4.  A Brief General Analysis:

The above postulate means that at every instance of the “real flow of time” there is only one metaphysical point, that is the unit of space-time geometry, and the Universe is a result of its perpetual recurrence in the “inner levels of time”, that is continuously re-creating the dimensions of space and what it may contain of matter particles, which then kinetically evolve throughout the outer (normal) level of time, that we encounter.

To understand this complex flow of time, we need to define at least two frames of reference. The first is our normal  “space” container which evolves in the outer time, that is the normal time that we encounter. And the second frame is the inner flow of time, that is creating the dimensions of space and matter. This inner frame is also composed of more inner levels to account for the creation of  and  space, but we shall not discuss them at this point.

From our point to view, as observers situated in the outer frame, the creation process is instantaneous, because we only see the Universe after it is created, and we don’t see it in the inner frames when it is being created, or perpetually re-created, at every instance. Nevertheless, the speed of creation, in the innermost level (or real flow) of time, is indefinite, rather than infinite, because there is nothing to compare to it at this level of absolute oneness. We shall show that this speed of creation is the same speed of light, and the reason why individual observers, situated in the outer frame, measure a finite value of it is because they are subject to the time lag during which the spatial dimensions are being re-created.

Therefore, in our outer frame, the speed of creation, that is the speed of light, is simply equal to the ratio of the outer to inner times, so it is a unit-less number whose normalized value corresponds to the fractal dimension of the genuinely-complex time-time geometry, rather than space-time, since space itself is created in the inner levels of time. The reason why this cosmological speed is independent of the observer is because creation is occurring in the inner real levels while physical motion is in the outer (normal) time that is flowing in the orthogonal dimension with relation to the real dimensions of space (or inner levels of time).

In other words, while the real time is flowing unilaterally in one continuous sequence, creating only one metaphysical point at every instance, individual observers witness only the discrete moments in which they are re-created, and during which they observe the dimensions of space and physical matter that have just been re-created in these particular instances; thus they only observe the collective (physical) evolution as the moments of their own time flows by, and that’s why it becomes imaginary, or latent, with relation to the original real flow of time that is creating space and matter.

Therefore, the speed of light in complete vacuum is the speed of its dynamic formation, and it is undefined in its own reference frame (as it can be also inferred from the current understanding of time dilation and space contraction of special relativity), because the physical dimensions are not yet defined at this metaphysical level. Observers in all other frames, when they are re-created, measure a finite value of this speed because they have to wait their turn until the re-creation process is completed, so any minimum action, or unitary motion, they can do is always delayed by an amount of time proportional to the dimensions of vacuum (and its matter contents if they are measuring in any other medium). Hence, this maximum speed that they can measure is also invariant because it is independent of any physical (imaginary) velocity, since their motion is occurring in the outer time dimension that is orthogonal onto the spatial dimensions which are being re-created in the inner (real) flow of time.

This also means that all physical properties, including mass, energy, velocity, acceleration and even the dimensions of space, are emergent properties; observable only on the outward level of time, as a result of the temporal coupling between at least two geometrical points or complex-time instances. Moreover, just like the complete dimensions of space, the outer time itself, which is a fractional dimension, is also emerging from the same real flow of time that is the perpetual recurrence of the original geometrical point. This metaphysical entity that is performing this creation is called “the Single Monad”, that has more profound characteristics which we don’t need to analyze in this paper (see Haj Yousef (2014) for more details) for more details); so we only consider it as a simple abstract or dimensionless point: .

It will be shown in section 7 how this single postulate leads at the same time to all the three principles of Special and General Relativity together, since there is no more any difference between inertial and non-inertial frames, because the instantaneous velocity in the imaginary time is always “zero”, whether the object is accelerating or not! This also means that both momentum and energy will be “complex” and “invariant” between all frames, as we shall discuss further in sections 6.3 and 6.4 below.

Henceforth, this genuinely-complex time, or time-time geometry will define a profound discrete symmetry that allows expressing the (deceitfully continuous) non-Euclidean space-time in terms of its granular and fractal complex-time space, whose granularity and fractality are expressed through the intrinsic properties of hyperbolic numbers (), i.e. without invoking Riemannian geometry, as discussed further in section 5.1. However, this hidden discrete symmetry is revealed only when we realize the internal chronological re-creation of spatial dimensions; otherwise if we suppose their continuous existence, space-time will still be hyperbolic but not discrete. Discreteness is introduced when the internal re-creation is interrupted to manifest in the outward normal time, because creation is processed sequentially by the perpetual recurrence of one metaphysical point, so the resulting complex-time is flowing either inwardly to create space, or outwardly as the normal time, and not both together.

Therefore, in accordance with the correspondence principle, we will see in section 5.3, that semi-Riemannian geometry on  is a special approximation of this discrete complex-time geometry on . This approximation is implicitly applied when we consider space and matter to be coexisting together in (and with) time, thus causing the deceptive continuity of physical existence, which is then best expressed by the non-Euclidean Minkowskian space-time continuum of General Relativity, or de Sitter/anti-de Sitter space, depending on the value of cosmological constant.

For the same reason, because we ideally consider the dimensions of space to be continuously existing, our observations become time-symmetric, since we can apparently-equally move in opposite directions. Therefore, this erroneous time-symmetry is reflected in most physics laws because they also do not realize the sequential metaphysical re-creation of space, and that is why they fail in various sensitive situations such as the second law of Thermodynamics (the entropic arrow of time), Charge-Parity violations in certain weak interactions, as well as the irreversible collapse of wave-function (or the quantum arrow of time).

In the Duality of Time Theory, the autonomous progression of the real flow of time provides a straightforward explanation of this outstanding historical problem. This will be explicitly expressed by equation 1, as discussed further in section 5.2, where we will also see that we can distinguish between three conclusive states for the flow of complex-time: either the imaginary time is larger than the real time, or the opposite, or they are equal. Each of the first two states forms one-directional arrow of time, which then become orthogonal, while the third state forms a two-directional dimension of space, that can be formed by or broken into the orthogonal time directions. This fundamental insight could provide an elegant solution to the problems of super-symmetry and matter-antimatter asymmetry at the same time, as we shall discuss in section 5.2.

Additionally, the genuine complex-time flow will be employed in section 6.2 to derive the mass-energy equivalence relation , in its simple and relativistic forms, directly from the principles of Classical Mechanics. This should provide a conclusive evidence to the Duality of Time hypothesis, because it will be shown that an exact derivation of this experimentally verified relation is not possible without considering the inner levels of time, since it incorporates motion at the speed of light which leads to infinities on the physical level. All current derivations of this critical relation suffer from unjustified assumptions or approximations (Einstein, 1905; Planck, 1906; Ives, 1952; Einstein, 1907; Ohanian, 2009), as was also repeatedly acknowledged by Einstein himself (Einstein, 1907; Capria, 2005).

Finally, as an additional support to the Duality of Time Theory, we will show in section 5.4 that the resulting dynamic quintessence will diminish the cosmological constant discrepancy by at least 117 orders of magnitude. This huge difference results simply from realizing that the modes of quantum oscillations of vacuum occur in chronological sequence, and not all at the same time. Therefore, we must divide by the number of modes included in the unit volume, to take the average, rather than the collective summation as it is currently treated in quantum field theories. The remaining small discrepancy could be also settled based on the new structure of physical vacuum, which is shown to be a perfect super-fluid. The Duality of Time Theory, therefore, brings back the same classical concept of aether but in a novel manner that does not require it to affect the speed of light, because it is now the background space itself, being granular and re-created dynamically in time, and not something in a fixed background continuum that used to be called vacuum. On the contrary, this dynamical aether provides a simple ontological reason for the constancy and invariance of the speed of light, which is so far considered an axiom that has not been yet proven in any theoretical sense.

The Duality of Time Theory provides a deeper understanding of time as a fundamental complex-scalar quantum field that reveals the discrete symmetry of space-time geometry. This revolutionary concept will have tremendous implications on the foundations of physics, philosophy and mathematics, including geometry and number theory; because complex numbers are now genuinely natural, while the reals are one of their extreme, or unrealistic, approximations. Many major problems in physics and cosmology can be resolved according to the Duality of Time Theory, but it would be too distracting to discuss all that in this introductory article. The homogeneity problem, for example, will instantly cease, since the Universe, no matter how large it could be, is re-created sequentially in the inner levels of time, so all the states are synchronized before they appear as one instance in the normal level. Philosophically also, since space-time is now dynamic and self-contained, causality itself becomes a consequence of the sequential metaphysical creation, and hence the fundamental laws of conservation are simply a consequence of the Universe being a closed system. This will also explain non-local and non-temporal causal effects, without breaking the speed of light limit, in addition to other critical quantum mechanical issues, some of which are outlined in other publications (Haj Yousef, 2014, 2017, 2019).