Canonical variation of a Lorentzian metric

TL;DR

This paper explores how to derive a Riemannian metric from a Lorentzian manifold using a timelike vector field, analyzing the curvature relations especially when the vector field has special properties.

Contribution

It introduces a canonical way to vary a Lorentzian metric to a Riemannian one and relates their curvatures, providing new insights especially for Killing or closed vector fields.

Findings

Derived relations between curvatures of Lorentzian and Riemannian metrics.

Identified special cases where the vector field is Killing or closed.

Provided applications to the geometry of Lorentzian manifolds.

Abstract

Given a Lorentzian manifold $(M,g_L)$ and a timelike unitary vector field $E$, we can construct the Riemannian metric $g_R=g_L+2\omega\otimes\omega$, being $\omega$ the metrically equivalent one form to $E$. We relate the curvature of both metrics, especially in the case of $E$ being Killing or closed, and we use the relations obtained to give some results about $(M,g_L)$.