Closed and broken electromagnetic orbits in Kerr--Newman spacetime
We study future-pointing timelike solutions of the Lorentz force equation in the sub-extremal Kerr--Newman spacetime, with special attention to the time-machine region $\mathfrak T$, where the axial Killing field $\partial_ϕ$ is timelike. We first construct smooth closed electromagnetic orbits tangent to $\partial_ϕ$ in the positive equatorial part of $\mathfrak T$: the radius of such a circle determines, and is determined by, the charge-to-mass ratio of the particle which must have opposite sign to that of the black hole charge. We then prove the existence of spherical electromagnetic orbits contained in the equatorial time-machine region and derive explicit relations between their radius, charge-to-mass ratio, energy and angular momentum. Next we give sufficient conditions ensuring that an equatorial electromagnetic orbit is a flyby orbit with radial turning point in $\mathfrak T$. Finally, to describe charged-particle decay processes whose fragments have different charge-to-mass ratios, we introduce the notion of a broken electromagnetic orbit: a continuous, piecewise smooth, future-pointing worldline whose smooth pieces solve the Lorentz force equation. Imposing conservation of kinetic four-momentum and electric charge at the decay vertices, we exhibit an energy extraction process followed by a causality-violating one. The latter is realized by a closed broken electromagnetic orbit, which we construct in both the $r$-positive and the $r$-negative regions inside the inner horizon, by concatenating two flyby branches sharing a common radial turning value $\bar r$, and having charge-to-mass ratios with opposite sign to that of the black hole charge, with a spherical electromagnetic orbit of radius $\bar r$.