III.2.9 Method IV (complex momentum and energy-momentum relation)

The equation: , that gives the relativistic energy-momentum, can be also derived directly from the fundamental definition of momentum: , when we include the metaphysical creation of mass in the inner levels of time, in addition to its physical motion in the outer level, and by taking into account the complex character of time. Thus we need to integrate  over the inner and outer levels, according to what happens in each stage; first by integrating between  and  on the inner real levels of time where the particle is created, or being perpetually re-created, at the speed of creation , and this term makes the real part of the complex momentum . Then we integrate between  and  on the outer imaginary level of time where the particle whose mass is  gains an apparent velocity , and thus its effective mass increases from  to , and this term makes the imaginary part of the complex momentum :

 (3.16)

The first term gives us the real momentum , while the second term gives the imaginary momentum . So the total complex momentum is: . Hence the modulus of this total complex momentum is given by:

 (3.17)

Again, we notice here that the contribution of the imaginary momentum  is negative with relation to the real momentum , just like the case of  and  as we have seen in sections 2.4.2 and 3.1.6, and as it will be also the case for complex energy as we shall see in section III.3.1 further below. All this is because the normal time, or physical motion, is interrupting the real creation, which is causing the disturbance and curvature of the otherwise infinite homogeneous Euclidean space that describes the vacuum state .

Therefore, to obtain the relativistic energy-momentum relation from equation 3.17, we simply multiply by :

 (3.18)

These equations 3.17 and 3.18 above, with the negative sign, do not contradict the equation:  in current Relativity, which treats energy as scalar and do not realize its complex dimensions (see also section III.3.1 below). Practically, in any mass-energy interaction or conversion, the negative term will be converted back to positive because when the potential energy  is released, in nuclear interactions for example, this means that it has been released from the inner levels of time where it is captured as mass, into the outer level to become kinetic energy or radiation. In other words: the absorption and emission of energy or radiation, nuclear interactions, or even the acceleration and deceleration of mass, are simply conversions between the inner and outer levels of time, or space and time, respectively. Eight centuries ago, Ibn al-Arabi described this amazing observation by saying: Space is rigid time, and time is liquid space. [12].

This derivation of the relativistic energy-momentum relation from the fundamental definition of momentum  is based on the Duality of Time concept, by taking into account the complex nature of time, as hyperbolic numbers, which is why the contribution of the imaginary term appears here as negative in equation 3.18. As we discussed in section IV.4.2, when we do not realize the discrete structure of space-time geometry that results from this genuinely-complex nature of time, the Duality of Time Theory is reduced to General Relativity, which considers both space and time to be real, and then we take the apparent rather than the complex velocity whose instantaneous value is always zero; so this negative sign in equations 3.17 and 3.18 above will appear positive, as if we are treating space-time to be spherical () rather than hyperbolic ().

Therefore, when we take into account the complex nature of time as we described in sections 1.4.5 and 3.1.6 above (or Figures: 1.4.4 and 3.1.6), energy and momentum will be also complex and hyperbolic. This significant conclusion, that is a result of the new discrete symmetry, will introduce an essential modification on the relativistic energy-momentum equation which will lead to the derivation of the equivalence principle and allows energy to be imaginary, negative and even multidimensional, as it will be discussed further in sections 3.3 and 3.3.1.