Instability of compact stars with a nonminimal scalar-derivative coupling

For a theory in which a scalar field $phi$ has a nonminimal derivative coupling to the Einstein tensor $G_{mu u}$ of the form $phi,G_{mu u} abla^{mu} abla^{ u} phi$, it is known that there exists a branch of static and spherically-symmetric relativistic stars endowed with a scalar hair in their interiors. We study the stability of such hairy solutions with a radial field dependence $phi(r)$ against odd- and even-parity perturbations. We show that, for the star compactness ${cal C}$ smaller than $1/3$, they are prone to Laplacian instabilities of the even-parity perturbation associated with the scalar-field propagation along an angular direction. Even for ${cal C}>1/3$, the hairy star solutions are subject to ghost instabilities. We also find that even the other branch with a vanishing background field derivative is unstable for a positive perfect-fluid pressure, due to nonstandard propagation of the field perturbation $delta phi$ inside the star. Thus, there are no stable star configurations in derivative coupling theory without a standard kinetic term, including both relativistic and nonrelativistic compact objects.