Analytic Solutions for Geodesic Motion in Static Axially Symmetric Spacetime

TL;DR

This paper develops a method to find exact solutions for geodesic paths in static axially symmetric spacetimes, providing multiple solutions and analyzing their properties in detail.

Contribution

It introduces a systematic procedure to derive analytic solutions for geodesics in such spacetimes, including general and particular cases, with a comprehensive analysis of their characteristics.

Findings

Derived two general solutions for static axially symmetric spacetimes.

Obtained five particular solutions based on motion conditions.

Provided a thorough analysis of all possible geodesic solutions.

Abstract

A procedure to find static axially symmetric solutions to the Einstein field equations is presented. We obtained two general solutions and five particular solutions, which depend on the existence conditions for circular and $z$ direction motion. Our endeavour consists making a thoroughrowly analysis of all the possible geodesics solutions stemming from this spacetime.