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

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.