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Hilbert cube model for fractal spacetime

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  • He, Ji-Huan

Abstract

A three-dimensional Hilbert cube has exactly three dimensions. It can mimic our spatial world on an ordinary observation scale. A four-dimensional Hilbert cube is equivalent to Elnaschie Cantorian spacetime. A very small distance in a very high observable resolution is equivalent to a very high energy spacetime which is inherently Cantorian, non-differentiable and discontinuous. This article concludes that spacetime is a fractal and hierarchical in nature. The spacetime could be modeled by a four-dimensional Hilbert cube. Gravity and electromagnetism are at different levels of the hierarchy. Starting from a simple picture of a four-dimensional cube, a series of higher dimensional polytops can be constructed in a self-similar manner. The resulting structure will resemble a Cantorian spacetime of which the expectation of the Hausdorff dimension equals to 4.23606799 provided that the number of hierarchical iterations is taken to infinity. In this connection, we note that Heisenberg Uncertainty Principle comes into play when we take measurement at different levels of the hierarchy.

Suggested Citation

  • He, Ji-Huan, 2009. "Hilbert cube model for fractal spacetime," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 2754-2759.
  • Handle: RePEc:eee:chsofr:v:42:y:2009:i:5:p:2754-2759
    DOI: 10.1016/j.chaos.2009.03.182
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    References listed on IDEAS

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    1. El Naschie, M.S., 2009. "An irreducibly simple derivation of the Hausdorff dimension of spacetime," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 1902-1904.
    2. He, Ji-Huan & Xu, Lan & Zhang, Li-Na & Wu, Xu-Hong, 2007. "Twenty-six dimensional polytope and high energy spacetime physics," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 5-13.
    3. Elokaby, A., 2009. "On the deep connection between instantons and string states encoder in Klein’s modular space," Chaos, Solitons & Fractals, Elsevier, vol. 42(1), pages 303-305.
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