In the context of general relativity and the canonical formulation of the theory, the Hamiltonian constraint is a fundamental equation that arises in the process of quantizing gravity. It plays a key role in the framework known as Hamiltonian formalism or the ADM (Arnowitt-Deser-Misner) formulation of general relativity.
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The Hamiltonian constraint is the normal-normal projection of the Einstein field equations onto a spatial hypersurface. With the extrinsic-curvature convention used here,