Lifting Klein-Gordon/Einstein Solutions to General Nonlinear Sigma-Models: the Wormhole Example

Abstract

We describe a simple technique for generating solutions to the classical field equations for an arbitrary nonlinear sigma-model minimally coupled to gravity. The technique promotes an arbitrary solution to the coupled Einstein/Klein-Gordon field equations for a single scalar field $\sigma$ to a solution of the nonlinear sigma-model for $N$ scalar fields minimally coupled to gravity. This mapping between solutions does not require there to be any target-space isometries and exists for every choice of geodesic computed using the target-space metric. In some special situations -- such as when the solution depends only on a single coordinate (e.g. for homogeneous time-dependent or static spherically symmetric configurations) -- the general solution to the sigma-model equations can be obtained in this way. We illustrate the technique by applying it to generate Euclidean wormhole solutions for multi-field sigma models coupled to gravity starting from the simplest Giddings-Strominger wormhole, clarifying why in the wormhole case Minkowski-signature target-space geometries can arise. We reproduce in this way the well-known axio-dilaton string wormhole and we illustrate the power of the technique by generating simple perturbations to it, like those due to string or $\alpha'$ corrections.