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.

当前的文章提供了一个数学工具包，用于研究在时空中以不消失的扭转传播的波。该工具包包括 Lichnerowicz-de Rham 和 Beltrami 波算子的广义版本，以及将它们与 Riemann-Cartan 几何相关的 Weitzenböck 恒等式。该构造适用于属于李（超）代数的矩阵表示的任何字段，其中包含\(\mathfrak {so} \left( \eta _{+}, \eta _{-} \right) \)子代数。这些工具使我们能够研究波在 Einstein-Cartan 背景上以 eikonal 参数中不同阶数的传播。它与更传统的方法形成鲜明对比，传统方法仅限于研究这种几何形状上波的前导顺序（“平面波”）。当前的文章仅关注数学方面，并为已经在物理应用中使用的一些结果提供证明和概括。特别是，次引导分析证明，在某些表示中，扭转会影响场的振幅和极化传播。这些结果表明人们可以如何使用引力波和多信使事件作为暗物质的扭转和自旋张量的探测器。