Stability of Schwarzschild-Like Spacetime 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 = const or partial derivative(2)f/partial derivative R partial derivative C2 = 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 and find that the no-ghost conditions and no-tachyon conditions are the same as those in f(R) theories.