En esta entrada os dejo el capitulo 2 del libro
2. Discrete space-time
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.
Planck force can be expressed as
where G is the gravitational constant, mp the Planck mass, c the speed of light, ħ the reduced Planck constant (h/2π), and rp the radius of Planck, which can be expressed
in terms of G, ħ and c as
Substituting all of
these into Equation (2.1), results in
2.2 Definition of time
What relates
time to speed? The answer is obvious, space. Here space is not the distance
between two points, but the space that fills the universe. We can thus define
time as: the variation of the
space-universe with speed. Planck’s particle expands in four
spatial dimensions at the speed of light, giving rise to the appearance of time
as a result of the variation of space with the speed of expansion, c.
Let us now
consider the gravitational force between the sun and the Earth. To simplify
matters, let us suppose that the earth describes a flat circular orbit. To
calculate this path we use the coordinate x-y axes. From the solar perspective,
the only important thing for the sun to exert a force on the Earth is the
distance r between them. It does not matter whether the Earth
is on the right, left, behind or in front, as this force does not vary with
position. The same happens with the expansion of the universe as only one of
the three spatial dimensions is important: the direction between the two points
or objects of reference considered. This direction is at all times
perpendicular to the fourth dimension (radial direction). The increase of space
is two dimensional and for an observer at rest time, will be given by:
If the surface
increase, ΔS, is positive, time will be real and positive. This is proper time,
pertaining to relativity and to the time we can observe and measure. However,
if the surface increase is negative, the time is imaginary. This is the time
that must be considered, for example, when we measure particles with energy
whose wavelength is less than that of the particle, or when we apply
Schrödinger’s equation.
2.3 Real and imaginary time
When we measure electron mass precisely, we need a lot of time, or conversely, little energy, which is equivalent to being at a distance greater than its wavelength. However, when we want to measure its position accurately, we use a lot of energy or short wavelength, which is equivalent to being at distances smaller than the electron’s wavelength. According to the given definition of time, the spherical surface will be formed by the fourth dimension (radial direction), and the direction of the observation (figure 2.1),
so that
will be real and positive for distances greater than the wavelength.
When we get closer to the particle (a distance shorter
than its wavelength) it turns out that the surface variation in the interior of
the surface of radius l=cto is negative, so that:
Here, ti,
is the time in the interior of the surface of radius l and te is the time on the exterior of the
surface. t0 is the time on the surface, or time at rest, or ct0,
which is the wavelength of the quantum mechanics.
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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