Clifford Valued Differential Forms, and Some Issues in Gravitation, Electromagnetism and 'Unified' Theories

TL;DR

This paper develops a Clifford algebra framework for linear metric compatible connections, reformulates Einstein's gravity as a gauge theory, critiques Sachs' unification attempt, and clarifies issues in energy-momentum conservation.

Contribution

It introduces a Clifford valued differential form approach to describe connections and gravity, and critically analyzes unification theories and conservation laws.

Findings

Einstein's gravity can be formulated as an Sl(2,C) gauge theory.

Sachs' unification of electromagnetism and gravity is invalid.

Clarifies misconceptions about energy-momentum conservation in gravity.

Abstract

In this paper we show how to describe the general theory of a linear metric compatible connection with the theory of Clifford valued differential forms. This is done by realizing that for each spacetime point the Lie algebra of Clifford bivectors is isomorphic to the Lie algebra of Sl(2,C). In that way the pullback of the linear connection under a local trivialization of the bundle (i.e., a choice of gauge) is represented by a Clifford valued 1-form. That observation makes it possible to realize immediately that Einstein's gravitational theory can be formulated in a way which is similar to a Sl(2,C) gauge theory. Such a theory is compared with other interesting mathematical formulations of Einstein's theory. and particularly with a supposedly "unified" field theory of gravitation and electromagnetism proposed by M. Sachs. We show that his identification of Maxwell equations within his formalism is not a valid one. Also, taking profit of the mathematical methods introduced in the paper we investigate a very polemical issue in Einstein gravitational theory, namely the problem of the 'energy-momentum' conservation. We show that many statements appearing in the literature are confusing or even wrong.