One of the interesting (to me) problems in physics is if one can admit continuous functions to the Lagrangian, and still maintain the constancy of c. In STR, the relation M0=CT (for a single physical system) implies that if C is the mass creation rate, then T provides the cutoff (in mass creation time) at which M0 is invariant.
The same is true of VT'; if V is independent of C for all values of V, then V is orthogonal to C; V can be considered a different mass creation rate for a different time T'. However, if a final result is defined as a relation between an initial state and a final state, then the "time dilation" equation obtains, which is actually a relation between mass creation times in terms of mass creation rates:
Note that a coordinate system is not referred to explicitly.
This means that CT is actually the LaGrangian (L = Ekinetic - Epotential
and can be interpreted as either a potential or a kinetic energy, but not both (since the energy defined this way is also independent of the coordinate system).
This means that the equation actually describes a change in state between two mass states M0 and M', where M' = M0*
Further analysis results in the Relativistic Energy equation, which is also discrete for discrete mass creation times, so that the relation between E' and E0 is (E')2=(PC)2 + (E0)2
One can then set Planck's constant H = (PC)2 which is again discrete, so that Edif = E' - E0 = HF where F = 1 (since the coordinate system is still not defined as a "frequency" in space-time)
So this is the fundamental problem Einstein had with quantum mechanics, since the latter description of a change in energy is discrete and quantized, and continuous system (a Riemannian Manifold) does not apply if energy and momentum are conserved. GTR requires a continuous system which changes the "speed" of light continuously as a space-only (contra-variant) transformation that changes xc = ctc in flat coordinate space-time. Such a transformation destroys gauge invariance (conservation of energy), since in such a metric, tv has exhausted all possible conserved mass transformations if an affine connection exists between space-time and energy-momentum domains.
If an affine connection does not exist, then the coordinate domain is that of "Ghosts", and the energy domain is that of "Machines".