Einstein's Equations and Equivalent Hyperbolic Dynamical Systems

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 ${\nabla }_{\beta }{G}^{\beta }{}_{\alpha }\equiv 0$ as a well-posed system for the evolution of the gravitational constraints in vacuum, and also (3) of ${\nabla }_{\beta }{T}^{\beta }{}_{\alpha }\equiv 0$ 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.

30 pages, LaTeX2e, to be published in the Proceedings of the 2nd Samos Meeting. (Errors in previous version corrected)

References in corpus (12)

• New Formalism for Numerical Relativity
• Conformal ``thin sandwich'' data for the initial-value problem of general relativity
• Fixing Einstein's equations
• On the Newtonian Limit of General Relativity
• First-order symmetric-hyperbolic Einstein equations with arbitrary fixed gauge
• NONLINEAR WAVE EQUATIONS FOR RELATIVITY
• Geometrical Hyperbolic Systems for General Relativity and Gauge Theories
• Hamiltonian Time Evolution for General Relativity
• Geometrical Well Posed Systems for the Einstein Equations
• A non-strictly hyperbolic system for the Einstein equations with arbitrary lapse and shift
• Curvature-based Hyperbolic Systems for General Relativity
• Hyperbolic Formulation of General Relativity