On the Nature of the Gravitational Field

Abstract

In this chapter we investigate the nature of the gravitational field. We first give a formulation for the theory of that field as a field in Faraday’s sense (i.e., as of the same nature as the electromagnetic field) on a 4-dimensional parallelizable manifold M. The gravitational field is represented through the 1-form fields \(\{\mathfrak{g}^{\mathbf{a}}\}\) dual to the parallelizable vector fields \(\{\boldsymbol{e}_{\mathbf{a}}\}\). The \(\mathfrak{g}^{\mathbf{a}}\)’s (a = 0, 1, 2, 3) are called gravitational potentials, and it is imposed that at least for one of them, \(d\mathfrak{g}^{\mathbf{a}}\neq 0\). A metric like field \(\boldsymbol{g} =\eta _{\mathbf{ab}}\mathfrak{g}^{\mathbf{a}} \otimes \mathfrak{g}^{\mathbf{b}}\) is introduced in M with the purpose of permitting the construction of the Hodge dual operator and the Clifford bundle of differential forms \(\mathcal{C}\ell(M,\mathtt{g})\), where \(\mathtt{g} =\eta ^{\mathbf{ab}}\boldsymbol{e}_{\mathbf{a}} \otimes e_{\mathbf{b}}\). Next a Lagrangian density for the gravitational potentials is introduced with consists of a Yang-Mills term plus a gauge fixing term and an auto-interacting term. Maxwell like equations for \(F^{\mathbf{a}} = d\mathfrak{g}^{\mathbf{a}}\) are obtained from the variational principle and a legitimate energy-momentum tensor for the gravitational field is identified which is given by a formula that at first look seems very much complicated. Our theory does not uses any connection in M and we clearly demonstrate that representations of the gravitational field as Lorentzian, teleparallel and even general Riemann-Cartan-Weyl geometries depend only on the arbitrary particular connection (which may be or not to be metrical compatible) that we may define on M. When the Levi-Civita connection of \(\boldsymbol{g}\) in M is introduced we prove that the postulated Lagrangian density for the gravitational potentials differs from the Einstein-Hilbert Lagrangian density of General Relativity only by a term that is an exact differential. The theory proceeds choosing the most simple topological structure for M, namely that it is \(\mathbb{R}^{4}\), a choice that is compatible with present experimental data. With the introduction of a Levi-Civita connection for the structure \((M = \mathbb{R}^{4},\boldsymbol{g})\) as a mathematical aid we can exhibit a nice short formula for the genuine energy-momentum of the gravitational field. Next, we introduce the Hamiltonian formalism and discuss possible generalizations of the gravitational field theory (as a field in Faraday’s sense) when the graviton mass is not null. Also we show using the powerful Clifford calculus developed in previous chapters that if the structure \((M = \mathbb{R}^{4},\boldsymbol{g})\) possess at least one Killing vector field, then the gravitational field equations can be written as a single Maxwell like equation, with a well defined current like term (of course, associated to the energy-momentum tensor of matter and the gravitational field). This result is further generalized for arbitrary vector fields generating one-parameter groups of diffeomorphisms of M in Chap. 14 Chapter 11 ends with another possible interpretation of the gravitational field, namely that it is represented by a particular geometry of a brane embedded in a high dimensional pseudo-Euclidean space. Using the theory developed in Chap. 5 we are able to write Einstein equation using the Ricci operator in such a way that its second member (of “wood” nature, according to Einstein) is transformed (also according to Einstein) in the “marble”nature of its first member. Such a form of Einstein equation shows that the energy momentum quantities \(-T^{\mathbf{a}} + \frac{1} {2}T\mathfrak{g}^{\mathbf{a}}\) (where \(T^{\mathbf{a}} = T_{\mathbf{b}}^{\mathbf{a}}\mathfrak{g}^{\mathbf{b}}\) are the energy momentum 1-form fields of matter and T = T a a ) which characterize matter is represented by the negative square of the shape operator (\(\mathbf{S}^{2}(\mathfrak{g}^{\mathbf{a}})\)) of the brane. Such a formulation thus give a mathematical expression for the famous Clifford “little hills” as representing matter.