Extrinsic Curvature and the Einstein Constraints

TL;DR

This paper unifies different formulations of Einstein's initial-value equations using a tensor decomposition with a weight function, revealing a natural choice in stationary spacetimes that simplifies extrinsic curvature.

Contribution

It introduces a tensor decomposition involving a weight function that aligns the extrinsic curvature and conformal thin sandwich formulations of Einstein's equations.

Findings

Complete conformity of formulations achieved

Natural weight function choice in stationary spacetimes

Transverse traceless part of extrinsic curvature vanishes in stationary spacetimes

Abstract

The Einstein initial-value equations in the extrinsic curvature (Hamiltonian) representation and conformal thin sandwich (Lagrangian) representation are brought into complete conformity by the use of a decomposition of symmetric tensors which involves a weight function. In stationary spacetimes, there is a natural choice of the weight function such that the transverse traceless part of the extrinsic curvature (or canonical momentum) vanishes.