  We study the characteristic structure of the Einstein-Hilbert (EH) action
when modifications of the form of $R^2,~ R_{\mu\nu}^2$,
$R_{\mu\nu\rho\sigma}^2$ and $C_{\mu\nu\rho\sigma}^2$ 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 (FRW) 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 (GR).
