The evolution of circular, non-equatorial orbits of Kerr black holes due to gravitational-wave emission

TL;DR

This study analyzes the radiative evolution of circular, non-equatorial Kerr orbits in the extreme mass-ratio limit, showing that the Carter constant can be inferred from gravitational-wave fluxes when orbits remain circular under adiabatic evolution. Using the Sasaki-Nakamura–Teukolsky formalism, it computes the energy and angular-momentum fluxes to infinity and into the horizon, and derives how the inclination angle and radius evolve from these fluxes. The results reveal that inclined circular orbits tend to increase their inclination in the strong field, with post-Newtonian predictions significantly underestimating the rate in some cases, and that waveforms become increasingly rich in harmonics as the black hole spin grows. The work underscores both the potential for flux-based evolution in this special case and the need for instantaneous radiation-reaction forces for broader applicability, with implications for modeling EMRIs for LISA and for interpreting strong-field Kerr spacetimes.

Abstract

A major focus of much current research in gravitation theory is on understanding how radiation reaction drives the evolution of a binary system, particularly in the extreme mass ratio limit. Such research is of direct relevance to gravitational-wave sources for space-based detectors (such as LISA). We present here a study of the radiative evolution of circular (i.e., constant Boyer-Lindquist coordinate radius), non-equatorial Kerr black hole orbits. Recent theorems have shown that, at least in an adiabatic evolution, such orbits evolve from one circular configuration into another, changing only their radius and inclination angle. This constrains the system's evolution in such a way that the change in its Carter constant can be deduced from knowledge of gravitational wave fluxes propagating to infinity and down the black hole's horizon. Thus, in this particular case, a local radiation reaction force is not needed. In accordance with post-Newtonian weak-field predictions, we find that inclined orbits radiatively evolve to larger inclination angles (although the post-Newtonian prediction overestimates the rate of this evolution in the strong field by a factor $\lesssim 3$). We also find that the gravitational waveforms emitted by these orbits are rather complicated, particularly when the hole is rapidly spinning, as the radiation is influenced by many harmonics of the orbital frequencies.

Figures (11)

• Figure 1: Comparison of the flux down the event horizon for orbits at $r = 25 M$ as a function of black hole spin $a/M$ for $l = 3$ modes. Agreement between the numerical and post-Newtonian fluxes is quite good, except for $m=2$; this is because the post-Newtonian expansions contain many terms, and usually are quite robust. The case $m=2$ is an example where the expansion is not as robust. The interesting upturn in the down-horizon flux for $m = 1$ and $a \ge 0.5 M$ is due to superradiant scattering --- some incoming radiation gets scattered by the black hole's ergosphere out to infinity.
• Figure 2: Typical result of validation test for inclined, Schwarzschild orbits. The dotted line is the modulus squared of the Wigner D-function for $l = 4$, $m = 2$, $k = 1$ as a function of inclination angle $\iota$; the large black points are the ratio ${\dot E}_{lmk}(\iota)/{\dot E}^{\rm eq}_{lm}$. The numerical data for the fluxes agrees with the analytical formula for the D-function to within $10^{-6} - 10^{-7}$. (For this plot, the numerical fluxes are evaluated at infinity; the results are identical to within the error when examining fluxes down the horizon. Also, these results are invariant --- within the error bounds --- as a function of orbital radius; this plot is generated for a particle orbiting at $r = 15 M$.)
• Figure 3: The gravitational waveform produced by orbits with $r = 7 M$, $\iota = 62.43^\circ$ about a black hole with $a = 0.95 M$. The observer is in the hole's equatorial plane, $\theta = 90^\circ$. The distance to the source is $D$. Notice that there are many sharp features in this waveform, indicating the strong presence of relatively large harmonics of the fundamental frequencies $\Omega_\phi$ and $\Omega_\theta$. This is consistent with the rather broad emission spectra produced by this orbit (cf. Fig. \ref{['fig:dEinfdt_r7_a0.95_l2']}). The low frequency modulation is due to Lense-Thirring precession ( i.e., the precession of the orbital plane due to dragging of inertial frames by the black hole's spin).
• Figure 4: The spectrum of energy for $l = 2$ modes radiated to infinity by orbits with $r = 7 M$, $\iota = 62.43^\circ$ about a black hole with $a = 0.95 M$. Of particular note in this case is that the distribution is rather broad with respect to $k$. This is primarily due to the fact that for very large spin, the Teukolsky potential [cf. Eq. (\ref{['eq:teukpotential']})] is fairly transmissive to high frequency modes. Notice that it is more transmissive to corotating modes ($m \omega > 0$) than it is to counterrotating modes: corotating modes are more readily scattered by the hole.
• Figure 5: The spectrum of energy radiated down the event horizon for $l = 2$ modes for orbits with $r = 7 M$, $\iota = 62.43^\circ$ about a black hole with $a = 0.95 M$. The distribution is, for the most part, sharply negative (particularly for corotating modes, $m\omega > 0$), indicating superradiant scattering. This is essentially a manifestation of the Penrose process --- radiation extracts energy from the black hole's ergosphere.