Geometrodynamics of Schwarzschild Black Holes

TL;DR

This paper reformulates Schwarzschild black hole dynamics using canonical variables, simplifying the Hamiltonian constraints and enabling a superposition of different black hole masses in quantum states.

Contribution

It introduces a new canonical variable framework for Schwarzschild black holes, reducing the Hamiltonian constraints to simple conditions and facilitating quantum superpositions of black hole states.

Findings

Canonical variables simplify black hole phase space

Quantum states are superpositions of different masses

New framework aids in studying collapsing matter systems

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.