Method III (the total relativistic energy):
The total relativistic energy in equation 13 can also be obtained by integrating the starting equation 8 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:
(15)
Thus we get the same equation 13:
(16)
This equation can also be given in the general form that relates the relativistic energy and momentum (see 9 for how to convert between these two equations):
(17)
This last equation, that is equivalent to equation 16, will be also derived in section 6.2.8 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 5.2 above.
Again, however, a fundamental derivation of this relativistic energy-momentum equation 17 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 and also 9), while finding this equation from the four-momentum expression, or space-time symmetry, relies on induction rather than rigorous mathematical formulation.


