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.
spatial part of the Synge general covariant equation.
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.
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
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.