Electric-magnetic duality in linearized Hořava-Lifshitz gravity

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 Hořava-Lifshitz gravity. Since this theory breaks diffeomorphism invariance, one finds that such implementation includes some peculiarities.

Figure 1

Role of the parent Lagrangian in $d$ dimensions for FP theory. $h_{ab}$ are the first order perturbations of the metric and the variables $T_{a_1\dots a_{d-3}\mid b}$ are the components of the dual graviton. The child theories $\mathcal{L}[e_{ab}]$ and $\mathcal{L}[{W^{abc\mid }}_d]$ depend on the vielbein and some potentials ${W^{abc\mid }}_d$, respectively (such that ${Y^{ab\mid }}_c={\partial }_d{W^{abd\mid }}_c$, ${Y^{ab\mid }}_c$ being a redefinition of the spin connection). They enjoy arbitrary shifts of $e_{[ab]}$ and of the trace ${W^{abc\mid }}_c$ as gauge symmetries, respectively. Those variables can be marginalized with gauge fixing (gf) conditions recovering the FP Lagrangian on the left branch and the Curtright theory on the right one. In $d=4$ the dual graviton is a rank-2 symmetric tensor and ${\mathcal{L}}_C={\mathcal{L}}_{\mathrm{FP}}$, revealing that FP theory is self-dual.

Figure 2

The Lagrangian density ${\mathcal{L}}_{\lambda }[e_{ab}]$ depends on the vielbeins and it is parametrized by $\lambda$, the same parameter appearing in linearized HL gravity defined with Lagrangian (1). This theory is diffeomorphism invariant, so it is not linearized HL gravity; the latter is obtained from ${\mathcal{L}}_{\lambda }[e_{ab}]$ by gauge fixing (gf) conditions that in fact eliminate all antisymmetric components of $e_{ab}$. Analogously on the dual side ${\mathcal{L}}_{\lambda }[{W^{abc\mid }}_d]$ has some variables that can be eliminated through gauge fixing conditions to get a Lagrangian depending only on the dual graviton components.

Figure 3

Plot of $\gamma$ vs $\lambda$.

Figure 4

Sample of three iterations starting in $\lambda =-0.15$.