Particle motion near high-spin black holes

Abstract

General relativity predicts that the Kerr black hole develops qualitatively new and surprising features in the limit of maximal spin. Most strikingly, the region of spacetime near the event horizon stretches into an infinitely long throat and displays an emergent conformal symmetry. Understanding dynamics in this NHEK (Near-Horizon Extreme Kerr) geometry is necessary for connecting theory to upcoming astronomical observations of high-spin black holes. We review essential properties of NHEK and its relationship to the rapidly rotating Kerr black hole. We then completely solve the geodesic equation in the NHEK region and describe how the resulting trajectories transform under the action of its enhanced symmetries. In the process, we derive explicit expressions for the angular integrals appearing in the Kerr geodesic equation and obtain a useful formula, valid at arbitrary spin, for a particle's polar angle in terms of its radial motion. These results will aid in the analytic computation of astrophysical observables relevant to ongoing and future experiments.

Near-horizon geodesics of high spin black holes

“…Since the signs ± r , ± θ are identical to the signs of respectively dr and dθ, the integral (2.11) is monotonous around each turning point, as it should in order to define an increasing Mino time λ along the geodesic. It is the same notation used in [17]. Note thatT θ ,Φ θ are normalized to be vanishing for equatorial motion.…”

“…This section aims to describe in full details the classification of the polar geodesic motion around a Kerr black hole. We will complete the classification given in the recent analysis [17] to all geodesics including in particular = 0 1 . Our taxonomy is depicted in Figures 1 and 2.…”

“…We have 0 < z + ≤ 1 and θ therefore belongs to the interval θ + ≤ θ ≤ π − θ + . The polar integral can be written (see [17])…”

“…we denote again as m the number of turning points at Mino time λ. The polar integral can be written (see [17] and Appendix A of [18]):…”

“…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.…”