Scaling up the extrinsic curvature in gravitational initial data

TL;DR

This paper explores how increasing extrinsic curvature in vacuum Einstein initial data affects the geometry, revealing that local volume expands instead of curvature blowing up, through analytic and numerical methods.

Contribution

It introduces a combined analytic and numerical analysis of the effects of scaling extrinsic curvature in cosmological vacuum initial data.

Findings

Local volume expands with increased extrinsic curvature.

Intrinsic curvature remains bounded despite increased kinetic energy.

Numerical simulations support the analytic results.

Abstract

Vacuum solutions to the Einstein equations can be viewed as the interplay between the geometry and the gravitational wave energy content. The constraints on initial data reflect this interaction. We assume we are looking at cosmological solutions to the Einstein equations so we assume that the 3-space is compact, without boundary. In this article we investigate, using both analytic and numerical techniques, what happens when the extrinsic curvature is increased while the background geometry is held fixed. This is equivalent to trying to magnify the local gravitational wave kinetic energy on an unchanged background. We find that the physical intrinsic curvature does not blow up. Rather the local volume of space expands to accommodate this attempt to increase the kinetic energy.