A fully relativistic radial fall

Abstract

Radial fall has historically played a momentous role. It is one of the most classical problems, the solutions of which represent the level of understanding of gravitation in a given epoch. A {\it gedankenexperiment} in a modern frame is given by a small body, like a compact star or a solar mass black hole, captured by a supermassive black hole. The mass of the small body itself and the emission of gravitational radiation cause the departure from the geodesic path due to the back-action, that is the self-force. For radial fall, as any other non-adiabatic motion, the instantaneous identity of the radiated energy and the loss of orbital energy cannot be imposed and provide the perturbed trajectory. In the first part of this letter, we present the effects due to the self-force computed on the geodesic trajectory in the background field. Compared to the latter trajectory, in the Regge-Wheeler, harmonic and all others smoothly related gauges, a far observer concludes that the self-force pushes inward (not outward) the falling body, with a strength proportional to the mass of the small body for a given large mass; further, the same observer notes an higher value of the maximal coordinate velocity, this value being reached earlier on during infall. In the second part of this letter, we implement a self-consistent approach for which the trajectory is iteratively corrected by the self-force, this time computed on osculating geodesics. Finally, we compare the motion driven by the self-force without and with self-consistent orbital evolution. Subtle differences are noticeable, even if self-force effects have hardly the time to accumulate in such a short orbit.

To appear in Int. J. Geom. Meth. Mod. Phys

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Cited by in corpus (7)

• Acceleration of particles to high energy via gravitational repulsion in the Schwarzschild field
• Motion of localized sources in general relativity: gravitational self-force from quasilocal conservation laws
• Gauge dependence and self-force from Galilean to Einsteinian free fall, compact stars falling into black holes, Hawking radiation and the Pisa tower at the general relativity centennial
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• Near-horizon geodesics for astrophysical and idealised black holes: Coordinate velocity and coordinate acceleration
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• Particle-without-Particle: a practical pseudospectral collocation method for linear partial differential equations with distributional sources