Strongly Hyperbolic Extensions of the ADM Hamiltonian

TL;DR

This paper presents an extension of the ADM Hamiltonian formulation of general relativity that achieves strong hyperbolicity while preserving its Hamiltonian structure, improving the mathematical well-posedness of the system.

Contribution

The paper introduces a Hamiltonian extension of the ADM formalism that incorporates gauge conditions as dynamical equations, enhancing hyperbolicity without losing the variational structure.

Findings

Extended ADM formulation achieves strong hyperbolicity with certain gauge conditions.

Maintains Hamiltonian and covariant structure of general relativity.

Some gauge choices, like 1+log slicing, remain weakly hyperbolic.

Abstract

The ADM Hamiltonian formulation of general relativity with prescribed lapse and shift is a weakly hyperbolic system of partial differential equations. In general weakly hyperbolic systems are not mathematically well posed. For well posedness, the theory should be reformulated so that the complete system, evolution equations plus gauge conditions, is (at least) strongly hyperbolic. Traditionally, reformulation has been carried out at the level of equations of motion. This typically destroys the variational and Hamiltonian structures of the theory. Here I show that one can extend the ADM formalism to (i) incorporate the gauge conditions as dynamical equations and (ii) affect the hyperbolicity of the complete system, all while maintaining a Hamiltonian description. The extended ADM formulation is used to obtain a strongly hyperbolic Hamiltonian description of Einstein's theory that is generally covariant under spatial diffeomorphisms and time reparametrizations, and has physical characteristics. The extended Hamiltonian formulation with 1+log slicing and gamma--driver shift conditions is weakly hyperbolic.