Dynamical determination of the basic Planck units

TL;DR

This paper derives the fundamental Planck units using dynamical principles based on quantum and gravitational length scales, providing a unified approach that relates these units to properties of Kerr-Newman black holes.

Contribution

It introduces a novel dynamical framework connecting Planck units to characteristic length parameters from quantum mechanics and general relativity, including black hole properties.

Findings

Planck charge derived from electrostatic potential energy

Planck mass and length from gravitational potential energy

Unified relations involving Kerr-Newman black hole parameters

Abstract

In this work we consider determination of the basic Planck units (mass, length and charge) using two dynamical principles. First one is definition of the (reduced) Compton length of the physical system (which, as it is well-known, can be considered as the critical parameter that characterizes necessity for application of the quantum field theoretical formalism). Second one is definition of the classical dimension of a physical system (that can be considered as a generalization of the concept of the classical radius of the electron) by equivalence of the system rest energy and potential energy for corresponding length. For electrostatic Coulomb potential energy it implies Planck charge, but without any explicit expression for system mass or (reduced) Compton length. For Newtonian gravitational potential energy it implies Planck mass and Planck length. It is shown that analogous results follow when second dynamical principle is changed by new dynamical principles in which Planck length corresponds to geometrical mean value between (reduced) Compton wavelength and Schwarzschild radius for arbitrary physical system. More generally, it is shown that basic Planck units can be obtained by analogous relations between three characteristic length parameters determined by general relativity dynamics for Kerr-Newman black hole with angular momentum equivalent to reduced Planck constant.