For a theory in which a scalar field $\phi$ has a nonminimal derivative coupling to the Einstein tensor $G_{\mu \nu}$ of the form $\phi\,G_{\mu \nu}\nabla^{\mu}\nabla^{\nu} \phi$, 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. Thus, the general relativistic branch with a vanishing field derivative is the only stable star configuration in such a derivative coupling theory, including both relativistic and nonrelativistic compact objects.

中文翻译：

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具有非最小标量导数耦合的致密星的不稳定性

对于标量场 $\phi$ 与爱因斯坦张量 $G_{\mu \nu}$ 的非最小导数耦合的理论，形式为 $\phi\,G_{\mu \nu}\nabla^{\ mu}\nabla^{\nu} \phi$，我们知道存在一个静态的、球对称的相对论恒星分支，它们的内部带有标量头发。我们研究了这种具有径向场依赖性 $\phi(r)$ 对抗奇偶校验扰动的毛状解的稳定性。我们表明，对于小于 $1/3$ 的恒星致密性 ${\cal C}$，它们容易出现与沿角方向的标量场传播相关的偶校验扰动的拉普拉斯不稳定性。即使对于 ${\cal C}>1/3$，毛状星解也受到鬼不稳定性的影响。因此，