Junction conditions in scalar-tensor theories

TL;DR

This paper investigates the conditions required for matching metrics and scalar fields across hypersurfaces in scalar-tensor theories, revealing possible configurations and clarifying relations between different frames and regularity conditions.

Contribution

It derives comprehensive junction conditions for scalar-tensor theories in arbitrary dimensions, including vacuum and regularity conditions, and compares these in Jordan and Einstein frames.

Findings

Continuity of metric and scalar field at hypersurfaces.

Possible vacuum thin-shell configurations in different frames.

Relations between $C^1$ regularity and vacuum conditions.

Abstract

We analyze junction conditions at a null or non-null hypersurface $\Sigma$ in a large class of scalar-tensor theories in arbitrary $n(\ge 3)$ dimensions. After showing that the metric and a scalar field must be continuous at $\Sigma$ as the first junction conditions, we derive the second junctions conditions from the Einstein equations and the equation of motion for the scalar field. Subsequently, we study $C^1$ regular matching conditions as well as vacuum conditions at $\Sigma$ both in the Jordan and Einstein frames. Our result suggests that the following configurations may be possible; (i) a vacuum thin-shell at null $\Sigma$ in the Einstein frame, (ii) a vacuum thin-shell at null and non-null $\Sigma$ in the Jordan frame, and (iii) a non-vacuum $C^1$ regular matching at null $\Sigma$ in the Jordan frame. Lastly, we clarify the relations between the conditions for $C^1$ regularity and also for vacuum $\Sigma$ in the Jordan and Einstein frames.