Kerr Geodesics in Terms of Weierstrass Elliptic Functions

Abstract

We derive novel analytical solutions describing timelike and null geodesics in the Kerr spacetime. The solutions are parameterized explicitly by constants of motion -- the energy, the angular momentum, and the Carter constant -- and initial coordinates. A single set of formulas is valid for all null and timelike geodesics, irrespectively of their radial and polar type. This uniformity has been achieved by applying a little-known result due to Biermann and Weierstrass, regarding solutions of a certain class of ordinary differential equations. Different from other expressions in terms of Weierstrass functions, our solution is explicitly real for all types of geodesics. In particular, for the first time the so-called transit orbits are now expressed by explicitly real Weierstrass functions.

38 pages, 11 figures. Corrected typos in eqs. (C15) and (C19)

References in corpus (15)

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• Exact Formulas for Spherical Photon Orbits Around Kerr Black Holes
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Cited by in corpus (16)

• Analytic Solution for the Motion of Spinning Particles in Kerr Space-Time
• Strong deflection of massive particles in spherically symmetric spacetimes
• Geodesic Motion of Test Particles around the Scalar Hairy Black Holes with Asymmetric Vacua
• Accretion of a Vlasov gas by a Kerr black hole
• Types of the geodesic motions in Kerr-Sen-AdS${}_4$ spacetime
• Spinoptics in the Kerr Spacetime: Polarized Wave Scattering
• Gravitational Lensing in the Kerr Spacetime: An Analytic Approach for Light and High-Frequency Gravitational Waves
• Kerr black hole in the formalism of isolated horizons
• The Repulsive Effect of Covariant Effective Quantum Gravity
• Revisiting Einstein's analogy: black holes as gradient-index lenses
• Massive particle surfaces and black hole shadows from intrinsic curvature
• Coupling constant metamorphosis, Fermat principle and light propagation in Kerr metric
• Revisiting critical orbits of test particles traveling in a black hole background
• Kerr isolated horizon revisited: Caustic-free congruence and adapted tetrad
• Null integrability and photon phenomenology in the accelerated Schwarzschild black hole
• Particles with precessing spin in Kerr spacetime: analytic solutions for eccentric orbits and homoclinic motion near the equatorial plane