Electric-magnetic duality in linearized Ho\v{r}ava-Lifshitz gravity

TL;DR

This paper explores electric-magnetic duality in linearized Hořava-Lifshitz gravity, extending the concept from electromagnetism and relativistic gravity to a nonrelativistic, diffeomorphism-breaking context using a first-order Lagrangian formalism.

Contribution

It introduces a formalism to implement electric-magnetic duality in nonrelativistic Hořava-Lifshitz gravity, highlighting unique features due to broken diffeomorphism invariance.

Findings

Duality can be extended to nonrelativistic gravity theories.

Implementation involves peculiarities due to broken diffeomorphism invariance.

Formalism connects Fierz-Pauli theory with its dual in this context.

Abstract

Known as a symmetry of vacuum Maxwell equations, the electric-magnetic duality can be lifted actually to a symmetry of an action. The Lagrangian of this action is written in terms of two vector potentials, one electric and one magnetic, and while it is manifestly invariant under duality rotations, it is not manifestly Lorentz covariant. This duality symmetry exists also in linearized gravity in four dimensions, and can be lifted off shell too. In $d$ dimensions, the link between linearized gravity and its dual can also be seen from the point of view of a parental action. This is defined by a first order Lagrangian (with the help of some auxiliary variables) that delivers both Fierz-Pauli theory and its dual. In this work we use this formalism to implement the electric-magnetic duality in the nonrelativistic deviation of Fierz-Pauli theory arising from Ho\v{r}ava-Lifshitz gravity. Because this theory breaks diffeomorphism invariance, one finds that such implementation includes some peculiarities.