In this paper we investigate the three-dimensional (3D) motion of a test particle in a stationary, axially symmetric spacetime around a central compact object, under the influence of a radiation field. To this aim we extend the two-dimensional (2D) version of the Poynting-Robertson effect in General Relativity (GR) that was developed in previous studies. The radiation flux is modeled by photons which travel along null geodesics in the 3D space of a Kerr background and are purely radial with respect to the zero angular momentum observer (ZAMO) frames. The 3D general relativistic equations of motion that we derive are consistent with the classical (i.e. non-GR) description of the Poynting-Robertson effect in 3D. The resulting dynamical system admits a critical hypersurface, on which radiation force balances gravity. Selected test particle orbits are calculated and displayed, and their properties described. It is found that test particles approaching the critical hypersurface at a finite latitude and with non-zero angular moment are subject to a latitudinal drift and asymptotically reach a circular orbit on the equator of the critical hypersurface, where they remain at rest with respect to the ZAMO. On the contrary, test particles that have lost all their angular momentum by the time they reach the critical hypersurface do not experience this latitudinal drift and stay at rest with respects to the ZAMO at fixed non-zero latitude. * vittorio.defalco@physics.cz † pavel.bakala@fpf.slu.cz ‡ emmanuelebattista@gmail.com 1 Note however that Robertson mentions general relativistic corrections to inertial terms in describing the perihelion shift in the quasi-Newtonian orbits [2].

We investigate the three-dimensional, general relativistic Poynting-Robertson (PR) effect in the case of rigidly rotating spherical source which emits radiation radially in the local comoving frame. Such radiation field is meant to approximate the field produced by the surface of a rotating neutron star, or by the central radiating hot corona of accreting black holes; it extends the purely radial radiation field that we considered in a previous study. Its angular momentum is expressed in terms of the rotation frequency and radius of the emitting source. For the background we adopt a Kerr spacetime geometry. We derive the equations of motion for test particles influenced by such radiation field, recovering the classical and weak-field approximation for slow rotation. We concentrate on solutions consisting of particles orbiting along circular orbits off and parallel to the equatorial plane, which are stabilized by the balance between gravitational attraction, radiation force and PR drag. Such solutions are found to lie on a critical hypersurface, whose shape may morph from prolate to oblate depending on the Kerr spin parameter and the luminosity, rotation and radius of the radiating sphere. For selected parameter ranges, the critical hypersurface intersects the radiating sphere giving rise to a bulging equatorial region or, alternatively, two lobes above the poles. We calculate the trajectories of test particles in the close vicinity of the critical hypersurface for a selected set of initial parameters and analyze the spatial and angular velocity of test particles captured on the critical hypersurface.

One of the outstanding problems of classical celestial mechanics was the restricted 3-body problem, in which a planetoid of small mass is subject to the Newtonian attraction of two celestial bodies of large mass, as it occurs, for example, in the sun-earth-moon system. On the other hand, over the last decades, a systematic investigation of quantum corrections to the Newtonian potential has been carried out in the literature on quantum gravity. The present paper studies the effect of these tiny quantum corrections on the evaluation of equilibrium points. It is shown that, despite the extreme smallness of the corrections, there exists no choice of sign of these corrections for which all qualitative features of the restricted 3-body problem in Newtonian theory remain unaffected.Moreover, first-order stability of equilibrium points is characterized by solving a pair of algebraic equations of fifth degree, where some coefficients depend on the Planck length. The coordinates of stable equilibrium points are slightly changed with respect to Newtonian theory, because the planetoid is no longer at equal distance from the two bodies of large mass. The effect is conceptually interesting but too small to be observed, at least for the restricted 3-body problems available in the solar system.

No abstract

We propose the Lagrangian formulation for describing the motion of a test particle in a general relativistic, stationary, and axially symmetric spacetime. The test particle is also affected by a radiation field, modeled as a coherent flux of photons traveling along the null geodesics of the background spacetime, including the general relativistic Poynting-Robertson effect. The innovative part of this work is to prove the existence of the potential linked to the dissipative action caused by the Poynting-Robertson effect in General Relativity through the help of an integrating factor, depending on the energy of the system. Generally such kinds of inverse problems involving dissipative effects might not admit a Lagrangian formulation, especially in General Relativity there are no examples of such attempts in literature so far. We reduce this general relativistic Lagrangian formulation to the classic case in the weak field limit. This approach facilitates further studies in improving the treatment of the radiation field and it contains for example some implications for a deeper comprehension of the gravitational waves.