In his work, the Dirac equation was extended by applying 8-dimensional spinors for the decomposition of the square root in the

covariant equation of special relativity.

Francis Fer [6] successfully extended the double solution theory by building a non-linear and

covariant equation wherein the "fluid" is taken as a physical entity.

As well as any general covariant equation, the geodesic deviation equation (2.8) can be projected onto the observer's time line and spatial section (his three-dimensional space) as given in [42, 43] or on page 40 herein.

Equation (8.14) is like the chr.inv.-spatial deviation equation for two free rest-particles (7.30)--the chr.inv.-spatial part of the Synge general covariant equation. The difference is that (8.14) contains derivatives d/d[tao] = 1/[square root of [g.sub.00]] [partial derivative]/[partial derivative]t, while (7.30) contains [partial derivative]/[partial derivative]t.

Using the identity [S.sup.[mu]v] [S.sup.[mu]v] = 8([[omega].sup.2][[pi].sup.2]- [([omega][pi]).sup.2]) together with (151), we obtain five

covariant equations which determine the spin-surface [S.sup.2] in an arbitrary Lorentz frame

If it is then assumed that the tensor forms of the Maxwell equations are the

covariant equations for electromagnetics, the corresponding coordinate transformation that leaves these equations covariant is the coordinate transformation that satisfies the Relativity Principle.

A particle's four-dimensional impulse vector is [P.sup.[alpha]] = [m.sub.0] d[x.sup.[alpha]]/ds, so the general

covariant equations of free motion are

In such a form, the general

covariant equations (2) are represented by the three sorts of their observable (chronometrically invariant) projections: the projection onto an observer's time line, the mixed (space-time) projection, and the projection onto the observer's spatial section [3, 4]

We then express our general

covariant equations in terms of the chosen reference frame.

This condition, being divided by the interval ds, gives general covariant equations of motion of the particle

In particular, the chr.inv.-Maxwell equations, which are chr.inv.-projections of the Maxwell general covariant equations (43), had first been obtained for an arbitrary field potential by del Prado and Pavlov [9], Zelmanov's students, at Zelmanov's request.

Proceeding from the general covariant equations of motion along only time lines, we are going to deduce the energy-momentum tensor for time density fields.