Electromagnetic radiation-reaction near black holes: orbital widening and the role of the tail

Abstract

We investigate the orbital evolution of a classical charged particle around a Schwarzschild black hole immersed in an external, uniform magnetic field, taking into full account both local radiation-reaction and the nonlocal tail self-force arising in curved spacetime. Starting from the DeWitt-Brehme equation and its Landau-Lifshitz reduction, we derive analytic expressions for the conservative and dissipative components of the electromagnetic self-force in both the weak-field (Newtonian) and strong-field regimes. By implementing backward-in-time integration of the third-order DeWitt-Brehme equation alongside the second-order Landau-Lifshitz equation, we demonstrate that the so-called orbital widening effect persists even when the tail term is included, and that for astrophysically realistic charge-to-mass ratios the tail contribution to the trajectory is negligible. We further show that this widening is directly controlled by the product of the magnetic field and radiation-reaction parameters and can be captured in the Newtonian limit. Finally, we identify a scaling symmetry showing that simulations with moderate parameter values can accurately represent the dynamics in realistic astrophysical conditions, confirming that orbital widening is a robust phenomenon that can persist even in astrophysical black hole environments.

Figures (8)

• Figure 1: Comparison of trajectories obtained by integrating the DeWitt-Brehme (DB) Eqs. (\ref{['EOM1LD']})--(\ref{['EOM3LD']}) and the Landau-Lifshitz (LL) Eqs. (\ref{['EOM1LD']})--(\ref{['EOM3LD']}) for both negative $\mathcal{B}$ (top two rows) and positive $\mathcal{B}$ (bottom two rows). The tail term is neglected in all cases. DB is integrated backwards in time, so that the starting point of the LL trajectory is the ending point of the DB trajectory in our numerical settings. Particles are spiraling out in case of $\mathcal{B}>0$.
• Figure 2: Influence of the tail term on the charged particle motion for different values of the radiation parameter $k$. The starting point of the particle is $r_{0}=7$. Lorentz and radiation-reaction forces are set to zero.
• Figure 3: The effect of the tail term. Charged particle trajectories for different ${\cal{B}}$ and $k$, all starting at $r_{0}=7$. Top row: conservative Lorentz force; second row: pure tail effect; third and fourth rows: full radiation-reaction formalism including the tail term. Subfigures show the corresponding numerically calculated quantities.
• Figure 4: Radiating particle in a circular orbit within the Newtonian approximation, shown without (black) and with (red) the contribution of the tail term. The starting point of the particle is indicated by the black dot at $r_0 = 1350$. The parameter values $B$, $m$, and $q$ are taken from Table I of San-Car-Nat:2023:PRD:.
• Figure 5: Same as in Fig. \ref{['Fig16']} within the Newtonian approximation, but with an increased magnetic field strength of $B = 2.2 \times 10^{-11}$ instead of $B = 1.2 \times 10^{-11}$ in Fig. \ref{['Fig16']}. The stronger field leads to occurence of the OW effect even in the presence of the tail term.