Geometry is Wavy: Curvature Wave Equations for Generic Affine Connections

TL;DR

In the most general metric-affine spacetimes, where the affine connection carries torsion and nonmetricity, curvature becomes a dynamical, wavy object. The authors derive a covariant wave-type equation for the Riemann tensor, $\Box R_{\gamma\lambda\rho\sigma}$, whose source and transport terms are organized by the torsion- and nonmetricity-induced tensors $X^{\mu}{}_{\lambda\nu\rho\sigma}$ and $Y_{\gamma\lambda\nu\rho\sigma}$, alongside standard quadratic-curvature interactions. The framework recovers the familiar Levi-Civita wave equation in the Riemannian limit and specializes to well-known theories (Einstein spaces, Einstein–Cartan, teleparallel, symmetric teleparallel), illustrating how torsion and nonmetricity modify curvature propagation. Overall, the work provides a unifying geometric picture: gravity can be described via curvature, torsion, or nonmetricity, but once these structures are present, geometry itself supports wave-like dynamics with potentially observable implications for metric-affine gravity models.

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.

Figures (1)

• Figure 1: Schematic classification of gravitational theories according to the geometric properties of spacetime. The table displays the presence or absence of curvature (Riemann tensor $R^{\rho}{}_{\mu\sigma\nu}$), torsion ($T^{\sigma}{}_{\mu\nu}$), and nonmetricity ($Q_{\mu\nu\sigma}$), together with the corresponding form of the affine connection $\Gamma^{\sigma}{}_{\mu\nu}$. Here $^{c}\Gamma^{\sigma}{}_{\mu\nu}$ denotes the Levi--Civita connection, $K^{\sigma}{}_{\mu\nu}$ the contortion tensor associated with torsion, and $L^{\sigma}{}_{\mu\nu}$ the disformation tensor associated with nonmetricity. General relativity is recovered when both torsion and nonmetricity vanish. Einstein--Cartan theory allows for nonvanishing torsion while preserving metric compatibility. Teleparallel theories are characterized by the vanishing of the curvature, with gravitation encoded entirely in torsion and/or nonmetricity.