Stars disformally coupled to a shift-symmetric scalar field

We investigate static and spherically symmetric stars disformally coupled to a scalar field. The scalar field is assumed to be shift symmetric, and hence the conformal and disformal factors of the metric coupled to matter are dependent only on the kinetic term of the scalar field. Assuming that the scalar field is linearly dependent on time, we consider a general shift-symmetric scalar-tensor theory and a general form of the matter energy-momentum tensor that allows for the anisotropic pressure and the heat flux in the radial direction. This is a natural starting point in light of how the gravitational field equations and the energy-momentum tensor transform under a disformal transformation. By inspecting the structure of the hydrostatic equilibrium equation in the presence of the derivative-dependent conformal and disformal factors, we show that the energy density and tangential pressure must vanish at the surface of a star. This fact is used to prove the disformal invariance of the surface of a star, which was previously subtle and unclear. We then focus on the shift-symmetric k-essence disformally coupled to matter, and study the interior and exterior metric functions and scalar-field profile in more detail. It is found that there are two branches of the solution depending on the velocity of the scalar field. The disformally-related metric functions of the exterior spacetime are also discussed.