A new definition of space – key to ToE
If we look at the history of physics, we will see that it progresses together with our understanding of the concept of space. At the beginning, space was seen as three-dimensional Euclidean space. Its usage was obvious since it is expressed as the inverse square law in Coulomb’s law and Newton’s laws of gravitation. Later, in Special Theory of Relativity (STR), three- dimensional Euclidean space was combined with time and reformulated as “flat” 4-dimensional Minkowski spacetime. This solved the inconsistency between classical mechanics and electromagnetism. Then, in General Theory of Relativity, the concept of spacetime was further modified to enable the geometric definition of gravitational force. The “flat” spacetime of CTR was replaced by a “curved” spacetime directly dependent on the energy and momentum of matter or radiation. This explained some gravitational effects. Today there are two distinct theories – the general theory of relativity and the quantum theory, and the hope of mainstream physics is to unify these theories into a Theory of Everything (ToE) – “a hypothetical single, all-encompassing, coherent theoretical framework of physics that fully explains and links together all physical aspects of the universe” (wiki). It is also called a final theory of physics. The dominant ToE model currently is string theory, which postulates 10 dimensional space. However, for the last three decades there has been no experimentalevidence speaking for it. Furthermore, the string theory has not been able to be formulated in a final form, and instead there is a set of variants.
In my view, the current crisis in developing a ToE is not the result of inadequate mathematical tools but an incorrectly formulated concept of space itself. We should start by closely examining what we mean by the concept of space in philosophy (metaphysics) while the past experimental data should be re-interpreted in a new light, without the constraints of existing theories.
If we work through a pile of books on ontology we will see that in metaphysics the concepts of space and time are considered in unity as attributes of matter. In the proposed theory, I have replaced time by motion and downgraded it to a secondary, technical concept derived from motion. Why? The reason is simple. From the beginning of philosophy, time was associated with change and could not be separated from motion. If so, then one must discard Minkowski spacetime from further analysis and return to the time free concept of three-dimensional Euclidean space. Even though mainstream physicists have called for the concept of spacetime to be replaced with something better (here, here ), there has, so far, been no progress.
What brings this change from time to motion in our understanding of the space concept? Time has one direction, from past to future, and it is not dependent on space. On the contrary, motion has directions inseparable from space. First, there are two opposite directions, which can be seen as an increase or decrease of distance between two points in space, or increase/decrease of length. Second, the directed motion can be independent or dependent from each other. The geometrical symbol for directed motion is a vector and a set of vectors can be defined as n- dimensional linear space. Such linear space is n-dimensional if a linearly independent system consists of n vectors, and any system consisting of a large number of vectors is linearly dependent. So, in this case the dimension of space is gaining the definition of number of directed motions, which are independent from each other. In turn, if we use the concept of motion we need also to include definitions of the related concepts of quantity, discreteness and length. These notions imply the definition of set of objects, since a motion of one object is not possible. Accordingly, n-dimensional space can be defined as a set of moving objects, where the number of independent motions of these objects corresponds to the n- dimension of space. In this definition, the concept of space has become equal to the concept of matter, which was always defined as a set of moving objects. Thus, space is not an attribute of matter anymore. It is matter itself. And again, as mentioned above, time is not an attribute of matter, but a secondary, technical concept that originates from motion.
This reformulation of the concepts of space, time and matter opens up a completely new approach to the construction of a ToE. The evolution of matter can now be seen as the evolution of n- dimensional space and the mathematical description of this evolution is what a ToE should be about. I.e. we should consider our universe as evolving from zero-dimension through 1-, and 2- dimesions to three dimension today and four-dimension in future. All that we know about dimensions has come from the framework of three-dimensional Euclidean geometry. With zero-dimensional space, we immediately strike a problem because if we take its definition directly from modern geometry it cannot be real. It is defined as a single vector of zero length. A vector of zero length does not match anything. A physical object of zero length does not exist. This means that either zero-dimensional space does not exist and is only a mathematical abstraction, or the definition of zero- dimensional space requires revision. A revision is obviously needed because the definition of zero-dimensional space does not follow from the general definition of the dimension of space. Zero- dimensional space is introduced as a separate definition complying with the eight axioms of linear space. This forces us to redefine zero-dimensional space using the common definition of dimension where “the dimension of space – is the largest number of linearly independent vectors”. Following this definition, three-dimensional space requires three linearly independent vectors, two-dimensional space requires 2, one-dimensional space requires 1, and zero- dimensional space has 0. Accordingly, zero-dimensional space is characterized by the absence of linearly independent vectors, and that also means that all vectors of zero-dimensional space are linearly dependent. As real objects of zero-dimensional space, these vectors should have non-zero length; the same as all other vectors in spaces of any dimensions.
How a new concept of space was developed into a ToE see the next posts or the book h-space theory