An invariant joint alternative, by frames, to Einstein and Schroedinger equations

TL;DR

This paper introduces an invariant, frame-based modification of the wave equation extending the Hodge-de Rham Laplacean, providing a unified approach to gravity and quantum mechanics with novel solutions and experimental consistency.

Contribution

It proposes a new invariant Laplacean operator on frames, offering alternative solutions to Einstein's equations and a Schrödinger-like quantum equation with hyperbolic dynamics.

Findings

Derived a closed, intrinsically different solution from Schwarzschild that matches classical tests.

Extended the equation to time-dependent frames and provided explicit linearized solutions.

Established a hyperbolic quantum-like equation as an alternative to Schrödinger's equation.

Abstract

The Hodge-de Rham Laplacean is an extension to forms of the wave equation. A frame is a quartuple of 1-forms. The Hodge-de Rham Laplacean is modified to model it on the frame itself (not on the standard frame $dx$). This modified Laplacean is invariant. The basic equation is: The modified Laplacean operating on the frame is equal to a source term times the frame. Kaniel and Itin (Il Nuovo Cimento vol 113B,N3,1998) analyzed the equation for steady state and spherically symmetric frame (General Relativity). They computed a closed solution for which the derived metric is Rosen's. This closed solution is intrinsically different than Schwarzschild solution. Yet it passes the three classical experimental tests to the same accuracy. The same basic equation is,also, the alternative to Schroedinger equation, where the source term is the electromagnetic potential. The same quantization as in Schroedinger equation is attained. The suggested system is hyperbolic, in contrast to the time dependent Schroedinger equation which is parabolic. For time dependent frames the equation is quite complicated. Thus, the linearized equation is explicitly solved .