New Minimal Distortion Shift Gauge

TL;DR

This paper introduces a refined minimal distortion shift gauge in general relativity that enhances the initial value problem formulation by linking the shift vector to the extrinsic curvature's longitudinal component, improving the evolution process.

Contribution

It proposes a slight modification to the minimal distortion shift gauge, enabling the shift vector to act as a potential for the extrinsic curvature's longitudinal part in the evolution.

Findings

The new shift vector minimizes conformal 3-metric changes with respect to spacetime measure.

The approach extends the initial value decomposition to influence the evolution.

It aligns the shift vector with the densitized lapse function for better formulation.

Abstract

Based on the recent understanding of the role of the densitized lapse function in Einstein's equations and of the proper way to pose the thin sandwich problem, a slight readjustment of the minimal distortion shift gauge in the 3+1 approach to the dynamics of general relativity allows this shift vector to serve as the vector potential for the longitudinal part of the extrinsic curvature tensor in the new approach to the initial value problem, thus extending the initial value decomposition of gravitational variables to play a role in the evolution as well. The new shift vector globally minimizes the changes in the conformal 3-metric with respect to the spacetime measure rather than the spatial measure on the time coordinate hypersurfaces, as the old shift vector did.