The Keyhole Geometry of Spacetime
Boundaries, Pathways, Field Convergence, and the Hourglass of Time
John Swygert
November 9, 2026
DOI: To be assigned
Abstract
This paper develops the hypothesis that recurring keyhole and opposed-keyhole (hourglass) geometries can serve as a cross-domain language for boundaries, pathways, constrictions, and transitions. It distinguishes visual silhouettes from actual transport bottlenecks, state-space constraints, observational limits, and hypothetical spacetime throats. A dimensionless constriction ratio is defined and applied to a rectangular electromagnetic waveguide, where Maxwell’s equations predict a measurable cutoff change under narrowing. The result demonstrates how geometry can matter without implying a universal mechanism. Nikola Tesla’s Wardenclyffe Tower, the receding-traveler horizon analogy, and Einstein–Rosen bridges are considered at explicitly different levels of physical evidence. The proposed program asks whether common dimensionless descriptors can predict transport behavior within specified classes after governing equations, material properties, and boundary conditions are controlled.
1. Introduction: Why the Recurrence Matters
The keyhole profile—an extended region narrowing toward a throat—appears in architecture, engineered transport systems, visual perspective, and mathematical representations of transitions. Two opposed profiles form an hourglass, a geometry in which connection and separation are descriptions of the same transition viewed in opposite directions. This recurrence is not assumed to be mere coincidence; it may reflect the repeated appearance of constraints on access, flow, and coupling. Whether it expresses a shared mathematical structure is a question to test, not a conclusion established by silhouette.
The organizing claim is structural and relational: boundaries can define admissible pathways, while pathways can determine where otherwise distinct domains interact. A portal in this sense need not be a science-fiction doorway; nor does the definition rule out hypothetical spacetime connections in advance. It requires the domain and the nature of the connection to be specified.
2. Definitions and an Early Taxonomy
Let D₁ and D₂ denote domains (physical regions or defined state sets), ∂D their relevant boundaries, P an admissible pathway, and T a local constriction or transition. The schematic D₁ ↔ T ↔ D₂ is direction-neutral; it does not assert that transport is possible or that the two domains are physically separate universes.
These classes must not be silently exchanged. A shared drawing can be informative even when the quantities and mechanisms are entirely different.
3. A Measurable Geometric Descriptor
For a physical transport channel with well-defined cross-sections, define C = Aₜ / min(A₁, A₂), where Aₜ is the minimum throat area and A₁ and A₂ are representative cross-sectional areas in the adjoining regions. For a true narrowing, 0 < C ≤ 1. The descriptor is undefined unless the relevant surfaces, metric, and measurement convention are specified. For a projected silhouette, image area is not a substitute for transport area.
C alone does not determine transmission: wavelength, shape, length, impedance mismatch, material response, polarization, and excitation matter. A useful research program must therefore pair C with dimensionless dynamic quantities such as wavelength-to-width ratio, normalized throat length, and impedance contrast.
4. Worked Electromagnetic Example: Waveguide Cutoff
Consider an ideal, air-filled, perfectly conducting rectangular waveguide operating in its dominant TE₁₀ mode. For broad-wall dimension a, the cutoff frequency is fᶜ = c₀/(2a), where c₀ is the speed of light in vacuum. Above cutoff, the mode propagates; below cutoff, its amplitude decays evanescently. This is a direct Maxwell-equation prediction, not an analogy.
For these defined surfaces, C = 100/200 = 0.50. At 10 GHz, narrowing changes the local TE₁₀ mode from propagating to evanescent. In a uniform narrow segment, the ideal below-cutoff amplitude decay constant is α = √[(π/a)² − (2πf/c₀)²], approximately 234 m⁻¹ for a = 10 mm and f = 10 GHz. Over a 20 mm segment, the isolated evanescent factor e^(−αL) is about 0.0093 in amplitude. This is NOT the complete device transmission coefficient: discontinuity reflections, higher-order evanescent modes, finite-length tunneling, and matching at both interfaces require full mode matching or numerical Maxwell simulation.
Testable prediction: holding materials, frequency, excitation, and external guide dimensions fixed, a 10 mm-wide constriction will strongly suppress TE₁₀ transmission near 10 GHz relative to an otherwise matched 20 mm-wide uniform guide, with the measured S₂₁ depending on throat length and transition profile. A smooth taper and an abrupt step with the same C need not yield the same reflection S₁₁ or transmission S₂₁. This supplies a falsifiable comparison and demonstrates why the constriction ratio is necessary but insufficient.
5. Convergence, Divergence, and the Hourglass
Two opposed keyhole profiles produce an hourglass: D₁ → T → D₂ in one direction, D₂ → T → D₁ in the other. The throat is a shared transition, not inherently an entrance or an exit. Physical hourglasses illustrate granular transport through a constriction, but the familiar “hourglass of time” is a symbol, not a derived spacetime metric. The useful scientific question is whether a common bottleneck description predicts measurable behavior within a specified physical class.
6. Observer and Traveler: The Vanishing-Point Example
A traveler driving toward the horizon becomes progressively smaller in the observer’s image and may eventually become invisible. The traveler continues into a region not accessible to that observer’s present view. This separates the limits of observation from the limits of existence. The image’s vanishing point is a projection property, not a physical throat crossed by the vehicle. Occlusion horizons, cosmological horizons, and relativistic causal boundaries have different definitions and must not be identified merely because they can limit accessibility.
7. Wardenclyffe: Geometry, Resonance, and Grounding
Tesla’s Wardenclyffe Tower combined a large elevated terminal, a tapered support, tuned electrical apparatus, and an extensive grounding arrangement. The silhouette suggests a dome and narrowing transition over an outward-spreading framework. Tesla’s actual engineering questions concerned electrical potentials, resonant coupling, grounding, and proposed transmission through the Earth and surrounding environment. The visible structural taper is not itself evidence of electromagnetic focusing or spacetime curvature.
A legitimate extension would reconstruct the electrical boundary conditions and compare calculated field intensity, current paths, and resonant response with geometrically different supports while preserving the electrically relevant terminal, ground, and circuit parameters. That would test whether the visible keyhole-like profile contributes anything beyond the functional electrical elements.
8. Spacetime Bridges: Similar Geometry, Different Physics
Einstein’s field equations, Gμν + Λgμν = (8πG/c₀⁴)Tμν, link spacetime curvature to stress-energy. The original Einstein–Rosen bridge in the extended Schwarzschild solution is not traversable. Traversable wormhole models, such as those studied by Morris and Thorne, describe throats in specified metrics but face severe energy-condition and stability questions. Embedding diagrams may resemble an hourglass; that visual similarity is not a derivation of a bridge from a keyhole-shaped object.
Ordinary relativistic time dilation can produce different elapsed proper times for different observers without a wormhole. Hypothetical time-offset wormhole mouths introduce additional causal complications. Any proposed accelerated transition into another observer’s future through a “focal throat” must specify a metric, a stress-energy source, traversability, tidal constraints, stability, and causal consistency. Electromagnetic field energy gravitates in principle, but no demonstrated apparatus can collapse spacetime into a traversable shortcut.
9. The Philadelphia Experiment: A Hypothetical Scenario Only
The Philadelphia Experiment is not offered as historical evidence. Used strictly as a fictionalized thought experiment, it asks what physical conditions would be necessary for an electromagnetic apparatus to change spatial or temporal connectivity. The answer must come from a quantitative gravitational model, not from the story or from visual analogy.
10. Research Program and Criteria for Failure
Stage 1 — Classification: build a dataset of keyhole/hourglass candidates and matched non-keyhole controls, tagged by the taxonomy in Section 2. Record whether the geometry is merely projected or represents an actual transport cross-section.
Stage 2 — Measurement: define C and supplementary dimensionless variables for each domain. For waveguides, record frequency relative to cutoff, normalized throat length, transition profile, material properties, and measured S-parameters.
Stage 3 — Prediction: derive within-domain results from Maxwell’s equations, continuum mechanics, optical wave theory, or explicitly stated dynamical equations. The primary electromagnetic hypothesis is that throat dimensions and profile predict a reproducible change in mode propagation and scattering compared with matched controls.
Stage 4 — Discrimination: determine whether the keyhole descriptor adds predictive power beyond standard quantities such as cutoff, impedance, and aperture size. If it does not, the geometry remains a useful descriptive language rather than a new physical principle. A proposed cross-domain law fails if no invariant mapping or predictive relationship survives controlled comparison.
Stage 5 — Relativistic extension: only after the mathematical structure is specified, examine candidate metrics for energy conditions, geodesic completeness, causal structure, and perturbative stability. No experimental waveguide result alone establishes a wormhole mechanism.
11. Relation to Broader Boundary–Pathway Thinking
The proposed geometry is compatible at the conceptual level with broader investigations of reality as structured by constraints, boundaries, and accessible pathways, including the author’s TSTOEAO research sequence. This is a conceptual bridge, not an assertion that those frameworks already supply a derivation of the physical results presented here.
12. Conclusion
The keyhole and hourglass profiles are worth investigating because constrictions and transitions repeatedly arise wherever access, transmission, and coupling are constrained. Their recurrence need not be dismissed as meaningless coincidence; nor does it establish a universal portal mechanism. The worked waveguide example shows that a throat can have quantitatively predictable physical consequences while making clear that its effect follows from governing equations and boundary conditions. The central proposal is therefore both expansive and testable: seek structural commonalities, keep distinct mechanisms distinct, and require the geometry to earn explanatory power through predictions that survive controls.
References
Einstein, A., & Rosen, N. (1935). The particle problem in the general theory of relativity. Physical Review, 48, 73–77. https://doi.org/10.1103/PhysRev.48.73
Morris, M. S., & Thorne, K. S. (1988). Wormholes in spacetime and their use for interstellar travel. American Journal of Physics, 56, 395–412. https://doi.org/10.1119/1.15620
Visser, M. (1995). Lorentzian Wormholes: From Einstein to Hawking. AIP Press.
Tesla, N. (1900). The problem of increasing human energy. The Century Magazine.
Pozar, D. M. (2012). Microwave Engineering (4th ed.). Wiley.
Collin, R. E. (1991). Field Theory of Guided Waves (2nd ed.). IEEE Press.
Misner, C. W., Thorne, K. S., & Wheeler, J. A. (1973). Gravitation. W. H. Freeman.
Research and publishing websites: https://tstoeao.com | https://ivorytowerjournal.com
No comments:
Post a Comment