Almost Universal Metrics

Abstract

A metric is called universal if every symmetric conserved rank-two tensor constructed locally from the metric, the curvature, and covariant derivatives of the curvature is proportional to the metric. Such a metric solves arbitrary metric-based higher-curvature field equations after, at most, a change in the effective cosmological constant. A metric is called almost universal if the same field equations reduce instead to a finite set of algebraic conditions and a single linear scalar equation for a profile function. Universal and almost universal metrics are useful because they keep higher-curvature gravity under analytic control. Plane waves are a well-known example of universal metrics, while the Kerr-Schild-Kundt class of metrics including the $pp$-waves is almost universal. Here, we add a new member to this class of metrics and show that nonzero constant curvature $pp$-wave metrics are also almost universal. They reduce the generic gravity field equations to those of cosmological Einstein-Maxwell theory with null dust. The background of the $pp$-waves has the topology ${\mathbb{R}}^{1,1}\times S^2$ and provides the missing partner to the Nariai metric with ${\mathrm{dS}}^2\times S^2$ and the Bertotti-Robinson metric with ${\mathrm{AdS}}^2\times S^2$ topologies. These quantum-protected metrics are of clear interest. We exemplify our results by using the quadratic and cubic gravity theories.

10 pages, title has changed, matches the published version

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