Non-static spherically symmetric spacetimes and their conformal Ricci collineations
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Fawad Khan · Tahir Hussain Sumaira Saleem Akhtar
Arabian Journal of Mathematics
· Ashfaque Hussain Bokhari ·
Non-static spherically symmetric spacetimes and their conformal Ricci collineations
Received: 3 November 2018 / Accepted: 18 February 2019 © The Author(s) 2019
Abstract For a perfect fluid matter, we present a study of conformal Ricci collineations (CRCs) of non-static spherically symmetric spacetimes. For non-degenerate Ricci tenor, a vector field generating CRCs is found subject to certain integrability conditions. These conditions are then solved in various cases by imposing certain restrictions on the Ricci tensor components. It is found that non-static spherically symmetric spacetimes admit 5, 6 or 15 CRCs. In the degenerate case, it is concluded that such spacetimes always admit infinite number of CRCs. Mathematics Subject Classification
83C15
1 Introduction The Einstein’s general theory of relativity is a fascinating theory of gravitation, published by Albert Einstein in 1915. In this theory, Einstein stated that spacetimes become curved due to the presence of mass and energy. In this way, this theory replaced the notion of force with the presence of spacetime curvature. The governing equations of this theory are the following nonlinear partial differential equations, known as the Einstein’s field equations (EFEs) [22]: R G ab = Rab − gab = G Tab , (1.1) 2 where G ab , Rab , gab and Tab are Einstein, Ricci, metric and energy–momentum tensors, respectively, R the Ricci scalar and G is the gravitational constant. The exact solutions of the EFEs are Lorentzian metrics which are obtained by solving the EFEs in closed form that are compatible with a physically realistic energy–momentum tensor. There are usually two complementary methods to deal with the exact solutions of EFEs. For the first method, one chooses a specific form of the energy–momentum tensor and studies the corresponding exact solutions of the EFEs. In the second F. Khan · T. Hussain (B) · S. S. Akhtar Department of Mathematics, University of Peshawar, Khyber Pakhtunkhwa, Pakistan E-mail: [email protected] F. Khan E-mail: [email protected] S. S. Akhtar E-mail: [email protected] A. H. Bokhari Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia E-mail: [email protected]
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approach, one focusses on some geometric properties admitted by a spacetime given by symmetries, and then look for a matter source that depicts these properties. For the present investigation, we focus on the second approach. Spacetime symmetries are the vector fields which preserve certain physical properties of spacetimes, such as geodesics, metric, curvature, Ricci or energy–momentum tensor. According to an approach followed by Hall, a smooth tensor T is invariant under a smooth vector field X on a spacetime M if φt∗ (T) = T, for any smooth local flow diffeomorphism φt associated with X. Equivalently, this states that the Lie derivative of T under X vanishes, tha
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