Revisiting timelike and null geodesics in the Schwarzschild spacetime: general expressions in terms of Weierstrass elliptic functions

Abstract

The theory of Schwarzschild geodesics is revisited. Basing on a result by Weierstrass and Biermann, we derive a formula describing all non radial, timelike and null trajectories in terms of Weierstrass elliptic functions. Quite remarkably, a single formula works for an entire geodesic trajectory, even if it passes through turning points. Using this formula, we derive expressions for the proper and coordinate time along the geodesic.

Matches the version accepted by Classical and Quantum Gravity

References in corpus (10)

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Cited by in corpus (10)

• Quantum Schwarzschild geometry in effective-field-theory models of gravity
• Kerr Geodesics in Terms of Weierstrass Elliptic Functions
• Strong deflection of massive particles in spherically symmetric spacetimes
• Timelike and null geodesics in the Schwarzschild space-time: Analytical solutions
• Particle dynamics in spherically symmetric electro-vacuum instantons
• Geodesics on metrics of Eguchi-Hanson type
• Optical properties of null geodesics emerging from dynamical systems
• Null integrability and photon phenomenology in the accelerated Schwarzschild black hole
• Revisiting critical orbits of test particles traveling in a black hole background
• Incorporating curved geometry in cosmological simulations