Remarks on the "non-canonicity puzzle": Lagrangian symmetries of the Einstein-Hilbert action

TL;DR

This paper investigates the symmetries of the Einstein-Hilbert action in different formulations, revealing that only diffeomorphism symmetry forms a group, which clarifies the non-canonicity puzzle and emphasizes the importance of canonical symmetries.

Contribution

It demonstrates that ADM transformations do not form a group, unlike diffeomorphisms, highlighting the unique canonical nature of diffeomorphism symmetry in the Einstein-Hilbert action.

Findings

Diffeomorphism transformations form a group.

ADM transformations do not form a group.

Canonical symmetry preservation is essential for Lagrangian equivalence.

Abstract

Given the non-canonical relationship between variables used in the Hamiltonian formulations of the Einstein-Hilbert action (due to Pirani, Schild, Skinner (PSS) and Dirac) and the Arnowitt-Deser-Misner (ADM) action, and the consequent difference in the gauge transformations generated by the first-class constraints of these two formulations, the assumption that the Lagrangians from which they were derived are equivalent leads to an apparent contradiction that has been called "the non-canonicity puzzle". In this work we shall investigate the group properties of two symmetries derived for the Einstein-Hilbert action: diffeomorphism, which follows from the PSS and Dirac formulations, and the one that arises from the ADM formulation. We demonstrate that unlike the diffeomorphism transformations, the ADM transformations (as well as others, which can be constructed for the Einstein-Hilbert Lagrangian using Noether's identities) do not form a group. This makes diffeomorphism transformations unique (the term "canonical" symmetry might be suggested). If the two Lagrangians are to be called equivalent, canonical symmetry must be preserved. The interplay between general covariance and the canonicity of the variables used is discussed.