A Construct of Dynamics, Space and Gravity from Loops

TL;DR

This paper develops a novel formalism using Loop Algebras to derive space, dynamics, and Einstein's equations, emphasizing structure over length or time dependence.

Contribution

It introduces new loop products and constructs a framework where metric and dynamics emerge from algebraic structures without predefined spacetime.

Findings

Derived Einstein Field Equations in a new algebraic form

Established that dynamics and structure are independent of length and time

Induced metric from Lie algebra properties rather than arbitrary assumptions

Abstract

An attempt is made to construct space and obtain dynamics from Loop Algebras and their elements. We define three new products between loops namely 'Vertical Product', 'Horizontal Product' and 'Total Product'. As for the dynamics, we obtain corresponding "velocity" and "canonical momenta" from it. Also, we build a new "Energy Variable" that is dependent on the velocity and momentum alone. Then, we apply the loop constructs to General Relativity and arrive at the Einstein Field Equations, although presented in a different form. The key feature of this formalism is that the metric is not arbitarized as prevalent on the space beforehand but is rather induced by restricting the Killing Form to the Cartan Sub-algebra of the underlying Lie Algebra. Then we go on to show that "Dynamics is Structure" and that both do not depend on length or time.