Knot Topology of Vacuum Space-Time and Vacuum Decomposition of Einstein's Theory

TL;DR

This paper classifies vacuum solutions in Einstein's theory using knot topology and presents a gauge-independent decomposition, offering insights into quantum gravity.

Contribution

It introduces a novel classification of vacuum space-times via knot topology and provides a gauge-independent decomposition of Einstein's theory.

Findings

Vacuum space-time classified by knot topology $ o \, ext{pi}_3(S^3) \\simeq \\pi_3(S^2)$

Constructed general vacuum connections with zero curvature

Discussed implications for quantum gravity

Abstract

Viewing Einstein's theory as the gauge theory of Lorentz group, we construct the most general vacuum connections which have vanishing curvature tensor and show that the vacuum space-time can be classified by the knot topology $\pi_3(S^3)\simeq \pi_3(S^2)$ of $\pi_3(SO(3,1))$. With this we obtain the gauge independent vacuum decomposition of Einstein's theory to the vacuum and gauge covariant physical parts. We discuss the physical implications of our result in quantum gravity.