Conformal ``thin sandwich'' data for the initial-value problem of general relativity

TL;DR

This paper introduces a conformal 'thin sandwich' approach to the initial-value problem in general relativity, reformulating the equations as elliptic problems and revealing their geometric roots through a new viewpoint.

Contribution

It presents a novel conformal 'thin sandwich' formulation that simplifies the initial-value problem in general relativity by expressing it as elliptic equations.

Findings

Reformulation of the initial-value problem as elliptic equations

Identification of the geometric roots via conformal 'thin sandwich' viewpoint

Unified approach for two nearby spacelike hypersurfaces

Abstract

The initial-value problem is posed by giving a conformal three-metric on each of two nearby spacelike hypersurfaces, their proper-time separation up to a multiplier to be determined, and the mean (extrinsic) curvature of one slice. The resulting equations have the {\it same} elliptic form as does the one-hypersurface formulation. The metrical roots of this form are revealed by a conformal ``thin sandwich'' viewpoint coupled with the transformation properties of the lapse function.