Hamiltonian Formalism for Black Holes and Quantization

Abstract

Starting from the Lagrangian formulation of the Einstein equations for the vacuum static spherically symmetric metric, we develop a canonical formalism in the radial variable $r$ that is time--like inside the Schwarzschild horizon. The Schwarzschild mass turns out to be represented by a canonical function that commutes with the $r$--Hamiltonian. We investigate the Wheeler--DeWitt quantization and give the general representation for the solution as superposition of eigenfunctions of the mass operator.

13 pages, Plain Tex, no figures, Mail Adrress: Dipartimento di Fisica Teorica dell'Universita` di Torino, Via Giuria 1, I-10125 Torino, Italy; TWO REFERENCES ADDED