Unconstrained Hamiltonian formulation of General Relativity with thermo-elastic sources

TL;DR

This paper introduces a new unconstrained Hamiltonian formulation of General Relativity coupled with thermo-elastic matter, simplifying the dynamics and potentially aiding numerical and quantum gravity research.

Contribution

It presents a gauge-based formulation where the gravitational and thermo-mechanical degrees of freedom are encoded in unconstrained variables, with a universal Hamilton-Jacobi system governing the dynamics.

Findings

Hamiltonian equals total matter entropy and generates the system's dynamics.

The Hamilton-Jacobi equations are universal, depending only on boundary conditions.

The vacuum limit of the Hamiltonian is recovered as a limit of a deep potential well.

Abstract

A new formulation of the Hamiltonian dynamics of the gravitational field interacting with(non-dissipative) thermo-elastic matter is discussed. It is based on a gauge condition which allows us to encode the six degrees of freedom of the ``gravity + matter''-system (two gravitational and four thermo-mechanical ones), together with their conjugate momenta, in the Riemannian metric q_{ij} and its conjugate ADM momentum P^{ij}. These variables are not subject to constraints. We prove that the Hamiltonian of this system is equal to the total matter entropy. It generates uniquely the dynamics once expressed as a function of the canonical variables. Any function U obtained in this way must fulfil a system of three, first order, partial differential equations of the Hamilton-Jacobi type in the variables (q_{ij},P^{ij}). These equations are universal and do not depend upon the properties of the material: its equation of state enters only as a boundary condition. The well posedness of this problem is proved. Finally, we prove that for vanishing matter density, the value of U goes to infinity almost everywhere and remains bounded only on the vacuum constraints. Therefore the constrained, vacuum Hamiltonian (zero on constraints and infinity elsewhere) can be obtained as the limit of a ``deep potential well'' corresponding to non-vanishing matter. This unconstrained description of Hamiltonian General Relativity can be useful in numerical calculations as well as in the canonical approach to Quantum Gravity.