Waveforms produced by a scalar point particle plunging into a Schwarzschild black hole: Excitation of quasinormal modes and quasibound states

With the possibility of testing massive gravity in the context of black hole physics in mind, we consider the radiation produced by a particle plunging from slightly below the innermost stable circular orbit into a Schwarzschild black hole. In order to circumvent the difficulties associated with black hole perturbation theory in massive gravity, we use a toy model in which we replace the graviton field with a massive scalar field and consider a linear coupling between the particle and this field. We compute the waveform generated by the plunging particle and study its spectral content. This permits us to highlight and interpret some important effects occurring in the plunge regime which are not present for massless fields, such as (i) the decreasing and vanishing, as the mass parameter increases, of the signal amplitude generated when the particle moves on quasicircular orbits near the innermost stable circular orbit; and (ii) in addition to the excitation of the quasinormal modes, the excitation of the quasibound states of the black hole.

Figure 1

The plunge trajectory obtained from Eq. (30). Here, we assume that the particle starts at $r=r_{\mathrm{ISCO}}(1-\epsilon )$ with $\epsilon ={10}^{-2}$, 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

Quadrupolar waveforms produced by the plunging particle. The results are obtained for a massless scalar field $(\tilde{\alpha }=0)$, and the observer is located at (a) $r=10M$, (b) $r=20M$, and (c) $r=50M$.

Figure 3

Quadrupolar waveforms produced by the plunging particle. The results are obtained for a massive scalar field ($\tilde{\alpha }=0.25$ is below the threshold ${\tilde{\alpha }}_c\approx 0.2722$), and the observer is located at (a) $r=10M$, (b) $r=20M$, and (c) $r=50M$.

Figure 4

Quadrupolar waveforms produced by the plunging particle. The results are obtained for a massive scalar field ($\tilde{\alpha }=0.35$ is above the threshold ${\tilde{\alpha }}_c\approx 0.2722$), and the observer is located at (a) $r=10M$, (b) $r=20M$, and (c) $r=50M$.

Figure 5

Quadrupolar waveforms produced by the plunging particle (blue line) and by a particle orbiting the BH on the ISCO (red dashed line). The reduced mass $(\tilde{\alpha }=0.25)$ is below the threshold $({\tilde{\alpha }}_c\approx 0.2722)$.

Figure 6

Quadrupolar waveforms produced by the plunging particle (blue line) and by a particle orbiting the BH on the ISCO (red dashed line). The reduced mass $(\tilde{\alpha }=0.35)$ is above the threshold $({\tilde{\alpha }}_c\approx 0.2722)$.

Figure 7

Comparison of the quadrupolar waveform produced by the plunging particle (blue line) and the quadrupolar quasinormal waveform (red dashed line). The results are obtained for an observer at $r=10M$.

Figure 8

Comparison of the quadrupolar waveform produced by the plunging particle (blue line) and the quadrupolar quasinormal waveform (red dashed line). The results are obtained for an observer at $r=50M$.

Figure 9

Spectral content of the adiabatic phase of the quadrupolar waveform produced by the plunging particle. The results are obtained for a massive scalar field ($\tilde{\alpha }=0.25$ is below the threshold ${\tilde{\alpha }}_c\approx 0.2722$), and the observer is located at (a) $r=20M$ and (b) $r=50M$. We observe the signature of the quasicircular motion of the plunging particle.

Figure 10

Spectral content of the adiabatic phase of the quadrupolar waveform produced by the plunging particle. The results are obtained for a massive scalar field ($\tilde{\alpha }=0.35$ is above the threshold ${\tilde{\alpha }}_c\approx 0.2722$), and the observer is located at (a) $r=20M$ and (b) $r=50M$. We observe, in addition to the signature of the quasicircular motion of the plunging particle, that of the first long-lived QBS.

Figure 11

Spectral content of the late-time phase of the quadrupolar waveform produced by the plunging particle. The results are obtained for a massive scalar field ($\tilde{\alpha }=0.25$ is below the threshold ${\tilde{\alpha }}_c\approx 0.2722$), and the observer is located at (a) $r=20M$ and (b) $r=50M$. We observe the signature of the first long-lived QBS.

Figure 12

Spectral content of the late-time phase of the quadrupolar waveform produced by the plunging particle. The results are obtained for a massive scalar field ($\tilde{\alpha }=0.35$ is above the threshold ${\tilde{\alpha }}_c\approx 0.2722$), and the observer is located at (a) $r=20M$ and (b) $r=50M$. We observe the signature of the first long-lived QBSs.

Figure 13

Waveform amplitudes for a particle on the ISCO and reduced masses in the range $\tilde{\alpha }\in [0,2]$. The results are obtained from (a7) for the particular $(\ell =2,m=2)$ mode. We study the role of the location of the observer. For an observer at a large distance from the BH, the transition between the propagating (but dispersive) and evanescent behaviors of the response occurs at ${\tilde{\alpha }}_c\approx 0.2722$.

Figure 14

Waveform amplitudes for a particle on the ISCO and reduced masses in the range $\tilde{\alpha }\in [0,2]$. The results are obtained from (a7) for some $(\ell ,m=\ell )$ modes and an observer at (a) $r=10M$, (b) $r=50M$, and (c) $r=100M$. The transition between the propagating (but dispersive) and evanescent behaviors of the response occurs at ${\tilde{\alpha }}_c\approx 0.2722$ for the $(\ell =2,m=2)$ mode, at ${\tilde{\alpha }}_c\approx 0.4082$ for the $(\ell =3,m=3)$ mode, and at ${\tilde{\alpha }}_c\approx 0.5443$ for the $(\ell =4,m=4)$ mode.

Figure 15

Waveform amplitudes for a particle on the ISCO and reduced masses in the range $\tilde{\alpha }\in [0,2]$. The results are obtained from (a7) for modes with angular momentum $\ell =4$. We study the influence of the azimuthal number $m$ for an observer at (a) $r=10M$ and (b) $r=100M$.

Figure 16

The two Riemann sheets of the function $p(\omega )=({\omega }^2-{\mu }^2{)}^{1/2}$ and the branch cut we have chosen. We have introduced the polar coordinates $({\rho }_1,{\theta }_1)$ and $({\rho }_2,{\theta }_2)$ such that $\omega -\mu ={\rho }_1{e}^{i{\theta }_1}$ and $\omega +\mu ={\rho }_2{e}^{i{\theta }_2}$. Here $0\le {\theta }_1,{\theta }_2<2\pi$.