Gravitational waves emitted by a particle rotating around a Schwarzschild black hole: A semiclassical approach

We analyze the gravitational radiation emitted from a particle in circular motion around a Schwarzschild black hole using the framework of quantum field theory in curved spacetime at tree level. The gravitational perturbations are written in a gauge-invariant formalism for spherically symmetric spacetimes. We discuss the results, comparing them to the radiation emitted by a particle when it is assumed to be orbiting a massive object due to a Newtonian force in flat spacetime.

Figure 1

Total power emitted by the test particle rotating around the black hole, given by Eq. (72), plotted as a function of the angular velocity $\mathrm{\Omega }$. The summation in $l$ in Eq. (72) is truncated at a certain value of $l$, which we denote as $l_{\max }$.

Figure 2

Scalar-type power emitted by the test particle, given by Eq. (67), as a function of the angular velocity $\mathrm{\Omega }$. We show here the modes with $l=m$, which give the main contributions to the total emitted power.

Figure 3

Comparison between the $l=2$ scalar-type power given by Eq. (67) and the $l=2$ vector-type power emitted by the test particle given by Eq. (65), as a function of the angular velocity $\mathrm{\Omega }$.

Figure 4

Comparison between the contributions of the $l=2$ and $l>2$ modes (we considered contributions up to $l=20$). For most of the range of $\mathrm{\Omega }$ (up to $\mathrm{\Omega }\approx 0.164M^{-1}$), the $l=2$ mode is the dominant contribution.

Figure 5

Ratio $W_{\mathrm{em}}^{l=2}/W_{\mathrm{em}}$ between the emitted power of the $l=2$ modes and the total emitted power (we have considered contributions to the total emitted power up to $l=20$). The $l=2$ modes remain relevant contributions to the emitted power even for unstable orbits.

Figure 6

Ratio $W_{\mathrm{em}}^{\mathrm{obs}}/W_{\mathrm{em}}$ between the asymptotically observed and the total emitted power, as a function of $\mathrm{\Omega }$, plotted for stable orbits. We have considered contributions up to $l=20$. We see that almost all the energy is radiated away to infinity, in the case of stable orbits.

Figure 7

Ratio $W_{\mathrm{em}}^{\mathrm{obs}}/W_{\mathrm{em}}$ between the asymptotically observed and the total emitted power, as a function of $\mathrm{\Omega }$, plotted for stable as well as for unstable circular orbits. As in Fig. 6, we have considered contributions up to $l=20$.

Figure 8

Total power emitted for a given frequency ${\omega }_m$. Frequencies up to $m=30$ are shown in this plot. Since only gravitons with frequencies that are integer multiples of the particle’s angular velocity are emitted (${\omega }_m=m\mathrm{\Omega }$), the spectrum is discrete.

Figure 9

Emitted power for a given frequency ${\omega }_m$ in the high-frequency range ($100\le m\le 2500$).

Figure 10

Ratio $W_{\mathrm{em}}/W_{\mathrm{em}}^M$ as a function of $\mathrm{\Omega }$. We have considered contributions up to $l=20$. The maximum value considered for $\mathrm{\Omega }M$ is $(6\sqrt{6}{)}^{-1}$. As $\mathrm{\Omega }$ increases, the ratio decreases up to approximately 25%, until it starts to increase as a result of the ultrarelativistic motion.