Stability of Schwarzschild-like solutions in f(R,G) gravity models

We study linear metric perturbations around a spherically symmetric static spacetime for general f(R,G) theories, where R is the Ricci scalar and G is the Gauss-Bonnet term. We find that unless the determinant of the Hessian of f(R,G) is zero, even-type perturbations have a ghost for any multi-pole mode. In order for these theories to be plausible alternatives to General Relativity, the theory should satisfy the condition that the ghost is massive enough to effectively decouple from the other fields. We study the requirement on the form of f(R,G) which satisfies this condition. We also classify the number of propagating modes both for the odd-type and the even-type perturbations and derive the propagation speeds for each mode.