Haar Wavelets and the Origin of Gravitational Inertia

TL;DR

This paper introduces a novel approach using Haar wavelets to model spacetime interactions, revealing that graviton wavelength fundamentally influences gravitational attraction and supporting Mach's principle that inertia arises from the universe's matter.

Contribution

It applies Haar wavelet transformations to spacetime metrics, simplifying Einstein's equations and linking graviton wavelength to gravitational inertia, offering a new perspective on gravity and inertia.

Findings

Haar wavelet transforms reduce Einstein's equations to a Poisson form.

Graviton wavelength is identified as the fundamental source of gravity.

Supports Mach's principle that inertia results from the universe's matter.

Abstract

Spacetime is considered to be everywhere Minkowski except at the location where a signal wave of energy interacts with the gravitational field. The conformal metric f[k(x-vt)]Nuv is suitably chosen to represent this interaction, where f[k(x-vt)]is a generalized wave or signal function. Parametrized and Taylor expanded at zero, the spacetime metric is transformed into a Haar wavelet having parameter width tau. Applying the Haar metric to the time component of General Relativistic wave equation reduces it from a second ordered covariant differential equation to a first ordered partial differential equation that allows the Einstein Tensor to be easily be expressed in the familiar Poisson form for gravitation. By comparison with the matter density of this equation, to the Haar-Einstein result, shows that the wavelength of a graviton becomes the fundamental source for gravitational attraction. Since the signal wave is unidirectional, it strongly supports Machs assumption that inertia arises from all the matter in the universe. Furthermore, because the Haar metric is conformal, the signal metric is solved for exactly.