Shape dependence of spontaneous scalarization

Spontaneous scalarization is an interesting mechanism for modification of gravity by nonminimal coupling of a scalar field to matter or curvature invariants in the context of scalar-tensor theories, and its onset is signaled by linear instability of the scalar field around the corresponding general relativity solution. We thus perform the linear stability analysis of the scalar field about general relativity solutions and highlight a crucial difference between a spherically symmetric profile and a planar symmetric profile. We clarify that the critical value for the instability is sensitive to the morphology and that the spontaneous scalarization occurs much more easily with the planar symmetric shape than with the spherically symmetric shape.