# Wave Operators, Torsion, and Weitzenb\"ock Identities

## TL;DR

This paper develops mathematical tools to analyze wave propagation in spacetimes with torsion, showing that massless waves travel at light speed along null geodesics and proposing gravitational waves as torsion probes.

## Contribution

It introduces generalized wave operators and identities for Riemann-Cartan geometries, enabling model-independent analysis of wave propagation with torsion.

## Key findings

- Massless waves propagate at light speed along null torsionless geodesics.
- The tools apply to fields in Lie (super) algebra representations.
- Gravitational waves could serve as probes for spacetime torsion.

## Abstract

We offer a mathematical toolkit for the study of waves propagating on a background manifold with nonvanishing torsion. Examples include electromagnetic and gravitational waves on a spacetime with torsion. The toolkit comprises generalized versions of the Lichnerowicz-de Rham and the Beltrami wave operators, and the Weitzenb\"ock identity relating them on Riemann-Cartan geometries. The construction applies to any field belonging to a matrix representation of a Lie (super) algebra containing an $\mathfrak{so}$ subalgebra. Using these tools, we analyze the propagation of different massless waves in the eikonal (geometric optics) limit in a model-independent way and find that they all must propagate at the speed of light along null torsionless geodesics, in full agreement with the multimessenger observation GW170817/GRB170817A. We also discuss how gravitational waves could be used as a probe to test for torsion.