Charged Rotating Disc of Dust

• Charged rotating disc of dust is a theoretical model defined by rigid rotation, constant specific charge, and pressureless matter, linking plasma physics with general relativity.
• The model employs Einstein-Maxwell theory and post-Newtonian expansion to determine equilibrium, stability, and multipole moments, converging to Kerr–Newman limits as relativity intensifies.
• Laboratory experiments and molecular dynamics simulations validate the model through observed vortex formations, phase transitions, and precise charge-dipole interactions in dusty plasmas.

A charged rotating disc of dust is a configuration in plasma physics and general relativity consisting of an infinitesimally thin layer of dust (pressureless matter) that rotates rigidly about a central axis and carries a constant specific electric charge. This system is studied both in laboratory dusty plasma environments, exhibiting aggregate and vortex phenomena, and as an exact or post-Newtonian solution to the coupled Einstein-Maxwell equations, revealing equilibrium, stability, multipole, and black hole transition properties. Its investigation illuminates fundamental aspects of gravitating, rotating, and charged matter in both non-relativistic and relativistic regimes.

1. Theoretical Frameworks for Charged Rotating Discs

Charged rotating discs of dust are systematically modeled within the Einstein-Maxwell theory as stationary, axisymmetric, reflection-symmetric configurations, with the dust undergoing rigid rotation with angular velocity Ω and possessing a uniform specific charge ϵ. The spacetime metric in Weyl–Lewis–Papapetrou coordinates takes the form

$$ds^2 = f^{-1}\left[h(d\rho^2 + d\zeta^2) + \rho^2 d\phi^2\right] - f(dt + a\,d\phi)^2$$

where $f$, $h$, and $a$ are functions determined by the coupled field equations, subject to boundary conditions on the disc and asymptotic flatness. The electromagnetic field is incorporated via a four-potential $A_\alpha = (0, 0, A_\phi, A_t)$, and the charge-to-mass ratio of the dust is constant: $ϵ = \rho_{el}/\mu$. The external spacetime is described as an electro-vacuum (no matter outside the disc).

The system admits a post-Newtonian (PN) expansion in a dimensionless relativity parameter $g = \sqrt{\gamma}$, where $\gamma = 1 - \sqrt{f_c}$ ($f_c$ is the value of $f$ at the center). This enables solutions to be obtained perturbatively, connecting the Newtonian limit ($g \ll 1$) to the ultra-relativistic (near-black-hole; $g \rightarrow 1$) regime (Palenta et al., 2013, Rumler et al., 2022).

2. Rotational Dynamics and Aggregation Effects

In laboratory plasma settings, charged dust particles exposed to external electric fields, such as in the sheath region, aggregate into clusters with asymmetric charge distributions, giving rise to non-zero electric dipole moments:

$\mathbf{p} = \sum_i q_i \mathbf{r}_i$

where $q_i$ and $\mathbf{r}_i$ are individual grain charges and positions relative to the aggregate center. Immersion in an external field $\mathbf{E}$ generates a torque

$\boldsymbol{\tau} = \mathbf{p} \times \mathbf{E}$

causing observable aggregate rotation (Matthews et al., 2011). This phenomenon facilitates 3D structure reconstruction and informs the charge-dipole-moment estimation via

$\boldsymbol{\tau} = I\alpha$

($I$: moment of inertia; $\alpha$: angular acceleration). Complex aggregates form "dust molecules," electrostatically bound but not in physical contact, stabilized by the combined Coulomb attraction and ion drag forces, with interaction potential

$U(d) = \frac{QQ'}{4\pi \epsilon_0 d}$

Debye screening further modifies this in the plasma environment.

3. Relativistic Solutions and Black Hole Limits

Charged rotating dust discs yield asymptotically flat solutions to the Einstein-Maxwell equations with post-Newtonian expansion up to high order. Global quantities such as gravitational mass ($M$), angular momentum ($J$), charge ($Q$), and baryonic mass ($M_0$) are determined from the far-field metric and electromagnetic potentials. A pivotal quantity is the relative binding energy

$E_B^{(rel)} = \frac{M_0 - M}{M_0}$

which, as $g \rightarrow 1$, approaches values closely matching analytic uncharged disc limits (Palenta et al., 2013).

As the system attains the ultra-relativistic limit ($g \rightarrow 1$), the normalized disc radius shrinks to zero, and the surface invariants approach those of an extreme Kerr–Newman black hole. The electric potential on the disc in the corotating frame becomes constant,

$$\sqrt{f'} + \epsilon \alpha' = \mathcal{D} = \text{const}$$

A set of black hole limit relations is satisfied:

• $M = 2\Omega J + \alpha' Q$
• $J^2 / [ (1-\psi^2) M^4 ] = 1$
• $\alpha'|_{\cal H} = \frac{\psi}{2-\psi^2}$ with $\psi = Q/M$

The multipole moments likewise converge precisely to the Kerr–Newman values for $g\to 1$ (Breithaupt et al., 2015).

4. Stability and Structure of Orbits

Stability analysis of particle orbits in the disc proceeds through effective potential methods. For the charged rotating disc of dust, circular orbits are examined using conditions on the effective potential $\mathcal{U}$:

$$\frac{1}{2} g_{\rho\rho} \left(\frac{d\rho}{d\tau} \right)^2 + \mathcal{U} = 0$$

with $\mathcal{U}$ combining metric coefficients and the electromagnetic potentials. Stability is ensured if

$$\mathcal{U} = 0, \quad \mathcal{U}_{,\rho} = 0, \quad \mathcal{U}_{,\rho\rho} > 0$$

For $\epsilon < 1$, all interior dust orbits ($0 \leq \rho < \rho_0$) are stable; for $\epsilon = 1$ (electrically counterpoised dust), all orbits are marginally stable ($\mathcal{U}_{,\rho\rho} = 0$), and no mechanical restoring force arises against equatorial perturbations (Rumler, 3 Aug 2025). The boundary at the rim ($\rho = \rho_0$) is singular (vanishing surface density) and does not host physical orbits.

Analytical closed-form expressions for circular geodesics in the exterior field—including angular velocity, specific energy, and angular momentum—hold for the charged rotating disc as well as for generic asymptotically flat axisymmetric vacuum spacetimes (Rumler, 3 May 2024). The qualitative behavior of orbits, photon, marginally bound, and marginally stable orbits matches the Kerr–Newman solution for large radii but diverges near the disc, reflecting the absence of an event horizon.

5. Multipole Moments and Physical Interpretation

The gravitational and electromagnetic multipole moments $(P_n, Q_n)$ for the charged rotating disc provide a complete characterization of the disc’s external field. Expanding suitable Ernst potentials along the axis yields the moments via power series coefficients $(m_n, q_n)$ with reflective symmetry ensuring real even moments and imaginary odd moments:

$P_n = M_n + iJ_n, \quad Q_n = E_n + iB_n$

A key finding—labeled the generalized multipole conjecture—is that for all $n\geq 2$, the absolute values of the disc's moments exceed those of the corresponding Kerr or Kerr–Newman black hole for the same $(M, J, Q)$, with equality only at the black hole limit:

$$|P_n| \geq |J^n/M^{n-1}|, \quad |Q_n| \geq |QJ^n/M^n|$$

This richer multipolar structure reflects the presence of matter (as opposed to pure vacuum) and is robust across the full range of expansion parameters (Rumler et al., 2023).

6. Laboratory and Astrophysical Contexts

In laboratory dusty plasma experiments, collections of charged dust grains in external fields reveal rotating aggregate clusters, vortex structures, and phase transitions akin to those observed in charged discs. The interplay of charge gradients, ion drag, and sheath electric fields drives rotation via torque and dissipative instabilities (Matthews et al., 2011, Choudhary et al., 2017, Choudhary et al., 2019). The transition from multiple to single vortices is governed by the cloud size relative to characteristic vortex diameter and is confirmed by quantitative agreement between PIV measurements and theoretical models.

Molecular dynamics simulations of finite clusters under magnetic confinement reveal first-order phase transitions from disordered to coherent rotational motion as the magnetic field or Coulomb coupling parameter exceeds a threshold. These collective behaviors model aspects of disc stability and structure in both laboratory and astrophysical settings (Sarma et al., 2023, Maity et al., 2020).

In astrophysical scenarios, charged rotating discs inform the structure of equatorial accretion flows and galactic nuclei, and the interplay between charge, rotation, and gravity elucidates possible transitions from disc matter to black holes, as well as the detailed multipole emissions relevant to gravitational wave astronomy (Schroven et al., 2017, Rumler et al., 2022).

7. Summary of Key Properties

• Charged, rotating discs of dust constitute exact or perturbative solutions of the Einstein–Maxwell system, parametrized by specific charge and rotation.
• Rigid rotation and constant charge yield stable equilibria except in the marginally bound, electrically counterpoised ($\epsilon = 1$) case.
• PN expansion techniques enable explicit calculation of global, geometric, and stability properties up to high order.
• Multipole moments of the disc model are systematically larger than those of Kerr–Newman solutions except in the black hole limit, indicating a richer structure.
• Experimental dusty plasma analogs share dynamic features—electric dipole-induced rotation, vortex formation, and phase transitions—with theoretical models.
• The disc system bridges laboratory plasma physics, general relativity, and astrophysical phenomena and provides a testbed for studying transitions to black hole states and the effects of charge and rotation in strong-field contexts.