Effective Field Theory of Black Hole Perturbations with Timelike Scalar
  Profile: Formulation
  We formulate the Effective Field Theory (EFT) of perturbations within
scalar-tensor theories on an inhomogeneous background. The EFT is constructed
while keeping a background of a scalar field to be $\textit{timelike}$, which
spontaneously breaks the time diffeomorphism. We find a set of consistency
relations that are imposed by the invariance of the EFT under the 3d spatial
diffeomorphism. This EFT can be generically applied to any inhomogeneous
background metric as long as the scalar profile is everywhere timelike. For
completeness, we report a dictionary between our EFT parameters to those of
Horndernski theories. Finally, we compute background equations for a class of
spherically symmetric, static black hole backgrounds, including a stealth
Schwarzschild-de Sitter solution.
