Lie-algebra expansions, Chern-Simons theories and the Einstein-Hilbert lagrangian

TL;DR

This paper explores how Lie algebra expansions applied to five-dimensional Chern-Simons gravity lead to a modified Einstein-Hilbert action with nonminimal matter couplings, resulting in significant deviations from standard General Relativity.

Contribution

It introduces a novel deformation of gravity theories via Lie algebra expansion, producing a gauge-invariant system with nonminimal matter couplings that differ fundamentally from GR.

Findings

Deformed theory includes Einstein-Hilbert plus nonminimally coupled matter.

In torsionless, matter-free case, field equations impose strong geometric restrictions.

Standard solutions like 5D Schwarzschild are not valid; brane-world configurations are possible.

Abstract

Starting from gravity as a Chern-Simons action for the AdS algebra in five dimensions, it is possible to deform the theory through an expansion of the Lie algebra that leads to a system consisting of the Einstein-Hilbert action plus nonminimally coupled matter. The deformed system is gauge invariant under the Poincare group enlarged by an Abelian ideal. Although the resulting action naively looks like General Relativity plus corrections due to matter sources, it is shown that the nonminimal couplings produce a radical departure from GR. Indeed, the dynamics is not continuously connected to the one obtained from Einstein-Hilbert action. In a matter-free configuration and in the torsionless sector, the field equations are too strong a restriction on the geometry as the metric must satisfy both the Einstein and pure Gauss-Bonnet equations. In particular, the five-dimensional Schwarzschild geometry fails to be a solution; however, configurations corresponding to a brane-world with positive cosmological constant on the worldsheet are admissible when one of the matter fields is switched on. These results can be extended to higher odd dimensions.