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.
