Black hole perturbation in parity violating gravitational theories

We study linear perturbations around static, spherically symmetric spacetimes in $f(R,C)$ gravitational theories whose Lagrangians depend on the Ricci scalar $R$ and the parity violating Chern-Simons term $C$. By an explicit construction, we show that the Hamiltonian for the perturbation variables is not bounded from below, 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 show that either $R=\mathrm{const}$ or $\frac{{\partial }^{2}f}{\partial R\partial C}=0$ is a necessary and sufficient condition for the stability. We then implement in detail the perturbation analysis for such theories which satisfy the stability conditions. For $\ell \ge 2$, where $\ell$ is the usual integer for the multipole expansion, the number of propagating modes is three, one from the odd and the other two from the even, all of which propagate at the speed of light. Unlike in the case of $f(R)$ theories, these modes are coupled to each other, which can be used as a distinctive feature to test the parity violating theories from observations. The no-ghost conditions and no-tachyon conditions 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 black hole solution whose metric is linearized in its angular momentum. We provide an integral expression of this solution. On the other hand, the even mode propagates at the speed of light. For the monopole perturbation, in addition to a mode which simply shifts the mass of the background black hole, there also exists one even mode that propagates at the speed of light.