Rotating Perforated Torus: From One-Aperture and N-Aperture Toy Models to Global All-Gamma Curved Slits and Multi-Species Plasma Realization

This monograph provides a comprehensive, self-contained theoretical treatment of the Casimir interaction in a rotating toroidal cavity with perforated or dielectrically modulated walls, extending the baseline small-aperture dipolar toy model to arbitrary Lorentz factors γ ∈ [1, ∞) and to a relativistic multi-species plasma realization. Key results derived in this work include:1. Born–Langevin Curved Slit Geometry and Ehrenfest Invariance: At high γ, a circular laboratory aperture of radius b on the upper annular face of radius R stretches in the local comoving frame along the cylinder circumference into a curved annular slit of proper length 2a = 2γb. Because the proper circumference of the torus simultaneously stretches to C'(γ) = 2πγR, both the single-hole angular fraction b/(πR) and the N-hole comoving open area fraction f'(γ) = f_lab are exact Lorentz invariants, preventing self-wrapping and inter-hole overlap at any γ.2. Exact 3D-Dipole to 2D-Slit Dimensional Crossover: By integrating the two-point Casimir Green kernel over a finite slot of length 2a opposite an intact mirror plate at gap d, we derive from first principles the transition from the 3D Bethe dipole law (U' ∝ γ³ / ln γ) to the extensive 1D narrow-slit waveguide law (U' ∝ γ), fixing the dimensionless crossover parameter μ² = 8π / [81 ζ(3)] (b/d) ≈ 0.25806 (b/d) with zero free parameters.3. Relativistic Bulk-Cancellation Theorem and End-Cap Quasi-Particles: For γ ≫ d/b, the extensive linear bulk energy of the elongated slit (∝ γ) obeys the parallel-plate vacuum equation of state Σ'_{x,bulk} = -U'_{bulk} and cancels in the laboratory frame via γ(1 - β²) = γ⁻¹ to a constant +U₀/μ². The ultra-relativistic laboratory energy is governed instead by the localized negative end-cap deficit at the two ends of the slit, which carries zero tangential stretching tension and transforms as ΔE(γ) ~ -U₀ [κ_end γ + |α_tip| γ (ln γ - 1)] → -∞.4. Multi-Species Relativistic Plasma Torus Realization: To overcome centrifugal mechanical fracture of solid rotors and enforce the comoving stretching law a = γb via Alfvén frozen-in flux, we formulate a physical model using a confined two-species plasma torus (a high-density reflective plasma matrix S₁ with ω'_{p1} ≫ πc/d hosting discrete low-density transparent plasma bubbles S₂ with ω'_{p2} ≪ πc/d stratified radially by centrifugal buoyancy). We derive the two-species spectral contrast factor, Ehrenfest comoving charge dilution (Mode A vs. Mode B), and the dynamics of an internal asymmetric (M₊, M₋) mass-energy dipole for spacetime swimming.