Constraints on Covariance

TL;DR

This paper explores the constraints on general covariance in field theories, showing that non-flat spacetimes restrict covariance groups and imply a universal interaction speed, based on principles of relativity.

Contribution

It demonstrates that general covariance requires more than local Lagrangian invariance, especially in curved spacetimes, and links covariance to universal propagation speed.

Findings

Covariance group reduces to linear transformations in curved spacetime.

Universal propagation speed is implied for linear field equations.

Constraints are derived from the principle of relativity and tensor parallel transport.

Abstract

Based on the principle of relativity, we find that the sufficient and necessary condition for the general covariance of a field theory actually requires more than the invariance of its local Langrangian density. If the spacetime is not a flat one, its derivative requirement from the analysis of the parallel transportation of tensor fields over spacetime restricts the generally supposed covariance group, the group of differmorphisms, to the group of linear coordinate transformations. Moreover, for any a field theory with linear equations of motion, it stipulates for a universal physical propagation speed of interaction over the spacetime manifold.