fields are consid- try. Vector fields which preserve a certain feature or quantity of a ered. Section 3 carries a study of collineation vector fields on spacetime along their local flow lines are called collineations. The f −associated standard static spacetimes. Finally, we study Ricci Lie derivative of aforesaid feature or quantity vanishes in direction soliton structures on f −associated standard static spacetimes ad- of a collineation vector field. The most important collineations are mitting conformal vector fields in Section 4 .

2. Preliminaries ∗ Corresponding author. E-mail addresses: [email protected] (H.K. El-Sayied), − [email protected] , [email protected] (S. Shenawy), A standard static spacetime (also called f associated SSST) is ¯ = × [email protected] (N. Syied). a Lorentzian warped product manifold M I f M furnished with http://dx.doi.org/10.1016/j.joems.2017.07.002 1110-256X/© 2017 Egyptian Mathematical Society. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license. ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

Please cite this article as: H.K. El-Sayied et al., Symmetries of f −associated standard static spacetimes and applications, Journal of the Egyptian Mathematical Society (2017), http://dx.doi.org/10.1016/j.joems.2017.07.002 ARTICLE IN PRESS JID: JOEMS [m5G; July 22, 2017;16:58 ]

2 H.K. El-Sayied et al. / Journal of the Egyptian Mathematical Society 0 0 0 (2017) 1–5 the metric g¯ = − f 2 dt 2 g, where ( M, g ) is a Riemannian manifold, for some smooth function ρ on M . Finally, a spacetime M is said ζ ∈ ( ) f : M → (0, ∞ ) is smooth and I = (t 1 , t 2 ) with −∞ ≤ t 1 < t 2 ≤∞ . We to admit a matter collineation if there is a vector field X M use the same notation for a vector field X ∈ X (M) and for its lift such that L ζ T = 0 , where T is the energy-momentum tensor. For to the f −associated SSST M¯ . Likewise, a function ω on M will be ( n + 1 )− dimensional spacetime, the Einstein field equation is ω◦π ¯ , π × → identified with on M where : If M M is the natural pro- given by jection map of I × M onto M . Note that we use gradω ∈ X (M) for r ω ¯ Ric − g = k n T , the gradient vector field of on M and for its lift to X M [2,3]. 2 The submanifold { t } × M and M are isomorphic for every t ∈ M . Ac- where k is called the multidimensional gravitational constant, r is cordingly, we refer to this factor submanifold as M . n the scalar curvature and λ is the cosmological constant [12] . Sup- Let M¯ = I × M be a standard static spacetime equipped with f pose that ζ is a Killing vector field, then L ζ T = 0 , i.e., ζ is a matter the metric tensor g¯ = − f 2 dt 2 g. Then the Levi–Civita connection collineation field. Note that a matter collineation is not necessarily D¯ on M¯ is given by Killing. D¯ ∂ ∂ = f grad f D¯ ∂ X = D¯ ∂ = X ( ln f )∂ t t t X t t (2.1) 3. Symmetries of a standard static spacetime D¯ X Y = D X Y

, ∈ ( ) , for any vector fields X Y X M where D is the Levi–Civita con- In this section, we explore several types of collineations on an nection on M . The Riemannian curvature tensor R¯ is given by − ¯ = × f associated SSST M I f M equipped with the metric tensor g¯ = − f 2 dt 2 g. Necessary and sufficient conditions are derived for R¯ ( ∂t , ∂t )∂ t = R¯ ( ∂t , ∂t )X = R¯ ( X, Y )∂ t = 0 a standard static spacetime to admit a conformal vector field or a R¯ ( X, ∂ )∂ = − fD grad f R¯ ( ∂ , X )Y = 1 H f ( X, Y )∂ t t X t f t collineation. R¯ ( X, Y )Z = R ( X, Y )Z ,

(2.2) 3.1. Conformal vector ﬁelds f where R is the curvature tensor of M and H ( X, Y ) = g ( D X grad f, Y ) Assume that h∂ , x∂ , y∂ ∈ X (I) and ζ , X, Y ∈ X (M) , then is the Hessian of f . Finally, the Ricci curvature tensor, R¯ ic , of the t t t f − associated SSST M¯ is as follows 2 ˙ L¯ ζ¯ g¯ X¯ , Y¯ = L ζ g ( X, Y ) − 2 xy f h + ζ ( ln f ) , (3.1)

¯ ( ∂ , ∂ ) = ¯ ( , ∂ ) = Ric t t f f Ric X t 0 where ζ¯ = h∂t + ζ , X¯ = x∂t + X and Y¯ = y∂t + Y . This formula

¯ ( , ) = ( , ) − 1 f ( , ), Ric X Y Ric X Y f H X Y (3.1) is a particular case of a notable one on warped product man-

(2.3) ifolds. The following result yields immediately from Eq. (3.1). ¯ = × where f denotes the Laplacian of f on M . Theorem 1. Let M I f Mbea standard static spacetime equipped = − 2 2 The Lie derivative L ζ in direction of ζ is given by with the metric tensor g¯ f dt g. Then a time-like vector ﬁeld

ζ¯ = h∂t ∈ X ( M¯ ) is a Killing vector field on M¯ if and only if h˙ = 0 . ¯ L ζ g (X, Y ) = g(D X ζ , Y ) + g(X, DY ζ ) (2.4) Moreover, assume that ζ ( f ) = 0 . Then a space-like vector field ζ = ζ on M¯ is Killing if and only if ζ ∈ X (M) is a Killing vector field on M. for any X, Y ∈ X ( M ). ¯ = × Now, we will recall the definitions of conformal vector fields Corollary 1. Let M I f Mbea standard static spacetime equipped and some collineations on an arbitrary pseudo-Riemannian man- with the metric tensor g¯ = − f 2 dt 2 g. Then a time-like vector field ifold (M, g, D) with metric g and the Levi-Civita connection D ζ¯ = h∂t ∈ X ( M¯ ) is a matter collineation on M¯ if h˙ = 0 . Also, a space- on M . A vector field ζ ∈ X ( M ) is called a conformal vector field like vector field ζ¯ = ζ ∈ X ( M¯ ) is a matter collineation on M¯ if ζ ∈ if L ζ g = ρg for some smooth function ρ : M → R , where L ζ is X (M) is a Killing vector field on M and ζ ( f ) = 0 . the Lie derivative in direction of ζ . In particular, ζ ∈ X ( M ) is ζ¯ = ∂ + ζ called homothetic if ρ is constant and Killing if ρ = 0 . The sym- Theorem 2. A vector field h t on a standard static spacetime ¯ = × ζ metry of Eq. (2.4) implies that ζ is a Killing vector field if and M I f Misa conformal vector field if and only if is a conformal ρ = ˙ + ζ ( ) only if g(D X ζ , X ) = 0 for any vector field X ∈ X ( M ). A pseudo- vector field on M with conformal factor ¯ 2 h ln f . Riemannian n − dimensional manifold has at most n ( n + 1 )/ 2 in- ζ¯ = ∂ + ζ ¯ = × dependent Killing vector fields and at most ( n + 1 ) ( n + 2 )/ 2 in- Proof. Let h t be a conformal vector field on M I f M with factor ρ¯, then Eq. (3.1) implies that dependent conformal vector fields. The symmetry generated by a Killing vector field ζ on M is called isometry. A pseudo-Riemannian 2 2 −ρ¯ f xy + ρ¯g ( X, Y ) = L ζ g ( X, Y ) − 2 xy f h˙ + ζ ( ln f ) . manifold which permits a maximum aforementioned symmetry ζ Thus has a constant curvature. Also, is called a concircular vector field if D ζ = ρX for any X ∈ X ( M )[11] . A concircular vector field X L ζ g ( X, Y ) = ρ¯g ( X, Y ) ζ ∈ X ( M ) on M is a conformal vector field with conformal factor

2 ρ . A concircular vector field is also a parallel vector field if ρ = 0 . −ρ¯ f 2xy = −2 xy f 2 h˙ + ζ ( ln f ) Moreover, for a constant factor ρ, we have R ( X, Y )ζ = 0 . A vector and consequently ζ is a conformal vector field on M with confor- field ζ on a pseudo-Riemannian manifold ( M, g ) is called a cur- mal factor ρ¯ = 2 h˙ + ζ ( ln f ) . The converse is direct. vature collineation if the Lie derivative of the curvature tensor R ζ ∈ , L = vanishes in the direction of X( M) that is, ζ R 0. Similarly, It is noted that the metric g¯ = − f 2dt 2 g on M¯ can be ex- M is said to admit a Ricci curvature collineation if there is a vec- pressed as a conformal metric to a product one on I × M . The met- ζ ∈ L = , tor field X( M) such that ζ Ric 0 where Ric is the Ricci cur- ric g¯ may be rewritten as follows vature tensor. One may notice that every Killing field is a curva- 1 ture collineation and every curvature collineation is a Ricci curva- g¯ = f 2 −dt 2 + g = f 2g˜, 2 ture collineation. The converse is not generally true. A vector field f ζ ∈ X ( M ) is called a conformal Ricci collineation if 2 1 where g˜ = −dt + gˆ and gˆ = g. Now, we examine the effects f 2

L ζ Ric ( X, Y ) = ρg ( X, Y ) of replacing g¯ on M by g˜ = −dt 2 + gˆ. Similar discussions on

H.K. El-Sayied et al. / Journal of the Egyptian Mathematical Society 0 0 0 (2017) 1–5 3

− 4 dimensional spacetimes and on some warped spacetimes are i.e., ζ¯ = h∂t + ζ is a concircular vector field on a standard static ¯ = × provided in [7, Chapter 11] and [13,14] respectively. Suppose that spacetime M I f M. ¯ ζ = h∂t + ζ is a conformal vector field on M¯ , g˜ with conformal Conversely, we assume that ρ is a scalar function. Then factor ρ˜ , then ¯ ζ¯ − ρ ¯ = ˙ + ( ) + ζ ( ) − ρ ∂ DX¯ X xh hX ln f x ln f x t L¯ ζ¯ g¯ = [2 ζ ( ln f ) + ρ˜ ]g ¯. + xh f grad f + D X ζ − ρX Thus, ζ¯ is a conformal vector field on M¯ , g¯ with factor ρ¯ = for any vector field X¯ = x∂ + X ∈ X M¯ . Suppose that ζ¯ is concir- ρ˜ +2 ζ ( ln f ). Likewise, a conformal vector field ζ on ( M, g ) with t ¯ , conformal factor ρ is a conformal vector field on ( M, g¯ ), where cular on M then ρ = ρˆ + 2 ζ ( ln f ). The above discussions and results in [15, Theo- x h˙ + hX ( ln f ) + xζ ( ln f ) − xρ = 0 , rem 1] imply the following. xh f grad f + D X ζ − ρX = 0 . ¯ = × Theorem 3. Let M I f Mbea standard static spacetime equipped If f is constant, we get that with the metric tensor g¯ = − f 2 dt 2 gand let g˜ = −dt 2 + gˆ and gˆ = ˙ 1 x h − ρ = 0 , 2 g. Then, f D X ζ − ρX = 0 , ζ , , , 1. a Killing vector field on M gˆ is a Killing vector field on M¯ g˜ i.e., ζ is a concircular vector field on M with factor ρ = h˙ . 2. M¯ , g˜ admits a homothetic vector field if and only if M, gˆ admits a homothetic vector field, 3.2. Ricci collineations 3. each conformal vector field on M¯ , g˜ is a conformal vector field ¯ , on M g¯ . Let us consider Ricci collineations on a standard static space- The following result is a direct consequence of the above result. time.

Theorem 4. Let ζ ∈ X ( M ) be a homothetic vector field on ( M, g ) with Proposition 1. Let ζ¯ = h∂t + ζ be a vector field on a standard static ζ = ζ¯ = + ∂ + ζ ¯ = × , factor c and let ( f ) 0. Then c( at b) t 2a is a homothetic spacetime M I f M then vector field on M¯ , g¯ with factor 2 ac . Moreover, ζ¯ is a Killing vector ˙ L¯ ζ¯ R¯ ic X¯ , Y¯ = L ζ Ric ( X, Y ) + 2 xy h f f + xyζ ( f f ) field on M¯ , g¯ if a = 0 or c = 0 .

1 f 1 f 1 f The study of Killing vector fields of constant length is re- −ζ H ( X, Y ) + H ( [ζ , X ], Y )+ H ( X, [ζ , Y ] ) f f f markable in that they correspond to isometries of constant dis- placement. Consequently, these vector fields are in relation with for any X¯ , Y¯ ∈ X ( M¯ ) . Clifford-Wolf translation in Riemannian manifolds [16] . In the fol- Proof. Let ζ¯ = h∂t + ζ ∈ X M¯ , then lowing, Killing vector fields of constant length on a standard static spacetime M¯ = I × M are considered. ¯ ¯ ¯ f L¯ ζ¯ R¯ ic X¯ , Y¯ = ζ R¯ ic X¯ , Y¯ − R¯ ic ζ , X¯ , Y¯ − R¯ ic X¯ , ζ , Y¯ Theorem 5. A Killing vector field ζ¯ = h∂ + ζ on a standard static 1 t = ζ¯ Ric ( X, Y ) − H f ( X, Y ) + xy f f + 2 xy h˙ f f ¯ = × ζ spacetime M I f Mhasa constant length if and only if satisfies f 1 − , ζ , + f , ζ , ζ + 2 = ˙ + ζ ( ) = . Ric( X [ Y ] ) H ( X [ Y ] ) Dζ h f grad f 0 and hh 2h ln f 0 (3.2) f Corollary 2. Let ζ¯ = h∂ + ζ be a Killing vector field of constant 1 t −Ric ( [ ζ , X ], Y ) + H f ( [ ζ , X ], Y ) ¯ = × length on a standard static spacetime M I f M. Then the flow lines f of ζ are geodesics on M if and only if f is constant or h = 0 . = L ζ Ric ( X, Y ) + 2 xy h˙ f f + xyζ ( f f ) Theorem 6. Let ζ¯ = h∂t + ζ be a Killing vector field on a standard 1 f 1 f 1 f ¯ = × , α ∈ R , −ζ H ( X, Y ) + H ( [ζ , X ], Y )+ H ( X, [ζ , Y ] ) static spacetime M I f M where f is constant, and let (s), s f f f be a geodesic on ( M¯ , g¯ ) with tangent vector field X¯ = x∂t + X. Then ¯ , ¯ ∈ ( ¯ ) h is constant and ζ ∈ X ( M ) is a Jacobi vector field along the integral for any vector fields X Y X M . curves of X. The above proposition leads directly to the following results. Theorem 7. Let ζ¯ = h∂ + ζ be a conformal vector field along a curve t Theorem 9. Let ζ¯ = h∂t ∈ X M¯ be a vector field on a standard static α( s ) with unit tangent vector U¯ = u∂t + Uon a standard static space- spacetime M¯ = I × M. Then, ζ¯ is a Ricci collineation on M¯ if and only ¯ = × ρ ζ¯ f time M I f M. Then the conformal factor ¯ of is given by if h˙ = 0 or f = 0 . 2 ˙ 2 ρ¯ = 2 −u h f + f ζ ( f ) + g ( DU ζ , U ) . Theorem 10. Let ζ¯ = ζ ∈ X M¯ be a vector field on a standard static

Now, the structure of concircular vector ﬁelds on a standard ¯ = × f = ζ¯ spacetime M I f Mandassume that H 0. Then, is a Ricci static spacetimes is considered. collineation on M¯ if and only if ζ is a Ricci collineation on M. Theorem 8. A vector ﬁeld ζ¯ ∈ X M¯ on a standard static spacetime

¯ = × ζ 4. Ricci soliton on standard static spacetimes M I f Misa concircular vector field if and only if is a concircular vector field on M with factor ρ = h˙ and f is constant. A smooth vector field ζ on a pseudo-Riemannian manifold ( M,

Proof. It is clear that g) is said to define a Ricci soliton if D¯ ζ¯ = x h˙ + hX ( ln f ) + xζ ( ln f ) ∂ + xh f grad f + D ζ 1 X¯ t X L ζ g ( X, Y ) + Ric ( X, Y ) = λg ( X, Y ), 2 for any vector field X¯ = x∂t + X ∈ X M¯ . Suppose that f is constant where Ric is the Ricci curvature, L ζ denotes the Lie derivative ζ ρ = ˙ , and is a concircular vector field on M with factor h then of the metric tensor g and λ is a constant [17,18] . In this sec- ¯ ζ¯ = ρ ¯ , DX¯ X tion, we consider Ricci solitons on standard static spacetimes. Let

4 H.K. El-Sayied et al. / Journal of the Egyptian Mathematical Society 0 0 0 (2017) 1–5 ¯ , , ζ¯, λ ¯ = × ¯ , , ζ¯, λ ¯ = × M g¯ be a Ricci soliton, where M I f M is a standard static Corollary 4. Let M g¯ be a Ricci soliton, where M I f Misa ζ¯ = ∂ + ζ ∈ ¯ ζ¯ = ∂ + ζ ∈ ¯ , spacetime and h t X M . It isclear that a potential field standard static spacetime and h t X M and assume that ζ¯ is conformal on M¯ , g¯ if and only if M¯ , g¯ is an Einstein mani- H f = 0 and ( M, g ) is Ricci flat. Then ζ¯ is conformal with factor 2 λ. fold. ( ¯ , , ζ¯, λ) ¯ = × Proof. Let M g¯ be a Ricci soliton where M I f M is a ¯ , , ζ¯, λ ¯ = × ζ¯ = ∂ + ζ ∈ ( ¯ ) Theorem 11. Let M g¯ be a Ricci soliton where M I f Mis standard static spacetime and h t X M . Then ζ¯ = ∂ + ζ ∈ a standard static spacetime and h t X M¯ and assume that 1 ( L )( , ) + ( , ) = λ ( , ) f ¯ζ¯ g¯ X¯ Y¯ R¯ ic X¯ Y¯ g¯ X¯ Y¯ H = 0 . Then ( M, g, ζ , λ) is a Ricci soliton. 2 ¯ , , ζ¯, λ for any vector fields X¯ , Y¯ ∈ X M¯ . Eqs. (2.3) imply that Proof. Let M g¯ be a Ricci soliton, then L ¯ , ¯ = λ ¯ , ¯ , 1 ¯ζ¯ g¯ X Y 2 g¯ X Y L¯ ζ¯ g¯ X¯ , Y¯ + R¯ ic X¯ , Y¯ = λg¯ X¯ , Y¯ , 2 i.e., ζ¯ is a conformal vector field with factor 2 λ. where X¯ = x∂t + X and Y¯ = y∂t + Y are vector fields on M¯ . By using Theorem 14. Let ζ¯ = h∂ + ζ ∈ X M¯ be a vector field on a standard Eqs. (3.1) and (2.3) , we get t ¯ = × ¯ , , ζ¯, λ static spacetime M I f M. Then M g¯ is a Ricci soliton if 1 1 L ζ g (X, Y ) + Ric ( X, Y ) − H f ( X, Y ) = λg ( X, Y ). 2 f 1. ζ is a conformal vector field on M with conformal factor 2 ρ, 2. ( M, g ) is Einstein manifold with factor μ, f = , Suppose now that H 0 then 3. H f = 0 , 1 4. h˙ + ζ ( ln f ) = ρ + μ = λ. L ζ g (X, Y ) + Ric ( X, Y ) = λg ( X, Y ). 2 Proof. Let ζ¯ = h∂t + ζ ∈ X M¯ , then Eqs. (3.1) and (2.3) imply Thus ( M, g, ζ , λ) is a Ricci soliton. that 1 1 Let M¯ , g¯, ζ¯, λ be a Ricci soliton, where M¯ = I × M is a stan- f L¯ ζ¯ g¯ X¯ , Y¯ + R¯ ic X¯ , Y¯ = L ζ g ( X, Y ) + Ric ( X, Y ) 2 2 ζ¯ = ∂ + ζ ∈ ¯ dard static spacetime and h t X M . Then − xy f 2 h˙ + ζ ( ln f ) 1

L¯ ζ¯ g¯ X¯ , Y¯ + R¯ ic X¯ , Y¯ = λg¯ X¯ , Y¯ . 1 2 + xy f f − H f ( X, Y ). f Thus Since ζ is a conformal vector field with conformal factor 2 ρ, H f = 1 1 f μ L ζ g (X, Y ) + Ric ( X, Y ) − H ( X, Y ) = λg ( X, Y ), (4.1) 0 and ( M, g ) is Einstein manifold with factor , 2 f 1 2 ˙ L¯ ζ¯ g¯ X¯ , Y¯ + R¯ ic X¯ , Y¯ = ( ρ + μ)g ( X, Y ) − xy f h + ζ ( ln f ) . 2 ˙ + ζ ( ) − = λ . h f f f f (4.2) The last condition implies that Suppose that grad f is a concircular vector field with factor ρ, then 1 L g¯ X¯ , Y¯ + R¯ ic X¯ , Y¯ = λg¯ X¯ , Y¯ ζ¯ 1 ρ 2 L ζ g (X, Y ) + Ric ( X, Y ) = λ + g ( X, Y ). 2 f and the proof is complete. ¯ , , ζ¯, λ ¯ = × A similar discussion leads to the following result. Theorem 12. Let M g¯ be a Ricci soliton, where M I f M is a standard static spacetime, andassume that grad f is a concircu- Theorem 15. Let ζ¯ = h∂t + ζ ∈ X M¯ be a vector field on a standard ρ ρ , , ζ , λ + ¯ = × ¯ , , ζ¯, λ lar vector field with factor . Then M g f is a Ricci soliton static spacetime M I f M. Then M g¯ is a Ricci soliton if ρ whenever is constant. f 1. ( M, g, ζ , λ) is a Ricci soliton, 2. H f = 0 , Theorem 13. Let M¯ , g¯, ζ¯, λ be a Ricci soliton, where M¯ = I × M f 3. h˙ + ζ ( ln f ) = λ. f is a standard static spacetime, H = 0 and ζ¯ = h∂t + ζ ∈ X M¯ is a conformal vector field on M¯ with factor 2 ρ. Then ( M, g ) is Ricci flat Acknowledgement and λ = ρ. We would like to thank the referees for their careful review and ¯ , , ζ¯, λ ¯ = × Proof. Let M g¯ be a Ricci soliton where M I f M is a valuable comments, which provided insights that helped us to im- standard static spacetime and ζ¯ = h∂t + ζ ∈ X M¯ be a conformal prove the article quality. vector field on M¯ . Then References R¯ ic X¯ , Y¯ = ( λ − ρ)g ¯ X¯ , Y¯ [1] J.K. Beem , P.E. Ehrlich , K.L. Easley , Global Lorentzian Geometry, second ed., and hence f = −( λ − ρ) f and Marcel Dekker, New York, 1996 . [2] R.L. Bishop , B. O’Neill , Manifolds of negative curvature, Trans. Amer. Math. Soc.

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