Integration of the geodesic equations via Noether Symmetries

TL;DR

This paper reviews how Noether symmetry methods can be used to integrate geodesic equations in various spacetimes, demonstrating their efficiency through specific examples and approximate symmetries.

Contribution

It introduces the application of Noether symmetry approach to solve geodesic equations, including approximate symmetries for complex spacetimes like Schwarzschild, Reissner-Nordström, and Kerr.

Findings

Successfully derived first integrals for geodesic equations in multiple spacetimes.

Demonstrated the efficiency of Noether symmetry approach in finding solutions.

Provided general solutions of geodesic equations in terms of arc length s.

Abstract

Through this article, I will overview the use of Noether symmetry approach in discussing the integration of geodesic equations for the geodesic Lagrangians of spacetimes. I will also give some examples to reveal the efficiency of Noether symmetry approach by finding the first integrals related for the geodesic Lagrangians of the G\"{o}del-type, Schwarzschild, Reissner-Nordstr\"{o}m and Kerr spacetimes. After obtaining the approximate Noether symmetries of the Schwarzschild, Reissner-Nordstr\"{o}m and Kerr spacetimes, the first integrals associated with each of approximate Noether symmetries have been integrated to find a general solution of geodesic equations in terms of the arc length $s$.