Periapsis and gravitomagnetic precessions of stellar orbits in Kerr and Kerr–de Sitter black hole spacetimes

Abstract

The exact solution for the motion of a test particle in a non-spherical polar orbit around a Kerr black hole is derived. Exact novel expressions for frame dragging (Lense-Thirring effect), periapsis advance and the orbital period are produced. The resulting formulae, are expressed in terms of Appell's first hypergeometric function F1, Jacobi's amplitude function, and Appell's F1 and Gauß hypergeometric function respectively. The exact expression for frame dragging is applied for the calculation of the Lense-Thirring effect for the orbits of S-stars in the central arcsecond of our Galaxy assuming that the galactic centre is a Kerr black hole, for various values of the Kerr parameter including those supported by recent observations. In addition, we apply our solutions for the calculation of frame dragging and periapsis advance for stellar non-spherical polar orbits in regions of strong gravitational field close to the event horizon of the galactic black hole, e.g. for orbits in the central milliarcsecond of our galaxy. Such orbits are the target of the GRAVITY experiment. We provide examples with orbital periods in the range of 100min -54 days. Detection of such stellar orbits will allow the possibility of measuring the relativistic effect of periapsis advance with high precision at the strong field realm of general relativity. Further, an exact closed form formula for the orbital period of a test particle in a non-circular equatorial motion around a Kerr black hole is produced. We also derive exact expressions for the periapsis advance and the orbital period for a test particle in a non-circular equatorial motion in the Kerr field in the presence of the cosmological constant in terms of Lauricella's fourth hypergeometric function FD.

Optical Phenomena in the Field of Braneworld Kerr Black Holes

“…We have shown that these phenomena could be measured and used to put limits on the tidal charge in case of Galaxy Center Sgr A * supermassive black hole. We generalized the approaches based on the transfer-function method as introduced and developed in Schwarzchild and Kerr backgrounds 16,27,5,42,25,14,17,32 where equations of photon motion are solved in terms of the elliptic integrals (see 32,23,24 ). For purposes of the present work, the transfer-function method seems to be most efficient.…”

Measuring Distance and Properties of the Milky Way’s Central Supermassive Black Hole with Stellar Orbits

“…Ever since the discovery of fast-moving (v > 1000 km s À1 ) stars within 0:3 00 (0.01 pc) of our Galaxy's central supermassive black hole Ghez et al 1998), the prospect of using stellar orbits to make precision measurements of the black hole's mass (M bh ) and kinematics, the distance to the Galactic center (R 0 ), and, more ambitiously, to measure postNewtonian effects has been anticipated (Jaroszyński 1998(Jaroszyński , 1999Salim & Gould 1999;Fragile & Mathews 2000;Rubilar & Eckart 2001;Weinberg et al 2005;Zucker & Alexander 2007;Kraniotis 2007;). An accurate measurement of the Galaxy's central black hole mass is useful for putting the Milky Way in context with other galaxies through the apparent relationship between the mass of the central black hole and the velocity dispersion, , of the host galaxy (e.g., Ferrarese & Merrit 2000;Gebhardt et al 2000;Tremaine et al 2002).…”

Photon motion in Kerr–de Sitter spacetimes

“…The motion of massive test particles was studied in [2,19,20,29,31,32,[35][36][37]48,51,59,66,67,72,74,86]. The KdS geometry can be relevant also for the so-called Kerr superspinars representing an alternative to black holes [14,26,27,83], breaking the blackhole bound on the dimensionless spin and exhibiting a variety of unusual physical phenomena [21,22,30,64,76,80,83,84].…”