Lorentzian spectral zeta functions on asymptotically Minkowski spacetimes

Abstract

In this note, we consider perturbations of Minkowski space as well as more general spacetimes on which the wave operator ${\square }_g$ is essentially self-adjoint. We review a recent result which gives the meromorphic continuation of the Lorentzian spectral zeta function density, i.e. of the trace density of complex powers $\alpha \mapsto ({\square }_g-i\epsilon {)}^{-\alpha }$. In even dimension $n\ge 4$, the residue at $\frac{n}{2}-1$ is shown to be a multiple of the scalar curvature in the limit $\epsilon \to 0^{+}$. This yields a spectral action for gravity in Lorentzian signature.

12 pages, proceedings paper based on arXiv:2012.00712