Near-horizon geodesics of high spin black holes

Abstract

We provide an exhaustive and illustrated classification of timelike and null geodesics in the near-horizon region of near-extremal Kerr black holes. The classification of polar motion extends to Kerr black holes of arbitrary spin. The classification of radial motion leads to a simple parametrization of the separatrix between bound and unbound motion. Furthermore, we prove that each timelike or null geodesic is related via conformal transformations and discrete symmetries to spherical orbits and we provide the explicit mappings. We detail the high spin behavior of both the innermost stable and the innermost bound spherical orbits.

Photon Emission from Circular Equatorial Kerr Orbiters

“…The quantities (µ, E, L, C) are conserved along the trajectory, ensuring that NHEK geodesic motion is completely integrable. The NHEK geodesic equation was only recently solved analytically in complete generality [17,44].…”

Classification of radial Kerr geodesic motion

“…The decoupling of radial and polar motion was accomplished by Mino using what is now called Mino time [29]. The geodesics in the near-horizon region of high-spin Kerr were analyzed in [30][31][32][33][34][35][36][37][38][39][40][41][42][43]. Part of the complete separatrix as defined below, namely the separatrix between plunging and bounded orbits, was reduced to a fourth order polynomial in terms of semi-latus rectum and eccentricity [44,45] and was further described in [46][47][48].…”

“…In the last two years, several novel analytic results on Kerr geodesics were achieved [40,41,[51][52][53][54][55][56][57][58][59][60]. Explicitly real, fully explicit, "initial data-dependent" analytical solutions in terms of elliptic functions were given for (i) radial and polar motion for timelike bounded orbits [52]; (ii) generic (i.e.…”

“…Explicitly real, fully explicit, "initial data-dependent" analytical solutions in terms of elliptic functions were given for (i) radial and polar motion for timelike bounded orbits [52]; (ii) generic (i.e. excluding zero measure sets) polar motion for null or timelike orbits [40,41]; (iii) generic radial motion for null orbits [53] and (iv) general (i.e. including zero measure sets) radial near-horizon motion in the high-spin limit [40,41].…”

“…excluding zero measure sets) polar motion for null or timelike orbits [40,41]; (iii) generic radial motion for null orbits [53] and (iv) general (i.e. including zero measure sets) radial near-horizon motion in the high-spin limit [40,41]. The only missing piece of information in order to complete such a state-of-the-art analytical standard for all Kerr orbits is the non-generic polar motion, and, general (generic and non-generic) radial motion for timelike geodesics, which is the main interest of this paper.…”