All $2D$ generalised dilaton theories from $d\geq 4$ gravities

TL;DR

This paper shows how generic 2D Horndeski theories can originate from higher-dimensional pure gravity, establishing a link between 2D solutions and higher-dimensional vacuum solutions, and introduces quasi-topological gravities with a Birkhoff theorem.

Contribution

It demonstrates the derivation of 2D Horndeski theories from higher-dimensional gravities and establishes a Birkhoff theorem for a class of these theories, termed quasi-topological gravities.

Findings

2D Horndeski theories can be obtained from $d extgreater= 4$ dimensional pure gravities.

A Birkhoff theorem is established for theories reducing to integrable 2D Horndeski models.

Regular black holes like Bardeen spacetime can be reconstructed as vacuum solutions in these theories.

Abstract

We demonstrate that generic two-dimensional Horndeski theories can arise from the reduction of pure gravities in $d \geq 4$ dimensions, and therefore generic onshell configurations for the two-dimensional metric and scalar field correspond to genuine $d$-dimensional gravitational vacuum solutions. We discuss separately the two-dimensional Horndeski theories which can arise from the reduction of $d$-dimensional generally covariant gravitational actions built only from curvature invariants without covariant derivatives and possessing second-order equations of motion on $2 + (d-2)$ warped-product backgrounds. The discussion is subsequently extended to generic $d$-dimensional gravitational actions with this latter property. We establish a Birkhoff theorem for all gravitational theories whose reduction yields an integrable two-dimensional Horndeski theory, in which case static spherically symmetric solutions satisfy $g_{tt} g_{rr} = -1$ in Schwarzschild gauge whereby the metric function $g_{tt} = -f$ is determined by an algebraic equation. We therefore propose to refer to all such theories as quasi-topological gravities. These results can be used to show in reverse that any $d$-dimensional static spherically symmetric and asymptotically flat spacetime satisfying $g_{tt} g_{rr} = -1$ in Schwarzschild gauge with an invertible dependence of $f$ on the ADM mass can be reconstructed explicitly as a vacuum solution to a $d$-dimensional gravitational theory. We discuss examples of regular black holes such as the Bardeen spacetime, which could not be obtained from polynomial and non-polynomial quasi-topological gravities involving only curvature invariants without covariant derivatives.