Lorentz-violating gravitoelectromagnetism

The well-known analogy between a special limit of general relativity and electromagnetism is explored in the context of the Lorentz-violating standard-model extension. An analogy is developed for the minimal standard-model extension that connects a limit of the $CPT$-even component of the electromagnetic sector to the gravitational sector. We show that components of the post-Newtonian metric can be directly obtained from solutions to the electromagnetic sector. The method is illustrated with specific examples including static and rotating sources. Some unconventional effects that arise for Lorentz-violating electrostatics and magnetostatics have an analog in Lorentz-violating post-Newtonian gravity. In particular, we show that even for static sources, gravitomagnetic fields arise in the presence of Lorentz violation.

Figure 1

The gravitomagnetic field ${\overrightarrow{B}}_G$ (dark arrows) from a static point mass $m$ (center). The field curls around the direction of $\overrightarrow{s}$ (light gray arrows) and falls off as the inverse square of the distance. An approaching test body is deflected opposite $\overrightarrow{s}$.

Figure 2

A depiction of the effective dipole moment that develops for a rotating sphere in the presence of the coefficients for Lorentz violation $\overrightarrow{s}$. The dipole moment (medium gray arrow) is proportional to the cross product of $\overrightarrow{s}$ (light gray arrows) with the angular momentum of the sphere (dark arrow).