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Change and local spatial variation are missing in canonical GeneralRelativity's observables as usually defined, part of the problem of time.Definitions can be tested using equivalent formulations, non-gauge and gauge,because they must have equivalent observables and everything is observable inthe non-gauge formulation. Taking an observable from the non-gauge formulationand finding the equivalent in the gauge formulation, one requires that theequivalent be an observable, constraining definitions. For massive photons, thede Broglie-Proca non-gauge formulation observable A_{\mu} is equivalent to theStueckelberg-Utiyama gauge formulation quantity A_{\mu}+\partial_{\mu} \phi.Thus observables must have 0 Poisson bracket not with each first-classconstraint, but with the Rosenfeld-Anderson-Bergmann-Castellani gauge generatorG, a tuned sum of first-class constraints, in accord with thePons-Salisbury-Sundermeyer definition of observables.  The definition for external gauge symmetries can be tested using massivegravity, where one can install gauge freedom by parametrization with clockfields X^A. The non-gauge observable g^{\mu\nu} has the gauge equivalentX^A,_{\mu} g^{\mu\nu} X^B,_{\nu}. The Poisson bracket of X^A,_{\mu} g^{\mu\nu}X^B,_{\nu} with G turns out to be not 0 but a Lie derivative. This non-zeroPoisson bracket refines and systematizes Kuchar's proposal to relax the 0Poisson bracket condition with the Hamiltonian constraint. Thus observablesneed covariance, not invariance, in relation to external gauge symmetries.  The Lagrangian and Hamiltonian for massive gravity are those of GeneralRelativity + Lambda + 4 scalars, so the same definition of observables appliesto General Relativity. Local fields such as g_{\mu\nu} are observables. Thusobservables change. Requiring equivalent observables for equivalent theoriesalso recovers Hamiltonian-Lagrangian equivalence.

Equivalent Theories Redefine Hamiltonian Observables to Exhibit Change in General Relativity