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The question of consistency of the canonical reduction of general relativity (obtained by eliminating constraints and also imposing coordinate conditions in the action or generator) is examined. It is shown that the equations of motion obtained from this ``reduced'' formalism agree with the original Einstein equations. Agreement is also established for the generators of space-time translations. In order to establish consistency, it is necessary to discard certain well-defined divergence terms in the original Lagrangian. These would otherwise appear as nondivergences in the reduced Lagrangian, incorrectly altering the equations.
Arnowitt et al. (Thu,) studied this question.