Lie Symmetries of the Energy–Momentum Tensor for Plane Symmetric Static Spacetimes (original) (raw)
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Symmetries of the Energy-Momentum Tensor for Static Plane Symmetric Spacetimes
This article explores matter collineations (MCs) of static plane-symmetric spacetimes, considering the stress-energy tensor in its contravariant and mixed forms. We solve the MC equations in two cases: when the energy-momentum tensor is nondegenerate and degenerate. For the case of degenerate energy-momentum tensor, we employ a direct integration technique to solve the MC equations, which leads to an infinite-dimensional Lie algebra. On the other hand, when considering the nondegenerate energy-momentum tensor, the contravariant form results in a finite-dimensional Lie algebra with dimensions of either 4 or 10. However, in the case of the mixed form of the energy-momentum tensor, the dimension of the Lie algebra is infinite. Moreover, the obtained MCs are compared with those already found for covariant stress-energy.
Theoretical and Mathematical Physics, 2018
Considering the degenerate and non-degenerate cases, we provide a complete classification of static plane symmetric spacetimes according to conformal Ricci collineations (CRCs) and conformal matter collineations (CMCs). In case of non-degenerate Ricci tensor, a general form of vector field generating CRCs is found in terms of unknown functions of t and x, subject to some integrability conditions. The integrability conditions are then solved in different cases depending upon the nature of Ricci tensor and it is concluded that static plane symmetric spacetimes possess 7, 10 or 15-dimensional Lie algebra of CRCs. Moreover, it is found that these spacetimes admit infinite number of CRCs when the Ricci tensor is degenerate. A similar procedure is adopted for the study of CMCs in degenerate and non-degenerate matter tensor cases. The exact form of some static plane symmetric spacetimes metrics is obtained admitting non-trivial CRCs and CMCs. Finally, we present some physical implications of our obtained results by considering a perfect fluid as a source of the energy-momentum tensor.
Matter Symmetries of Non-Static Plane Symmetric Spacetimes
2020
The matter collineations of plane symmetric spacetimes are studied according to the degenerate energy-momentum tensor. We have found many cases where the energy-momentum tensor is degenerate but the group of matter collineations is finite. Further we obtain different constraint equations on the energy-momentum tensor. Solving these constraints may provide some new exact solutions of Einstein field equations.
RICCI and Matter Collineations of SOM-ROY Chaudhary Symmetric Space Time
Mehran University Research Journal of Engineering and Technology
This paper is devoted to explore the RICCI and MCs (Matter Collineations of the Som-Ray Chaudhary spacetime. The spacetime under consideration is one of the spatially homogeneous and rotating spacetimes. Collineations are the some kinds of the Lie symmetries. To discuss the required collineations we have used the RICCI and energy momentum tensors. As the RICCI tensor is formulated from the metric tensor, it must possess its symmetries. RCs (RICCI Collineations) leads to conservation laws. On the other hand for the distribution of matter in the spacetimes, the symmetries of energy momentum tensor or MCs provides conservation laws on matter field. Throughout this paper, these collineations are discussed by vanishing Lie derivative of RICCI and energy momentum tensors respectively. Complete solution of the RCs and MCs equations, which are formed in the result of vanishing Lie derivative are explored. Studying all these collineations in the said spacetime, it has been shown that RCs of the spacetime form an infinite dimensional vector space where as MCs are Killing vector fields.
Matter Collineations of Plane Symmetric Spacetimes
arXiv: General Relativity and Quantum Cosmology, 2008
This paper is devoted to the study of matter collineations of plane symmetric spacetimes (for a particular class of spacetimes) when the energy-momentum tensor is non-degenerate. There exists many interesting cases where we obtain proper matter collineations. The matter collineations in these cases are {\\it four}, \\emph{five}, {\\it six}, \\emph{seven} and {\\it ten} with some constraints on the energy-momentum tensor. We have solved some of these constraints to obtain solutions of the Einstein field equations.
Classification of Spherically Symmetric Static Spacetimes According to Their Matter Collineations
General Relativity and Gravitation, 2003
The spherically symmetric static spacetimes are classified according to their matter collineations. These are studied when the energy-momentum tensor is degenerate and also when it is non-degenerate. We have found a case where the energy-momentum tensor is degenerate but the group of matter collineations is finite. For the non-degenerate case, we obtain either four, five, six or ten independent matter collineations in which four are isometries and the rest are proper. We conclude that the matter collineations coincide with the Ricci collineations but the constraint equations are different which on solving can provide physically interesting cosmological solutions.
Symmetries of Energy-Momentum Tensor: Some Basic Facts
Communications in Theoretical Physics, 2007
It has been pointed by Hall et al. [1] that matter collinations can be defined by using three different methods. But there arises the question of whether one studies matter collineations by using the L ξ T ab = 0, or L ξ T ab = 0 or L ξ T b a = 0. These alternative conditions are, of course, not generally equivalent. This problem has been explored by applying these three definitions to general static spherically symmetric spacetimes. We compare the results with each definition.
Ricci and matter collineations of locally rotationally symmetric space-times
A new method is presented for the determination of Ricci Collineations (RC) and Matter Collineations (MC) of a given spacetime, in the cases where the Ricci tensor and the energy momentum tensor are non-degenerate and have a similar form with the metric. This method reduces the problem of finding the RCs and the MCs to that of determining the KVs whereas at the same time uses already known results on the motions of the metric. We employ this method to determine all hypersurface homogeneous locally rotationally symmetric spacetimes, which admit proper RCs and MCs. We also give the corresponding collineation vectors.
Matter collineations of spacetime homogeneous G del-type metrics
Classical and Quantum Gravity, 2003
The spacetime homogeneous Gödel-type spacetimes which have four classes of metrics are studied according to their matter collineations. The obtained results are compared with Killing vectors and Ricci collineations. It is found that these spacetimes have infinite number of matter collineations in degenerate case, i.e. det(T ab ) = 0, and do not admit proper matter collineations in non-degenerate case, i.e. det(T ab ) = 0. The degenerate case has the new constraints on the parameters m and w which characterize the causality features of the Gödel-type spacetimes.
Non-static spherically symmetric spacetimes and their conformal Ricci collineations
Arabian Journal of Mathematics, 2019
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.