Fixing Einstein's equations

TL;DR

This paper introduces a new hyperbolic formulation of Einstein's equations that ensures well-posedness and maintains the physical characteristics of the original theory, improving the mathematical robustness of general relativity's dynamical evolution.

Contribution

It presents the most economical first-order symmetrizable hyperbolic formulation of Einstein's equations with only physical characteristic speeds, closely related to the original 3+1 form.

Findings

Ensures well-posedness of Einstein's equations as a hyperbolic system

Clarifies relationships between different hyperbolic formulations

Maintains physical characteristic speeds for all variables

Abstract

Einstein's equations for general relativity, when viewed as a dynamical system for evolving initial data, have a serious flaw: they cannot be proven to be well-posed (except in special coordinates). That is, they do not produce unique solutions that depend smoothly on the initial data. To remedy this failing, there has been widespread interest recently in reformulating Einstein's theory as a hyperbolic system of differential equations. The physical and geometrical content of the original theory remain unchanged, but dynamical evolution is made sound. Here we present a new hyperbolic formulation in terms of $g_{ij}$, $K_{ij}$, and $\bGam_{kij}$ that is strikingly close to the space-plus-time (``3+1'') form of Einstein's original equations. Indeed, the familiarity of its constituents make the existence of this formulation all the more unexpected. This is the most economical first-order symmetrizable hyperbolic formulation presently known to us that has only physical characteristic speeds, either zero or the speed of light, for all (non-matter) variables. This system clarifies the relationships between Einstein's original equations and the Einstein-Ricci and Frittelli-Reula hyperbolic formulations of general relativity and establishes links to other hyperbolic formulations.