Wave operators, torsion, and Weitzenböck identities

Abstract

The current article offers a mathematical toolkit for the study of waves propagating on spacetimes with nonvanishing torsion. The toolkit comprises generalized versions of the Lichnerowicz–de Rham and the Beltrami wave operators, and the Weitzenböck 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 \(\mathfrak {so} \left( \eta _{+}, \eta _{-} \right) \) subalgebra. These tools allow us to study the propagation of waves on an Einstein–Cartan background at different orders in the eikonal parameter. It stands in strong contrast with more traditional approaches that are restricted to studying only the leading order for waves on this kind of geometry (“plane waves”). The current article focuses only on the mathematical aspects and offers proofs and generalizations for some results already used in physical applications. In particular, the subleading analysis proves that torsion affects the propagation of amplitude and polarization for fields in some representations. These results suggest how one may use gravitational waves and multimessenger events as probes for torsion and the spin tensor of dark matter.