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

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

2004-07-13

arxiv.org/abs/math-ph/0407025v8

Int.J.Mod.Phys.D13:1879-1915,2004

Physics

Mathematical Physics

Misprints and errors in some equations of the printed version have been corrected

Scientific paper

10.1142/S021827180400581X

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.

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