Physics in deformable spacetime: Physical laws emerging from the surface minimality principle and the masses of particles

In this work, the study was focused on how deformable spacetime gives rise to physical laws, particularly quantum and special relativity. It was shown that both the Lorentz factor and quantum mechanics can emerge from the dynamical behavior of minimal surfaces in deformable spacetime, where the Möbius transformations prevail rather than the Lorentz transformations. The motion of deformable surfaces under minimality conditions creates a helicoidal pattern, a minimal surface with a metric designated by AdS (anti-de Sitter) space. Its isometry is a catenoid of which metric belongs to dS (de Sitter) space. When the two ends of the helicoid join it becomes a Möbius strip with multi-twists (i.e. windings), the so-called Möbius helicoid. It represents the most elementary mass with the attributes of charge and spin. It was shown that the number of windings increases with velocity according to the Lorentz factor. In other words, the Lorentz factor emerged from the transformation equations between the helicoid and catenoid, and both of them are isometric minimal surfaces. This fact serves as a means of transforming the metrics of the two isometric surfaces in AdS and dS spaces. There is an intimate relation between helicoidal motion, Möbius helicoid, and the Lorentz transformations. The zitterbewegung behavior of Dirac electron can be explained by the internal dynamics of the Möbius helicoid. Instead of using electromagnetic theory which led to the birth of special relativity, one can use the electrical circuit theory for deformable surface dynamics. A master equation was derived by using the resistive, capacitive, and inductive properties of deformable spacetime. The electromagnetic wave, Schrödinger, Klein–Gordon, Dirac, quantum telegrapher, and quantum torsion equations could be easily obtained from this master equation by making some simplifying assumptions for the parameters involved. The existing torsion due to helicoidal motion modifies the energy/mass equation (i.e.E=mc2) of special relativity. The Higgs potential can be obtained from the dynamic potential term of the master equation. Deformable space–time allows the emergence of a rich number of physical laws and relations. The masses of particles were calculated using the Möbius helicoid model for the very elementary structure of mass. The mass of a particle can be visualized as a three-dimensional network of helicoids with relatively higher stability. The stability is particularly associated with the phase changes of the catenoidal structure of the Möbius helicoid and the torsional vibrational modes of windings. Both the phase (or shape) change and/or the increase of the vibrational energies of windings produce a new configuration that represents a new particle. The masses of 187 particles were determined from this mechanism with an accuracy of ±3 MeV. It turned out that excitation energy states which could be expressed in terms of prime number multiples of a precursor mass play an important role in providing relatively higher stability.