[Paper Review] Stability of equatorial circular geodesics in static axially symmetric spacetimes

Guillermo González, Framsol López-Suspes|arXiv (Cornell University)|Apr 3, 2011

Advanced Differential Geometry Research1 references3 citations

TL;DR

This paper presents a general analysis of the stability of equatorial circular geodesics in static axially symmetric spacetimes, deriving analytical expressions for radius, specific energy, angular momentum, and the marginally stable orbit radius for both timelike and null geodesics. It shows that all null circular orbits are unstable with no marginally stable null geodesics, while timelike geodesics can exhibit bounded, unbounded, or circular orbits depending on the effective potential, with explicit stability ranges derived for solutions in cylindrical, oblate, and prolate spheroidal coordinates.

ABSTRACT

A general study of the stability of equatorial circular orbits in static axially symmetric gravitating systems is presented. Important circular geodesics as the marginally stable orbit, the marginally bounded orbit and the photon orbit are analyzed. We found general expressions for the radius, specific energy, specific angular momentum and the radius of the marginally stable orbit, both for null and timelike circular geodesics. Solutions expressed in cylindrical coordinates, oblate spheroidal coordinates, and prolate spheroidal coordinates are considered. We show that all null circular orbits are unstable and that there are not marginally stable null geodesics, whereas that for timelike geodesics the orbits can be unbounded, bounded or circulars.

Motivation & Objective

• To develop a general framework for analyzing the stability of equatorial circular geodesics in static axially symmetric spacetimes.
• To derive analytical expressions for key orbital parameters: radius, specific energy, specific angular momentum, and marginally stable orbit radius.
• To examine the stability of null and timelike geodesics across different coordinate systems—cylindrical, oblate spheroidal, and prolate spheroidal.
• To determine the conditions under which circular orbits are stable, bounded, or unbounded using the effective potential formalism.
• To apply the formalism to specific solutions, including the Chazy-Curzon field, Morgan-Morgan disks, and the Erez-Rosen solution, to illustrate stability ranges.

Proposed method

• Derives the geodesic equations and effective potential from the Weyl line element for static axially symmetric spacetimes.
• Uses conserved quantities—specific energy E and specific angular momentum ℓ—from the Lagrangian due to time and axial symmetry.
• Applies the effective potential method to analyze orbital stability in the equatorial plane (z=0), focusing on extrema and curvature to determine stability.
• Transforms the formalism into cylindrical, oblate spheroidal, and prolate spheroidal coordinates to analyze specific solutions.
• Derives stability conditions via second derivatives of the effective potential, identifying marginally stable orbits as where the potential curvature vanishes.
• Solves for critical radii and stability boundaries numerically and analytically for specific solutions like the Erez-Rosen metric.

Experimental results

Research questions

• RQ1 What are the general conditions for the existence and stability of equatorial circular geodesics in static axially symmetric spacetimes?
• RQ2 Why are all null circular geodesics unstable, and do marginally stable null geodesics exist?
• RQ3 How do the stability properties of timelike geodesics differ from those of null geodesics in such spacetimes?
• RQ4 What is the radius of the marginally stable circular orbit for timelike geodesics in prolate spheroidal coordinates, particularly for the Erez-Rosen solution?
• RQ5 How do coordinate system choices (cylindrical, oblate, prolate) affect the analytical form and physical interpretation of orbital stability conditions?

Key findings

• All null circular orbits in static axially symmetric spacetimes are unstable, and no marginally stable null geodesics exist.
• For timelike geodesics, orbits can be unbounded, bounded, or circular, depending on the effective potential's curvature and energy-angular momentum configuration.
• In prolate spheroidal coordinates, the Erez-Rosen solution exhibits a stable orbit range defined by 0 ≤ u ≤ 4.77 and ℓ/m ≥ 11.62, with the marginally stable orbit at (ℓ/m, u) = (11.62, 4.77).
• The radius of the marginally stable circular orbit for timelike geodesics is derived in closed form for cylindrical, oblate, and prolate coordinate systems using the effective potential and its second derivative.
• Specific angular momentum and energy are expressed analytically in terms of metric functions and their derivatives, with constraints ensuring real, physical values (e.g., u − 2(u²−1)ψ,u ≥ 0).
• The stability condition for timelike geodesics in prolate coordinates is given by a complex inequality involving ψ,u and ψ,uu, which reduces to a solvable form when metric functions are expanded in Legendre functions.