An Analogue of the Laplace-Runge-Lenz Vector for Timelike Geodesics in Schwarzschild Spacetime

Abstract

In Schwarzschild spacetime, the timelike geodesics are the trajectories of free, massive particles, orbiting a singularity at the origin r = 0. In this work we derive four scalar first integrals of timelike geodesics in Schwarzschild spacetime. Two of the first integrals, corresponding to energy and angular momentum, are well-known. The other two first integrals, an angular quantity and a temporal quantity, are not as well-known. Using the freedom to shift first integrals by a constant value we set a ‘zero-point’ for each of the four first integrals. By choosing a natural point on a non-circular trajectory such as a turning point or inertial point to set the zero-point value, the angular and temporal first integrals will correspond respectively to the angle and time of the chosen zero-point. We then take the Newtonian limit of the angular and temporal first integrals, and show that using a natural choice of zero-point they provide a generalization of the classical Laplace-Runge-Lenz (LRL) vector. We then evaluate the angular first integral for each type of timelike geodesic in Schwarzschild spacetime. In most cases we are able to choose a turning or inertial point to set a zero-point. For an unbound or asymptotic trajectory which falls into the singularity of the metric at r = 0, however, we find that we must take a different point, such as the point where the trajectory crosses the horizon at r = 2M, which we call the ‘horizon point.’ For the case of a precessing elliptic orbit we find that the angular first integral is multi-valued, with the zero-point jumping each time the trajectory crosses an apoapsis. It is found that the angular and temporal first integrals provide a relativistic generalization of the classical LRL vector, where we the first integrals correspond to a larger class of physically meaningful points compared to Newtonian orbits and where the LRL vector and angular and temporal first integrals may always correspond to the periapsis.