Equivalent Theories and Changing Hamiltonian Observables in General Relativity

TL;DR

This paper explores different definitions of observables in Hamiltonian General Relativity, proposing a systematic approach that reconciles gauge invariance and change, with implications for understanding gravitational observables.

Contribution

It introduces a combined framework using the gauge generator G and non-zero Lie derivative brackets to define observables, addressing limitations of previous definitions in GR.

Findings

The Pons-Salisbury-Sundermeyer definition with G succeeds in gauge electromagnetism.

The usual definition fails for gauge electromagnetism, but the new approach succeeds.

Lie derivative as Poisson bracket aligns with gauge invariance in GR.

Abstract

Change and local spatial variation are missing in Hamiltonian General Relativity according to the most common definition of observables (0 Poisson bracket with all first-class constraints). But other definitions have been proposed. Seeking Hamiltonian-Lagrangian equivalence, Pons, Salisbury and Sundermeyer use the Anderson-Bergmann-Castellani gauge generator G, a tuned sum of first-class constraints. Kucha\v{r} waived the 0 Poisson bracket condition for the Hamiltonian constraint to achieve changing observables. A systematic combination of the two reforms might use the gauge generator but permit non-zero Lie derivative Poisson brackets. One can test definitions by calculation using two formulations of a theory, one without gauge freedom and one with it, which must have equivalent observables. For de Broglie-Proca non-gauge massive electromagnetism, all constraints are second-class, so everything is observable. Demanding equivalent observables from gauge Stueckelberg-Utiyama electromagnetism, one finds that the usual definition fails while the Pons-Salisbury-Sundermeyer definition with G succeeds. This definition does not readily yield change in GR, however. Should GR's external gauge freedom of General Relativity share with internal gauge symmetries the 0 Poisson bracket (invariance), or is covariance (a transformation rule) sufficient? A graviton mass breaks the gauge symmetry (general covariance), but it can be restored by parametrization with clock fields. By requiring equivalent observables, one vindicates the Lie derivative as the Poisson bracket of observables with the gauge generator G.