5.3.  General Relativity Approximation:

The representation of space-time with imaginary time was used in the early formulation of Special Relativity, by Poincare (Poincaré, 1906), and even Minkowski (Einstein, 2010), but because there were no substantial reasons to treat time as imaginary, Minkowski had to introduce the four-dimensional space-time: , with Lorentzian metric , in which time and space are treated equally, except for the minus. This four-dimensional space later became necessary for General Relativity, due to the presence of gravity, which required Riemannian geometry to evaluate space-time curvatures.

In the split-complex hyperbolic geometry, Lorentz transformations become rotations in the imaginary plane (Fjelstad, 1986), and according to the new discrete symmetry of the time-time frame, this transformation will be equally valid between inertial and non-inertial frames alike, because the dynamic relation between the real and imaginary parts of time implies that the instantaneous velocity in the imaginary time is always zero (see also Figure 5), whether the object is accelerating or not. Therefore, in addition to Lorentz transformations, this essential characteristic of the dynamic complex-time geometry allows direct derivation of the equivalence principle that lead to General Relativity. This will be discussed further in sections 7, 6.1 and 6.3 below.

In the Theory of Relativity, we need to differentiate between inertial and non-inertial frames, because we are considering the “apparent velocity”, since the observer is measuring the change of position (i.e. space coordinates) with respect to time, thus implicitly assuming their real co-existence and continuity; so considering motion to be real transmutation, and that is why space and time are considered continuous and differentiable. The observer is therefore not realizing the fact that the dimensions of space are being sequentially re-created within the inner levels of time, as we described above. This sequential re-creation is what makes space-time complex and granular, in which case the instantaneous velocity is always zero, while the apparent physical velocity is a result of the superposition of all the velocities of the individual geometrical points , which constitute the object of observation, and each of which is either zero, in the outer time, or , in the inner time, as can be calculated by equation 5. So in this hidden discrete symmetry of space, motion is a result of re-creation in the new places rather than gradual and infinitesimal transmutation from one place to the other. Moving objects do not leave their places to occupy new adjacent positions, but they are successively re-created in them, so they are always at rest in any position along the path.

When we realize the re-creation of space at the only real speed , and thus consider the apparent velocity of physical objects to be genuinely imaginary, we will automatically obtain Lorentz transformations, equally for velocity, momentum and energy (which will become also complex, as explained further in sections 6.3 and 6.4), without the need for introducing the principle of invariance of physics laws, so we do not need to differentiate between inertial and non-inertial frames, because the instantaneous velocity is zero in either case. As an extra bonus, we will also be able to derive the mass-energy equivalence relation  without introducing any approximation or un-mathematical induction, and this relation is indeed the same equivalence between gravitational and inertial masses. All this is treated in section 7 below.

Therefore, the non-Euclidean Minkowski space-time continuum is the first global approximation of the metaphysical reality (of Oneness, or sequential re-creation from one single point), just as the Euclidean Minkowski space-time is a local approximation when the effect of gravity is neglected, while the Galilean space is the classical approximation for non-relativistic velocities. These three relative approximations are still serving very well in describing the respective physical phenomena, but they can not describe the actually metaphysical reality of the Universe, which is dynamically re-creating the geometry of space-time itself, and what it contains of matter particles. As Hawking had already noticed: “In fact one could take the attitude that quantum theory, and indeed the whole of physics, is really defined in the Euclidean region and that it is simply a consequence of our perception that we interpret it in the Lorentzian regime.” (Hawking, 1979). The Duality of Time explains exactly that the source of this deceptive perception is the fact that we do not witness the metaphysical perpetual re-creation process, but, being part of it, we always see the Universe after it is re-created, so we “imagine” that this existence is continuous, and thus describe it with the various laws of Calculus and Differential Geometry, that implicitly suppose the continuity of space and the co-existence of matter particles in space and time.

In other words, normal observers, since they are part of the Universe, are necessarily approximating the reality, at best in terms of non-Euclidean Minkowskian space, and this approximation is enough to describe the macroscopic physical phenomena from the point of view of observers (necessarily) situated inside the Universe. However, this will inevitably lead to singularities at extreme conditions because, being inside the Universe, observers are trying to fit the surrounding infinite spatial dimensions in one instance of time, which would have been possible only if they are moving at the speed of light, or faster, and in this case a new spatial dimension is formed and the Universe would become confined but now observed from a higher dimension.

For example, we normally see the Earth flat and infinite when we are confined to a small region on its surface, but we see it as a finite semi-sphere when we view it from outer space. In this manner, therefore, we always need one higher dimension to describe the (deceptive, and apparently infinite) physical reality, in order to contain the curvatures (whether they are intrinsic or extrinsic), and that is why Riemannian geometry is needed to describe General Relativity.

Therefore, since using higher dimensions to describe the reality behind physical existence will always lead to space-time singularities, the Duality of Time Theory is working with this same logic, but backward, by penetrating inside the dimensions of space, as they are dynamically formed in the inner levels of time, down to the origin that is the zero-dimensional metaphysical point, which is the unit of space-time geometry. The Duality of Time Theory is therefore penetrating beyond the apparently-continuous physical existence, into its instantaneous or perpetual dynamic formation through the real flow of time, whose individual discrete instances can accommodate only one metaphysical or geometrical point at a time, that then correlate, or entangle, into physical objects that are kinetically evolving in the normal level of time that we encounter.

At the level of this (unreal) physical multiplicity, any attempt to quantize space-time is destined to fail, because we always need a predefined background geometry, or topology, to accommodate multiplicity and define the respective relations between its various entities. In contrast, the background geometry of the Duality of Time Theory is “void”, which is an absolute mathematical vacuum that has no structure or reality, while also explaining how the physical vacuum is dynamically formed by simple chronological recurrence. So, apart from natural counting, the Duality of Time does not rely on any predefined geometrical structure, but it explains how space-time geometry itself is re-created as dynamic and genuinely-complex structure.

The fact that each frame of the inner time (which constitutes space) appears as one instance on the outward time is what justifies treating time as imaginary with relation to space, thus orthogonal on it. In this dynamic creation of space in the complex time, the outward time is discrete and imaginary, while space becomes continuous with relation to this outer time, but this is only relative to the dimension in which the observer is situated, so for example: the  plane is itself continuous with relation to its inner dimensions but it forms one discrete instance with relation to the flow of time in the encompassing , which then appears internally continuous but discrete with regard to the encompassing outward time. For this reason perhaps, although representing Minkowski space-time in terms of Clifford geometric algebra  employing bivectors (Pavšič, 2005), or even the spinors of complex vector space (Lounesto, 2001), allowed expressing the equations in simple forms, but it could not discover the intrinsic granularity of space-time without any background, since it is still working on the multiplicity level, and not realizing the sequential re-creation process.