Geodesic motion in Euclidean Schwarzschild geometry

Area of Science:

• Theoretical physics
• General relativity
• Differential geometry

Background:

• Geodesic motion describes the path of objects in spacetime.
• Schwarzschild geometry is a solution to Einstein's field equations describing a non-rotating, uncharged black hole.
• Euclidean geometry differs from Lorentzian geometry in its fundamental properties.

Purpose of the Study:

• To systematically investigate geodesic motion within the equatorial plane of Euclidean Schwarzschild geometry.
• To explicitly derive the mathematical form of these geodesics.
• To compare the characteristics of these geodesics with those in Lorentzian spacetime.

Main Methods:

• The study focuses on the equatorial plane of the Euclidean Schwarzschild geometry.
• Mathematical derivations are performed to obtain the explicit form of geodesic motion.
• Analysis involves the use of incomplete elliptic integrals of the first, second, and third kinds.

Main Results:

• The explicit form of geodesic motion is determined using incomplete elliptic integrals.
• No elliptic-like orbits are found in Euclidean Schwarzschild geometry, contrasting with Lorentzian patterns.
• Among unbounded orbits, only those of the first kind are permissible.

Conclusions:

• The nature of geodesic motion in Euclidean Schwarzschild geometry is fundamentally different from that in Lorentzian spacetime.
• The absence of elliptic-like orbits and the restriction on unbounded orbits highlight unique characteristics of this geometry.