In a one-dimension Mauldin-Williams Random Cantor Set Universe, the Sigalotti topological speed of light is where . It follows then that the corresponding topological acceleration must be a golden mean downscaling of c namely . Since the maximal height in the one-dimensional universe must be where is the unit interval length and note that the topological mass ( m) and topological dimension ( D) where m = D = 5 are that of the largest unit sphere volume, we can conclude that the potential energy of classical mechanics translates to . Remembering that the kinetic energy is , then by the same logic we see that when m = 5 is replaced by for reasons which are explained in the main body of the present work. Adding both expressions together, we find Einstein’s maximal energy . As a general conclusion, we note that within high energy cosmology, the sharp distinction between potential energy and kinetic energy of classical mechanics is blurred on the cosmic scale. Apart of being an original contribution, the article presents an almost complete bibliography on the Cantorian-fractal spacetime theory.
Space, time, matter and energy are concepts far from being trivial or obvious even within Newtonian classical mechanics [
The crucial turning point for E-infinity was when the Author’s basic work came in touch with the work on non-commutative geometry [
where
In addition this dimensional function is generic and can be used to understand some of the most complex and difficult problems in Physics and Astrophysics [
while the quantum wave maybe modeled by the empty set given by the bi-dimension [
In other words, the zero set quantum particle is described by a bi-dimension, zero for the topological dimension and
From this simple mental and mathematical picture, we were able to show that the volume of the quantum particle zero set in Kaluza-Klein spacetime is simply
where c is the speed of light. In that way, we were able to show that [
By contrast in the present work, we will take another route to arrive at the same result by stressing an optional separation between kinetic energy and potential energy in fractal spacetime.
The following is a “post-modern” and quite novel approach to the same fundamental problems connected to the total accepted theoretical energy density of the universe versus that which was measured and which gave rise to the new concepts of dark energy and dark matter. This problem was previously solved using a plethora of mathematical techniques. However and as we anticipated in the previous section, we are making in the present analysis a strict although optional distinction between potential energy and kinetic energy [
For this reason we start from a one-dimensional Cantor set. For this set everything is zero with the exception of one fundamental thing. The bi-dimension indicated already that the topological dimension is zero. The only thing which is not zero is the Hausdorff dimension which is equal to
or equivalently
In addition the measure i.e. the length of the complementary set is a trivial 1 − 0 = 1. In other words this empty set is a fat Cantor set [
Now let us look at the velocity in D(0). This was established by the work of the notable Italian physicist L. Sigalotti to be
Now in E-Infinity we have a technique similar to non standard analysis were differentiation is equivalent to golden mean down scaling while integration is a golden mean scaling up [
In this case we have to down scale
Again, not surprisingly this corresponds in elasticity to a torsional term and is numerically equal to the Hausdorff dimension of the empty set quantum wave [
Our next step is to determine the height of the mass in the gravity field which is endowed with a positive energy i.e. a potential energy. Since the edges of the unit Cantor interval corresponding to the limit of the universe at a nominal infinity, then the maximum length of the unit interval is simply one half (1/2).
Now we can write heuristically a fractal expression for conventional potential energy for
provided we know what m is. This is easily reasoned if we get access or an insight into the real meaning of mass. This is clearly connected to energy and energy is related to entropy. On the other hand entropy maybe measured via the Hausdorff dimension which is
exactly as shown previously using various other methods. Let us stress this point again. We have just established the potential energy nature of dark energy and squared it with the energy of the quantum particle via a mathematical tautology. This is because at the end of the day it is completely the same thing to say
(12)
or to say
Thus
Returning to the kinetic energy, this is relatively simpler because real energy of real zero set quantum particles is sensibly interpreted as a 3D mass. In this case we have then
Inserting in Newton’s Kinetic energy, we find the expected result
In agreement with expectations, the total energy which is the sum of the Kinetic and the potential energy is equal to Einstein’s maximal energy density [
In concluding this part of our analysis we stress the subtlety of various interpretations of E(D) which could be potentially confusing. This is because
in equal measure as the quantum wave kinetic energy
the spacetime potential energy
We have come a long way in a relatively short time to recognize the depth and beauty involved in the discovery of the so-called missing dark energy of the cosmos. Dark energy is simply potential energy latent in the five-dimensional empty set spacetime. However, one could equally say that dark energy is the energy of quantum wave. Since it may be seen as a product of minus one dimensional empty set, it has a different sign to that of the ordinary energy. Consequently, the topological acceleration
El Naschie, M.S. (2016) Cantorian-Fractal Kinetic Energy and Potential Energy as the Ordinary and Dark Energy Density of the Cosmos Respectively. Natural Science, 8, 511-540. http://dx.doi.org/10.4236/ns.2016.812052