Junction conditions in scalar-Tensor theories

We analyze junction conditions at a null or non-null hypersurface in a large class of scalar-Tensor theories in arbitrary dimensions. After showing that the metric and a scalar field must be continuous at 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 both in the Jordan and Einstein frames. Our result suggests that the following configurations may be possible; (i) a vacuum thin-shell at null in the Einstein frame, (ii) a vacuum thin-shell at null and non-null in the Jordan frame, and (iii) a non-vacuum C ¹ regular matching at null in the Jordan frame. Lastly, we clarify the relations between the conditions for C ¹ regularity and also for vacuum in the Jordan and Einstein frames.