Curvature Tensor for the Spacetime of General Relativity

TL;DR

This paper investigates the properties of the W-curvature tensor in general relativity spacetimes, analyzing conditions for vanishing W-tensor and its implications for symmetries and cosmological models.

Contribution

It provides a detailed study of the W-curvature tensor in Einstein spacetimes, including conditions for vanishing tensor and its relation to symmetries and cosmological solutions.

Findings

Vanishing W-tensor leads to Einstein or Friedmann-Robertson-Walker spacetimes.

Existence of Killing and conformal Killing vectors in certain W-tensor conditions.

Divergence of W-tensor relates to conserved quantities in perfect fluid spacetimes.

Abstract

In the differential geometry of certain F-structures, the role of W-curvature tensor is very well known. A detailed study of this tensor has been made on the spacetime of general relativity. The spacetimes satisfying Einstein field equations with vanishing W-tensor have been considered and the existence of Killing and conformal Killing vector fields has been established. Perfect fluid spacetimes with vanishing W-tensor have also been considered. The divergence of W-tensor is studied in detail and it is seen, among other results, that a perfect fluid spacetime with conserved W-tensor represents either an Einstein space or a Friedmann-Robertson-Walker cosmological model.