Metrics With Vanishing Quantum Corrections

Abstract

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 $T_{\mu \nu }$ 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, $T_{\mu \nu }$ is a multiple of the metric. A Ricci flat classical solution is called {\it strongly universal} if, when evaluated on that Ricci flat metric, $T_{\mu \nu }$ 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 $\mathrm{Sim}(n-2)$ 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 $\mathrm{Sim}(2)$ Einstein metrics. We also discuss generalizations to higher dimensions.

23 pages

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