  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.
