On the action principle as a guide to substantive general covariance

TL;DR

This paper critically examines proposals to distinguish substantive from formal general covariance, showing that many formulations with auxiliary variables are physically equivalent, and that additional criteria like corner charges are also insufficient.

Contribution

The work demonstrates that diffeomorphism-invariant actions with trivial auxiliary variables do not differ physically from Kretschmannian formulations, challenging existing criteria for substantive general covariance.

Findings

Diffeomorphism-invariant actions with trivial auxiliaries are equivalent to Kretschmannian formulations.

Actions with non-trivial auxiliaries may represent distinct physical theories.

Corner charge criteria are insufficient to establish substantive general covariance.

Abstract

While Einstein was guided by the principle of general covariance in formulating general relativity, Kretschmann later argued that this principle lacks physical significance, since any space-time theory can be reformulated in a generally covariant form. This critique has prompted an ongoing debate over how to distinguish substantive general covariance from mere formal general covariance. Some proposals for defining substantive general covariance are based on the requirement that a theory be derivable from a diffeomorphism-invariant action. The present work aims to critically assess these proposals by examining canonical examples of Kretschmannian formulations of special relativistic theories. It will be shown that these formulations -- which seem merely formally generally covariant -- can always be derived from a diffeomorphism-invariant action. Although these actions involve auxiliary variables, these variables are trivial in the sense that they are either pure gauge or dynamically fixed by the other variables. Consequently, the theories described by these actions are arguably equivalent to the original Kretschmannian formulations. This stands in contrast to the actions proposed by Rosen and Sorkin, which include non-trivial auxiliary variables and hence seem to describe distinct physical theories. More recently, Freidel and Teh have suggested an additional criterion for substantial general covariance, namely that the action should also yield a non-trivial corner charge associated to diffeomorphism invariance. However, this too appears insufficient, since such actions can always be constructed.