On the Weyl and Ricci tensors of Generalized Robertson-Walker space-times

TL;DR

This paper investigates the properties of Ricci and Weyl tensors in generalized Robertson-Walker space-times, revealing conditions for harmonic Weyl tensors and characterizing when these space-times are quasi-Einstein or Robertson-Walker.

Contribution

It provides new theorems relating the Ricci and Weyl tensors to Chen's vector, characterizes harmonic Weyl tensors, and describes the Riemann tensor structure in GRW space-times.

Findings

Ricci tensor decomposes into perfect fluid and Weyl terms via Chen's vector.

Weyl tensor is harmonic iff annihilated by Chen's vector.

GRW space-time with null conformal divergence in 4D is Robertson-Walker.

Abstract

We prove theorems about the Ricci and the Weyl tensors on generalized Robertson-Walker space-times of dimension $n\ge 3$. In particular, we show that the concircular vector introduced by Chen decomposes the Ricci tensor as a perfect fluid term plus a term linear in the contracted Weyl tensor. The Weyl tensor is harmonic if and only if it is annihilated by Chen's vector, and any of the two conditions is necessary and sufficient for the GRW space-time to be a quasi-Einstein (perfect fluid) manifold. Finally, the general structure of the Riemann tensor for Robertson-Walker space-times is given, in terms of Chen's vector. A GRW space-time in n = 4 with null conformal divergence is a Robertson-Walker space-time.