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# Curvature properties of ( t − z ) -type plane wave metric

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## Abstract

The objective, in this paper, is to obtain the curvature properties of (t−z)-type plane wave metric studied by Bondi et al. (1959). For this a general (t−z)-type wave metric is considered and the condition for which it obeys Einstein’s empty spacetime field equations is obtained. It is found that the rank of the Ricci tensor of (t−z)-type plane wave metric is 1 and is of Codazzi type. Also it is proved that it is not recurrent but Ricci recurrent, conformally recurrent and hyper generalized recurrent. Moreover, it is semisymmetric and satisfies the Ricci generalized pseudosymmetric type condition P⋅P=−13Q(Ric,P). It is interesting to note that, physically, the energy momentum tensor describes a radiation field with parallel rays and geometrically it is a Codazzi tensor and semisymmetric. As special case, the geometric structures of Taub’s plane symmetric spacetime metric are deduced. Comparisons between (t−z)-type plane wave metric and pp-wave metric with respect to their geometric structures are viewed.

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... We note that Morris-Thorne spacetime [103] and Gödel spacetime [28] are Ricci simple manifolds, Robertson-Walker spacetime [26] and Siklos spacetime [29] are quasi-Einstein manifolds, Kantowski-Sachs spacetime [84] and Som-Raychaudhuri spacetime [25] are 2-quasi Einstein manifolds and Kaigorodov spacetime [29] is an Einstein manifold. For curvature properties of Robinson-Trautman metric, Melvin magnetic metric and generalized pp-wave metric, etc., we refer the reader to see [30,[104][105][106]. ...
... We note that the Ricci tensor of Gödel spacetime [28] is cyclic parallel and the Ricci tensor of (t − z)-type plane wave metric [104] is of Codazzi type. Let ζ be a (0, 4)-type tensor on M. Then a symmetric (0,2)-type tensor ν corresponding to the endomorphism I ν is said to be ζ-compatible if ...
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... Also, it is noteworthy to mention that Sabina et al. studied various curvature properties of (t − z)-type plane wave spacetime [22], Morris-Thorne wormhole spacetime [21]. ...
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... It may be mentioned that Gödel spacetime [26], Som-Raychaudhury spacetime [75], Defrise spacetime [68] are equipped with cyclic parallel Ricci tensor while the (t − z)type plane wave metric [31] has Codazzi Ricci tensor. ...
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