Exact solutions in gravity with a sigma model source

Abstract

We consider a D -dimensional model of gravity with non-linear "scalar fields" as a matter source. The model is defined on the product manifold M , which contains n Einstein factor spaces. General cosmological type solutions to the field equations are obtained when n − 1 factor spaces are Ricci-flat, e.g. when one space M 1 of dimension d 1 > 1 has nonzero scalar curvature. The solutions are defined up to solutions to geodesic equations corresponding to a sigma model target space. Several examples of sigma models are presented. A subclass of spherically symmetric solutions is studied and a restricted version of "no-hair theorem" for black holes is proved. For the case d 1 = 2 a subclass of latent soliton solutions is singled out.

Exact holographic RG flows and the A1 × A1 Toda chain

“…The Lagrangian (2.7) with the help of (2.8)-(2.11) can be taken to the following form (see [17] and refs. therein)…”

Elliptic solutions of generalized Brans–Dicke gravity with a non-universal coupling

“…Here we focus our attention on the exact solutions for the couplings given by (31). The metric h given by Eq.…”

“…For the constant h AB (ϕ) = h AB the reduction to the sigma model was proved (for a more general setup) in [39]. The case of diagonal h AB with arbitrary dependence on scalar fields in D dimensions, D ≥ 3 , was considered in [40]. The variables σ A , A = 2, 3 , are cyclic and the corresponding equations of motion read…”

“…The complexity having scalar fields makes difficulties for finding exact solutions. Nevertheless, in the Einstein frame it can be shown that under the cosmological ansatz for required solutions the gravitational equations are trivial and scalar fields equations correspond to geodesic equations on the target space of a nonlinear sigma model [29][30][31][32][33][34]. We show that using the sigma-model approach yields an effective one-component Lagrangian with a potential.…”