5.1.  The Genuinely-Complex Time-Time Frame:

Mathematically, the nested levels of time can be represented by imaginary or complex numbers where space is treated as a plane or spherical wave, and time is the orthogonal imaginary axis. However, in addition to the normal complex number plane: , that can describe Euclidean space, split-complex, or hyperbolic, numbers: , are required to express the relation between space-like and time-like dimensions, which are the inner and outer levels of time, respectively. Normal complex numbers can describe homogeneous or isomorphic space, without (the outer) time, where each number  defines a circle, or sphere, because its modulus is given by , while in split-complex numbers the modulus is given by , so , which defines a hyperbola. This negative sign in calculating the modulus of complex time  reflects the essential fact that the perpetual re-creation of space and matter particles in the inner levels of time  is interrupted and re-newed every instance of the outward time , which produces kinetic motions on the physical level, as dynamic local deformations of the otherwise flat and homogeneous Euclidean space.

Therefore the non-Euclidean Minkowski space-time coordinates  are an approximation of the complex space-time coordinates , or complex time-time coordinates:  where  or  represent the inner real part of time: , and  represents the outer imaginary time, so the total complex time is , where  is the split-imaginary unit which defines hyperbolic numbers, thus: , and its modulus is . These hyperbolic numbers have been also called: tessarines, motors, bireal, perplex, semi-complex, or split-complex, but in this article, unless otherwise stated, imaginary and complex refer to these hyperbolic numbers  (Rochon and Shapiro, 2004).

In this abstract complex frame, space and time are absolute, or mathematical, just as they had been originally treated in the classical Newtonian Mechanics, but now empty space is void, which is a pure mathematical space, because it does not have any material reality, to differentiate it from the physical vacuum, which is the dynamic aether, or the ground state of matter. So, for void both the real and imaginary parts are null: , while for vacuum only the imaginary part is null: , which indicates infinite and inert space that is the ground state of matter particles, that are then described by , or: , which means that they are internally being re-created at the speed of , that is the real part, and moving outwardly at the apparent velocity , that is the imaginary part; given by equation 5.

The physical vacuum, which is the dynamic aether, is therefore an extreme state which may be achieved when the apparent velocity, or momentum, becomes absolutely zero, both as the object’s total velocity and any vector velocities of its constituents, and this corresponds to absolute zero temperature (0K). This dynamic vacuum state is therefore a super-fluid, which is a perfect Bose-Einstein condensate (BEC), since it consists of indistinguishable geometrical points that all share the same state. In quantum field theory, complex-scalar fields are employed to describe superconductivity and superfluidity (Lan, 1965). The Higgs field itself is complex-scalar, and it is the only fundamental scalar quantum field that has been observed in nature, but there are other effective field theories that describe various physical phenomena. Indeed, some cosmological models have already suggested that vacuum could be a kind of yet-unknown super-fluid, which would explain all the four fundamental interactions and provide mass generation mechanism that replaces or alters the Higgs mechanism that only partially solves the problem of mass. In BEC models, masses of elementary particles can arise as a result of interaction with the super-fluid vacuum, similarly to the gap generation mechanism in superconductors (Zloshchastiev, 2011), in addition to other anticipated exotic properties that could explain many problems in the current models, including dark matter and dark energy (HUANG, 2013; HUANG et al., 2012). Therefore, the new complex-time geometry is the natural complex-scalar quantum field that explains the dynamic generation of space, mass and energy. We will discuss the origin of mass in sections 5.5 and 6.2.3 below.

Actually, according to this genuinely-complex time-time geometry, there can be four absolute or “super” states: super-mass , super-fluid , super-gas , and super-energy , which can be compared with the classical four elements: earth, water, air, and fire, respectively. These four extreme or elemental states, which the ancient Sumerians employed in their cosmology to explain the complexity of Nature, are formed dynamically, in the inner levels of time, by the Single Monad that is their “quint-essence”. We will see, in section 5.4 below, that this new concept of aether and quintessence is essential for understanding dark matter and energy, and solving the cosmological constant discrepancy.

Moreover, the super-fluid and super-gas states,  and , are in orthogonal time directions, so if  describes matter that is kinetically evolving in the normal level of time with  velocity,  would similarly describe anti-matter in the orthogonal direction. This could at once solve the problems of super-symmetry and matter-antimatter asymmetry, because fermions in one time direction are bosons in the orthogonal dimension, and vice versa, and of course these two dimensions do not naturally interact because they are mutually orthogonal. This could also provide some handy tests to verify the Duality of Time Theory, but this requires prolonged discussion beyond the scope of this article, as outlined in other literature (Haj Yousef, 2017, 2019). Super-symmetry and its breaking will be also discussed further in section 7.1.