The wave resolvent for compactly supported perturbations of static spacetimes

TL;DR

This paper studies the wave operator on perturbed static spacetimes, proving self-adjointness and microlocal estimates for the resolvent, facilitating spectral analysis relevant to Einstein equations.

Contribution

It provides an elementary proof of self-adjointness and uniform microlocal estimates for the wave operator on perturbed static spacetimes, offering a simple model for Lorentzian spectral analysis.

Findings

Proves essential self-adjointness of the wave operator.

Establishes uniform microlocal estimates for the resolvent.

Provides a framework for deriving Einstein equations from spectral data.

Abstract

In this note, we consider the wave operator $\square_g$ in the case of globally hyperbolic, compactly supported perturbations of static spacetimes. We give an elementary proof of the essential self-adjointness of $\square_g$ and of uniform microlocal estimates for the resolvent in this setting. This provides a model for studying Lorentzian spectral zeta functions which is particularly simple, yet sufficiently general for locally deriving Einstein equations from a spectral Lagrangian.