Geodesic motion on the group of boundary diffeomorphisms from Einstein's equations

TL;DR

This paper explores how vacuum Einstein equations in a boundary region can be reformulated as geodesic equations on boundary diffeomorphism groups, extending Arnold's geometric approach to continuum mechanics.

Contribution

It revisits and clarifies the geometric interpretation of Einstein's equations as geodesic flows on boundary diffeomorphism groups, especially in two spatial dimensions.

Findings

Reformulation of Einstein equations as boundary geodesic equations.

Comparison with classical continuum mechanics models.

Detailed analysis for 2D boundary case.

Abstract

In arXiv:1904.12869 it was shown how in an adiabatic limit the vacuum Einstein equations on a compact spatial region can be re-expressed as geodesic equations on the group of diffeomorphisms of the boundary. This is reminiscent of the program initiated by V. Arnold to reformulate models of continuum mechanics in terms of geodesic motion on diffeomorphism groups. We revisit some of the results of arXiv:1904.12869 in this light, pointing out parallels and differences with the typical examples in geometric continuum mechanics. We work out the case of 2 spatial dimensions in some detail.