2. Discrete space-time


2.1 Introduction. 13 .............................................................................. 13
2.2 Definition of time ....................................................................... 14
2.3 Real and imaginary time ............................................................ 15
2.4 Conclusion ................................................................................ 16

The current theories, based on a space-time continuum, sometimes give rise to the appearance of infinites masked with renormalization, and the denomination of singularity, as in the case of black holes. Instead, in the discrete 4D model, the infinities disappear because at no time can the space-time be zero.

2.1 Introduction

General relativity implies that space-time is a continuum. However, there is no experimental evidence for this. Are space and time a continuum or are they composed of indivisible discrete units? We're probably convinced of continuity as a result of education. In recent years however, both physicists and mathematicians have asked if it is possible that space and time are discrete? Smolin states that space is formed from “irreducible pieces of volume that cannot be broken into anything smaller” that he calls “Atoms of Space and Time” [3].
Minimum volume, length or area are measured in units of Planck [3]. Planck's constant, h, which represents the elementary quantum of action, has an important role in quantum mechanics. There are several theories that predict the existence of a minimum length [4,5]. These theories are related to quantum gravity, such as string theory and double special relativity, as well as black hole physics [6–8].
Quantification of space-time maintains relativistic invariance [9] and causation, and allows us to distinguish elementary particles from each other in a simple and natural way [10].
There is evidence of discrete structures on the largest scales, for example superclusters and the redshift [11]. Cowan already said in 1969 that the redshift can only occur with discrete values [12]. This was subsequently confirmed by Karlsson [13].
Heisenberg said that physics must have a fundamental length scale, and with Planck's constant, h, and the speed of light, c, allow derivation of the masses of the particles [14, 15]. Planck’s length can be considered as the shortest distance having any physical meaning. To Sprenger, “a fundamental (minimal) length scale naturally emerges in any quantum theory in the presence of gravitational effects that accounts for a limited resolution of space-time.” The Planck scale appears to combine gravity (G), quantum mechanics (h), and special relativity (c) [16]. Padmanabhan shows that the Planck length provides a lower limit of length in any suitable physical space-time [17, 18]. Also, Messen starts from a minimum length he calls a, and a four-dimensional space, which allows him to characterize the different types of particles by quantum numbers. Then, different states of the particles correspond to different excitations of space-time [19].
From Planck units Planck force is derived, which is associated with the gravitational potential energy and electromagnetic energy.

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2.4 Conclusion

Time is a two-dimensional physical quantity. The fourth dimension we measure is time due to the expansion of the universe. There is also another time, which is the distance between two points in space. The expansion of the universe generates real and positive time. In addition, when we measure particles with long wavelengths, we have a real and positive time. But when we measure particles with short wavelengths, smaller than the wavelength of the particle, then the time is imaginary

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