Geometrodynamics of Schwarzschild Black Holes

TL;DR

Geometrodynamics of Schwarzschild black holes develops a canonical chart by turning curvature coordinates into phase-space variables, revealing that $P_T(r)=0$ and $P_{\sf R}(r)=0$ on-shell and leaving a mass parameter $m$ with conjugate $p$ as constants of motion. This reduction yields a trivial quantum evolution with state $\Psi(m)$, a superposition of Schwarzschild masses, while exposing boundary-term structure and horizon-crossing behavior in a horizon-penetrating foliation. The approach provides a clear reconstruction of spacetime data from canonical data and extends naturally to collapsing matter via the same canonical map, including a description in Kruskal coordinates. The results offer a soluble, geometrically transparent framework for quantum black holes and the dynamical study of gravitational collapse using canonical methods.

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 $Ψ(m; T, {\sf R}] = Ψ(m)$ 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.