On the curvature dynamics for metric gravitational theories

Abstract

Any ’’metric gravitation theory’’ (including general relativity) is shown to determine transport equations for the connection and curvature of the Lorentz frame bundle P₄ defined by the metric g. Observers are generally defined as curves in P₄ which project down to timelike trajectories in space–time. The transport of curvature along an observer trajectory is then given by a Lorentz Lie algebra‐valued current composed of an internal and external part. Einstein’s equations are shown to define one part of the self‐dual limit of the usual Yang–Mills gauge equations, here called a particular form of curvature dynamics. As a consequence, the Yang–Mills‐like energy–momentum tensor, introduced for the Lorentz connection, vanishes identically under Einstein’s vacuum conditions.