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

Johanna Borissova

Abstract

We show that all two-dimensional Horndeski theories can arise from the reduction of pure gravities in $d \geq 4$ dimensions and therefore all 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 call all such theories 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.

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

Abstract

We show that all two-dimensional Horndeski theories can arise from the reduction of pure gravities in dimensions and therefore all onshell configurations for the two-dimensional metric and scalar field correspond to genuine -dimensional gravitational vacuum solutions. We discuss separately the two-dimensional Horndeski theories which can arise from the reduction of -dimensional generally covariant gravitational actions built only from curvature invariants without covariant derivatives and possessing second-order equations of motion on warped-product backgrounds. The discussion is subsequently extended to generic -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 in Schwarzschild gauge whereby the metric function is determined by an algebraic equation. We therefore propose to call all such theories quasi-topological gravities. These results can be used to show in reverse that any -dimensional static spherically symmetric and asymptotically flat spacetime satisfying in Schwarzschild gauge with an invertible dependence of on the ADM mass can be reconstructed explicitly as a vacuum solution to a -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.
Paper Structure (13 sections, 50 equations, 1 figure)

This paper contains 13 sections, 50 equations, 1 figure.

Table of Contents

  1. Introduction
  2. 2D Horndeski theory and equations of motion
  3. Action and covariant equations of motion
  4. Equations of motion in extended Schwarzschild gauge and Birkhoff theorem
  5. Reverse construction of solutions to integrable theories
  6. Example: 2D dilaton theories
  7. 2D Horndeski theory from $d$-dimensional gravity
  8. Reduced action on $2 + (d-2)$ warped-product backgrounds
  9. Invariants for $2+ (d-2)$ warped-product spacetimes
  10. Curvature invariants
  11. Curvature-derivative invariants
  12. Expressing 2D Horndeski densities as reduced $d$-dimensional covariant densities
  13. Curvature Lagrangian densities

Figures (1)

  • Figure 1: Landscape of $d\geq 4$ gravitational theories with second-order equations of motion on $2+(d-2)$ warped-product backgrounds.