A Gutzwiller trace formula for stationary space-times

TL;DR

This paper extends the Gutzwiller trace formula to stationary space-times in a relativistic setting, linking spectral properties of the wave operator to periodic orbits of the spacetime's geodesic flow.

Contribution

It introduces a relativistic generalization of the trace formula for wave groups on stationary space-times with compact Cauchy surfaces, connecting spectral data to geometric periodic orbits.

Findings

Spectrum of the Killing vector field Z is discrete.

Singularities in the trace occur at periods of periodic orbits.

Provides a Weyl law for eigenvalues of Z on the null space.

Abstract

We give a relativistic generalization of the Gutzwiller-Duistermaat-Guillemin trace formula for the wave group of a compact Riemannian manifold to globally hyperbolic stationary space-times with compact Cauchy hypersurfaces. We introduce several (essentially equivalent) notions of trace of self-adjoint operators on the null-space $\ker \Box$ of the wave operator and define $U(t)$ to be translation by the flow $e^{t Z}$ of the timelike Killing vector field $Z$ on $\Box$. The spectrum of $Z$ on $\ker \Box$ is discrete and the singularities of $\rm{Tr}\;e^{t Z} |_{\ker \Box}$ occur at periods of periodic orbits of $\exp t Z$ on the symplectic manifold of null geodesics. The trace formula gives a Weyl law for the eigenvalues of $Z$ on $\ker \Box$.