All the solutions of the form M2(warped)x\Sigma(d-2) for Lovelock gravity in vacuum in the Chern-Simons case

TL;DR

This paper classifies a family of warped Lovelock gravity solutions in odd dimensions greater than four, revealing degeneracies and showing that higher curvature terms do not create new solution branches in the Chern-Simons case.

Contribution

It provides a comprehensive classification of Lovelock solutions with warped product metrics in the Chern-Simons case across arbitrary odd dimensions, extending previous five-dimensional results.

Findings

Solutions are classified by constraints on the base manifold.

Higher curvature terms do not generate new solution branches.

Degeneracy exists in metric functions not being fully determined.

Abstract

In this note we classify a certain family of solutions of Lovelock gravity in the Chern-Simons (CS) case, in arbitrary (odd) dimension greater than four. The spacetime is characterized by admitting a metric that is a warped product of a two-dimensional spacetime M2 and an (a priori) arbitrary Euclidean base manifold Sigma(d-2) of dimension d-2. We show that the solutions are naturally classified in terms of the equations that restrict the base manifold. According to the strength of such constraints we found the following branches in which Sigma(d-2) has to fulfill: a Lovelock equation with a single vacuum (Euclidean Lovelock Chern-Simons in dimension d-2), a single scalar equation that is the trace of an Euclidean Lovelock CS equation in dimension d-2, or finally a degenerate case in which the base manifold is not restricted at all. We show that all the cases have some degeneracy in the sense that the metric functions are not completely fixed by the field equations. This result extends the static five-dimensional case previously discussed in Phys.Rev. D76 (2007) 064038, and it shows that in the CS case, the inclusion of higher powers in the curvature does not introduce new branches of solutions in Lovelock gravity. Finally we comment on how the inclusion of a non-vanishing torsion and matter fields may modify this analysis.