Equatorial stability analysis of dust particle orbits within a charged rotating disc of dust

Technical Summary: Equatorial Stability Analysis of Dust Particle Orbits within a Charged Rotating Disc of Dust

Problem Statement
The paper addresses the stability of circular orbits for dust particles within a charged, rotating disc of dust, a specific solution to the Einstein-Maxwell equations. While the stability of physical solutions is a prerequisite for their physical relevance, a comprehensive stability analysis of the entire disc structure is analytically intractable. Consequently, the study focuses on a necessary condition for stability: the equatorial stability of individual dust particle orbits against infinitesimal perturbations within the disc's equatorial plane. The analysis investigates how this stability depends on the disc's specific charge ($\epsilon$) and the relativity parameter ($g$), which characterizes the strength of the gravitational field via a post-Newtonian expansion.

Methodology
The study utilizes the Einstein-Maxwell theory, where the charged rotating disc of dust is modeled as an axisymmetric, stationary, and reflection-symmetric configuration. The solution is defined by a rigid rotation with constant angular velocity $\Omega$ and a constant specific charge $\epsilon$ (the ratio of electric charge density to baryonic mass density).

1. Metric and Expansion: The global spacetime is described using Weyl-Lewis-Papapetrou coordinates. The metric functions and electromagnetic four-potential are expressed as post-Newtonian expansions up to the tenth order in the relativity parameter $g$, defined via the redshift at the disc's center.
2. Equations of Motion: The motion of charged dust particles is governed by the Lorentz force equation coupled with geodesic motion. Due to the rigid rotation, the boundary condition ensures that all particles follow circular equatorial orbits.
3. Effective Potential and Stability Criterion: The authors employ the effective potential formalism for charged test particles. By deriving the specific angular momentum ($\tilde{L}$) and specific energy ($\tilde{E}$) from the known angular velocity, they construct an effective potential $U$.
   • Circular orbits correspond to extrema of $U$ ($U_{,\rho} = 0$).
   • Stability is determined by the second derivative of the potential ($U_{,\rho\rho}$). A minimum ($U_{,\rho\rho} > 0$) indicates stability, while a maximum indicates instability.
   • The analysis introduces a dimensionless function $S$, related to $U_{,\rho\rho}$, to evaluate stability conditions explicitly.
4. Boundary Analysis: Special attention is paid to the center of the disc ($\rho=0$) and the rim ($\rho=\rho_0$), where standard limits and continuity conditions require careful treatment due to the transition between interior and exterior spacetimes.

Key Contributions and Results
The paper provides an analytical derivation of the stability conditions for dust orbits within the disc, expanding upon previous work on uncharged discs and general charged spacetimes.

• Dependence on Specific Charge ($\epsilon$):
  • Case $\epsilon = 1$ (Static/Maximally Charged): For a static disc where the specific charge equals unity, all dust particles are found to be in a marginally stable state. In this configuration, the specific angular momentum vanishes ($\tilde{L}=0$) and the specific energy is unity ($\tilde{E}=1$). There is no restoring mechanism for displaced particles, nor is there a mechanism for perturbations to grow; particles remain in equilibrium everywhere within the disc.
  • Case $\epsilon < 1$ (Rotating): For rotating discs, the analysis reveals that all circular orbits within the interior of the disc ($0 \le \rho < \rho_0$) are stable. The effective potential exhibits a minimum for these radii.
• The Rim ($\rho = \rho_0$):
  • For $\epsilon < 1$, the orbit at the very rim of the disc is found to be unstable. The second derivative of the effective potential ($U_{,\rho\rho}$) is discontinuous at the rim, with the limit from the exterior approaching $-\infty$.
  • However, the paper notes that the proper surface mass density of the disc vanishes at the rim. Therefore, no actual dust particles occupy this unstable orbit. The disc effectively maximizes its size by occupying all stable orbits up to, but not including, the rim.
• Convergence: The results are derived using a post-Newtonian expansion. The authors demonstrate that the series for the stability function $S$ and the potential derivatives exhibit good convergence behavior even at high relativity parameters ($g \approx 0.9$), validating the analytical approach within the studied regime.

Significance and Claims
The paper claims that the equatorial stability of dust particle orbits serves as a necessary and important condition for the physical relevance of the charged rotating disc of dust solution. The findings confirm that:

1. The solution is physically meaningful in the sense that the constituent particles do not spontaneously destabilize under equatorial perturbations (except for the theoretical rim orbit, which is unoccupied).
2. The transition from marginal stability ($\epsilon=1$) to stable orbits ($\epsilon < 1$) is consistent with the dynamics of the system, where rotation provides the necessary centrifugal support against gravitational collapse.
3. The results align with and extend previous studies on uncharged rotating discs, confirming that the introduction of charge (up to the limit $\epsilon=1$) does not introduce equatorial instabilities for the occupied orbits.

The authors explicitly state that this analysis does not constitute a full stability analysis of the disc (which would require studying collective perturbations and numerical treatments) but establishes a foundational analytical result regarding the stability of the individual orbits that constitute the disc. Future work is suggested to investigate stability against perturbations in the $\zeta$-direction (vertical to the disc) and collective modes.