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Symmetries of Associated Standard Static Spacetimes and Applications ARTICLE IN PRESS JID: JOEMS [m5G; July 22, 2017;16:58 ] Journal of the Egyptian Mathematical Society 0 0 0 (2017) 1–5 Contents lists available at ScienceDirect Journal of the Egyptian Mathematical Society journal homepage: www.elsevier.com/locate/joems 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 grad ω ∈ 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 .
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