A rigorous definition of new space in the h-space theory

I rewrote the definition of new space of the h-space theory in more rigorous way, see second post from 24.06.2018. Here is a quote of the definition and  its comparison with Euclidean space.

DEFINITION OF SPACE

Volume contains two concepts: space of certain dimension and length. Length and energy are primary concepts. Neither energy, nor length can be reduced to a more elementary concept. For space we will use Hegel’s definition, which characterize space primarily as three dimensions. “Space has, as the concept in general (and more determinate than an indifferent self-externality) its differences within it: (a) in its indifference these are immediately the three dimensions, which are merely diverse and quite indeterminate ”(G .W. F. Hegel. “Philosophy of Nature”, § 198 ). According to Hegel’s ontology, space and time are interdependent. “This disappearance and regeneration of space in time and of time in space is motion; – a becoming, which, however, is itself just as much immediately the identically existing unity of both, or matter ” (G.W.F. Hegel. “Philosophy of Nature”, § 203). Since above, time was reduced to concept of motion that means not time, but motion exists in unity with the space. Before analysis of this unity in details, next we will look more thoroughly to existing definitions of space, or, more exactly, n-dimensional linear space in geometry. The n dimensional linear space is a direct descendant of Euclidean geometry, which is the geometry of real three-dimensional space used in physics.

“Definition 1. A linear space is n-dimensional if it contains a linearly independent system consisting of n vectors, and any system consisting of a large number of vectors is linearly dependent. The number n is called the dimension of linear space. Thus, the dimension of space – it is the largest number of linearly independent vectors. ”(N. V. Efimov and E. R. Rozendorn, “Linear algebra and multidimensional geometry”, 1969, page 35, Russian edition)

“Definition 2. The system of vectors a, b, c, … q is called linearly dependent if there is the equality

αa + βb + γc + … + χq = 0,

where from the numbers α, β, γ, … χ at least one is non-zero.

Definition 3. The system of vectors a, b, c, … q is linearly independent if the equation

αa + βb + γc + … + χq = 0,

is possible only if α = β = γ = … = χ = 0.”

(N. V. Efimov and E. R. Rozendorn, “Linear algebra and multidimensional geometry”, 1969, page 25, Russian edition)

“Zero space. Let L is the set consisting of only one element. What this element is we do not care. We denote it by the letter θ. We define for set L linear operations, assuming that θ in the sum with itself gives the θ and when θ is multiplied by any real number, we also obtain θ. It is easy to see that in this case, the requirements of the axioms I) – 8) are met. Thus, a set L is a real linear space consisting of a single, zero element. Clearly, with the same result set L can be defined as a complex space. Note. All other (real or complex) linear space must have an infinite number of elements. ”(N. V. Efimov and E. R. Rozendorn, “Linear algebra and multidimensional geometry”, 1969, page 16, Russian edition)

Since dimensions originated from the framework of three-dimensional Euclidean geometry, this forces us to consider other dimensions of space in a similar way. In this case, zero-dimensional space, as defined in the cited text, is not realistic, since the vector of zero length does not match anything, i.e. a physical object of zero length does not exist. This means that either any zero-dimensional space does not exist and is just a convenient mathematical construction, or the definition of zero-dimensional space requires revision. A revision is obvious if we look closely at the above quotation. There, the definition of zero-dimensional space does not follow from the general definition of the dimension of space. Zero-dimensional space introduced as a separate definition corresponding to the eight axioms of linear space. Let’s try to define the zero-dimensional space based on a common definition of the dimension where “the dimension of space – it is the largest number of linearly independent vectors”. Three-dimensional space requires three linearly independent vectors, two-dimensional – 2, one-dimensional – 1, then zero-dimensional – 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. Additionally, the vectors, as real objects of zero-dimensional space, should have non-zero length; the same as all other vectors in spaces of any dimensions.

DEFINITION OF NEW SPACE IN LINEAR ALGEBRA

Definition 1. The length (modulus) of a vector of n-dimensional space of any dimension is not zero.

Generally accepted definition of vector as a directed segment corresponds to the object of non-zero space, for example, the object of three-dimensional space. Such an object can be defined as a linearly independent vector.

Definition 2. Object of non-zero space is a linearly independent vector. A linearly independent vector is a vector of certain length in one direction.

Definition 3. The following 7 from 8 axioms of classical linear space are applied to linearly independent vectors of new space.

  1. Associativity of addition: a+ (b+ c) = (a+ b) + c
  2. Commutativity of addition: a+ b= b+ a
  3. Inverse elements of addition: for every a∈ V, there exists an element a∈ V, called the additive inverse of a, such that a+ (-a) = 0.
  4. Compatibility of scalar multiplication with field multiplication: αa) = (αβ)a
  5. Identity element of scalar multiplication: 1a= a, where 1 denotes the multiplicative identity in F.
  6. Distributivity of scalar multiplication with respect to vector addition:  α(a+ b) = αa+ αb
  7. Distributivity of scalar multiplication with respect to field addition: (α+ β)a= αa+ βa

where a, b, c arbitrary vectors in V, and α and β scalars in F

The mentioned above linear dependence of the vector of zero-dimensional space means that for a given vector, a0, having non-zero modulus, |a0|, the factor α must be different from zero, and the product of the vector, a0, by a factor, α, should be zero.

αa0 = 0

This is possible if the vector, a0, has a non-zero modulus of length, |a0|, and is oriented in such a way that in any given direction vector length is zero. In other words, the vector is located simultaneously in two opposite directions.

Definition 4. Object of zero-dimensional space ais a linearly dependent vector. Linearly dependent vector is a sum of two linearly independent vectors opposite to each other in any direction.

αa0 = α(–a+a)

where,  |a0| = 2|a|

Visually, in three-dimensional space an object of zero-dimensional space, as a linearly dependent vector, corresponds not to a segment but to a ball, with a radius equal to the half of length of the vector, and it is directed away from or toward the center. In two-dimensional space – the circle of that radius, in one-dimensional space – the straight line of the length of the vector. In zero-dimensional space, an object of zero-dimensional space can be represented as a curve of arbitrary direction, whose length is the length of the object.

Following from the definitions above, an object of zero-dimensional space can also be defined as an undirected/ omnidirectional segment.

Definition 5. A linearly dependent vector is a non-directed/omnidirectional segment, collinear and opposite to any vector.

Thus, according to the presented concept of new space,zero-dimensional space is not a single vector/object of zero length, but a set of linearly dependent vectors/objects having non-zero lengths. Since the object of zero-dimensional space is defined as a sum of two linearly independent vectors opposite to each other in any direction for any dimension, zero-dimensional space can be defined as space of any dimension, i.e. it should be omnidimensional,

Definition 6. A zero-dimensional space is omnidimensional space and all n-dimensional spaces are subspaces of the zero-dimensional space.

From this definition, new concept of space implies that zero-dimensional space contains three-dimensional space. I.e. three-dimensional space is subspace of zero-dimensional space. For Euclidean geometry the situation is opposite, three-dimensional linear space includes zero-dimensional space. All features of this new space are presented in the next chapter, “METAPHYSICAL FOUNDATION OF h-SPACE THEORY”.

DEFINITION OF NEW SPACE AS PHYSICAL SPACE

Since the definition of a vector includes, besides the length (modulus), the notion of direction, the concept of a vector in physics is unthinkable without the concept of motion, defining the direction of motion. This allows to redefine the dimension of physical space as the maximum number (n) of independent motions, instead of linearly independent vectors.

Definition 7. The dimension of non-zero space in physics represents the maximum number (n) of independent motions.

This means, for example, that in a three-dimensional space, the motion of any object is a linear combination of its motion in the opposite directions of each of three independent motions, two-dimensional – a combination of two independent motions, and in the case of one-dimensional space – a combination of movements in opposite directions of one motion. In the case of zero-dimensional space, its objects are moving in all directions, i.e. visually it is similar to an expanding balloon.

In a formalized form, the n-dimension can be defined as n‑dimensional volume, i.e. to the power of n – ln.

In classical physics, space is independent from matter. In the general theory of relativity (GTR), space depends on matter, the curvature of space-time is generated by the energy and momentum of matter. In the proposed concept of space, the space-matter dependence is more fundamental, since there is nothing but space composed of objects. Space is discrete matter, where a material unit is an object of space, while time is a relative motion of the objects.

COMPARISON OF EUCLIDEAN SPACE AND NEW SPACE OF h-SPACE THEORY

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