Geodesic equation in Schwarzschild--(anti-)de Sitter space--times: Analytical solutions and applications

Abstract

The complete set of analytic solutions of the geodesic equation in a Schwarzschild--(anti-)de Sitter space--time is presented. The solutions are derived from the Jacobi inversion problem restricted to the set of zeros of the theta function, called the theta divisor. In its final form the solutions can be expressed in terms of derivatives of Kleinian sigma functions. The different types of the resulting orbits are characterized in terms of the conserved energy and angular momentum as well as the cosmological constant. Using the analytical solution, the question whether the cosmological constant could be a cause of the Pioneer Anomaly is addressed. The periastron shift and its post--Schwarzschild limit is derived. The developed method can also be applied to the geodesic equation in higher dimensional Schwarzschild space--times.

arXiv comment23 pages, 12 figures

References in corpus (18)

• Tests of general relativity from timing the double pulsar
• Study of the anomalous acceleration of Pioneer 10 and 11
• A massive binary black-hole system in OJ287 and a test of general relativity
• General Non-Extremal Rotating Black Holes in Minimal Five-Dimensional Gauged Supergravity
• The Contribution of the Cosmological Constant to the Relativistic Bending of Light Revisited
• The geodesic structure of the Schwarzschild Anti-de Sitter black hole
• Complete Integrability of Geodesic Motion in General Kerr-NUT-AdS Spacetimes
• Complete analytic solution of the geodesic equation in Schwarzschild--(anti) de Sitter space--times
• Precise relativistic orbits in Kerr and Kerr-(anti) de Sitter spacetimes
• Standard Clocks, Orbital Precession and the Cosmological Constant
• Compact calculation of the Perihelion Precession of Mercury in General Relativity, the Cosmological Constant and Jacobi's Inversion problem
• Scales Set by the Cosmological Constant
• Separability of the Hamilton-Jacobi and Klein-Gordon Equations in Kerr-de Sitter Metrics
• Two-body problem with the cosmological constant and observational constraints
• Using Early Data to Illuminate the Pioneer Anomaly
• Integrability and the Kerr-(A)dS black hole in five dimensions
• Integrability of Some Charged Rotating Supergravity Black Hole Solutions in Four and Five Dimensions
• OJ 287 binary black hole system

Cited by in corpus (105)

• Calculating black hole shadows: Review of analytical studies
• Influence of global cosmological expansion on local dynamics and kinematics
• Analytical solution of the geodesic equation in Kerr-(anti) de Sitter space-times
• The Pioneer Anomaly
• Gravitational self-force on generic bound geodesics in Kerr spacetime
• Geodesics of electrically and magnetically charged test particles in the Reissner-Nordström space-time: analytical solutions
• Analytic solutions of the geodesic equation in higher dimensional static spherically symmetric space-times
• Quantum Schwarzschild geometry in effective-field-theory models of gravity
• The Application of Weierstrass elliptic functions to Schwarzschild Null Geodesics
• High precision thermal modeling of complex systems with application to the flyby and Pioneer anomaly
• Shadow of a black hole at cosmological distance
• Gravitational lensing and frame dragging of light in the Kerr-Newman and the Kerr-Newman-(anti) de Sitter black hole spacetimes
• Gravitational self-force on eccentric equatorial orbits around a Kerr black hole
• Black hole shadow as a standard ruler in cosmology
• Geodesic Motion in Schwarzschild Spacetime Surrounded by Quintessence
• Analytic solutions for parallel transport along generic bound geodesics in Kerr spacetime
• Inversion of hyperelliptic integrals of arbitrary genus with application to particle motion in General Relativity
• Analytic solutions of the geodesic equation in axially symmetric space-times
• Geodesic motion in Euclidean Schwarzschild geometry
• Motion of charged particles on the Reissner-Nordström (Anti)-de Sitter black holes
• Analytical solutions of the geodesic equation in the spacetime of a black hole in f(R) gravity
• Test particle motion in the space-time of a Kerr black hole pierced by a cosmic string
• The complete set of solutions of the geodesic equations in the space-time of a Schwarzschild black hole pierced by a cosmic string
• Cyclic and heteroclinic flows near general static spherically symmetric black holes
• Light paths of normal and phantom Einstein-Maxwell-dilaton black holes
• Geodesic equations in the static and rotating dilaton black holes: Analytical solutions and applications
• Gravitational redshift/blueshift of light emitted by geodesic test particles, frame-dragging and pericentre-shift effects, in the Kerr-Newman-de Sitter and Kerr-Newman black hole geometries
• A Symmetric Integrator for non-integrable Hamiltonian Relativistic Systems
• Motion of the charged test particles in Kerr-Newman-Taub-NUT spacetime and analytical solutions
• Orbits of light rays in scale-dependent gravity: Exact analytical solutions to the null geodesic equations
• Detailed study of geodesics in the Kerr-Newman-(A)dS spactime and the rotating charged black hole spacetime in $f(R)$ gravity
• A study of geodesic motion in a (2+1)-dimensional charged BTZ black hole
• Study of null and time-like geodesics in the exterior of a Schwarzschild black hole with quintessence and cloud of strings
• Observables for bound orbital motion in axially symmetric space-times
• Analytic solutions of the geodesic equation for Einstein-Maxwell-dilaton-axion black holes
• Analytical study of light ray trajectories in Kerr spacetime in the presence of an inhomogeneous anisotropic plasma
• Geodesic Motion in the Singly Spinning Black Ring Spacetime
• Wormholes in conformal gravity
• Rindler Modified Schwarzschild Geodesics
• Geodesic motion in the (rotating) black string spacetime
• Approximating light rays in the Schwarzschild field
• Geodesic Motion in the (Charged) Doubly Spinning Black Ring Spacetime
• Study of the geodesic equations of a spherical symmetric spacetime in conformal gravity
• Revisiting timelike and null geodesics in the Schwarzschild spacetime: general expressions in terms of Weierstrass elliptic functions
• Analytic solutions of the geodesic equation for Reissner-Nordström-(anti-)de Sitter black holes surrounded by different kinds of regular and exotic matter fields
• Revisiting timelike geodesics in the Fisher-Janis-Newman-Winicour-Wyman spacetime
• Analytic treatment of complete geodesics in a static cylindrically symmetric conformal spacetime
• Simulation of geodesic trajectory of charged BTZ black holes in massive gravity
• Strong deflection of massive particles in spherically symmetric spacetimes
• Circular orbits on a warped spandex fabric
• Null Geodesics and Repulsive Behavior of Gravity in (2+1)-dimensional Massive Gravity
• Investigation the geodesic motion of three dimensional rotating black holes
• Structure of geodesics in the regular Hayward black hole space-time
• Matter Accretion onto a Conformal Gravity Black Hole
• Coupling a Point-Like Mass to Quantum Gravity with Causal Dynamical Triangulations
• The local and global geometrical aspects of the twin paradox in static spacetimes: I. Three spherically symmetric spacetimes
• Particles motion on topological Lifshitz black holes in 3+1 dimensions
• Orbits of particles with magnetic dipole moment around magnetized Schwarzschild black holes: Applications to S2 star orbit
• Isofrequency pairing of spinning particles in Schwarzschild-de Sitter spacetime
• Geodesic motion in R- charged black hole space-times
• Constraints on Scalar-Tensor Theory of Gravity by Solar System Tests
• Influence of cosmological expansion in local experiments
• Modeling approaches for precise relativistic orbits: Analytical, Lie-series, and pN approximation
• Quantum Yang-Mills theory in de Sitter ambient space formalism
• Geodesic Motions near an improved Schwarzschild black hole
• Velocity and velocity bounds in static spherically symmetric metrics
• Gravitational Rutherford scattering and Keplerian orbits for electrically charged bodies in heterotic string theory
• Analytical solutions of the geodesic equation in the space-time of a black hole surrounded by perfect fluid in Rastall theory
• Analytic solutions of the geodesic equation for U(1)^2 dyonic rotating black holes
• Elliptical-like orbits on a warped spandex fabric: A theoretical/experimental undergraduate research project
• Particle motion in the field of a five-dimensional charged black hole
• Jacobi fields, conjugate points and cut points on timelike geodesics in special spacetimes
• On the Applicability of the Geodesic Deviation Equation in General Relativity
• Electrogeodesics in the di-hole Majumdar-Papapetrou spacetime
• Geodesic equations for particles and light in the black spindle spacetime
• Analytical study of particle geodesics around a scale-dependent de Sitter black hole
• Conformal geodesics in spherically symmetric vacuum spacetimes with Cosmological constant
• Local Hubble Expansion: Current State of the Problem
• Geodesic Motion in the Spacetime Of a SU(2)-Colored (A)dS Black Hole in Conformal Gravity
• Probing the Information-Probabilistic Description
• Motion of charged particles around a scalarized black hole in Kaluza-Klein theory
• Displaced orbits and electric-magnetic black hole binaries
• The geodesics structure of Schwarzschild electromagnetic black hole
• Geodesic motion of a test particle around a noncommutative Schwarzchild Anti-de Sitter black hole
• On cosmological expansion and local physics
• The role of elliptic integrals in calculating the gravitational lensing of a charged Weyl black hole surrounded by plasma
• Null geodesics around a black hole with weakly coupled global monopole charge
• Motion of the charged test particle in the spinning nonlinear electromagnetic black hole
• Types of the geodesic motions in Kerr-Sen-AdS${}_4$ spacetime
• Light propagation around a Kerr-like black hole immersed in an inhomogeneous anisotropic plasma in Rastall gravity: Analytical solutions to the equations of motion
• Matter environments around black holes: geodesics, light rings, and ultracompact configurations
• Deflection and gravitational lensing of null and timelike signals in general asymptotically (anti-)de Sitter spacetimes
• On Geodesic Congruences and the Raychaudhuri Equations in ${\text{SAdS}}_4$ Spacetime
• Geometrodynamics based on geodesic equation with Cartan structural equation on Riemannian manifolds
• Precisely computing bound orbits of spinning bodies around black holes II: Generic orbits
• Particles with precessing spin in Kerr spacetime: analytic solutions for eccentric orbits and homoclinic motion near the equatorial plane
• Deviation of trajectories in nonrelativistic and relativistic cases and Shirokov effect
• Revisiting critical orbits of test particles traveling in a black hole background
• Precisely computing bound orbits of spinning bodies around black holes I: General framework and results for nearly equatorial orbits
• Thermodynamics, Phase Transitions, and Geodesic Structure of F(R)-Phantom Banados-Teitelboim-Zanelli (BTZ) Black Holes
• Periapsis shift and deflection of light by hairy black holes
• An exact solution of the orbit equation for a massive particle in Schwarzschild metric
• Hyperelliptic functions and motion in general relativity
• Geodesics and Bending of Light around a BTZ Black Hole Surrounded by Quintessential Matter
• Elliptic solutions of generalized Brans-Dicke gravity with a non-universal coupling