Lorentzian spectral zeta functions on asymptotically Minkowski spacetimes

TL;DR

This paper studies the spectral zeta functions of wave operators on asymptotically Minkowski spacetimes, revealing a connection between residues at specific poles and scalar curvature, which informs a Lorentzian spectral action for gravity.

Contribution

It demonstrates the meromorphic continuation of Lorentzian spectral zeta functions and links residues to scalar curvature, advancing spectral approaches to Lorentzian gravity.

Findings

Residue at rac{n}{2}-1 is proportional to scalar curvature.

Spectral zeta functions admit meromorphic continuation in Lorentzian spacetimes.

Results provide a spectral action principle for Lorentzian gravity.

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 $\alpha \mapsto (\square_g-i \varepsilon)^{-\alpha}$. In even dimension $n\geq 4$, the residue at $\frac{n}{2}-1$ is shown to be a multiple of the scalar curvature in the limit $\varepsilon\to 0^+$. This yields a spectral action for gravity in Lorentzian signature.