Geometrodynamics of Schwarzschild black holes

Abstract

The curvature coordinates T, R of a Schwarzschild spacetime are turned into canonical coordinates T(r), ssR(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 ${\mathit{P}}_{\mathit{T}}$(r), ${\mathit{P}}_{\mathit{s}\mathit{s}\mathit{R}}$(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 \ensuremath{\Psi}(m;T,ssR]=\ensuremath{\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.