The vacuum weighted Einstein field equations

TL;DR

This paper introduces and analyzes the vacuum weighted Einstein field equations on spacetimes with a density function, characterizing critical metrics and classifying specific solutions in various dimensions.

Contribution

It defines the vacuum weighted Einstein equations, links them to critical metrics of an action, and classifies solutions in conformally flat and harmonic curvature cases.

Findings

Characterization of critical metrics via an action principle

Classification of conformally flat solutions in arbitrary dimensions

Classification of four-dimensional solutions with harmonic curvature

Abstract

On a spacetime $(M,g)$ endowed with a density function $h$, we consider the vacuum weighted Einstein field equations: \[h\rho-\operatorname{Hes}_h+\Delta h g=0.\] First, it is shown that the equation characterizes critical metrics for an appropriate action. Then, after describing locally conformally flat solutions in arbitrary dimension, four-dimensional solutions with harmonic curvature are classified.