Einstein's Equations and Equivalent Hyperbolic Dynamical Systems

TL;DR

This paper explores hyperbolic formulations of Einstein's equations emphasizing the slicing function's role, leading to clearer Hamiltonian dynamics, well-posed constraint evolution, and new initial value problem approaches.

Contribution

It introduces a unified hyperbolic framework for Einstein's equations highlighting the slicing function's fundamental role in various formulations and constraint evolutions.

Findings

Explicit causal hyperbolic formulations of Einstein's equations.

Demonstration of the slicing function's role in well-posed constraint evolution.

Development of a conformal thin sandwich initial value formulation.

Abstract

We discuss several explicitly causal hyperbolic formulations of Einstein's dynamical 3+1 equations in a coherent way, emphasizing throughout the fundamental role of the ``slicing function,'' $\alpha$---the quantity that relates the lapse $N$ to the determinant of the spatial metric $\bar{g}$ through $N = \bar{g}^{1/2} \alpha$. The slicing function allows us to demonstrate explicitly that every foliation of spacetime by spatial time-slices can be used in conjunction with the causal hyperbolic forms of the dynamical Einstein equations. Specifically, the slicing function plays an essential role (1) in a clearer form of the canonical action principle and Hamiltonian dynamics for gravity and leads to a recasting (2) of the Bianchi identities $\nabla_\beta G^\beta\mathstrut_\alpha \equiv 0$ as a well-posed system for the evolution of the gravitational constraints in vacuum, and also (3) of $\nabla_\beta T^\beta\mathstrut_\alpha \equiv 0$ as a well-posed system for evolution of the energy and momentum components of the stress tensor in the presence of matter, (4) in an explicit rendering of four hyperbolic formulations of Einstein's equations with only physical characteristics, and (5) in providing guidance to a new ``conformal thin sandwich'' form of the initial value constraints.