Two conserved angular momenta in Schwarzschild spacetime geodesics

Abstract

Abstract The present study investigated the geodesic paths in the 3+1 dimensional Schwarzschild spacetime. Four conserved parameters were found: the first is the conserved total energy: the second is the coordinate-invariant metrics: and the final two are the angular momenta (Pθ and Pφ ) in the spherical coordinate. For θ = π/2 and when excluding a 1/c 2 term in the equation of motion, we recover the orbit equation of the two-body problem. But when not excluding that term, we recover the orbit precession, e.g. the perihelion precession of Mercury. When the value θ is not fixed, we found the equation of motion to be the radius r(θ) as a function of θ, which is similar to the function r(φ) for a fixed value of θ.