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

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