Self-Consistent Orbital Evolution of a Particle around a Schwarzschild Black Hole

The motion of a charged particle is influenced by the self-force arising from the particle’s interaction with its own field. In a curved spacetime, this self-force depends on the entire past history of the particle and is difficult to evaluate. As a result, all existing self-force evaluations in curved spacetime are for particles moving along a fixed trajectory. Here, for the first time, we overcome this long-standing limitation and present fully self-consistent orbits and waveforms of a scalar charged particle around a Schwarzschild black hole.

Figure 1

Orbits for neutral and charged particles starting at $p=7.2$, $e=0.5$. The orbital evolution is started close to apastron (at $t=t_1$), and the dots represent events at times $t_1=400M$, $t_2=600M$, $t_3=1100M$, $t_4=1300M$, $t_5=1800M$, and $t_6=2043.8M$, the instant of plunge. The coordinates $\{x,y\}=\{r\cos \varphi ,r\sin \varphi \}$ are Cartesian coordinates in the equatorial plane.

Figure 2

Waveforms from self-consistently evolved orbits as detected by an observer located in the orbital plane at ${\mathcal{I}}^{+}$.

Figure 3

Orbital evolution in $p\mathrm{\text{−}}e$ space, for an orbit that begins with $p=7.2$, $e=0.5$. Each oscillation in these tracks corresponds to one full radial cycle.

Figure 4

Positional element $w$ interpreted as a measure of the periapsis shift relative to its geodesic value. The curves end when the particle crosses the separatrix.

Figure 5

Acceleration along a trajectory that starts at $p=7.2$, $e=0.5$ for $q=1/32$. The blue curve indicates $M{a}^r$, the red curve $M^2{a}^{\varphi }$, and the gray curve $\frac{dm}{d\tau }$. The bottom plot shows the corresponding radial position of the particle.