Hidden symmetry group for particle orbits (timelike geodesics) in Schwarzschild spacetime

TL;DR

This paper identifies a hidden symmetry group for timelike geodesics in Schwarzschild spacetime, revealing new conserved quantities analogous to the Laplace-Runge-Lenz vector through Noether's theorem.

Contribution

It provides a natural symmetry interpretation for three hidden conserved quantities in Schwarzschild geodesics, completing the Noether symmetry group for these equations.

Findings

Discovered three hidden symmetry transformations for geodesics.

These transformations commute with known Killing symmetries.

The transformations involve shifts and scaling of energy and angular momentum.

Abstract

For the timelike geodesic equations in Schwarzschild spacetime, three hidden conserved quantities were found recently, which are analogues of dynamical quantities related to the well-known Laplace-Runge-Lenz (LRL) vector in Newtonian gravity. In particular, the geodesic equations possess an LRL angle, an LRL Killing-vector time and an LRL proper-time, each of which is a conserved quantity for all timelike geodesics. The present work provides a natural symmetry interpretation for these three quantities by applying Noether's theorem in reverse to the geodesic Lagrangian. This yields three hidden symmetry transformations. They are shown to commute with the Killing isometries and act on the equatorial geodesics by separate shifts and scaling of the geodesic energy and angular momentum. Together with the Killing symmetries, these transformations comprise the complete Noether symmetry group of the timelike equatorial geodesic equations.