Analytic self-force effects on radial infalling particles in the Schwarzschild spacetime: the radiated energy
Donato Bini, Giorgio Di Russo
TL;DR
This work delivers the first analytic computation of the radiated energy from a radially infalling particle into a Schwarzschild black hole at first self-force order, for both scalar and gravitational perturbations, with consistency checks from Post-Newtonian theory. The authors develop a unified framework: in the scalar case they solve the wave equation with a delta-function source, extract the self-force and radiated energy, and obtain a PN-expanded result $E_{\rm rad}=\frac{q^2}{M}\left[ -\frac{\eta^3}{90}-\frac{739\eta^5}{1512}+\left(\frac{2133\sqrt{3}}{14000}-\frac{49}{360}\right)\pi\eta^6+\cdots\right]$, while in the gravitational case they compute the leading Newtonian flux and extend to 2PN order using the Teukolsky/MST formalism, yielding explicit expressions for the energy and flux $\mathcal{E}=\mathcal{E}_{\rm N}+\eta^2\mathcal{E}_{\rm 1PN}+\eta^4\mathcal{E}_{\rm 2PN}$ and $\mathcal{F}=\mathcal{F}_{\rm N}+\eta^2\mathcal{F}_{\rm 1PN}+\eta^4\mathcal{F}_{\rm 2PN}$; the 1SF waveform is computed in the frequency domain, with the radiated energy $E_{\rm rad}=\frac{1}{\mu^2}\sum_l\int d\omega\,\frac{|\widehat Z_{lm\omega}|^2}{\omega^2}$ including contributions up to $l=6$ and detailed expressions for $dE/d\omega$.
Abstract
We compute, at the first self force accuracy level, the radiated energy from a radially infalling particle released from rest in a Schwarzschild spacetime. We examine both the cases of a scalar particle and that of a massive particle, in the context of gravitational perturbations. Our findings are accompanied by Post-Newtonian checks. In spite of the specific interest for this kind of computations, we outline the building blocks for future higher-order Post-Newtonian computations as well as for extending these results to other interesting situations out of the black hole case.