An Analogue of the Laplace-Runge-Lenz Vector for Timelike Geodesics in Schwarzschild Spacetime
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.
Schwarzschild, geodesic, LRL, first integral, General Relativity
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