Hamiltonian Time Evolution for General Relativity

Hamiltonian time evolution in terms of an explicit parameter time is derived for general relativity, even when the constraints are not satisfied, from the Arnowitt-Deser-Misner-Teitelboim-Ashtekar action in which the slicing density $\alpha (x,t)$ is freely specified while the lapse $N=\alpha g^{1/2}$ is not. The constraint “algebra” becomes a well-posed evolution system for the constraints; this system is the twice-contracted Bianchi identity when $R_{\mathrm{ij}}=0\mathrm{}$. The Hamiltonian constraint is an initial value constraint which determines $g^{1/2}$ and hence $N$, given $\alpha$.