5.2.1  Genuinely-Complex Time-Time Flow

The Duality of Time implies that normal time is actually imaginary or latent with relation to the real flow of time that is creating space and matter. Mathematically, this complex time can be represented by imaginary or complex numbers where space is treated as a plane or spherical wave, and time is the imaginary axis perpendicular to it. This complex metaphysical time replaces the non-Euclidean Minkowski space-time coordinatesby Euclidean space coordinates, or time coordinates:whererepresent the inner (real part of) time, andrepresents the outer imaginary or latent time.

In the continuous creation of the Duality of Time hypothesis, the physical dimensions of space are sequentially being re-created in the inner levels of time, which makes the outward time genuinely imaginary with respect to the inner time, and thus easily expressed in terms of Euclidean geometry by using hyperbolic split-complex numbers, as explained further in section 2.2. This will also have deep implications on the various fields of mathematics, such as geometry and number theory, because complex numbers are now genuinely natural while the reals are one of their special approximation.

Without postulating the Duality of Time and the resulting continuous creation of space, this concept of imaginary time does not have any genuine reality or justification outside the mathematical formulation, because both the Galilean space and Minkowski space-time consider space and time to be coexisting together, i.e. they both are real. 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 perpendicular to 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 the outer time, but this is only relative to the dimension in which the observer is situated, so for example: theplane 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 in terms of Clifford geometric algebraemploying bivectors, or even the spinors of complex vector space, allowed expressing the equations in simple forms, but it could not discover the intrinsic granularity of space-time without any background.

Discreteness implies interruption or discontinuity, and this is what the outer time is doing to the continuous flow of the inner time that is creating space and matter. Mathematically, this is achieved by multiplying with the imaginary unit which produces an abrupt rotation by, creating a new dimension that is perpendicular on the previous level. Multiplying with the imaginary unit again causes time to become real again, i.e. like space. Each space-time point, therefore, is the product of seven instances of time, the first six of which make the three spatial dimensions and the seventh is the outer time (see also Figure 5.2). It is not possible otherwise to imagine granular space-time without any background in which the topology is defined, although LQG is one way to get around this obstacle.

Henceforth, the Duality of Time will be based on simple Euclidean geometry, but on complex-time, as it will be explained in section 2.2. This genuinely complex-time frame will allow expressing the (deceitfully continuous) non-Euclidean space-time in terms of this granular Euclidean complex-time space, whose granularity is expressed intrinsically by using the hyperbolic split-complex numbers (), i.e. without invoking Riemannian geometry. This will be discussed further in section 2.2, where it will be shown that Riemannian geometry onis an approximation of this complex-time Euclidean geometry on. This approximation, that lead to the non-Euclidean Minkowski space-time continuum of General Relativity, is necessary only when we consider the dimensions of space to be existing together in (and with) time, thus causing the deceptive continuity of physical existence.

In this complex representation, Lorentz transformation becomes a rotation, as it was originally shown by Poincaré Poincaré (1906), but now this applies to transformations 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 outer level of time is always zero, whether the object is accelerating or not.

This essential characteristic of the dynamic and granular complex time, that results from the Duality of Time Postulate, leads to the three principles of Relativity all at once. Therefore, in addition to making gravity a true quantum field theory, other fundamental interactions could be also represented in terms of this space-time geometry, but in lower dimensions:,,,, as we shall discuss further in section 4.

The concept of imaginary time is already being used widely in various mathematical formulations in quantum physics and cosmology, without any actual justification apart from the fact that it is a quite convenient mathematical trick that is useful in solving many problems. As Hawking stated: “It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed but also effects we have not been able to measure yet nevertheless believe in for other reasons. So what is real and what is imaginary? Is the distinction just in our minds?” Hawking (1998).

Hawking, however, considers the imaginary time as something that is perpendicular to normal time that exists together with space, and that is how it is usually treated in physics and cosmology. According to the postulate of the Duality of Time, however, since space is now the real time, the normal time itself becomes genuinely imaginary.

Employing imaginary time is very useful because it provides a method for connecting Quantum Mechanics with statistical mechanics by using a Wick rotation, by. In this manner we can find a solution to dynamics problems indimensions, by transposing their descriptions indimensions, i.e. by trading one dimension of space for one dimension of time, which means substituting a mathematical problem in Minkowski space-time into a related problem in Euclidean space.

Schroedinger equation and the heat equation are also related by Wick rotation. This method is also used in Feynman’s path integral, which was extended in 1966 by DeWitt into gauge invariant functional-integral DeWitt (1967). For this reason, there has been many attempts to describe quantum gravity in terms of Euclidean geometry, because in this way it is possible to avoid singularities which are unavoidable in General Relativity.

The Duality of Time does not replace or contradict the main exiting theories, but it exposes a deeper understanding of time that leads to metaphysical (non-background) Euclidean complex-time space withsignature, which approximates to the non-Euclidean Minkowski spacewhen we suppose continuous background space with physical dimensions. It is this approximation that lead to General Relativity, but Quantum Gravity can not be achieved without taking into account the new symmetry of this granular complex-time geometry, which is also capable of explaining the other fundamental interactions in lower dimensions, as it will be outlined in section 3 below.