The Lattice World, Quantum Foam and the Universe-Wide Metamaterial

David Thomas Crouse1 1The City University of New York, USA [email protected] [email protected]

Abstract— The concept of a universe-wide gravity crystal that combines Heisenberg´sLattice World and Wheeler´sQuantum Foam is described. It is assumed that space is a crystal with a lattice constant equal to the and a basis of a Planck mass. Inertial anomalies are calculated that include a parameter that connects the external gravitational field and gravita- tional flux. Similar to the electric permittivity and magnetic permeability of metamaterials, this parameter can take on positive, zero and negative values.

In order to provide a solution to the self-energy of the electron, in 1930 3 introduced the concept of space being discretized in cells of volume ro, with ro = ~/cMproton [1]. In correspondence with Niels Bohr, Heisenberg also states that he believes that the discreteness of space underlies the quantum mechanical uncertainty relations [2]. However, he rather quickly sets aside this concept because of its obvious and inherent problems - one being the breaking of continuous rotational and translational symmetries of space. Another problem pointed out by Bohr was that an absolute “minimum” possible length in one frame of reference is length-contracted to a smaller value in a different frame of reference, leading to a contradiction. However the concept did not disappear - being further discussed by Hiesenberg, Wolfgang Pauli, Gleb Wataghin, and recently by a whole host of physicist in the field of [3]. From Heisenberg in the 1930’s, one jumps ahead in time to 1957, when John Wheeler introduced the concept of quantum foam as part of a theory emerging at this time called quantum geometro- dynamics (QG), or quantum gravity as it is more widely known by today [4]. His objective for the paper was to introduce the theory of QG and show that all of classical physics is purely geometrical and based throughout on the most firmly established principles of electromagnetism and general relativity [4]. Using concepts within the field of , Wheeler argued that for a possible history to contribute to the transition probability, the phase of the exponent, which is dependent on the metric g, should be small (∼ 1 rad) to avoid destructive interference [4]. This limits fluctuations in g (i.e., ∆g) that can occur over a volume of space L3 to be on the order of −35 ∆g ∼ Lp/L, where Lp is the Planck length (Lp = 1.61x10 m). The fluctuations in the metric ∆g remains small relative to g until L approaches Lp, at which point, ∆g ∼ g and “the character of the space undergoes an essential change... and multiple connectedness develops” [4] that re- sults in an array of “” in the topology of space with a lattice constant of approximately Lp; he calls this array a ”quantum foam”. These wormholes have an electric charge of qfluct = 12e that produces an intense electromagnetic field energy E that has associated with it a −2 q hc −5 mass of mp = c E = G = 2.17 × 10 g. In this paper, the spatial order of Heisenberg’s lattice world is imposed upon the quantum foam such that it forms a gravity crystal (GC) throughout all of space. It is assumed that the “local compensation” of electromagnetic energy and gravitational energy discussed by Wheeler either does not occur or is incomplete. It is also assumed that the particles either are stable, and do not come into and out of existence as Wheeler describes, or that they are present over most of a time period −44 given by the Planck time tp = 5.39x10 s and only fleeting into and out of existence every tp step in time so as to satisfy Heisenberg’s . With these assumptions, the emperical pseudopotential method (EPM) [5] and tight binding method (TBM)[6] are used to calculate the dispersion curves of non-relativistic and relativistic particles within the GC, as well as the particles’ effective mass as a function of momentum and gravitational mass. It is shown that particles in this crystal can appear to violate Einstein’s Equivalence Principle equating inertial and gravitational masses, but only because the the system includes not only the particle, but the crystal as well. The inertial mass (mi) can diverge from the gravitational mass (mg) as the particle’s momentum h aclto ftedseso uvsi efre,a ela h ffciemse fparticles field the of to masses connection effective its the and REFERENCES produces as crystal well the crystal. discussed. as that gravity are anomalies performed, universe-wide metamaterials inertial a is of The of curves , form crystal. dispersion beyond the this the in not within space is of of it calculation discretization - The the all concerning at Wheeler meta John not is itself. metamaterial nature new of this fabric then the true, but be anything to proves metamaterials concept optical for for and Values via gravity (i.e., done of materials is of case negative effects constant gravitational the a double the in with than between materials) (unity other in relation encountered fluxes constitutive normally magnetic a those and permeability electric gravitational introducing to discussed. 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