Analytic solutions of the geodesic equation in higher dimensional static spherically symmetric spacetimes

2008/12/12 by Eva Hackmann, Valeria Kagramanova, Jutta Kunz +2 · 120 citations
Mathematics · Physics and Astronomy · #Angular momentum #Astrophysical Phenomena and Observations #Black Holes and Theoretical Physics #Classical mechanics #Cosmology and Gravitation Theories #De Sitter universe #Geodesic #Gravitation #Mathematical analysis #Mathematical physics #Mathematics #Physics #Quantum mechanics #Schwarzschild radius #gr-qc

paper · pdf · doi:10.1103/physrevd.78.124018

published in Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields 78(12) (American Physical Society) · accepted for publication in PRD

arxiv created 2008/12/12 · openalex publication_date 2008/12/23 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

The complete analytical solutions of the geodesic equation of massive test particles in higher dimensional Schwarzschild, Schwarzschild--(anti)de Sitter, Reissner-Nordstr"om and Reissner--Nordstr"om--(anti)de Sitter spacetimes are presented. Using the Jacobi inversion problem restricted to the theta divisor the explicit solution is given in terms of Kleinian sigma functions. The derived orbits depend on the structure of the roots of the characteristic polynomials which depend on the particle's energy and angular momentum, on the mass and the charge of the gravitational source, and the cosmological constant. We discuss the general structure of the orbits and show that due to the specific dimension-independent form of the angular momentum and the cosmological force a rich variety of orbits can emerge only in four and five dimensions. We present explicit analytical solutions for orbits up to 11 dimensions. A particular feature of Reissner-Nordstr"om spacetimes is that bound and escape orbits traverse through different universes.

Citations

• Charged–rotating black holes and black strings in higher dimensional Einstein–Maxwell theory with a positive cosmological constant
• Schwarzschild field inn dimensions and the dimensionality of space problem
• Hidden symmetries of higher-dimensional black holes and uniqueness of the Kerr-NUT-(A)dS spacetime
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• Charged Rotating Black Holes in Odd Dimensions
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• Large Mass Hierarchy from a Small Extra Dimension
• General Kerr–NUT–AdS metrics in all dimensions
• An Alternative to Compactification
• The Hierarchy Problem and New Dimensions at a Millimeter
• Classical black hole production in high-energy collisions
• The Large-N Limit of Superconformal Field Theories and Supergravity
• Rotating Black Holes in Higher Dimensions
• Considerationes generales de transcendentibus Abelianis.
• The large N limit of superconformal field theories and supergravity

Cited by

• McGehee regularization of general SO(3)-invariant potentials and applications to stationary and spherically symmetric spacetimes
• Analysis of accretion disk around a black hole in dRGT massive gravity
• Study of motion around a static black hole in Einstein and Lovelock gravity
• Motion of the charged test particles in Kerr-Newman-Taub-NUT spacetime and analytical solutions