Geometrodynamics of Schwarzschild Black Holes

Abstract

The curvature coordinates $T,R$ of a Schwarz\-schild spacetime are turned into canonical coordinates $T(r), {\sf R}(r)$ on the phase space of spherically symmetric black holes. The entire dynamical content of the Hamiltonian theory is reduced to the constraints requiring that the momenta $P_{T}(r), P_{\sf R}(r)$ vanish. What remains is a conjugate pair of canonical variables $m$ and $p$ whose values are the same on every embedding. The coordinate $m$ is the Schwarzschild mass, and the momentum $p$ the difference of parametrization times at right and left infinities. The Dirac constraint quantization in the new representation leads to the state functional $\Psi (m; T, {\sf R}] = \Psi (m)$ which describes an unchanging superposition of black holes with different masses. The new canonical variables may be employed in the study of collapsing matter systems.