Heat and Wave kernel expansions for stationary spacetimes

TL;DR

This paper explores the spectral analysis of wave operators on stationary spacetimes, relating wave-trace expansions to heat kernel coefficients and extending classical spectral geometry.

Contribution

It computes the second wave-trace coefficient for stationary spacetimes, generalizing heat kernel coefficients and linking spectral properties to spacetime geometry.

Findings

Derived a formula for the second wave-trace coefficient in stationary spacetimes.

Connected wave-trace expansion terms to heat kernel coefficients and scalar curvature.

Extended classical spectral geometry results to stationary spacetime contexts.

Abstract

The generator of time-translations on the solution space of the wave equation on stationary spacetimes specialises to the square root of the Laplacian on Riemannian manifolds when the spacetime is ultrastatic. Its spectral analysis therefore constitutes a generalization of classical spectral geometry. If the spacetime is spatially compact the spectrum is discrete and admits a wave-trace expansion at time zero. A Weyl law for the eigenvalues and a wave-trace formula was shown in a previous paper and related to the geometry of the space of null-geodesics. In this paper we investigate the relation to heat kernel coefficients and residues of zeta functions in this context and compute the second non-zero term in the wave-trace expansion. This second coefficient is an analogue in the category of stationary spacetimes of the second heat kernel coefficient of the Laplace operator. The general formula is quite involved but reduces to the usual term involving the scalar curvature when specialised to ultra-static spacetimes.