Hamiltonian constraint

Summary

The Hamiltonian constraint is the equation $\mathcal{H} = 0$ that arises in a theory invariant under reparametrisation of time. In such a theory the Hamiltonian itself must vanish on physical states. In general relativity, in the ADM formalism, it is the component of the Einstein equations along the normal of a spatial slice. It relates the scalar curvature of space to the extrinsic curvature and the energy density of matter. In canonical quantum gravity, imposing it as an operator equation gives the Wheeler–DeWitt equation.