By means of a canonical transformation, the Hamiltonian of general relativity can be made homogeneous in the momenta. Solutions of the corresponding Hamilton-Jacobi equations yield functionals of the configuration-space variables which are constant over equivalence classes generated by the constraints. The constancy of such solutions of the Hamilton-Jacobi equations is also a sufficient condition for infinitesimally neighboring sets of configuration variables to lie within the same four-dimensional space-time of the given family of Ricciflat manifolds.
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Arthur Komar (1971) studied this question.
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