Pitts, J. Brian (2018) Equivalent Theories and Changing Hamiltonian Observables in General Relativity. Foundations of Physics.

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Abstract
Change and local spatial variation are missing in Hamiltonian General Relativity according to the most common definition of observables as having 0 Poisson bracket with all firstclass constraints. But other definitions of observables have been proposed. In pursuit of HamiltonianLagrangian equivalence, Pons, Salisbury and Sundermeyer use the AndersonBergmannCastellani gauge generator $G$, a tuned sum of firstclass 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 nonzero Lie derivative Poisson brackets for the external gauge symmetry of General Relativity.
Fortunately one can test definitions of observables by calculation using two formulations of a theory, one without gauge freedom and one with gauge freedom. The formulations, being empirically equivalent, must have equivalent observables. For de BroglieProca nongauge massive electromagnetism, all constraints are secondclass, so everything is observable. Demanding equivalent observables from gauge StueckelbergUtiyama electromagnetism, one finds that the usual definition fails while the PonsSalisburySundermeyer 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 can test whether observables should have 0 or the Lie derivative as the Poisson bracket with the gauge generator G. The latter definition is vindicated by calculation. While this conclusion has been reported previously, here the calculation is given in some detail.
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