Codazzi tensors and their space-times, and Cotton gravity

TL;DR

This paper explores the geometric properties of Codazzi tensors in space-times and their role in Cotton gravity, identifying specific conditions and solutions like Stephani, Nariai, and Bertotti-Robinson metrics that satisfy the theory.

Contribution

It provides new characterizations of Codazzi tensors in space-times and applies these to analyze solutions in Cotton gravity, including conditions for static, spherically symmetric, and De Sitter space-times.

Findings

Perfect-fluid tensor is Codazzi iff the metric is a generalized Stephani universe.

Certain space-times like Nariai and Bertotti-Robinson solve Cotton gravity with realistic energy-momentum tensors.

Conditions for space-times to host current-flow Codazzi tensors are established.

Abstract

We study the geometric properties of certain Codazzi tensors for their own sake, and for their appearance in the recent theory of Cotton gravity. We prove that a perfect-fluid tensor is Codazzi if and only if the metric is a generalized Stephani universe. A trace condition restricts it to a warped space-time, as proven by Merton and Derdzinski. We also give necessary and sufficient conditions for a space-time to host a current-flow Codazzi tensor. In particular, we study the static and spherically symmetric cases, which include the Nariai and Bertotti-Robinson metrics. The latter are a special case of Yang Pure space-times, together with spatially flat FRW space-times with constant curvature scalar. We apply these results to the recent Cotton gravity by Harada. The equations have the freedom of choosing a Codazzi tensor, that constrains the space-time where the theory is staged. The tensor, chosen in forms significative for physics, implies the form of the Ricci tensor, and the two specify the energy-momentum tensor, which is the source in Cotton gravity for the chosen metric. For example, the Stephani, Nariai and Bertotti-Robinson space-times solve Cotton gravity with physically sensible energy-momentum tensors. Finally, we discuss Cotton gravity in De Sitter space-times.