Canonical Theory of 2+1 Gravity

TL;DR

This paper develops a canonical formalism for 2+1 Einstein gravity, introducing local conserved quantities and solving constraints analytically, providing a foundation for canonical quantum theory in this setting.

Contribution

It presents a canonical approach to 2+1 Einstein gravity, distinct from the Chern-Simons formulation, with explicit solutions for spherically symmetric spacetimes.

Findings

Defined local conserved quantities like mass and angular momentum.

Solved the constraints analytically for the quantum states.

Established a framework for canonical quantum theory in 2+1 gravity.

Abstract

Recently 2+1 dimensional gravity theory, especially ${\rm AdS_3}$ has been studied extensively. It was shown to be equivalent to the 2+1 Chern-Simon theory and has been investigated to understand the black hole thermodynamics, i.e. Hawking temperature and others. The purpose of this report is to investigate the canonical formalism of the original 2+1 Einstein gravity theory instead of the Chern-Simon theory. For the spherically symmetric space-time, local conserved quantities(local mass and angular momentum) are introduced and using them canonical quantum theory is defined. Constraints are imposed on state vectors and solved analytically. The strategy to obtain the solution is followed by our previous work.