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We find stealth Schwarzschild solutions with a nontrivial profile of thescalar field regular on the horizon in the Einstein gravity coupled to thescalar field with the k-essence and/or generalized cubic galileon terms, whichis a subclass of the Horndeski theory breaking the shift symmetry, where thepropagation speed of gravitational waves coincides with the speed of light.After deriving sufficient conditions for the shift symmetry breaking theory toallow a general Ricci-flat metric solution with a nontrivial scalar fieldprofile, we focus on the stealth Schwarzschild solution with the scalar fieldwith or without time dependence. For the profile $\phi=\phi_0(r)$, weexplicitly obtain two types of stealth Schwarzschild solutions, one of which isregular on the event horizon. The linear perturbation analysis clarifies thatthe kinetic term of the scalar mode identically vanishes, indicating that thescalar mode is strongly coupled. The absence of the kinetic term of the scalarmode in the quadratic action would inevitably arise for the stealthSchwarzschild solutions in the theory with a general scalar field profiledepending only on the spatial coordinates. On the other hand, for thetime-dependent scalar field profile, we clarify that there does not exist astealth Schwarzschild solution in the shift symmetry breaking theories.

Stealth Schwarzschild solution in shift symmetry breaking theories