Electromagnetic radiation generated by a charged particle plunging into a Schwarzschild black hole: Multipolar waveforms and ringdowns

Electromagnetic radiation emitted by a charged particle plunging from slightly below the innermost stable circular orbit into a Schwarzschild black hole is examined. Both even- and odd-parity electromagnetic perturbations are considered. They are described by using gauge invariant master functions and the regularized multipolar waveforms as well as their unregularized counterparts constructed from the quasinormal-mode spectrum are obtained for arbitrary directions of observation and, in particular, outside the orbital plane of the plunging particle. They are in excellent agreement and the results especially emphasize the impact of higher harmonics on the distortion of the waveforms.

Figure 1

The plunge trajectory obtained from Eq. (23). Here, we assume that the particle starts at $r=r_{\mathrm{ISCO}}(1-\epsilon )$ with $\epsilon ={10}^{-3}$ and we take ${\phi }_0=0$. The red dashed line at $r=6M$ and the red dot-dashed line at $r=2M$ represent the ISCO and the horizon, respectively.

Figure 2

Components $E_{\theta /\phi }^{(e/o)}$ of the electric field observed at infinity in the direction ($\theta =\pi /3$, $\phi =0$) above the orbital plane of the plunging particle. We study the influence of the number of modes on the distortion of the waveforms.

Figure 3

Electric field $E_{\theta }^{(e)}$ observed at infinity for various directions above the orbital plane of the plunging particle. We consider $\phi =0$ and we study the distortion of the multipolar waveform and of the associated quasinormal ringdown when $\theta$ varies between $-\pi /2$ and $+\pi /2$. We note that $E_{\theta }^{(e)}$ vanishes for $\theta =\pm \pi /2$ and that, for $\theta =0$, only the ($\ell =1$, $m=\pm 1$) modes contribute to the signal.

Figure 4

Electric field $E_{\phi }^{(e)}$ observed at infinity for various directions above the orbital plane of the plunging particle. We consider $\phi =0$ and we study the distortion of the multipolar waveform and of the associated quasinormal ringdown when $\theta$ varies between $-\pi /2$ and $+\pi /2$. We note that $E_{\phi }^{(e)}$ vanishes for $\theta =0$.

Figure 5

Electric field $E_{\theta }^{(o)}$ observed at infinity for various directions above the orbital plane of the plunging particle. We consider $\phi =0$ and we study the distortion of the multipolar waveform and of the associated quasinormal ringdown when $\theta$ varies between $-\pi /2$ and $+\pi /2$. We note that $E_{\theta }^{(o)}$ vanishes for $\theta =\pm \pi /2$ and that, for $\theta =0$, only the ($\ell =2$, $m=\pm 1$) modes contribute to the signal.

Figure 6

Electric field $E_{\phi }^{(o)}$ observed at infinity for various directions above the orbital plane of the plunging particle. We consider $\phi =0$ and we study the distortion of the multipolar waveform and of the associated quasinormal ringdown when $\theta$ varies between $-\pi /2$ and $+\pi /2$. We note that, for $\theta =0$, only the ($\ell =2$, $m=\pm 1$) modes contribute to the signal.

Figure 7

Semilog graphs of some multipolar waveforms showing the dominance of the quasinormal ringing at intermediate times and the agreement of the regularized waveforms with the unregularized quasinormal responses.