The Initial Value Problem Using Metric and Extrinsic Curvature

TL;DR

This paper presents a comprehensive reformulation of the initial value problem in general relativity using metric and extrinsic curvature, employing conformal transformations to convert constraints into an elliptic system.

Contribution

It introduces a complete elliptic reformulation of the initial value constraints using conformal techniques and extrinsic curvature, aligning with the conformal thin sandwich approach.

Findings

Reformulation of initial value equations into elliptic form.

Use of conformal transformations and extrinsic curvature.

Equivalence to the conformal thin sandwich formulation.

Abstract

The initial value problem is introduced after a thorough review of the essential geometry. The initial value equations are put into elliptic form using both conformal transformations and a treatment of the extrinsic curvature introduced recently. This use of the metric and the extrinsic curvature is manifestly equivalent to the author's conformal thin sandwich formulation. Therefore, the reformulation of the constraints as an elliptic system by use of conformal techniques is complete.