Gravitational radiation from radial infall of a particle into a Schwarzschild black hole: A numerical study of the spectra, quasinormal modes, and power-law tails

The computation of the gravitational radiation emitted by a particle falling into a Schwarzschild black hole is a classic problem that was already studied in the 1970s. Here we present a detailed numerical analysis of the case of radial infall starting at infinity with no initial velocity. We compute the radiated waveforms, spectra, and energies for multipoles up to $l=6$, improving significantly on the numerical accuracy of existing results. This is done by integrating the Zerilli equation in the frequency domain using the Green’s function method. The resulting wave exhibits a “ring-down” phase whose dominant contribution is a superposition of the quasinormal modes of the black hole. The numerical accuracy allows us to recover the frequencies of these modes through a fit of that part of the wave. Comparing with direct computations of the quasinormal modes, we reach a $\sim {10}^{-4}$ to $\sim {10}^{-2}$ accuracy for the first two overtones of each multipole. Our numerical accuracy also allows us to display the power-law tail that the wave develops after the ring-down has been exponentially cut off. The amplitude of this contribution is $\sim {10}^2$ to $\sim {10}^3$ times smaller than the typical scale of the wave.

Figure 1

Top panels: the energy spectra $1/m^{2}d{E}_l/d\omega$ on linear and logarithmic scales. Middle panels: The amplitude spectra, modulus (logarithmic scale), and phase. Bottom panels: low frequency asymptotic behavior of the phases and energy spectra.

Figure 2

Top left panel: the amplitude $A_{2,\mathrm{out}}/Mm$. Top right panel: the waveform ${\psi }_{2,\mathrm{out}}/m$. Bottom left panel: the curve corresponding to the first mode of the QNM contribution. Bottom right panel: the power-law tail of ${\psi }_{2,\mathrm{out}}/m$ compared to the QNM contribution.

Figure 3

Top left panel: the amplitude $A_{3,\mathrm{out}}/Mm$. Top right panel: the waveform ${\psi }_{3,\mathrm{out}}/m$. Bottom left panel: the curves corresponding to the QNM contribution up to the second overtone $n=2$. Bottom right panel: the power-law tail of ${\psi }_{3,\mathrm{out}}/m$ compared to the QNM contribution.

Figure 4

Top left panel: the amplitude $A_{4,\mathrm{out}}/Mm$. Top right panel: the waveform ${\psi }_{4,\mathrm{out}}/m$. Bottom left panel: the curves corresponding to the QNM contribution up to the second overtone $n=2$. Bottom right panel: the power-law tail of ${\psi }_{4,\mathrm{out}}/m$ compared to the QNM contribution.

Figure 5

Top left panel: the amplitude $A_{5,\mathrm{out}}/Mm$. Top right panel: the waveform ${\psi }_{5,\mathrm{out}}/m$. Bottom panel: the curves corresponding to the QNM contribution up to the second overtone $n=2$.

Figure 6

Top left panel: the amplitude $A_{6,\mathrm{out}}/Mm$. Top right panel: the waveform ${\psi }_{6,\mathrm{out}}/m$. Bottom panel: the curves corresponding to the QNM contribution up to the second overtone $n=2$.

Figure 7

Top panel: the QNMs computed through the waveforms fit. The dotted lines’ intersections are the expected values. Vertical lines link QNMs of the same $l$ while horizontal lines link QNMs with the same $n$. $n$ increases by going downwards. Middle panels: the amplitudes $B_{l,n}/Mm$ and phases ${\theta }_{l,n}$ of the first two overtones. Bottom left panel: example of the 3D plot obtained when fitting the waveform for QNMs on all intervals $[a,b]$ in [0, 30] of $u/2M$. Bottom right panel: the corresponding histogram giving the number of points in each horizontal slice of the 3D plot.