# Spacetimes admitting W2-curvature tensor

16 Apr 2014-International Journal of Geometric Methods in Modern Physics (World Scientific Publishing Company)-Vol. 11, Iss: 04, pp 1450030

TL;DR: In this article, it was shown that a W2-flat spacetime is conformally flat and hence it is of Petrov type O, and if the perfect fluid spacetime with vanishing W 2-curvature tensor obeys Einstein's field equation without cosmological constant, then the spacetime has vanishing acceleration vector and expansion scalar and the ideal fluid always behaves as a cosmologically constant.

Abstract: The object of this paper is to study spacetimes admitting W2-curvature tensor. At first we prove that a W2-flat spacetime is conformally flat and hence it is of Petrov type O. Next, we prove that if the perfect fluid spacetime with vanishing W2-curvature tensor obeys Einstein's field equation without cosmological constant, then the spacetime has vanishing acceleration vector and expansion scalar and the perfect fluid always behaves as a cosmological constant. It is also shown that in a perfect fluid spacetime of constant scalar curvature with divergence-free W2-curvature tensor, the energy-momentum tensor is of Codazzi type and the possible local cosmological structure of such a spacetime is of type I, D or O.

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TL;DR: In this paper, the W2-curvature tensor on warped product manifolds and on generalized Robertson-Walker and standard static space-times has been studied and the geometry of the base and fiber of these warped product space-time models has been investigated.

Abstract: The purpose of this paper is to study the W2-curvature tensor on (singly) warped product manifolds as well as on generalized Robertson–Walker and standard static space-times. Some different expressions of the W2-curvature tensor on a warped product manifold in terms of its relation with W2-curvature tensor on the base and fiber manifolds are obtained. Furthermore, we investigate W2-curvature flat warped product manifolds. Many interesting results describing the geometry of the base and fiber manifolds of a W2-curvature flat warped product manifold are derived. Finally, we study the W2-curvature tensor on generalized Robertson–Walker and standard static space-times; we explore the geometry of the fiber of these warped product space-time models that are W2-curvature flat.

23 citations

### Cites background from "Spacetimes admitting W2-curvature t..."

...Among such applications, we want to mention that a W2-curvature flat 4dimensional space-time is an Einstein manifold [2, 5]....

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...In [2, 5], the authors study the properties of flat space-time under some conditions regarding the W2-curvature tensor and W2-flat space-times....

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TL;DR: In this article, a necessary and sufficient condition for a spacetime with pseudo symmetric energy-momentum tensor to be a pseudo Ricci symmetric spacetime was given, and several interesting results were obtained.

Abstract: The object of the present paper is to characterize spacetimes with different types of energy–momentum tensor. At first we consider spacetimes with pseudo symmetric energy–momentum tensor T . We obtain a necessary and sufficient condition for a spacetime with pseudo symmetric energy–momentum tensor to be a pseudo Ricci symmetric spacetime. Next we consider the spacetimes with Codazzi type of energy–momentum tensor and several interesting results are pointed out. Moreover, some results related to perfect fluid spacetimes with different forms of energy–momentum tensors have been obtained. We study spacetimes with quadratic Killing energy–momentum tensor T and show that a GRW spacetime with quadratic Killing energy–momentum tensor is an Einstein space. Finally, we have considered general relativistic spacetimes with semisymmetric energy–momentum tensor and obtained some important results.

9 citations

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TL;DR: In this paper, a new type of curvature tensor called H-curvature tensors of type (1, 3) was introduced, which is a linear combination of conformal and projective curvatures.

Abstract: In this paper, we introduce a new type of curvature tensor named H-curvature
tensor of type (1, 3) which is a linear combination of conformal and
projective curvature tensors. First we deduce some basic geometric
properties of H-curvature tensor. It is shown that a H-flat Lorentzian
manifold is an almost product manifold. Then we study pseudo H-symmetric
manifolds (PHS)n (n > 3) which recovers some known structures on Lorentzian
manifolds. Also, we provide several interesting results. Among others, we
prove that if an Einstein (PHS)n is a pseudosymmetric (PS)n, then the scalar
curvature of the manifold vanishes and conversely. Moreover, we deal with
pseudo H-symmetric perfect fluid spacetimes and obtain several interesting
results. Also, we present some results of the spacetime satisfying
divergence free H-curvature tensor. Finally, we construct a non-trivial
Lorentzian metric of (PHS)4.

8 citations

### Cites background from "Spacetimes admitting W2-curvature t..."

...Also, several authors studied spacetimes in different way such as ([21]-[24], [38], [50]) and many others....

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...Also, several authors studied spacetimes in different way such as ([21], [22], [36]-[38], [50]) and many others....

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TL;DR: In this article, mixed quasi-Einstein manifolds have been studied and some geometric properties of mixed QE2Einstein manifold have been discussed, and M(QE)4 spacetime with spac...

Abstract: The object of the present paper is to study mixed quasi-Einstein manifolds. Some geometric properties of mixed quasi-Einstein manifolds have been studied. We also discuss M(QE)4 spacetime with spac...

6 citations

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TL;DR: In this paper, the P -curvature tensor is introduced and investigated, which generalizes projective, conharmonic, M -projective and the set of W i curvature tensors introduced by Pokhariyal and Mishra.

Abstract: The object of the present paper is to introduce and investigate the P -curvature tensor that generalizes projective, conharmonic, M -projective and the set of W i curvature tensors introduced by Pokhariyal and Mishra. It is proven that pseudo-Riemannian manifolds admitting a traceless P -curvature tensor are Einstein and those admitting flat P -curvature tensor has a constant curvature. Classification theorems for pseudo-Riemannian manifolds admitting a divergence-free P -curvature tensor are given in each subspace of Gray’s decomposition of the covariant derivative of the Ricci tensor. Space–times having a flat P -curvature tensor or a divergence free P -curvature tensor are scrutinized. Finally, perfect fluid space–times admitting various features of the P -curvature tensor are considered.

5 citations

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TL;DR: In this article, the authors consider the case where the flow lines of a perfect fluid form a time-like shear-free normal congruence and show that all the degenerate fields admit at least a one-parameter group of local isometries with space-like trajectories.

Abstract: Flows of a perfect fluid in which the flow-lines form a time-like shear-free normal congruence are investigated. The space-time is quite severely restricted by this condition on the flow: it must be of Petrov Type I and is either static or degenerate. All the degenerate fields are classified and the field equations solved completely, except in one class where one ordinary differential equation remains to be solved. This class contains the spherically symmetric non-uniform density fields and their analogues with planar or hyperbolic symmetry. The type D fields admit at least a one-parameter group of local isometries with space-like trajectories. All vacuum fields which admit a time-like shear-free normal congruence are shown to be static. Finally, shear-free perfect fluid flows which possess spherical or a related symmetry are considered, and all uniform density solutions and a few non-uniform density solutions are found. The exact solutions are tabulated in section 7.

87 citations

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TL;DR: In this paper, the relativistic significance of concircular curvature tensors has been explored and the existence of Killing and conformal Killing vectors has been established for spacetimes satisfying Einstein field equations.

Abstract: In the differential geometry of certain F-structures, the importance of concircular curvature tensor is very well known. The relativistic significance of this tensor has been explored here. The spacetimes satisfying Einstein field equations and with vanishing concircular curvature tensor are considered and the existence of Killing and conformal Killing vectors have been established for such spacetimes. Perfect fluid spacetimes with vanishing concircular curvature tensor have also been considered. The divergence of concircular curvature tensor is studied in detail and it is seen, among other results, that if the divergence of the concircular tensor is zero and the Ricci tensor is of Codazzi type then the resulting spacetime is of constant curvature. For a perfect fluid spacetime to possess divergence-free concircular curvature tensor, a necessary and sufficient condition has been obtained in terms of Friedmann-Robertson-Walker model.

38 citations

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TL;DR: In this paper, it was shown that a general relativistic space-time with covariant-constant energy-momentum tensors is Ricci symmetric, and two particular types of such general space-times were considered and determined.

Abstract: It is shown that a general relativistic space-time with covariant-constant energy-momentum tensor is Ricci symmetric. Two particular types of such general relativistic space-times are considered and the nature of each is determined.

37 citations

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TL;DR: In this paper, new curvature tensors have been defined on the lines of Weyl's projective tensor and it has been shown that the order in which the vectors in question are arranged before being acted upon by the tensor in question plays an important role in shaping the various physical and geometrical properties of a tensor.

Abstract: In thi paper new curvature tensors have been defined on the lines of Weyl's projective curvature tensor and it has been shown that the distribution (order in which the vectors in question are arranged before being acted upon by the tensor in question) of vector field over the metric potentials and matter tensors plays an important role in shaping the various physical and geometrical properties of a tensor viz the formulation of gravitational waves, reduction of electromagnetic field to a purely electric field, vanishing of the contracted tensor in an Einstein Space and the cyclic property.

23 citations

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TL;DR: In this article, the authors studied the properties of flat spacetimes under some conditions regarding the W2-curvature tensor and showed that the energy momentum tensor satisfying the Einstein's equations with a cosmological constant is a quadratic conformal Killing tensor.

Abstract: The object of this paper is to study the properties of flat spacetimes under some conditions regarding the W2-curvature tensor. In the first section, several results are obtained on the geometrical symmetries of this curvature tensor. It is shown that in a spacetime with W2curvature tensor filled with a perfect fluid, the energy momentum tensor satisfying the Einstein’s equations with a cosmological constant is a quadratic conformal Killing tensor. It is also proved that a necessary and sufficient condition for the energy momentum tensor to be a quadratic Killing tensor is that the scalar curvature of this space must be constant. In a radiative perfect fluid, it is shown that the sectional curvature is constant. 2000 Mathematics Subject Classification: 53C15; 53C25; 53B15; 53B20

7 citations