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

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 two-step nilpotent and a Kundt spacetime if it is three-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.

Automated summary

In this paper, a weighted Einstein tensor is defined on a smooth metric measure spacetime, and its applications in isotropic solutions are studied. By investigating the nilpotent Ricci operator, specific types of manifold are obtained. In dimension 3, all isotropic solutions can be expressed as plane waves or Kundt spacetimes.