Mahdieh Gol Bashmani Moghadam

arXiv2026

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

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.

gr-qc↗

arXiv2026

Symmetry transformation group arising from the Laplace-Runge-Lenz vector

The Kepler problem in classical mechanics exhibits a rich structure of conserved quantities, highlighted by the Laplace--Runge--Lenz (LRL) vector. Through Noether's theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which is well known in the literature. However, the physically relevant part of the LRL vector is its direction angle in the plane of motion (since its magnitude is just a function of energy and angular momentum). The present work derives the infinitesimal dynamical symmetry corresponding to the direction part of the LRL vector, and obtains the explicit form of the symmetry transformations that it generates. When combined with the rotation symmetries,the resulting symmetry group is shown to be the semi-direct product of $SO(3)$ and $R^3$. This stands in contrast to the $SO(4)$ symmetry group generated by the LRL symmetries and the rotations. As a by-product, the action of the new infinitesimal symmetries on all of the conserved quanties is obtained. The results are given in terms of the physical kinematical variables in the Kepler problem, rather than in an enlarged auxiliary space in which the LRL symmetries are usually stated.

math-ph↗