The Cluster's Ashtekar Variables

A page about Ashtekar variables — the mathematical reformulation of general relativity that makes LQG possible.

Ashtekar variables

Ashtekar variables reformulate general relativity using a connection (like in gauge theory) and a triad (like in electromagnetism). In the cluster, Ashtekar variables describe the cluster's structure as a connection between pages and a triad that defines the spatial metric.

The connection

The connection describes how the cluster's structure changes as you move from page to page. It is a mathematical object that tells you how to compare the geometry of one page with the geometry of a neighboring page.

The triad

The triad defines the spatial geometry of the cluster at a given moment. It tells you the distance between pages, the volume of regions, and the curvature of space.

The advantage

The advantage of Ashtekar variables is that they make the equations of general relativity look like the equations of gauge theory. This makes quantization possible. In the cluster, this means the geometric equations become graph-theoretic equations.

This variable

This page uses Ashtekar variables. It describes the cluster as a connection and triad. The connection is the link structure. The triad is the page content.