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

Abstract

We explore the ultra-relativistic (UR) limit of a class of four dimensional gravity theories, known as Lovelock–Cartan (LC) gravities, in the first order formalism. First, we review the well known limit of the Einstein–Hilbert (EH) 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 Weitzenböck 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 UR limit of the more general LC 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 EH 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 Weitzenböck like manifolds are derived in the same lines of the EH case.