Revisiting timelike and null geodesics in the Schwarzschild spacetime: general expressions in terms of Weierstrass elliptic functions
Adam Cieślik, Patryk Mach
Abstract
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.
Topics & Concepts
GeodesicWeierstrass functionsPhysicsSchwarzschild radiusGeodesics in general relativityElliptic functionNull (SQL)SpacetimeSchwarzschild metricMathematical physicsElliptic integralSpace (punctuation)Mathematical analysisGeneral relativityQuantum mechanicsMathematicsDatabaseComputer scienceLinguisticsPhilosophyAstrophysical Phenomena and ObservationsRelativity and Gravitational TheoryPulsars and Gravitational Waves Research