Through detection by low gravitational wave space interferometers, the capture of stars by supermassive black holes will constitute a giant step forward in the understanding of gravitation in strong field. The impact of the perturbations on the motion of the star is computed via the tail, the back-scattered part of the perturbations, or via a radiative Green function. In the former approach, the self-force acts upon the background geodesic, while in the latter, the geodesic is conceived in the total (background plus perturbations) field. Regularisations (mode-sum and Riemann-Hurwitz $\zeta$ function) intervene to cancel divergencies coming from the infinitesimal size of the particle. The non-adiabatic trajectories require the most sophisticated techniques for studying the evolution of the motion, like the self-consistent approach.
Deep Dive into Self-force and motion of stars around black holes.
Through detection by low gravitational wave space interferometers, the capture of stars by supermassive black holes will constitute a giant step forward in the understanding of gravitation in strong field. The impact of the perturbations on the motion of the star is computed via the tail, the back-scattered part of the perturbations, or via a radiative Green function. In the former approach, the self-force acts upon the background geodesic, while in the latter, the geodesic is conceived in the total (background plus perturbations) field. Regularisations (mode-sum and Riemann-Hurwitz $\zeta$ function) intervene to cancel divergencies coming from the infinitesimal size of the particle. The non-adiabatic trajectories require the most sophisticated techniques for studying the evolution of the motion, like the self-consistent approach.
The relativistic two-body problem implies the emission of radiation and thus poses still formidable challenges even for radial fall a 1 and generally whenever adiabaticity can't be evoked. Indeed, adiabatic averaging intervenes if a sufficiently long period, in which energy-momentum balance may be applied, does exist. In curved spacetime, at any time the emitted radiation may backscatter off the spacetime curvature, and interact back with the particle later on: the instantaneous conservation of energy is not applicable and the momentary self-force acting on the particle depends on the particle's entire history 3 . Thus, the computation and the application of the back-action all along the trajectory and the continuous correction of the background geodesic, it is the only semi-analytic way to determine motion in non-adiabatic cases. Non-adiabatic gravitational waveforms are one of the original aims of the self-force community, since they express i) the physics closer to the black hole horizon ii) the most complex trajectories iii) the most tantalising theoretical questions.
a Back-action shows itself even in Newtonian physics, as the uniqueness of acceleration holds as long as the masses of the falling bodies are negligible. For free fall, the implications on the equivalence principle have been discussed. With the appearance of general relativity, the radial fall of a test particle into a Schwarzschild-Droste black hole has fueled a vivacious controversy in the first seventy years of general relativity but still echoing today on the existence of repulsion and on the velocity of a particle at the horizon 2 .
Perturbations were first dealt in 1957 4 , when a Schwarzschild-Droste black hole was shown to regain stability after undergoing small vibrations about its spherical form, if subjected to a small perturbation. In the following forty years of analysis in the frequency domain, the captured mass radiates energy (the second time derivative of the quadrupole moment is different than zero), but its motion is still unaffected by the radiation emitted 5 . As analysis in the frequency domain is inherently limited, the breakthrough arrived thanks to a specifically tailored finite differences method, consisting of the numerical integration of the inhomogeneous wave equation in time domain 6 and thanks to the determination of the self-force.
It is only slightly more than a decade, that we possess methods for the evaluation in strong field of the self-force for point particles, thanks to concurring situations. On one hand, theorists progressed in understanding radiation reaction and obtained formal prescriptions for its determination and, on the other hand, the appearance of requirements from the LISA (Laser Interferometer Space Antenna) project for the detection of captures of stars by supermassive black holes (EMRI, Extreme Mass Ratio Inspiral), notoriously affected by radiation reaction.
Before the appearance of the self-force equation and of the regularisation methods, the main theoretical unsolved problem was represented by the infinities of the perturbations at the position of the particle, represented by a Dirac delta distribution. In 1997, via the conservation of the total stress-energy tensor 7 , or via the matched asymptotic expansion 7 , or via the axiomatic approach 8 , but all yelding the same formal expression of the self-force, baptised MiSaTaQuWa from the surname first two initials of its discoverers, was found. On the footsteps of Dirac’s definition of radiation reaction, in 2003, a fourth approach was presented 9 , hence MiSaTaQuWa-DeWh 1 . A more rigorous way of deriving a gravitational self-force has been attempted without the step of Lorenz gauge relaxation 10 ; an alternative approach and a new derivation of the self-force have been proposed 11,12 . A comprehensive living review online 13 and an upcoming book introduce the self-force (the entire volume previously cited 1 ).
One pictorial description of the self-force refers to a particle that crosses the curved spacetime and thus generates gravitational waves. These waves are partly radiated to infinity (the instantaneous part) and partly scattered back by the black hole potential (the non-local part) thus forming tails which impinge on the particle and give origin to the self-force. Alternatively, the same phenomenon is described by an interaction particle-black hole generating a field which behaves as outgoing radiation in the wave-zone and thereby extracts energy from the particle. In the near-zone, the field acts on the particle and determines the self-force which impedes the particle to move on the geodesic of the background metric. The self-force is written as (l represent the mode):
where F α f ull represents the total contributions, F α inst. the contributions that propagate along the past light cone, and F α tail the contributions from inside the past light cone, product of the scattering of perturbations due to the motion of the pa
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