III.2.8 Method III (the total relativistic energy):
The total relativistic energy in equation 3.11 can also be obtained by integrating the starting equation 3.6 over the inner and outer time together, since in the inner time the rest energy , or mass , is generated at the speed of creation as a result of the instantaneous coupling between the geometrical points which constitute the particle of mass (thus and in the inner time), and in the outer time the kinetic energy is generated as this mass moves gradually so its apparent velocity changes by , which corresponds to increasing the effective mass from to , thus we can integrate:
(3.13)
Thus we get the same equation 3.11:
(3.14)
This equation can also be given in the general form that relates the relativistic energy and momentum:
(3.15)
This last equation, that is equivalent to equation 3.14, will be also derived in section III.2.9 starting from the definition of momentum as , but because it is genuinely complex, and hyperbolic, the imaginary part of momentum will have negative contribution just as we have seen for the outer time when we discussed the arrow of time in section II.4.2 above.
Again, however, this fundamental derivation of equation 3.15, which gives the relativistic energy-momentum relation, is not possible without the Duality of Time postulate. All the current derivations in the literature rely on the effective mass relation: , which is equivalent to the same relation we are trying to derive (see above), while finding this equation from the four-momentum expression, or space-time symmetry, relies on induction rather than rigorous mathematical formulation.