A Unified Invariant Formulation, by Frames, from General Relativity to the Atomic Scale

TL;DR

This paper develops a unified, invariant formulation of fundamental physical laws using frame-based differential forms, deriving classical and quantum equations without relying on traditional postulates like geodesics.

Contribution

It introduces a novel invariant frame-based operator and field equation that generalizes and unifies laws from General Relativity, Newtonian gravity, Coulomb, Maxwell, Schrödinger, and Dirac equations.

Findings

Derives Newtonian and Coulomb laws from the frame-field equation.

Provides invariant formulations of Maxwell, Schrödinger, and Dirac equations.

Proposes a complete alternative to Einstein's vacuum field equations.

Abstract

The aim of this article is the formulation of the basic laws of Physics by frames, i.e. quadruples of exterior differential one forms. The basic operator is a modification of the Hodge-de Rham Laplacian d*d*+*d*d, where * is the hyperbolic star. In this article it is modified depending on the frame. The modified * is invariant w.r. to any diffeomorphism. Consequently, the modified Laplavian is invariant. The field equation developed in this article is a complete alternative to the field equation of General Relativity in vacuum. The frame-field equation yields a derivation of Newtonian (Einstein) law of attraction without recourse to the geodesic postulate. Coulomb law is also derived. Invariant formulation of Maxwell equations is exhibited. Then first order linear approximation is considered. It is used to derive invariant formulation of Schroedinger equation (classical and relativistic) and Dirac equation all of which are linear. The lhs of the field equation, defined on a four dimensional manifold, is the same for all bodies. Thus hopefully, it may set the foundation for a field theory. The interaction of the particles has to be worked out. The basic equation of this article is motivated by the Einstein equation in nonempty space.