A Trace Formula on Stationary Kaluza-Klein Spacetimes

TL;DR

This paper establishes a trace formula for wave spectra on stationary Kaluza-Klein spacetimes, generalizing previous results and applying to the Klein-Gordon equation with large vector bundle representations.

Contribution

It extends ladder asymptotics to relativistic Kaluza-Klein spacetimes, linking spectral distribution to representation theory and wave equations.

Findings

Derived a relativistic trace formula for wave spectra.

Connected spectral asymptotics to representation weights.

Applied results to Klein-Gordon equation on vector bundles.

Abstract

We prove relativistic versions of the ladder asymptotics from V. Guillemin and A. Uribe [Journal of Differential Geometry, 32(2):315-347, 1990] on principal bundles over globally hyperbolic, stationary, spatially compact spacetimes equipped with a Kaluza-Klein metric. This involves understanding the distribution of the frequency spectrum for the wave equation on a Kaluza-Klein spacetime when restricted to the isotypic subspace of an irreducible representation of the structure group, in the limit that the weight of the representation approaches infinity in the Weyl chamber. This is a direct generalization of the results from A. Strohmaier and S. Zelditch [Indagationes Mathematicae 32 (2021), 323-363] and is closely related to Strohmaier-Zelditch [Advances in Mathematics, Volume 376, 2021, 107434] and O. Islam arXiv:2109.09219. Furthermore we show how to apply these results to frequency asymptotics for the massive Klein-Gordon equation on vector bundles as one takes the representation defining the vector bundle to infinity.