On the compatibility of Lorentz-metrics with linear connections on 4-dimensional manifolds

TL;DR

This paper investigates conditions under which a symmetric connection on a 4-dimensional Lorentzian manifold is compatible with a Lorentz metric, deriving criteria and relationships between the metric and connection.

Contribution

It provides new sufficient conditions involving curvature tensors for a connection to be the Levi-Civita connection of a Lorentz metric.

Findings

Derived explicit conditions on curvature tensors for metric compatibility.

Established relationships between given Lorentz metric and compatible metric g.

Provided examples illustrating the applicability of the conditions.

Abstract

This paper considers 4-dimensional manifolds upon which there is a Lorentz metric, h, and a symmetric connection and which are originally assumed unrelated. It then derives sufficient conditions on the metric and connection (expressed through the curvature tensor) for the connection to be the Levi-Civita connection of some (local) Lorentz metric, g, and calculates the relationship between g and h. Some examples are provided which help to assess the strength of the sufficient conditions derived.