Cauchy problem as a two-surface based `geometrodynamics'

TL;DR

This paper reformulates Einstein's equations using a two-surface foliation approach, transforming constraints into parabolic and hyperbolic systems, which simplifies solving the initial value problem in general relativity.

Contribution

It introduces a novel gauge fixing and variable choice that recasts Einstein's constraints into parabolic and hyperbolic equations, enabling a new approach to the Cauchy problem in gravity.

Findings

Constraints are expressed as parabolic and hyperbolic systems.

Solutions to the mixed hyperbolic-hyperbolic system solve Einstein's equations.

Initial data can be specified on a single two-surface foliation.

Abstract

Four-dimensional spacetimes foliated by a two-parameter family of homologous two-surfaces are considered in Einstein's theory of gravity. By combining a 1+(1+2) decomposition, the canonical form of the spacetime metric and a suitable specification of the conformal structure of the foliating two-surfaces a gauge fixing is introduced. It is shown that, in terms of the chosen geometrically distinguished variables, the 1+3 Hamiltonian and momentum constraints can be recast into the form of a parabolic equation and a first order symmetric hyperbolic system, respectively. Initial data to this system can be given on one of the two-surfaces foliating the three-dimensional initial data surface. The 1+3 reduced Einstein's equations are also determined. By combining the 1+3 momentum constraint with the reduced system of the secondary 1+2 decomposition a mixed hyperbolic-hyperbolic system is formed. It is shown that solutions to this mixed hyperbolic-hyperbolic system are also solutions to the full set of Einstein's equations provided that the 1+3 Hamiltonian constraint is solved on the initial data surface $\Sigma_0$ and the 1+2 Hamiltonian and momentum type expressions vanish on a world-tube yielded by the Lie transport of one of the two-surfaces foliating $\Sigma_0$ along the time evolution vector field. Whenever the foliating two-surfaces are compact without boundary in the spacetime and a regular origin exists on the time-slices---this is the location where the foliating two-surfaces smoothly reduce to a point---it suffices to guarantee that the 1+3 Hamiltonian constraint holds on the initial data surface. A short discussion on the use of the geometrically distinguished variables in identifying the degrees of freedom of gravity are also included.