Black hole perturbation in parity violating gravitational theories
  We study linear perturbations around the static and spherically symmetric
spacetime for the gravitational theories whose Lagrangian depends on Ricci
scalar and the parity violating Chern-Simons term. By an explicit construction,
we show that Hamiltonian for the perturbation variables is not bounded from
below in general, suggesting that such a background spacetime is unstable
against perturbations. This gives a strong limit on a phenomenological
gravitational model which violates parity. We also provide a necessary and
sufficient condition for the theory to belong to a special class in which no
such instability occurs. For such theories, the number of propagating modes for
$\ell \ge 2$ is three, one from the odd and the other two from the even. Unlike
in the case of $f(R)$ theories, those modes are coupled each other, which can
be used as a distinctive feature to test the parity violating theories from
observations. All the modes propagate at the speed of light. No-ghost condition
and no-tachyon condition are the same as those in $f(R)$ theories. For the
dipole perturbations, the odd and the even modes completely decouple. The odd
mode gives a slowly-rotating BH solution whose metric is linearized in its
angular momentum. We provide an integral expression of such a solution. On the
other hand, the even mode propagates at the speed of light. For the monopole
perturbation, in addition to a mode which just shifts the mass of the
background BH, there is also one even mode that propagates at the speed of
light.
