  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.
