Degree of arbitrariness directly from moving frames

TL;DR

This paper introduces a straightforward algorithm to compute the degree of arbitrariness in physical systems using moving frames, simplifying complex calculations for Einstein equations, gauge theories, and relativistic flows.

Contribution

The paper presents a novel, easy-to-apply algorithm for calculating the degree of arbitrariness in systems formulated with moving frames, streamlining previous complex methods.

Findings

Simplified calculation of arbitrariness in Einstein solutions across dimensions

Determined arbitrariness in gauge theories for arbitrary groups

Derived arbitrariness of relativistic rigid flows with constraints

Abstract

For physical theories, the degree of arbitrariness of a system is of great importance, and is often closely linked to the concept of degree of freedom, and for most systems this number is far from obvious. In this paper we present an easy to apply algorithm for calculating the degree of arbitrariness for large classes of systems formulated in the language of moving frames, by mere manipulation of the indices of the differential invariants. We then give several examples illustrating our procedure, including a derivation of the degree of arbitrariness of solutions of the Einstein equation in arbitrary dimensions, which is vastly simpler than previous calculations, the degree of arbitrariness of gauge theories of arbitrary groups under Yang-Mills type equations, for which until now only specific cases have been calculated by rather messy means, and the degree of arbitrariness of relativistic rigid flow with or without additional constraints, which has not been derived before. Finally we give the proof of our algorithm.