Local freedom in the gravitational field revisited

TL;DR

This paper revisits the concept of local freedom in the gravitational field within general relativity, providing a covariant characterization of curvature derivatives and relating different formalisms to clarify which parts are determined by matter and which are free.

Contribution

It extends previous work by explicitly relating covariant 1+3 formalism results to spinor approach decompositions for curvature derivatives at any order.

Findings

Identifies the locally free parts of curvature derivatives in the 1+3 formalism.

Establishes explicit relations between matter-dependent and free curvature components.

Connects the 24 locally free data to the $ abla ext{ extPsi}$ quantities in the spinor approach.

Abstract

Maartens {\it et al.}\@ gave a covariant characterization, in a 1+3 formalism based on a perfect fluid's velocity, of the parts of the first derivatives of the curvature tensor in general relativity which are ``locally free'', i.e. not pointwise determined by the fluid energy momentum and its derivative. The full decomposition of independent curvature derivative components given in earlier work on the spinor approach to the equivalence problem enables analogous general results to be stated for any order: the independent matter terms can also be characterized. Explicit relations between the two sets of results are obtained. The 24 Maartens {\it et al.} locally free data are shown to correspond to the $\nabla \Psi$ quantities in the spinor approach, and the fluid terms are similarly related to the remaining 16 independent quantities in the first derivatives of the curvature.