Kruskal coordinates as canonical variables for Schwarzschild black holes

TL;DR

This paper introduces a non-singular canonical transformation from ADM variables to Kruskal coordinates for Schwarzschild black holes, simplifying the constraints and providing a new perspective on black hole phase space.

Contribution

It presents a novel canonical transformation to Kruskal coordinates that remains regular at the horizon, extending previous work by Kuchar with a new scaling approach.

Findings

Transformation is non-singular at the horizon.

Constraints simplify to vanishing canonical momenta.

Provides a new canonical framework for Schwarzschild black holes.

Abstract

We derive a transformation from the usual ADM metric-extrinsic curvature variables on the phase space of Schwarzschild black holes, to new canonical variables which have the interpretation of Kruskal coordinates. We explicitly show that this transformation is non-singular, even at the horizon. The constraints of the theory simplify in terms of the new canonical variables and are equivalent to the vanishing of the canonical momenta. Our work is based on earlier seminal work by Kuchar in which he reconstructed curvature coordinates and a mass function from spherically symmetric canonical data. The key feature in our construction of a nonsingular canonical transformation to Kruskal variables, is the scaling of the curvature coordinate variables by the mass function rather than by the mass at left spatial infinity.