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General Metrics in Relativistic Alpha Field Theory

The notion of an alpha field has been associated to any potential field that can be presented by two dimensionless field parameters α and α′. For an example, to this category belong an electromagnetic field, a gravitational field and their combination (a twopotential field). These parameters should satisfy the field equations of the related potential field in each concrete case. The field parameters α and α′ are the functions of the potential energy of the related field. In the multipotential field the total potential energy that influences to a particle in that field can be calculated by adding up of all potential energies in the multipotential field. In that case the field parameters α and α′ become the functions of the total potential energies in the multipotential field. This fact opens the possibilities of derivation of form invariant mathematical descriptions that unify dynamics of one, or two, or more potential fields. Even vacuum (without any potential field) is included, because in that case the field parameters α and α′ are equal to one (α = α′ =1). Following this idea one can derive the form invariant features that unify the well known Einstein’s Special and General Theory of Relativity. The first step in that unification should be the derivation of the related unified metric forms. Thus, the problem in this paper is to derive the generalized metrics in the Relativistic Alpha Field Theory (RAFT). In that sense, it is started with the new General Lorentz Transformation model in an alpha field (GLTα – model) derived by employing the well known group postulates and isotropy of the space. It follows the derivation of the general line element and the related general metric tensor in an alpha field, both in the Minkowski and Riemannian metrics. The one section of the paper is devoted to derivation of a general diagonal form of a line element and metric tensor in an alpha field. It has been shown that there exists a simple coordinate transformation procedure in an alpha field that transforms the Riemannian metrics into the Minkowski one and vice versa. The presented line elements and metric tensors can be used in Special Relativity for α = α′ = 1 and in General Relativity by identification of the field parameters α and α′ in a gravitational potential field, using the well known Einstein’s field equations. Following the considerations in this paper, one can conclude that the unification of the line elements, metric tensors and coordinate transformations in the Einstein's Special and General Theory of Relativity is possible, if one employs the dimensionless field parameters α and α′.

General Metrics; Relativistic Alpha Field Theory; General Line Element; General Metric Tensors; Diagonal Forms in Metrics.

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