Metrics With Vanishing Quantum Corrections

We investigate solutions of the classical Einstein or supergravity equations that solve any set of quantum corrected Einstein equations in which the Einstein tensor plus a multiple of the metric is equated to a symmetric conserved tensor $T_{mu u}$ constructed from sums of terms the involving contractions of the metric and powers of arbitrary covariant derivatives of the curvature tensor. A classical solution, such as an Einstein metric, is called {it universal} if, when evaluated on that Einstein metric, $T_{mu u}$ is a multiple of the metric. A Ricci flat classical solution is called {it strongly universal} if, when evaluated on that Ricci flat metric, $T_{mu u}$ vanishes. It is well known that pp-waves in four spacetime dimensions are strongly universal. We focus attention on a natural generalisation; Einstein metrics with holonomy ${rm Sim} (n-2)$ in which all scalar invariants are zero or constant. In four dimensions we demonstrate that the generalised Ghanam-Thompson metric is weakly universal and that the Goldberg-Kerr metric is strongly universal; indeed, we show that universality extends to all 4-dimensional ${rm Sim}(2)$ Einstein metrics. We also discuss generalizations to higher dimensions.