Black hole perturbation in the most general scalar-tensor theory with   second-order field equations II: the even-parity sector
  We perform a fully relativistic analysis of even-parity linear perturbations around a static and spherically symmetric solution in the most general scalar-tensor theory with second-order field equations. This paper is a sequel to Kobayashi {\em et al.} (2012), in which the linear perturbation analysis for the odd-parity modes is presented. Expanding the Horndeski action to second order in perturbations and eliminating auxiliary variables, we derive the quadratic action for even-parity perturbations written solely in terms of two dynamical variables. The two perturbations can be interpreted as the gravitational and scalar waves. Correspondingly, we obtain two conditions to evade ghosts and two conditions for the absence of gradient instabilities. Only one in each pair of conditions yields a new stability criterion, as the conditions derived from the stability of the gravitational-wave degree of freedom coincide with those in the odd-parity sector. Similarly, the propagation speed of one of the two modes is the same as that for the odd-parity mode, while the other differs in general from them. Our result is applicable to all the theories of gravitation with an extra single scalar degree of freedom such as the Brans-Dicke theory, $f(R)$ models, and Galileon gravity. 
Tsutomu Kobayashi, Hayato Motohashi, Teruaki Suyama
gr-qc astro-ph.CO hep-th