Local freedom in the gravitational field

TL;DR

This paper investigates the locally free components of the gravitational field in cosmology, revealing their structure through covariant decomposition and their importance for gravitational wave existence.

Contribution

It provides a complete covariant decomposition of derivatives of the Weyl tensor's electric and magnetic parts, identifying the locally free curvature components.

Findings

Locally free curvature parts are symmetrised trace-free derivatives and curls of E_{ab} and H_{ab}

These parts are essential for gravitational wave existence

The decomposition clarifies the structure of gravitational fields in cosmology

Abstract

In a cosmological context, the electric and magnetic parts of the Weyl tensor, E_{ab} and H_{ab}, represent the locally free curvature - i.e. they are not pointwise determined by the matter fields. By performing a complete covariant decomposition of the derivatives of E_{ab} and H_{ab}, we show that the parts of the derivative of the curvature which are locally free (i.e. not pointwise determined by the matter via the Bianchi identities) are exactly the symmetrised trace-free spatial derivatives of E_{ab} and H_{ab} together with their spatial curls. These parts of the derivatives are shown to be crucial for the existence of gravitational waves.