Vacuum Einstein field equations in smooth metric measure spaces: the isotropic case

TL;DR

This paper extends Einstein field equations to smooth metric measure spaces, analyzing isotropic solutions and their geometric properties, including classifications as Brinkmann waves and Kundt spacetimes, especially in three dimensions.

Contribution

It introduces a weighted Einstein tensor in smooth metric measure spaces and classifies isotropic solutions, revealing their geometric structures and specific forms in three dimensions.

Findings

Isotropic solutions have nilpotent Ricci operators.

Manifolds are Brinkmann waves if 2-step nilpotent.

Manifolds are Kundt spacetimes if 3-step nilpotent.

Abstract

On a smooth metric measure spacetime $(M,g,e^{-f} dvol_g)$, we define a weighted Einstein tensor. It is given in terms of the Bakry-\'Emery Ricci tensor as a tensor which is symmetric, divergence-free, concomitant of the metric and the density function. We consider the associated vacuum weighted Einstein field equations and show that isotropic solutions have nilpotent Ricci operator. Moreover, the underlying manifold is a Brinkmann wave if it is $2$-step nilpotent and a Kundt spacetime if it is $3$-step nilpotent. More specific results are obtained in dimension $3$, where all isotropic solutions are given in local coordinates as plane waves or Kundt spacetimes.