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 → 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.
Highlights
A manifestly diffeomorphism invariant formulation of a gravitation theory based on the Carroll group was given in [5]
After a brief survey of the geometrical concepts adapted to the description of Carroll contractions and the underlying symmetry groups, we establish the conditions for a theory to be Carroll-invariant
How this arises in our approach will be discussed but first, we show that the same conclusions concerning transformation rules hold in the Hamiltonian formalism
Summary
Curved Carroll geometries were defined long ago in [5]. They were called there “zero Hamiltonian signature spacetimes” because the Hamiltonian signature = ±1, 0 is a parameter that distinguishes in the Hamiltonian formulation of general relativity between Euclidean signature ( = 1), Lorentzian signature ( = −1) and “zero Hamiltonian signature” ( = 0), which lie halfway between the Euclidean and Minkowskian cases [4]. A Carroll manifold is a manifold equipped with a Carroll structure in the tangent space at each point, which depends smoothly on the point In local coordinates, it is defined by a symmetric tensor gαβ(x) with the above properties and a density Ω(x), which. Since the one-form θα comes on top of the basic Carroll structure defined by the degenerate metric gαβ and the null vector nα, we shall insist that “Carrollian physics” should not depend on θα, i.e. should be invariant under (2.7) and (2.8). These transformations should appear as gauge transformations in any purely Carrollian action. The introduction of a metric-preserving, symmetric affine connection was found to be unconvenient for some purposes in [6] and the authors of [6, 20] reverted to the earlier definition of [5] without this extra connection, which turns out to be appropriate for the generalization to conformal Carroll structures and the link with the BMS group [6]
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