Flying Doughnuts (FDs): Toroidal Phenomena

• Flying Doughnuts (FDs) are toroidal entities characterized by a doughnut-shaped topology that appears in astrophysical accretion disks, electromagnetic pulses, and geometric flows.
• In astrophysics, FDs manifest as thick 'Polish doughnuts' with collimated radiation, whose efficiency and structure are governed by specific angular momentum profiles and advective effects.
• In optics and geometry, FDs describe focused toroidal light pulses and self-shrinking tori that underpin applications in imaging, communications, and advanced mathematical modeling.

Flying Doughnuts (FDs) refer to a family of entities characterized by toroidal, or doughnut-like, topology that arise in diverse domains such as astrophysical accretion physics, spatiotemporally structured electromagnetic pulses, nonlinear dynamics, and geometric analysis. Across the literature, the term encompasses thick accretion tori (notably “Polish doughnuts”), toroidal electromagnetic pulses (focused or toroidal light pulses), toroidal self-shrinking solutions in geometric flow, and topologically intricate sets in higher-dimensional dynamical systems. These systems are unified by their inherent toroidal geometry and, in many instances, by their capacity to localize, collimate, or structure energy or dynamical flow in nontrivial ways.

1. Toroidal Topologies and Defining Characteristics

FDs are structurally defined by a closed loop with a central void, resulting in a toroidal geometry. This manifests as:

• Geometrically thick accretion disks (Polish doughnuts) forming equilibrium structures around compact objects, exhibiting high thickness-to-radius ratios and often featuring narrow axial funnels.
• Focused electromagnetic pulses (also called toroidal or flying doughnut pulses) whose electromagnetic field configuration is spatially and temporally non-separable and encircles a central null, featuring toroidal or skyrmionic field topology.
• Self-similar toroidal solutions under mean curvature flow, which appear as self-shrinking tori (“shrinking doughnuts”).
• Topologically rich sets in quasiregular dynamics, such as spiders’ webs containing embedded genus-$g$ tori.

Toroidal FDs, by virtue of their topology, support distinct physical and mathematical behaviors: energy collimation along the symmetry axis, support for toroidal modes and non-radiating current patterns, symmetry breaking under confinement or curvature, and intricate dynamical or singularity structures.

2. Astrophysical Flying Doughnuts: Thick Accretion Tori

In high-energy astrophysics, FDs are closely associated with “Polish doughnuts” (PDs), which are geometrically thick accretion disks orbiting black holes or compact objects. Key features include:

• Maximizing Thickness and Efficiency: Extremal PD configurations are constructed via angular momentum profiles that are constant and as low as allowed by force balance, yielding analytic disk shapes:

$$H(R) = \left\{ \left[ \frac{GM (R_{in} - r_S)}{GM - (R_{in} - r_S)I(R)} + r_S \right]^2 - R^2 \right\}^{1/2}$$

where $I(R)$ is an integral over the angular momentum distribution. The relative geometrical thickness $h = H(R)/R$ can reach maximized values (e.g., $\chi \approx 1/\sqrt{8\rho}$ for small $\rho$).

• Radiative Efficiency and Upper Bounds: The radiative efficiency without advection is $$\epsilon_{\infty}(R_{in}) = GM (R_{in} - 2r_S) / [2 (R_{in} - r_S)^2]$$, which is linearly reduced by advection via $$\epsilon_{rad}(R_{in}, \xi) = (1-\xi)\epsilon_{\infty}(R_{in})$$.
• Collimation and Luminosity Scaling: The narrow funnel geometry collimates radiation with collimating factor $\beta \approx 1/(4\rho)$ for very thick PDs. The collimated, isotropic-equivalent luminosity scales linearly as $L_{col} \approx 0.0625 \dot{M}c^2$, not logarithmically as in slim disk solutions. This supports $L_{col}/L_{Edd} \approx 0.625\dot{m}$, where $\dot{m}$ is the mass accretion rate in Eddington units (Wielgus et al., 2015).
• Limits from Advective Cooling: High levels of advective loss ($\xi \rightarrow 1$) strongly suppress both geometrical thickness (limiting $\chi$ to $\sim 6$ even at $\dot{M} \gtrsim 10^5 \dot{M}_{Edd}$) and luminosity, constraining the ability of FDs to explain ultraluminous X-ray source (ULX) properties unless advection is minimal.

Magnetized Polish Doughnuts extend this model with toroidal magnetic fields and more general angular momentum profiles, supporting richer morphologies and serving as initial data for simulation of relativistic magnetohydrodynamics (Gimeno-Soler et al., 2017).

In spacetimes with exotic compact objects, such as rotating boson stars, FDs exhibit features not present in Kerr spacetimes. Notably, “static rings” (where test particles remain stationary with respect to distant observers) and double-centered, possibly cusped toroidal discs emerge, with retrograde tori supporting static surfaces inside which fluid motion reverses from retrograde to prograde (Teodoro et al., 2020).

3. Electromagnetic Flying Doughnuts: Toroidal Light Pulses

Toroidal or focused flying doughnut pulses represent exact Maxwellian solutions with doughnut-like topology:

• Field Characteristics: The electromagnetic fields of flying doughnut pulses (e.g., in the TM case) can be described by

$$E_{\rho} = -4i\omega_0\mu_0 \rho (q_2 - q_1 - 2iz) / [\rho^2 + (q_1 + it)(q_2 - ig)]^{3/2}$$

and analogous expressions for $E_z$ and $H_\phi$. The parameters $q_1$ and $q_2$ determine bandwidth and spatial extent. The spatial profile is intrinsically toroidal and few-cycle, with both transverse and longitudinal components (Vignjevic et al., 13 Sep 2025).

• Non-Separability: These pulses are rigorously non-separable in both space-polarization and space-time domains. The spatial polarization varies across the beam, and the spatial profile is frequency-dependent—higher-frequency components reside centrally, lower frequencies peripherally.
• Topological Features: FDs display skyrmion-like field configurations with winding polarization and extensive regions of local energy backflow.
• Quantifying Non-Separability: Measurements such as concurrence ($\sim 0.80$ for space-polarization, $\sim 0.91$ for space-time) and fidelity ($\sim 0.88$ and $0.72$ correspondingly) closely approach ideal models, confirming robust non-separability in experimentally realized FDs.

Experimentally, FDs are generated via ultrashort laser pulses shaped by spatial filtering and segmented waveplates, with state tomography characterizing modal decompositions and space-polarization-time coupling. The spatial and spatiotemporal state can be reconstructed via projective measurements using devices such as digital micromirror devices and waveplates for both spatial and polarization eigenstates. The observed pulses are isodiffracting, with angular divergence matched to theoretical FD behavior across the spectrum.

Applications span telecommunications (state multiplexing), nonlinear spectroscopy (excitation of toroidal and non-radiating modes in matter), metrology, imaging, and energy transport.

4. Geometric and Topological FDs: Mean Curvature Flow and Dynamical Systems

In geometric analysis, self-shrinking tori (“shrinking doughnuts”) exist as solutions to the mean curvature flow:

• Variational Construction: By evolving carefully constructed initial closed curves in a weighted Euclidean half-plane via a modified curvature flow ($V_g = k_g/K$), one obtains convergence to a simple closed geodesic that, upon rotation, yields a self-shrinking torus (Drugan et al., 2017).
• Energy Bounds: The variational approach rigorously establishes upper bounds for the weighted energy, ensuring stability and selection of the toroidal shrinker over degenerate solutions.
• Connections to FDs: Though purely geometric, the results provide mathematical underpinnings for the existence and stability of toroidal structures under dissipative flows, potentially mirroring physical scenarios where toroidal FDs act as attractors or stable singular configurations.

In the theory of uniformly quasiregular mappings in higher dimensions, escape sets (fast escaping sets $A(L)$ for linearizers $L$ at repelling points) can form spiders' web structures containing embedded genus $g$ tori—“webs of doughnuts”—imparting rich topological complexity unavailable in planar dynamics (Fletcher et al., 2018).

5. Curvature Effects and Symmetry Breaking in Soft Material Systems

The response of cholesteric liquid crystals to confinement in curved, especially toroidal (doughnut-like), geometries directly manifests the influence of curvature on toroidal structures:

• Landau-de Gennes Modeling: The tensorial order-parameter framework describes the local alignment and distortional energetics, with free energy including bulk, elastic, and cholesteric terms. The equilibrium pitch $P(r)$ is locally modulated by geometric constraints and quantified by

$$P(r)/P_0 = - \frac{9}{4} \cdot \frac{S^2}{S_{tw} q_0}$$

where $S_{tw}$ encodes twist deformation.

• Curvature-Induced Symmetry Breaking: In toroidal droplets, additional curvature along the major axis drives symmetry breaking not seen in cylinders or flat walls, leading to shifted, split, or “broken” cholesteric layers. This effect can be exploited to tune optical/mechanical properties in cholesteric-based devices.
• Potential Applications and Relevance: Control of toroidal confinement geometry may modulate reflection bands or sensor response, with theoretical insights guiding device design where curvature acts as an active parameter (Fialho et al., 2016).

6. Comparative Analysis and Interdisciplinary Connections

The term “Flying Doughnuts” is thus multivalent, referring to systems that share a doughnut-like topology and toroidal structure but span multiple disciplines:

• In accretion physics (astrophysical FDs), collimation, geometrical thickness, and radiative efficiency determine their capacity to explain observed ULX properties; strong advective cooling restricts their capabilities.
• In optics (electromagnetic FDs), the focus is on exact solutions, topological polarization, spatiotemporal entanglement, and robust modal structure with applications in advanced light-matter interactions.
• In mathematical physics and topology, FDs manifest as self-similar singularity models, novel attractors, or intricate dynamical sets (webs of tori) in higher dimensions.

A consistent theme is the interplay of geometry, topology, and dynamical or energetic constraints—manifest as collimation limits, symmetry breaking, energetic stability bounds, or non-separable structure.

7. Outstanding Issues and Prospects

• Limits of Collimation and Efficiency: Theoretical upper bounds on thickness (e.g., $\chi \sim 1/\sqrt{8\rho}$) and luminosity scaling ($L_{col} \sim 0.0625 \dot{M} c^2$) restrict the role of FDs in explaining extreme luminosities, unless advection of energy is negligible—a condition unlikely in high $\dot{M}$ flows (Wielgus et al., 2015).
• Higher-Dimensional and Exotic Generalizations: In both dynamical systems and accretion theory, the move beyond canonical black hole (Kerr) or planar (complex analytic) settings—toward exotic sources (boson stars), nontrivial topology, or high-dimensional nonlinear maps—enriches the spectrum of possible FD phenomena (Teodoro et al., 2020, Fletcher et al., 2018).
• Measurement and Characterization: Tomographic methods now allow for quantitative benchmarks (concurrence, fidelity) of optical FDs, bridging the gap from idealized mathematical solutions to realistic laboratory pulses (Vignjevic et al., 13 Sep 2025).
• Mathematical Foundations and Stability: Variational and dynamical analysis establish the existence and selectivity of toroidal FDs as attractors or singularity models in geometric evolution, supporting their physical and theoretical relevance (Drugan et al., 2017).

A plausible implication is that further interdisciplinary exploration—integrating optical, astrophysical, and mathematical approaches—may clarify universal FD behaviors and leverage their unique topological/energetic properties for advanced applications in sensing, energy transfer, and information processing.