Curvature Propagation for the 3+1 Dimensional U(1) Symmetric Einstein Spacetimes

TL;DR

This paper explores the structure of U(1) symmetric Einstein spacetimes in 3+1 dimensions, drawing analogies with Yang-Mills theory to aid in understanding the global behavior of Einstein's equations.

Contribution

It introduces a new perspective by relating U(1) symmetric Einstein spacetimes to Yang-Mills theory, extending previous work on dimensional reduction.

Findings

Identifies an analogy between Einstein equations and Yang-Mills theory in U(1) symmetric spacetimes

Provides a framework for analyzing the global structure of Einstein equations using this analogy

Reconciles the reduced field equations with the Yang-Mills analogy

Abstract

As it is well known, the global structure of the Einstein equations for general relativity in the context of the initial value problem, is a difficult and intricate mathematical problem. Therefore, any additional structure in their formulation is useful as a tool for studying the global behaviour of the initial value problem of the Einstein equations. In our previous works, we have used the additional structure provided by the dimensional reduction to $2+1$ dimensional Einstein-wave map system. In this work, we shall focus on yet another structure in the Einstein equations for the U(1) symmetric spacetimes, namely the analogy with the Yang-Mills theory and reconcile with the dimensionally reduced field equations.