Worldtube conservation laws for the null-timelike evolution problem

TL;DR

This paper investigates the worldtube constraints in the null-timelike initial-boundary value problem of Einstein's equations, revealing conservation laws that influence spacetime evolution and have implications for gravitational waveform extraction.

Contribution

It introduces a new understanding of worldtube constraints as conservation laws and formulates a well-posed initial value problem for the extrinsic curvature in this context.

Findings

Worldtube constraints act as conservation laws in null-timelike evolution.

A well-posed initial value problem for the extrinsic curvature is established.

Implications for gravitational waveform extraction are discussed.

Abstract

I treat the worldtube constraints which arise in the null-timelike initial-boundary value problem for the Bondi-Sachs formulation of Einstein's equations. Boundary data on a worldtube and initial data on an outgoing null hypersurface determine the exterior spacetime by integration along the outgoing null geodsics. The worldtube constraints are a set of conservation laws which impose conditions on the integration constants. I show how these constraints lead to a well-posed initial value problem governing the extrinsic curvature of the worldtube, whose components are related to the integration constants. Possible applications to gravitational waveform extraction and to the well-posedness of the null-timelike initial-boundary value problem are discussed.