Symmetries of distributional domain wall geometries

TL;DR

This paper investigates the symmetries of distributional domain wall geometries by extending Lie derivatives to distribution-valued tensors, revealing how symmetries behave in thin wall limits of scalar field configurations.

Contribution

It introduces a rigorous method to analyze symmetries of distributional curvature tensors in domain wall geometries, including non-reflection symmetric cases.

Findings

Distributional curvature tensor symmetries are the Killing symmetries of the metric pullback on the singular surface.

Symmetries are preserved in the thin wall limit for all considered geometries.

Non-reflection symmetric walls exhibit symmetries that are not symmetries of the entire spacetime.

Abstract

Generalizing the Lie derivative of smooth tensor fields to distribution-valued tensors, we examine the Killing symmetries and the collineations of the curvature tensors of some distributional domain wall geometries. The chosen geometries are rigorously the distributional thin wall limit of self gravitating scalar field configurations representing thick domain walls and the permanence and/or the rising of symmetries in the limit process is studied. We show that, for all the thin wall spacetimes considered, the symmetries of the distributional curvature tensors turns out to be the Killing symmetries of the pullback of the metric tensor to the surface where the singular part of these tensors is supported. Remarkably enough, for the non-reflection symmetric domain wall studied, these Killing symmetries are not necessarily symmetries of the ambient spacetime on both sides of the wall.