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  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 , 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.