Geometry is Wavy: Curvature Wave Equations for Generic Affine Connections

TL;DR

This paper generalizes the wave equation satisfied by curvature tensors from Riemannian geometry to spacetimes with arbitrary affine connections, including torsion and nonmetricity, across various geometric frameworks.

Contribution

It derives a covariant wave equation for the Riemann tensor in metric-affine geometries, extending previous results beyond Levi-Civita connections.

Findings

Wave equation for Riemann tensor in affine connections derived

Analysis of the equation in Einstein, teleparallel, and Einstein-Cartan geometries

Insights into curvature behavior in generalized geometric settings

Abstract

Geometry is wavy: even at the purely geometric level (no particular theory chosen), curvature satisfies a covariant quasilinear wave equation. In Riemannian geometry equipped with the Levi-Civita connection, the Riemann curvature tensor obeys a wave equation of the schematic form \[ \Box Riem=\mathcal{Q}(Riem,Riem), \] where $\mathcal{Q}(Riem,Riem)$ denotes the terms quadratic in the curvature arising from the Bianchi identities. In this work, we generalize this curvature wave equation to spacetimes endowed with a generic affine connection possessing torsion and nonmetricity. Working within the metric-affine framework, we derive the corresponding wave equation for the Riemann tensor and analyze its structure in several geometrically and physically distinguished settings, including Einstein spaces, teleparallel gravity, and Einstein-Cartan theory.