Carroll limit of four-dimensional gravity theories in the first order formalism

TL;DR

This paper investigates the ultra-relativistic limit of four-dimensional Lovelock-Cartan gravity theories in the first order formalism, revealing new solutions, symmetries, and extensions of classical results like Birkhoff's theorem.

Contribution

It extends the understanding of ultra-relativistic limits to Lovelock-Cartan gravities, introducing scale symmetries and explicit solutions including matter effects.

Findings

Verification of Birkhoff's theorem in torsionless cases

Derivation of explicit solutions with non-trivial geometry

Extension of solutions to include matter in Lovelock-Cartan gravity

Abstract

We explore the ultra-relativistic limit of a class of four dimensional gravity theories, known as Lovelock-Cartan gravities, in the first order formalism. First, we review the well known limit of the Einstein-Hilbert action. A very useful scale symmetry involving the vierbeins and the boost connection is presented. Moreover, we explore the field equations in order to find formal solutions. Some remarkable results are obtained: Riemann and Weitzenb\"ock like manifolds are discussed; Birkhoff's theorem is verified for the torsionless case; an explicit solution with non-trivial geometry is discussed; A quite general solution in the presence of matter is obtained. Latter, we consider the ultra-relativistic limit of the more general Lovelock-Cartan gravity. The previously scale symmetry is also discussed. The field equations are studied in vacuum and in the presence of matter. In comparison with the Einstein-Hilbert case, a few relevant results are found: Birkhoff's theorem is also verified for the torsionless case; A quite general solution in the presence of matter is obtained. This solution generalizes the previous case; Riemann and Weitzenb\"ock like manifolds are derived in the same lines of the Einstein-Hilbert case.