Gravitational waveforms from a point particle orbiting a Schwarzschild black hole

TL;DR

The paper develops a time-domain perturbative framework to compute gravitational waves from a point particle of mass $\mu$ orbiting a Schwarzschild black hole of mass $M$ by solving the inhomogeneous Zerilli-Moncrief and Regge-Wheeler equations with a delta-function source. It emits waveforms and energy/angular momentum fluxes to infinity and through the horizon for circular, eccentric, and parabolic orbits, and demonstrates that horizon absorption is typically a small correction (usually $<1\%$, rising for small periastra). The authors validate the time-domain results against frequency-domain calculations for circular orbits with sub-percent agreement and show the method efficiently handles highly eccentric motion where frequency-domain sums would be costly. The work provides a robust, generalizable approach for modeling gravitational radiation in extreme-mass-ratio encounters, with clear implications for capture-rate estimates and potential extension to rotating black holes.

Abstract

We numerically solve the inhomogeneous Zerilli-Moncrief and Regge-Wheeler equations in the time domain. We obtain the gravitational waveforms produced by a point-particle of mass $μ$ traveling around a Schwarzschild black hole of mass M on arbitrary bound and unbound orbits. Fluxes of energy and angular momentum at infinity and the event horizon are also calculated. Results for circular orbits, selected cases of eccentric orbits, and parabolic orbits are presented. The numerical results from the time-domain code indicate that, for all three types of orbital motion, black hole absorption contributes less than 1% of the total flux, so long as the orbital radius r_p(t) satisfies r_p(t)> 5M at all times.

Figures (13)

• Figure 1: In the left panel, we display the trajectories in the $x_{p}$-$y_{p}$ plane for a geodesic with $e=1$, and $p=8.001$. For this choice of parameters, the particle orbits the black hole approximately four times before leaving the central region. In the right panel, we display a $e=0.9$ and $p=7.8001$ geodesic. When the particle reaches the periastron, it orbits the black hole on a quasi-circular orbit for approximately six cycles. In both cases, the exact number of cycles is given by Eq. (\ref{['eqn:Ncycle']}).
• Figure 2: The dominant radiation modes for the Zerilli-Moncrief (left, $l=2$ and $m=2$) and Regge-Wheeler (right, $l=2$ and $m=1$) functions for a particle orbiting the black hole at $r_{p}=12M$. At early times, the waveforms are dominated by the initial data content. We calculate the energy and angular momentum fluxes after a time $t/(2M)=350.0$
• Figure 3: In the left panel, we display $c_{E}(FD)$, as well as $c_{E}(TD)$ and $c_{L}(TD)$, as functions of $p$. Both $c_{E}$ and $c_{L}$ slowly approach 1 from below for large $p$. For small values of $p$, the coefficients approach 1.15 as $p$ approaches 6. In the right panel, we display the residuals $R_{E}$ and $R_{L}$ as defined in the text. Using the time-domain method, the fluxes are calculated accurately to $0.7\%$ for $p=6.0001$, and to $0.2\%$ for large values of $p$.
• Figure 4: We display the energy and angular momentum fluxes through the event horizon normalized by the fluxes in the radiation zone. Even for highly relativistic motion, the horizon fluxes contribute less than $0.4\%$ of the total fluxes. For circular orbits, the theoretical prediction is that $\dot{E}^{\rm eh}/\dot{E}^{\infty}=\dot{L}^{\rm eh}/\dot{L}^{\infty}$. Numerically, this relation is only approximate, but nevertheless the two curves are indistinguishable. The right panel displays these ratios normalized by $(r_{p}/M)^{-4}$, the weak-field and slow-motion approximation.
• Figure 5: The Zerilli-Moncrief (left, $l=2, m=2$) and Regge-Wheeler (right, $l=2, m=1$) functions for $p=12$ and $e=0.2$. As in the case of circular orbits, early times are dominated by the initial data content.