Dynamics of Einstein's equation modified by a higher-order derivative term

The search for a renormalized stress-energy operator of a quantum field in curved spacetime has raised the question of whether one can add terms to Einstein's equation containing fourth-order derivatives of the metric, and still maintain a reasonable theory. We investigate this question by considering the simple case of a conformally invariant field in a conformally flat spacetime, where one has a well-defined prediction for the vacuum stress energy of the field. We find that if a certain fourth-order term (associated with the $\square R$ trace anomaly) enters with one sign, flat spacetime is unstable to conformally flat perturbations that grow exponentially on the Planck time scale. If this term enters with the other sign, conformally flat perturbations could cause spacetime to oscillate at the Planck frequency, resulting in high-energy radiation by charged test particles.