Orbital mechanics and quasiperiodic oscillation resonances of black holes in Einstein-{\AE}ther theory

In this paper, we study the motion of test particles around two exact charged black-hole solutions in Einstein-{\AE}ther theory. Specifically, we first consider the quasi-periodic oscillations (QPOs) and their resonances generated by the particle moving in the Einstein-{\AE}ther black hole and then turn to study the periodic orbits of the massive particles. For QPOs, we drop the usually adopted assumptions $\nu_U=\nu_\theta$, $\nu_L=\nu_r$, and $\nu_U/\nu_L=3/2$ with $\nu_U$ ($\nu_L$) and $\nu_r$ ($\nu_\theta$) being the upper (lower) frequency of QPOs and radial (vertical) epicyclic frequency of the orbiting particles, respectively. Instead, we put-forward a new working ansatz for which the Keplerian radius is much closer to that of the innermost stable circular orbit and explore in detail the effects of the {\ae}ther field on the frequencies of QPOs. We then realize good curves for the frequencies of QPOs, which fit to data of three microquasars very well by ignoring any effects of rotation and magnetic fields. The innermost stable circular orbits (isco) of timelike particles are also analyzed and we find the isco radius increases with increasing $c_{13}$ for the first type black hole while decreases with increasing $c_{14}$ for the second one. We also obtain several periodic orbits and find that they share similar taxonomy schemes as the periodic equatorial orbits in the Schwarzschild/Kerr metrics, in addition to exact solutions for certain choices of the Einstein-\AE ther parameters. The equations for null geodesics are also briefly considered, where we study circular photon orbits and bending angles for gravitational lensing.