Hyperbolicity constraints in extended gravity theories

We study the characteristic structure of the Einstein–Hilbert (EH) action when modifications of the form of R 2 , R μ ν 2 , R μ ν ρ σ 2 and C μ ν ρ σ 2 are included. We show that when these quadratic terms are significant, the initial value problem is generically ill-posed. We do so by demanding the hyperbolicity of the effective metric for propagation of perturbations. Here, we find a general expression for the effective metric in field space and calculate it explicitly about the cosmological Friedman–Robertson–Walker spacetime, and the spherically symmetric Schwarzschild solution. We find that when these quadratic contributions are non-negligible, the signature of the effective metric becomes non-Lorentzian and hence non-hyperbolic. As a consequence, we conclude that theories suggesting the inclusion of these terms can only be considered as a perturbative extension of the EH action and therefore cannot constitute a true alternative to general relativity.