Which action for brane worlds?

TL;DR

This paper clarifies that Lanczos' jump conditions for hypersurfaces in general relativity can be derived from the Einstein action but not from the Einstein-Hilbert action, highlighting a subtlety in the action principle.

Contribution

It demonstrates the specific conditions under which Lanczos' jump conditions can be obtained from the Einstein action, emphasizing the importance of the action choice in hypersurface dynamics.

Findings

Lanczos' jump conditions derive from Einstein action

Discontinuity of connection requires two spacetime regions

Einstein-Hilbert action does not produce these jump conditions

Abstract

In his pioneering work on singular shells in general relativity, Lanczos had derived jump conditions across energy-momentum carrying hypersurfaces from the Einstein equation with codimension 1 sources. However, on the level of the action, the discontinuity of the connection arising from a codimension 1 energy-momentum source requires to take into account two adjacent space-time regions separated by the hypersurface. The purpose of the present note is to draw attention to the fact that Lanczos' jump conditions can be derived from an Einstein action but not from an Einstein-Hilbert action.