Location of the last stable orbit in Kerr spacetime

Abstract

Black hole spacetimes, like the Kerr spacetime, admit both stable and plunging orbits, separated in parameter space by the separatrix. Determining the location of the separatrix is of fundamental interest in understanding black holes, and is of crucial importance for modeling extreme mass-ratio inspirals. Previous numerical approaches to locating the Kerr separatrix were not always efficient or stable across all of parameter space. In this paper we show that the Kerr separatrix is the zero set of a single polynomial in parameter space. This gives two main results. First, we thoroughly analyze special cases (extreme Kerr, polar orbits, etc.), finding strict bounds on the limits of roots, and unifying a number of results in the literature. Second, we pose a stable numerical method which is guaranteed to quickly and robustly converge to the separatrix. This new approach is implemented in the Black Hole Perturbation Toolkit, and results in a ∼ 45× speedup over the prior robust approach. II. TIME-LIKE GEODESICS IN KERR SPACETIMEGiven any spacetime, let us denote the trajectory of a timelike (non-spinning) test body of mass µ by a curve x α (τ ) where τ is the proper time as measured along the world line. The four-velocity of the body is given by arXiv:1912.07609v1 [gr-qc]

Near-horizon geodesics of high spin black holes

“…This critical angle was also previously obtained in [20,41]. Finally note that spherical photon orbits in the high spin limit were also discussed in [51,52].…”

“…The study of timelike and null geodesics of the Kerr metric has a long history which is still ongoing [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. There are at least two motivations for studying Kerr geodesics.…”

FastEMRIWaveforms: New tools for millihertz gravitational-wave data analysis

“…With the trajectories and amplitudes determined according to the following sections, this information was combined with the spin-weighted spherical harmonics and then put through mode selection, both of which are described in Section III D. Following mode selection, the waveform is built using the interpolated summation (see Section III C). This eccentric Schwarzschild adiabatic model is valid for p min ≤ p ≤ p s + 10 and 0 ≤ e ≤ 0.7, where p s is the separatrix [54] and p min = max(p s + 0.1, 7p s − 41.9).…”

“…In order to efficiently interpolate the fluxes with bicubic splines, it is helpful to place place the data on a grid with uniform spacing. As in Schwarzschild spacetime the separatrix is given by p s = 6 + 2e [54], we instead introduce u = ln(p − p s + 3.9) [36]. This allows us to build a uniform grid in (u, e) space with 1.37 ≤ u ≤ 3.82 in steps of 0.05 and 0.0 ≤ e ≤ 0.8 in steps of 0.025.…”

Rapid Generation of Fully Relativistic Extreme-Mass-Ratio-Inspiral Waveform Templates for LISA Data Analysis

“…It is useful to parametrize the system by an equivalent set of quasi-Keplerian orbital elements: the semilatus rectum (henceforth "separation") p and eccentricity e, with [32]. In this parametrization, stable bound orbits exist for p > p s ¼ 6þ2e and 0 ≤ e < 1, where p s denotes the separatrix [33]. Each instantaneous orbit ðp; eÞ is associated with a radial and azimuthal frequency, denoted by Ω r and Ω φ , respectively.…”