Three roads to the geometric constraint formulation of gravitational theories with boundaries

TL;DR

This paper explores various methods for formulating geometric constraints in gravitational theories with boundaries, comparing Dirac's approach with alternative techniques based on equations of motion and tangent bundle formulations.

Contribution

It introduces and analyzes alternative approaches to Hamiltonian constraint formulation in gravitational theories with boundaries, extending existing methods with new perspectives.

Findings

Compared Dirac and alternative methods for boundary gravitational theories.

Extended Pontryagin and Husain-Kuchař actions to manifolds with boundary.

Highlighted relations between different geometric constraint formulations.

Abstract

The Hamiltonian description of mechanical or field models defined by singular Lagrangians plays a central role in physics. A number of methods are known for this purpose, the most popular of them being the one developed by Dirac. Here, we discuss other approaches to this problem that rely on the direct use of the equations of motion (and the tangency requirements characteristic of the Gotay, Nester, Hinds method), or are formulated in the tangent bundle of the configuration space. Owing to its interesting relation with general relativity we will use a concrete example as a test bed: an extension of the Pontryagin and Husain-Kucha\v{r} actions to four dimensional manifolds with boundary.