Geodesic motion in Euclidean Schwarzschild geometry

TL;DR

This paper systematically analyzes geodesic motion in Euclidean Schwarzschild geometry, deriving explicit solutions and highlighting key differences from Lorentzian spacetime, such as the absence of elliptic-like orbits.

Contribution

It provides explicit elliptic integral solutions for geodesics in Euclidean Schwarzschild space and identifies unique orbital constraints not present in Lorentzian geometry.

Findings

No elliptic-like orbits exist in Euclidean Schwarzschild geometry.

Only unbounded first-kind orbits are possible among unbounded orbits.

Distinct differences from Lorentzian spacetime in orbit types and constraints.

Abstract

This paper performs a systematic investigation of geodesic motion in Euclidean Schwarzschild geometry, which is studied in the equatorial plane. The explicit form of geodesic motion is obtained in terms of incomplete elliptic integrals of first, second and third kind. No elliptic-like orbits exist in Euclidean Schwarzschild geometry, unlike the corresponding Lorentzian pattern. Among unbounded orbits, only unbounded first-kind orbits are allowed, unlike general relativity where unbounded second-kind orbits are always allowed.