Thick planar domain wall: its thin wall limit and dynamics

TL;DR

This paper investigates the properties and dynamics of thick planar domain walls in gravity, demonstrating that their core expands similarly to thin walls and analyzing how their thickness evolves over time.

Contribution

It introduces a coordinate transformation to describe the dynamics of thick domain walls and shows their core expansion mimics thin wall behavior.

Findings

Core of the thick wall expands like a thin wall.

Thickness varies over time in the new coordinate system.

Israel junction conditions are recovered in the thin limit.

Abstract

We consider a planar gravitating thick domain wall of the $\lambda \phi^4$ theory as a spacetime with finite thickness glued to two vacuum spacetimes on each side of it. Darmois junction conditions written on the boundaries of the thick wall with the embedding spacetimes reproduce the Israel junction condition across the wall in the limit of infinitesimal thickness. The thick planar domain wall located at a fixed position is then transformed to a new coordinate system in which its dynamics can be formulated. It is shown that the wall's core expands as if it were a thin wall. The thickness in the new coordinates is not constant anymore and its time dependence is given.