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III. An Invariantive Principle

From the point of view just illustrated it is to be expected that, for a particle in an arbitrary environment, the proportionality between the velocity and momentum 4-vectors fails. In this case equation (2) suggests that a particle in a general environment can be described by a 2-form which, in general, is not an exterior product. Let this differential form be

(4)

Thus, from the invariantive point of view we have to describe the non-free particle by two differential forms, namely the 1-form ω from (1), representing the momentum of particle and the 2-form Ω from (4), which must therefore contain the conditions of environment where the particle exists and moves. The case from equation (2) is then a consequence of the fact that the 2-form Ω is null over the field of vectors where the 1-form ω is null. In that case, according to another theorem of Cartan [3], Ω is necessarily of the form

(5)

where ω’ is a conveniently chosen differential 1-form. Thus, in this dynamics the velocity form is only a matter of convenience. One can say that the classical results are recovered when the 2-form Ω is zero, over the regions of Space-Time where ω is not zero. Only in those cases we have a field of velocities that can conveniently describe the momentum of particles. Otherwise, when both the momentum form and the 2-form (4) are nonzero, the invariantive equations of motion ought to be written some other way.

The first obvious idea is to treat the 2-form (4) as a kind of potential form. The invariantive equations of motion can, for once, be obtained from the equality between the exterior derivative of ω and Ω, i.e. from the equation

(6)

In view of this equation, Ω must be, physically, an action so that the coefficients ω_ij must be rates of change of mass (or energy) between the particle and the environment. Therefore, this model of particle has, for instance, the mechanical transport characteristics of that used by De Broglie in his “Hidden Thermodynamics” [4]. This suggests that ω_ij must also involve some thermodynamic properties, but let us not elaborate on the meanings for the moment being.

Instead, let us see what formal properties are prompted by our dynamical principle from equation (6). One of these properties follows directly from equation (6): namely, Ω must be closed, i.e. d^Ω = 0 , from which, in turn, it follows

(7)

This equation allows us to express ω_ij as a curl, like in the Classical Electrodynamics. It is equivalent to a first set of Maxwell’s equations. On the other hand, proceeding again directly from (6), we can write

(8)

and, by still another theorem of Cartan [3], we have

(9)

where the matrix (h) is symmetric in its indices. The equations of motion can now be written in the form

(10)

From these, we can obtain a second set of Maxwell equations, as follows [2]: define the tensor

(11)

which is the so-called Hilbert stress tensor associated to ω_ij. Here we put

(12)

and the indices are raised or lowered with the matrix (h_ij). One can see that we use this last matrix as a metric tensor. The conditions to be satisfied by (11) in order to be considered as a true energy tensor are

(13)

where symbol D_j denotes the covariant derivative in the natural connection of tensor (h_ij). For this derivative we have then [2]

(14)

where we used the notation

(15)

This last quantity vanishes in the virtue of (7) and the skew symmetry of ω_ij. Thus, the equation (14) becomes

(16)

and (13) takes place if, and only if,

(17)

in view of the fact that det (ω_j^k)≠ 0. These equations are the Born-Infeld equations of the Nonlinear Electrodynamics [2, 5]. Here they are equivalent to a conservation law representing an equilibrium equation, viz. equation (13), and play the part of a second group of equations, necessary to assure, at least in principle, the determination of the matrix (ω_ij) on one hand, and to define a correct stress-energy tensor on the other hand.

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