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

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 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 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 Weitzenböck like manifolds are derived in the same lines of the Einstein-Hilbert case.

22 pages; 1 figure

References in corpus (7)

• Conformal Carroll groups and BMS symmetry
• Conformal Carroll groups
• Dynamics of Carroll Particles
• Newton-Cartan Gravity and Torsion
• Generalized Birkhoff theorem in the Poincaré gauge gravity theory
• Non-relativistic limit of gravity theories in the first order formalism
• Covariant Poisson's equation in torsional Newton-Cartan gravity

Cited by in corpus (4)

• Carroll Expansion of General Relativity
• Carrollian hydrodynamics from symmetries
• Carrollian manifolds and null infinity: A view from Cartan geometry
• Hamiltonian form of `Carroll' gravity