Derivation of the Planck Units Based in a Membranes Model

TL;DR

This paper derives Planck units using a membrane model with specific assumptions, linking fundamental forces and proposing a unification parameter that influences force dominance at microscopic and macroscopic scales.

Contribution

It introduces a membrane-based derivation of Planck units and proposes a unification parameter connecting Coulomb and Newton laws.

Findings

Planck units are derived from membrane oscillation assumptions.

The parameter ξ links electromagnetic and gravitational forces.

Potential V is related to Yukawa's equation and influences force unification.

Abstract

In this study, the Planck units (mass, time and length) have only been derived, explained and attributed a physical meaning when they were deduced based on the concept of interacting membranes (membranes instead of strings of string theory). For this purpose, a set of five assumptions were proposed: (a) the existence of the interacting membranes; (b) the curvatures of the membranes oscillate according to the classical wave equation; (c) the spatial period of the wave that arise when the membranes oscillate is given by $\lambda = {\xi}{\pi}/k$; (d) the membranes oscillate with wavelength given by de Broglie relation and (e) $x=ct$ holds. The parameter $\xi$ determines the period of oscillation of the given membranes. In deriving the Planck units in this work, $\xi$ must take the value 2 and determines a period 2$\pi$, closely to minimum value 1 or to fundamental period $\pi$, respectively. In this context, Planck units must be fundamental. Moreover, the parameter $\xi$ was reported as a unification parameter between the formulas for the Coulomb$^{\prime}$s law and Newton$^{\prime}$s law of universal gravitation linking the forces of microworld and macroworld. Depending on the value $\xi$ takes, one force or another will be had. It is also shown that the potential $V = hc/{\xi}{\pi}x$ deduced from the above assumptions and which contributes to deduce the Planck units, can be derived from Yukawa$^{\prime}$s equation. Hence, the present work would be contributing to theoretical physics, since at the Planck scale predictions of some theories like Standard Model, quantum field theory and general relativity are not expected to be valid.