Carroll contractions of Lorentz-invariant theories

TL;DR

This work shows that Lorentz-invariant theories admit two distinct Carroll contractions—electric and magnetic—arising from the ultrarelativistic limit $c\to0$ with different rescalings of canonical variables. Using a Hamiltonian framework, the authors derive Carroll-invariant actions for a wide range of theories, including scalars, $p$-form gauge fields, higher-spin fields, and gravity, providing both the dynamics and the symmetry structure (Carroll generators) beyond the equations of motion. They also construct manifestly Carroll-invariant covariant actions for $p$-form gauge fields and explain how the magnetic contraction requires an auxiliary gauge field, which alters Carroll transformation rules after gauge fixing. Extending the procedure to gravity and higher spins clarifies how two Carroll limits persist in these theories and highlights potential implications for nonrelativistic holography and strong-coupling regimes near spacelike singularities, with future work aiming to connect to gauged Carroll algebras and asymptotic symmetries.

Abstract

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 $c \rightarrow 0$ 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.