We consider Carroll-invariant limits of Lorentz-invariant field theories. We show that just as in the case of electromagnetism, there are two inequivalent limits, one “electric” and the other “magnetic”. Each can be obtained from the corresponding Lorentz-invariant theory written in Hamiltonian form through the same “contraction” procedure of taking the ultrarelativistic limit c → 0 where c is the speed of light, but with two different consistent rescalings of the canonical variables. This procedure can be applied to general Lorentz-invariant theories (p-form gauge fields, higher spin free theories etc) and has the advantage of providing explicitly an action principle from which the electrically-contracted or magnetically-contracted dynamics follow (and not just the equations of motion). Even though not manifestly so, this Hamiltonian action principle is shown to be Carroll invariant. In the case of p-forms, we construct explicitly an equivalent manifestly Carroll-invariant action principle for each Carroll contraction. While the manifestly covariant variational description of the electric contraction is rather direct, the one for the magnetic contraction is more subtle and involves an additional pure gauge field, whose elimination modifies the Carroll transformations of the fields. We also treat gravity, which constitutes one of the main motivations of our study, and for which we provide the two different contractions in Hamiltonian form.

We construct finite-and infinite-dimensional non-relativistic extensions of the Newton-Hooke and Carroll (A)dS algebras using the algebra expansion method, starting from the (anti-)de Sitter relativistic algebra in D dimensions. These algebras are also shown to be embedded in different affine Kac-Moody algebras. In the three-dimensional case, we construct Chern-Simons actions invariant under these symmetries. This leads to a sequence of non-relativistic gravity theories, where the simplest examples correspond to extended Newton-Hooke and extended (post-)Newtonian gravity together with their Carrollian counterparts.

The solution space of three-dimensional asymptotically antide Sitter or flat Einstein gravity is given by the coadjoint representation of two copies of the Virasoro group in the former and the centrally extended BMS 3 group in the latter case. Dynamical actions that control these solution spaces are usually constructed by starting from the Chern-Simons formulation and imposing all boundary conditions. In this note, an alternative route is followed. We study in detail how to derive these actions from a group-theoretical viewpoint by constructing geometric actions for each of the coadjoint orbits, including the appropriate Hamiltonians. We briefly sketch relevant generalizations and potential applications beyond three-dimensional gravity.

We study a three-dimensional Chern-Simons gravity theory based on the Maxwell algebra. We find that the boundary dynamics is described by an enlargement and deformation of the bms 3 algebra with three independent central charges. This symmetry arises from a gravity action invariant under the local Maxwell group and is characterized by presence of Abelian generators which modify the commutation relations of the super-translations in the standard bms 3 algebra. Our analysis is based on the charge algebra of the theory in the BMS gauge, which includes the known solutions of standard asymptotically flat case. The field content of the theory is different than the one of General Relativity, but it includes all its geometries as particular solutions. In this line, we also study the stationary solutions of the theory in ADM form and we show that the vacuum energy and the vacuum angular momentum of the stationary configuration are influenced by the presence of the gravitational Maxwell field.