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Anti-Gravity from Vacancies in Fractal Space-Time: The Case of a Menger Sponge

Karl Svozil

Abstract

We explore the idea that anti-gravity, interpreted as matter-matter repulsion, may emerge as an effective description of spacetime with a reduced local ``substratum density,'' modeled heuristically by vacancies in a fractal lattice. Using the Menger Sponge as a prototypical vacancy-dominated fractal, we motivate a phenomenological identification of a net vacancy parameter with the mass parameter in the Schwarzschild solution. This yields an effective negative-mass Schwarzschild-like metric for embedded observers when vacancies dominate. We analyze curvature, geodesic motion, and energy-condition issues, and we emphasize that our construction is not a microscopic derivation from a specific stress-energy tensor but a re-interpretation of the negative-mass Schwarzschild geometry in terms of fractal vacancies. We discuss conceptual implications, stability challenges, and possible observational signatures.

Anti-Gravity from Vacancies in Fractal Space-Time: The Case of a Menger Sponge

Abstract

We explore the idea that anti-gravity, interpreted as matter-matter repulsion, may emerge as an effective description of spacetime with a reduced local ``substratum density,'' modeled heuristically by vacancies in a fractal lattice. Using the Menger Sponge as a prototypical vacancy-dominated fractal, we motivate a phenomenological identification of a net vacancy parameter with the mass parameter in the Schwarzschild solution. This yields an effective negative-mass Schwarzschild-like metric for embedded observers when vacancies dominate. We analyze curvature, geodesic motion, and energy-condition issues, and we emphasize that our construction is not a microscopic derivation from a specific stress-energy tensor but a re-interpretation of the negative-mass Schwarzschild geometry in terms of fractal vacancies. We discuss conceptual implications, stability challenges, and possible observational signatures.
Paper Structure (17 sections, 29 equations, 3 figures)

This paper contains 17 sections, 29 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Intuitive Analogies
  3. Fractals and Embedded Observers
  4. Analogy to Defects in Solids
  5. Semi-Quantitative Phenomenological Model
  6. Motivation from the Schwarzschild Metric
  7. Defining the Effective Defect Source
  8. Defect-Induced Metric as Reparameterized Schwarzschild
  9. Ricci Scalar and Vacuum Character
  10. Geodesic Motion and Repulsion
  11. Proper Acceleration of Static Observers
  12. Free-Fall Geodesics: Attraction vs. Repulsion
  13. Energy Conditions and Exotic Matter
  14. Vacuum Exterior ($r > 0$)
  15. Strong Energy Condition and Repulsion
  16. ...and 2 more sections

Figures (3)

  • Figure 1: The Menger Sponge, a universal curve with a topological dimension of 1 and a fractal dimension of approximately 2.73, shown at the third iteration of its construction (gray: material, transparent: vacancy).
  • Figure 2: Extended line which is not a universal curve: Koch Curve at resolution level three.
  • Figure 3: Carved out line: the Cantor set at resolution level three.