Curvature Properties of Interior Black Hole Metric

Abstract

A spacetime is a connected 4-dimensional semi-Riemannian manifold endowed with a metric tensor g with signature (− + ++). The geometry of a spacetime is described by the tensor g and the Ricci tensor S of type (0, 2) whereas the energy momentum tensor of type (0, 2) describes the physical contents of the spacetime. Einstein’s field equations relate g, S and the energy momentum tensor and describe the geometry and physical contents of the spacetime. By solving Einstein’s field equations for empty spacetime (i.e. S = 0) for a non-static spacetime metric, one can obtain the interior black hole solution, known as the interior black hole spacetime which infers that a remarkable change occurs in the nature of the spacetime, namely, the external spatial radial and temporal coordinates exchange their characters to temporal and spatial coordinates, respectively, and hence the interior black hole spacetime is a non-static one as the metric coefficients are time dependent. For the sake of mathematical generalizations, in the literature, there are many rigorous geometric structures constructed by imposing the restrictions to the curvature tensor of the space involving first order and second order covariant differentials of the curvature tensor. Hence a natural question arises that which geometric structures are admitted by the interior black hole metric. The main aim of this paper is to provide the answer of this question so that the geometric structures admitting by such a metric can be interpreted physically.

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Acknowledgement

The work was carried out when the first named author visited Department of Mathematics of the Sardar Patel University as a visiting fellow under their UGC-SAP-DRS programme (F-510/5/DRS/2009 (SAP-II)). He also greatfully acknowledges the financial support of CSIR, New Delhi, India [Project F. No. 25(0171)/09/EMR-II]. The second named author is supported by a grant of the Technische Universität Berlin (Germany).

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Correspondence to Absos Ali Shaikh or Ryszard Deszcz or Abdulvahid H. Hasmani or Vrajeshkumar G. Khambholja.

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Shaikh, A.A., Deszcz, R., Hasmani, A.H. et al. Curvature Properties of Interior Black Hole Metric. Indian J Pure Appl Math 51, 1779–1814 (2020). https://doi.org/10.1007/s13226-020-0497-2

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Key words

  • Einstein’s field equations
  • interior black hole metric
  • warped product metric
  • Tachibana tensor
  • quasi-Einstein manifold
  • 2-quasi-Einstein manifold
  • partially Einstein manifold
  • pseudosymmetric space
  • curvature condition of pseudosymmetry type

2010 Mathematics Subject Classification

  • 53B20
  • 53B25
  • 53B50
  • 53C25
  • 53C40
  • 83C57