Extending the rigidity of general relativity

TL;DR

This paper establishes the most general conditions under which the Hamiltonian formulation of general relativity remains unique, focusing on spatially covariant scalar constraints quadratic in momenta without specific algebraic assumptions.

Contribution

It provides the broadest set of criteria ensuring the uniqueness of the Hamiltonian in general relativity, allowing for general quadratic momentum dependence and removing previous algebraic restrictions.

Findings

All such covariant scalar constraints are second class.

The quadratic dependence on momenta is fully general with arbitrary local operators.

The results do not depend on specific Poisson bracket algebra.

Abstract

We give the most general conditions to date which lead to uniqueness of the general relativistic Hamiltonian. Namely, we show that all spatially covariant generalizations of the scalar constraint which extend the standard one while remaining quadratic in the momenta are second class. Unlike previous investigations along these lines, we do not require a specific Poisson bracket algebra, and the quadratic dependence on the momenta is completely general, with an arbitrary local operator as the kinetic term.