2026-08-25
The origin of the Lorentz factor from an infinite idempotent splitting
Split space theorem
In this paper, we will consider space not as a pre-existing system of independent dimensions, but as a structure capable of successively splitting into new mutually orthogonal states. Each subsequent level arises within the as-yet-undisclosed portion of the previous one and replicates the structure of the entire space. If this splitting continues indefinitely, a single space turns into an infinite sequence of related but independent geometric directions.
We will show that in such a space, a special self-similar state naturally arises: upon transition to each subsequent level, its contribution decreases by the same factor. The total geometric length of this infinite state turns out to be equal to the Lorentz factor. Thus, it can be viewed not only as a well-known coefficient of relativity theory, but also as an intrinsic metric property of an infinitely split space.
Introduction
In the paper "Theorem on the Transformation of a Scalar into a Vector," it was shown that the Lorentz factor can be represented as the length of an infinite-dimensional vector:
\[ \tag{1} \gamma(\beta) = \frac{1}{\sqrt{1-\beta^2}} = \left\| \mathbf j_0+\beta\mathbf j_1+\beta^2\mathbf j_2+\cdots \right\|, \] where \(\mathbf j_0,\mathbf j_1,\mathbf j_2,\ldots\) are mutually orthogonal unit vectors. However, in that work, this basis was introduced in a ready-made form. The question remained open: where does the infinite sequence of orthogonal directions geometrically arise, and why should their coefficients have the form
\[ 1,\quad \beta,\quad \beta^2,\quad \beta^3,\quad\ldots \] In the article "Multilevel Idempotent Splitting of Phase Planes," it was shown that introducing new independent hyperbolic units successively increases the number of mutually orthogonal channels in space. Each new unit creates an additional level of binary splitting.
In this paper, we combine these two results. It will be shown that an infinite successive splitting of unity creates a countable set of orthogonal states, and the only scale-self-similar state of such a space has length
\[ \tag{2} \boxed{ \left\|\boldsymbol{\Gamma}_\beta\right\| = \frac{1}{\sqrt{1-\beta^2}}. } \] After physical identification
\[ \beta=\frac{v}{c} \] This natural length of a state of space coincides with the Lorentz factor.
Thus, the Lorentz factor can be viewed not only as an external transformation coefficient for physical quantities, but also as an intrinsic metric characteristic of an infinitely split self-similar space.
1. Vector Representation of the Lorentz Factor
Consider the ordinary Lorentz factor:
\[ \tag{3} \gamma(\beta)=\frac{1}{\sqrt{1-\beta^2}}, \qquad |\beta|<1. \] The square of this expression can be factored into a geometric series:
\[ \tag{4} \gamma^2(\beta) = \frac{1}{1-\beta^2} = 1+\beta^2+\beta^4+\beta^6+\cdots. \] If we assign a separate orthogonal direction to each term of this series, we obtain the vector
\[ \tag{5} \boxed{ \boldsymbol{\Gamma}_\beta = \mathbf j_0 +\beta\mathbf j_1 +\beta^2\mathbf j_2 +\beta^3\mathbf j_3 +\cdots, } \] where the basis satisfies the conditions
\[ \tag{6} \mathbf j_n\cdot\mathbf j_m = \delta_{nm} = \begin{cases} 1,&n=m,\\ 0,&n\ne m. \end{cases} \] Due to orthogonality, mixed products disappear:
\[ \begin{aligned} \left\|\boldsymbol{\Gamma}_\beta\right\|^2 &= \left( \sum_{n=0}^{\infty}\beta^n\mathbf j_n \right) \cdot \left( \sum_{m=0}^{\infty}\beta^m\mathbf j_m \right)\\ &= \sum_{n,m=0}^{\infty} \beta^{n+m} \left(\mathbf j_n\cdot\mathbf j_m\right)\\ &= \sum_{n=0}^{\infty}\beta^{2n}\\ &= \frac{1}{1-\beta^2}. \end{aligned} \] Therefore,
\[ \tag{7} \boxed{ \left\|\boldsymbol{\Gamma}_\beta\right\| = \frac{1}{\sqrt{1-\beta^2}} = \gamma(\beta). } \] Formula (7) was already obtained in the previous work. But so far it is only an analytical expansion of a predetermined function. Now we need to show that vector (5) arises naturally from the internal structure of the multilevel space.
2. Complete and Successive Splitting
In complete multilevel splitting, each new independent hyperbolic unit separates all existing components. If \(N\) such units are introduced, then
\[ 2^N \] minimal complexical channels.
To construct the Lorentz factor vector, we need a special case of this general construction. At each level, one branch will be fixed as a new state, while the other will remain active and split further.
The resulting sequence is:
\[ \text{original unit} \longrightarrow \text{first state and remainder} \longrightarrow \text{second state and new remainder} \longrightarrow\cdots. \] This splitting does not abolish the complete binary tree. It identifies a single, infinitely continuing chain within it. It is this chain that will generate the states
\[ E_0,E_1,E_2,E_3,\ldots. \] 3. Idempotents of a Single Level
Let each splitting level correspond to an independent hyperbolic unit \(\j_n\):
\[ \tag{8} \j_n^2=1, \qquad \j_n\j_m=\j_m\j_n. \] We define two idempotents for it:
\[ \tag{9} p_n^+ = \frac{1+\j_n}{2}, \qquad p_n^- = \frac{1-\j_n}{2}. \] They satisfy the conditions
\[ \tag{10} (p_n^\pm)^2=p_n^\pm, \qquad p_n^+p_n^-=0, \qquad p_n^++p_n^-=1. \] In the sequential construction under consideration, the \(p_n^-\) branch will fix the state formed at a given level, and the \(p_n^+\) branch will denote the residual part, which continues to split.
The signs \(+\) and \(-\) here have no physical meaning of direction, velocity, or charge. They only distinguish two orthogonal branches of each splitting.
4. Sequential Chain of States
At the first level, the unit is expanded as follows:
\[ 1=p_1^-+p_1^+. \] The first term is fixed, and the second is further split:
\[ p_1^+ = p_1^+p_2^- + p_1^+p_2^+. \] Therefore,
\[ 1 = p_1^- + p_1^+p_2^- + p_1^+p_2^+. \] At the next level:
\[ p_1^+p_2^+ = p_1^+p_2^+p_3^- + p_1^+p_2^+p_3^+. \] As a result
\[ 1 = p_1^- + p_1^+p_2^- + p_1^+p_2^+p_3^- + p_1^+p_2^+p_3^+. \] Let's introduce the states:
\[ \begin{aligned} E_0&=p_1^-,\\ E_1&=p_1^+p_2^-,\\ E_2&=p_1^+p_2^+p_3^-,\\ E_3&=p_1^+p_2^+p_3^+p_4^-,\\ \dots \end{aligned} \] In general terms:
\[ \tag{11} \boxed{ E_n = \left( \prod_{k=1}^{n}p_k^+ \right)p_{n+1}^-, \qquad n=0,1,2,\ldots } \] For \(n=0\), there is no product before \(p_1^-\), therefore
\[ E_0=p_1^-. \] The residual branch after \(N+1\) splits has the form
\[ \tag{12} \boxed{ R_{N+1} = \prod_{k=1}^{N+1}p_k^+. } \] 5. Orthogonality of the Obtained States
We prove that the states \(E_n\) are idempotent and mutually orthogonal.
Lemma 1
For any \(n\) and \(m\):
\[ \tag{13} \boxed{ E_nE_m=\delta_{nm}E_n. } \] Proof
First, let's check idempotency. Since all projectors commute and each of them is idempotent,
\[ \begin{aligned} E_n^2 &= \left[ \left( \prod_{k=1}^{n}p_k^+ \right)p_{n+1}^- \right]^2\\ &= \left( \prod_{k=1}^{n}(p_k^+)^2 \right)(p_{n+1}^-)^2\\ &= \left( \prod_{k=1}^{n}p_k^+ \right)p_{n+1}^-\\ &=E_n. \end{aligned} \] Now let \(m>n\). The projection \(E_n\) contains the factor \(p_{n+1}^-\), while the projection \(E_m\) on the same level contains the factor \(p_{n+1}^+\). Therefore, their product contains
\[ p_{n+1}^-p_{n+1}^+=0. \] Therefore,
\[ E_nE_m=0, \qquad n\ne m. \] The lemma is proved.
Geometrically, this means that two states arising at different levels diverge at one of the splitting points and after that have no common component.
6. Decomposition of Unity
Lemma 2
After \(N+1\) successive splittings, unity is represented as
\[ \tag{14} \boxed{ 1 = \sum_{n=0}^{N}E_n+R_{N+1}. } \] Proof
For \(N=0\):
\[ 1=p_1^-+p_1^+=E_0+R_1. \] Suppose that for some \(N\) we have
\[ 1=\sum_{n=0}^{N}E_n+R_{N+1}. \] Split the residual branch using the following pair of idempotents:
\[ \begin{aligned} R_{N+1} &= R_{N+1}(p_{N+2}^-+p_{N+2}^+)\\ &= R_{N+1}p_{N+2}^- + R_{N+1}p_{N+2}^+. \end{aligned} \] By definitions (11) and (12):
\[ R_{N+1}p_{N+2}^-=E_{N+1}, \] \[ R_{N+1}p_{N+2}^+=R_{N+2}. \] Therefore,
\[ 1 = \sum_{n=0}^{N+1}E_n+R_{N+2}. \] The lemma is proved by mathematical induction.
For any finite number of levels, the residual branch is preserved. In an infinite metric completion, it extends to an unlimited depth, and the sequence of states
\[ E_0,E_1,E_2,\ldots \] forms a countable set of mutually orthogonal channels.
7. From Idempotent Channels to Multidimensional Space
Algebraic Condition
\[ E_nE_m=0,\qquad n\ne m \] shows the independence of the channels. However, to calculate the length of a state, it is necessary to additionally define a metric.
Assign a unit basis vector to each idempotent channel:
\[ \tag{15} E_n\longleftrightarrow\mathbf j_n. \] Introduce the dot product:
\[ \tag{16} \boxed{ \mathbf j_n\cdot\mathbf j_m=\delta_{nm}. } \] Then an arbitrary vector has the form
\[ \mathbf u = u_0\mathbf j_0+ u_1\mathbf j_1+ u_2\mathbf j_2+\cdots, \] and its squared length is
\[ \tag{17} \|\mathbf u\|^2 = u_0^2+u_1^2+u_2^2+\cdots. \] Thus, multilevel idempotent splitting creates a countable-dimensional orthogonal space.
\[ \mathcal H = \operatorname{span} \{\mathbf j_0,\mathbf j_1,\mathbf j_2,\ldots\}. \] It can be understood as an infinite-dimensional space in which directions, orthogonality, length, and convergence of states are defined.
8. A space that reproduces itself
Consider the complete space
\[ \mathcal H = \operatorname{span} \{\mathbf j_0,\mathbf j_1,\mathbf j_2,\ldots\}. \] After selecting the first direction \(\mathbf j_0\), the subspace remaining is
\[ \mathcal H_1 = \operatorname{span} \{\mathbf j_1,\mathbf j_2,\mathbf j_3,\ldots\}. \] It has the same structure as the original space: only the numbering of the directions is different. That's why
\[ \mathcal H_1\simeq\mathcal H. \] Let's introduce the transition operator to the next level:
\[ \tag{18} \boxed{ S\mathbf j_n=\mathbf j_{n+1}. } \] For an arbitrary vector:
\[ S \left( u_0\mathbf j_0+ u_1\mathbf j_1+ u_2\mathbf j_2+\cdots \right) = u_0\mathbf j_1+ u_1\mathbf j_2+ u_2\mathbf j_3+\cdots. \] The operator \(S\) preserves length:
\[ \tag{19} \|S\mathbf u\|=\|\mathbf u\|. \] Furthermore, the entire transferred state is orthogonal to the initial direction:
\[ \tag{20} \mathbf j_0\cdot S\mathbf u=0. \] Therefore, space has a recursive structure:
\[ \tag{21} \boxed{ \mathcal H = \operatorname{span}\{\mathbf j_0\} \oplus S\mathcal H. } \] Formula (21) means that space consists of an initial unit direction and a complete copy of itself, transferred to the next level.
We will call this property the geometric self-similarity of space.
9. The Natural State of Space
Now we need to determine the state consistent with the self-similar structure (21).
Let the state amplitude be multiplied by the same dimensionless coefficient \(\beta\) upon each transition to the next level. Then the complete state must consist of:
- the initial unit state \(\mathbf j_0\);
- a copy of itself, carried over by the operator \(S\);
- the scaling factor \(\beta\) between adjacent levels.
Therefore, the state must satisfy the recursive equation
\[ \tag{22} \boxed{ \boldsymbol{\Gamma}_\beta = \mathbf j_0+\beta S\boldsymbol{\Gamma}_\beta. } \] This equation is not a specially chosen representation of the Lorentz factor. It directly expresses the structure of the space under consideration:
\[ \boxed{ \text{integer state} = \text{initial state} + \text{scaled copy of the integer}. } \] We will call the solution of equation (22) the natural self-similar \(\beta\)-state of space.
10. Why this state is natural
Definition (22) identifies a state not by a predetermined formula, but by its intrinsic geometric properties.
Natural state:
- maintains its shape at all levels;
- does not require a separate choice of coefficient for each direction;
- differs at adjacent levels only by the common factor \(\beta\);
- is consistent with the infinite structure of space itself;
- for \(|\beta|<1\), it has a finite norm;
- is the only state with the listed properties.
It is the existence and uniqueness of such a state that allows us to speak, that it is a natural property of the space, and not an arbitrary construction.
11. The Theorem on the Natural State of a Split Space
Theorem
Let the space \(\mathcal H\) be formed by an infinite sequence of mutually orthogonal states
\[ \mathbf j_0,\mathbf j_1,\mathbf j_2,\ldots, \] and the operator \(S\) takes each state to the next level:
\[ S\mathbf j_n=\mathbf j_{n+1}. \] Then for \(|\beta|<1\), the self-similarity equation
\[ \boldsymbol{\Gamma}_\beta = \mathbf j_0+\beta S\boldsymbol{\Gamma}_\beta \] has a unique finite-norm solution:
\[ \tag{23} \boxed{ \boldsymbol{\Gamma}_\beta = \sum_{n=0}^{\infty}\beta^n\mathbf j_n. } \] The norm of this state is
\[ \tag{24} \boxed{ \left\|\boldsymbol{\Gamma}_\beta\right\| = \frac{1}{\sqrt{1-\beta^2}}. } \] 12. Proof of Existence
Let's start with the self-similarity equation:
\[ \boldsymbol{\Gamma}_\beta = \mathbf j_0+\beta S\boldsymbol{\Gamma}_\beta. \] Let's substitute the right side of the same equation instead of \(\boldsymbol{\Gamma}_\beta\):
\[ \begin{aligned} \boldsymbol{\Gamma}_\beta &= \mathbf j_0+ \beta S \left( \mathbf j_0+\beta S\boldsymbol{\Gamma}_\beta \right)\\ &= \mathbf j_0+ \beta\mathbf j_1+ \beta^2S^2\boldsymbol{\Gamma}_\beta. \end{aligned} \] After the following substitution:
\[ \boldsymbol{\Gamma}_\beta = \mathbf j_0+ \beta\mathbf j_1+ \beta^2\mathbf j_2+ \beta^3S^3\boldsymbol{\Gamma}_\beta. \] After \(N+1\) iterations:
\[ \tag{25} \boldsymbol{\Gamma}_\beta = \sum_{n=0}^{N}\beta^n\mathbf j_n + \beta^{N+1}S^{N+1}\boldsymbol{\Gamma}_\beta. \] Since the operator \(S\) preserves the norm,
\[ \left\| \beta^{N+1}S^{N+1}\boldsymbol{\Gamma}_\beta \right\| = |\beta|^{N+1} \left\|\boldsymbol{\Gamma}_\beta\right\|. \] For \(|\beta|<1\):
\[ \lim_{N\to\infty} |\beta|^{N+1} \left\|\boldsymbol{\Gamma}_\beta\right\| = 0. \] Therefore, the remainder in formula (25) vanishes, and we obtain
\[ \boxed{ \boldsymbol{\Gamma}_\beta = \mathbf j_0+ \beta\mathbf j_1+ \beta^2\mathbf j_2+ \beta^3\mathbf j_3+\cdots. } \] The series has finite norm, since
\[ \sum_{n=0}^{\infty}\beta^{2n} \] converges at \(|\beta|<1\). Therefore, the natural state exists.
13. Proof of Uniqueness
Suppose there are two finite norm states satisfying the same equation:
\[ \boldsymbol{\Gamma}_\beta^{(1)} = \mathbf j_0+ \beta S\boldsymbol{\Gamma}_\beta^{(1)}, \] \[ \boldsymbol{\Gamma}_\beta^{(2)} = \mathbf j_0+ \beta S\boldsymbol{\Gamma}_\beta^{(2)}. \] Subtract the second equation from the first and denote
\[ \boldsymbol{\Delta} = \boldsymbol{\Gamma}_\beta^{(1)} - \boldsymbol{\Gamma}_\beta^{(2)}. \] Then
\[ \boldsymbol{\Delta} = \beta S\boldsymbol{\Delta}. \] Take the norms of both parts:
\[ \|\boldsymbol{\Delta}\| = |\beta|\,\|S\boldsymbol{\Delta}\|. \] Since \(S\) preserves the norm,
\[ \|\boldsymbol{\Delta}\| = |\beta|\,\|\boldsymbol{\Delta}\|. \] For \(|\beta|<1\), this equality is only possible if
\[ \|\boldsymbol{\Delta}\|=0. \] Therefore,
\[ \boldsymbol{\Gamma}_\beta^{(1)} = \boldsymbol{\Gamma}_\beta^{(2)}. \] Thus, the natural self-similar state of space is unique.
14. First Proof of the Lorentz Factor
We obtained the state
\[ \boldsymbol{\Gamma}_\beta = \sum_{n=0}^{\infty}\beta^n\mathbf j_n. \] Let's calculate the square of its norm:
\[ \begin{aligned} \left\|\boldsymbol{\Gamma}_\beta\right\|^2 &= \left( \sum_{n=0}^{\infty}\beta^n\mathbf j_n \right) \cdot \left( \sum_{m=0}^{\infty}\beta^m\mathbf j_m \right)\\ &= \sum_{n,m=0}^{\infty} \beta^{n+m}\delta_{nm}\\ &= \sum_{n=0}^{\infty}\beta^{2n}\\ &= 1+\beta^2+\beta^4+\cdots\\ &= \frac{1}{1-\beta^2}. \end{aligned} \] That's why
\[ \tag{26} \boxed{ \left\|\boldsymbol{\Gamma}_\beta\right\| = \frac{1}{\sqrt{1-\beta^2}}. } \] This expression coincides with the Lorentz factor.
15. Second Proof: The Geometric Pythagorean Theorem
There is a shorter and more geometrically clearer derivation.
The natural state of space consists of two orthogonal parts:
\[ \boldsymbol{\Gamma}_\beta = \underbrace{\mathbf j_0}_{\text{initial state}} + \underbrace{\beta S\boldsymbol{\Gamma}_\beta}_{\text{scaled copy of space}}. \] Since
\[ \mathbf j_0\cdot S\boldsymbol{\Gamma}_\beta=0, \] according to the Pythagorean theorem:
\[ \begin{aligned} \left\|\boldsymbol{\Gamma}_\beta\right\|^2 &= \|\mathbf j_0\|^2 + \beta^2 \left\|S\boldsymbol{\Gamma}_\beta\right\|^2\\ &= 1+ \beta^2 \left\|\boldsymbol{\Gamma}_\beta\right\|^2. \end{aligned} \] Therefore,
\[ \left(1-\beta^2\right) \left\|\boldsymbol{\Gamma}_\beta\right\|^2 = 1. \] From this we again obtain
\[ \tag{27} \boxed{ \left\|\boldsymbol{\Gamma}_\beta\right\| = \frac{1}{\sqrt{1-\beta^2}}. } \] In this proof, the Lorentz factor arises without first expanding the function into a series. It is a direct consequence of the orthogonality and self-similarity of space.
That is why formula (27) can be considered a natural metric characteristic of the geometry under consideration.
16. Nested Form of State
The scalar idea underlying the construction can be written as
\[ 1+x\left[ 1+x\left( 1+x\left( 1+\cdots \right) \right) \right]. \] However, in our case, each subsequent term refers to a new orthogonal direction. Therefore, the vector nesting has the form
\[ \tag{28} \boxed{ \boldsymbol{\Gamma}_\beta = \mathbf j_0+ \beta S \left[ \mathbf j_0+ \beta S \left( \mathbf j_0+ \beta S(\cdots) \right) \right]. } \] After calculating the squared norm, operator transitions turn into factors of \(\beta^2\):
\[ \tag{29} \boxed{ \gamma^2 = 1+\beta^2 \left[ 1+\beta^2 \left( 1+\beta^2 \left( 1+\cdots \right) \right) \right]. } \] The entire expression inside the first square bracket is again equal to \(\gamma^2\), so
\[ \gamma^2=1+\beta^2\gamma^2. \] Hence
\[ \gamma^2=\frac{1}{1-\beta^2}. \] Thus, the nested construction
\[ 1+x(1+x(1+x+\cdots)) \] corresponds to the Lorentz factor when
\[ x=\beta^2, \] if each new unit represents a separate orthogonal state of space.
17. Why do powers of \(\beta^n\) arise?
It's important to clarify that space does not consist of an infinite number of states with the same amplitude \(\beta\). The vector
\[ \mathbf j_0+ \beta\mathbf j_1+ \beta\mathbf j_2+ \beta\mathbf j_3+\cdots \] would have an infinite norm for any \(\beta\ne0\).
In our construction, the coefficient \(\beta\) is applied not separately to each new state, but to the entire remaining copy of the space:
\[ \boldsymbol{\Gamma}_\beta = \mathbf j_0+\beta S\boldsymbol{\Gamma}_\beta. \] To reach the \(n\) level, it is necessary to perform \(n\) successive transitions. Therefore, the amplitude of this level is
\[ \beta^n. \] The resulting sequence is:
\[ \boxed{ 1,\quad \beta,\quad \beta^2,\quad \beta^3,\quad\ldots } \] Therefore, the degrees \(\beta\) are not manually assigned to each level. They arise automatically from the repetition of the same scale transition.
18. Finite-level space
If the splitting ends after \(N+1\) states, we obtain a finite-dimensional vector:
\[ \boldsymbol{\Gamma}_{\beta,N} = \sum_{n=0}^{N}\beta^n\mathbf j_n. \] Its squared norm:
\[ \left\|\boldsymbol{\Gamma}_{\beta,N}\right\|^2 = \sum_{n=0}^{N}\beta^{2n} = \frac{1-\beta^{2N+2}}{1-\beta^2}. \] Therefore, the finite-dimensional analogue of the Lorentz factor is
\[ \tag{30} \boxed{ \gamma_N(\beta) = \sqrt{ \frac{1-\beta^{2N+2}} {1-\beta^2} }. } \] The first approximations are as follows:
\[ \begin{aligned} \gamma_0&=1,\\ \gamma_1&=\sqrt{1+\beta^2},\\ \gamma_2&=\sqrt{1+\beta^2+\beta^4},\\ \gamma_3&=\sqrt{1+\beta^2+\beta^4+\beta^6}. \end{aligned} \] As the number of levels increases:
\[ \tag{31} \boxed{ \lim_{N\to\infty}\gamma_N(\beta) = \frac{1}{\sqrt{1-\beta^2}}, \qquad |\beta|<1. } \] Thus, the exact Lorentz factor corresponds to an infinitely deep splitting of space, and a finite number of levels gives its finite-dimensional approximation.
19. Geometric meaning of the parameter \(\beta\)
In a purely mathematical const\(\beta\) is the coefficient of scale transition between adjacent levels:
\[ \mathbf j_n \longrightarrow \beta\mathbf j_{n+1}. \] After physical identification
\[ \tag{32} \boxed{ \beta=\frac{v}{c} } \] the natural state takes the form
\[ \boldsymbol{\Gamma}(v) = \mathbf j_0 + \frac{v}{c}\mathbf j_1 + \left(\frac{v}{c}\right)^2\mathbf j_2 + \left(\frac{v}{c}\right)^3\mathbf j_3 +\cdots. \] Its length is
\[ \tag{33} \boxed{ \left\|\boldsymbol{\Gamma}(v)\right\| = \frac{1}{\sqrt{1-v^2/c^2}}. } \] With this approach, velocity determines the degree of manifestation of the internal levels of space.
At low velocity:
\[ |\beta|\ll1, \] higher components rapidly decrease:
\[ 1\gg|\beta|\gg\beta^2\gg|\beta|^3\gg\cdots. \] Therefore, the state of space is practically limited to the first directions.
As \(|\beta|\) increases, subsequent levels decrease more slowly, and more and more orthogonal components begin to participate in the full length of the state.
20. Effective Number of Manifested Levels
We can estimate the number of levels whose amplitude exceeds a certain small value \(\varepsilon\). The condition for the noticeability of the \(n\)th level is given by
\[ |\beta|^n\geq\varepsilon. \] After taking the logarithm:
\[ n\leq \frac{\ln\varepsilon}{\ln|\beta|}. \] Therefore, the effective number of manifested levels can be estimated as
\[ \tag{34} \boxed{ N_{\mathrm{eff}} \approx \frac{\ln\varepsilon}{\ln|\beta|}. } \] At \(|\beta|\to1\):
\[ \ln|\beta|\to0, \qquad N_{\mathrm{eff}}\to\infty. \] That is, as \(|\beta|\) approaches unity, more and more levels of split space become significantly more apparent.
21. Boundary States
Case \(\beta=0\)
If
\[ \beta=0, \] then
\[ \boldsymbol{\Gamma}_0=\mathbf j_0, \qquad \|\boldsymbol{\Gamma}_0\|=1. \] Only the initial state is manifested. All subsequent levels have zero amplitude.
Case \(0<|\beta|<1\)
The state involves an infinite number of levels:
\[ 1,\beta,\beta^2,\beta^3,\ldots, \] but their amplitudes decrease. Therefore, the total norm remains finite.
Limit \(|\beta|\to1\)
As the amplitude approaches unity, the suppression of higher levels disappears:
\[ \beta^n\to1. \] Then
\[ \left\|\boldsymbol{\Gamma}_\beta\right\|^2 = 1+\beta^2+\beta^4+\cdots \longrightarrow\infty. \] Geometrically, this means that an infinite number of orthogonal states receive practically the same weight. Such a state no longer has a finite norm.
Case \(|\beta|>1\)
The coefficients
\[ 1,\beta,\beta^2,\beta^3,\ldots \] do not decrease, but increase. Therefore, the state does not belong to the constructed space of finite norm.
Thus, the condition
\[ |\beta|<1 \] arises as a mathematical condition for the existence of a finite natural state of an infinitely split space.
22. The Lorentz Factor as the Intrinsic Length of Space
In ordinary scalar notation
\[ \gamma=\frac{1}{\sqrt{1-\beta^2}} \] The Lorentz factor appears as a coefficient that must be additionally applied to some physical quantity.
In the representation under consideration, it acquires a direct geometric meaning:
\[ \tag{35} \boxed{ \gamma = \left\|\boldsymbol{\Gamma}_\beta\right\|. } \] The coordinates of the natural state are
\[ \{1,\beta,\beta^2,\beta^3,\ldots\}, \] and the Lorentz factor is the total length of this state in all orthogonal directions.
The following logical sequence is obtained:
\[ \boxed{ \begin{aligned} &\text{multilevel idempotent splitting}\\ &\Downarrow\\ &\text{infinite sequence of orthogonal channels}\\ &\Downarrow\\ &\text{self-similar state with coefficient }\beta\\ &\Downarrow\\ &\boldsymbol{\Gamma}_\beta = \mathbf j_0+\beta\mathbf j_1+\beta^2\mathbf j_2+\cdots\\ &\Downarrow\\ &\left\|\boldsymbol{\Gamma}_\beta\right\| = \frac{1}{\sqrt{1-\beta^2}}. \end{aligned} } \] Therefore, the Lorentz factor does not need to be introduced separately into this geometry. It appears as the norm of the unique self-similar state, consistentwith the internal structure of space.
23. In what sense is this property natural?
The word "natural" here has a precise mathematical meaning.
We do not separately assign the coefficients
\[ 1,\beta,\beta^2,\ldots \] and do not select them specifically to obtain a known formula. Instead, only the basic properties of space are specified:
\[ \boxed{ \begin{aligned} &\text{1. Infinite splittability;}\\ &\text{2. Orthogonality of the emerging states;}\\ &\text{3. Self-similarity of the residual part;}\\ &\text{4. A single transition coefficient }\beta. \end{aligned} } \] From these follows the recursive equation
\[ \boldsymbol{\Gamma}_\beta = \mathbf j_0+\beta S\boldsymbol{\Gamma}_\beta. \] This equation has a unique solution of finite norm, and its length is inevitably equal to
\[ \frac{1}{\sqrt{1-\beta^2}}. \] Therefore, the Lorentz factor is not a random result of the chosen decomposition, but a necessary metric characteristic of the space with the listed properties.
24. Boundaries of the Obtained Result
For accuracy, it is necessary to distinguish between results obtained directly from algebra and additional properties of the spatial model under consideration.
Multilevel idempotent splitting yields:
- a system of mutually orthogonal channels;
- the ability to sequentially select states \(E_n\);
- an infinitely continuing splitting structure;
- an algebraic basis for a multidimensional space.
Additionally, the following are introduced:
- metric correspondence \(E_n\longleftrightarrow\mathbf j_n\);
- self-similarity of the remaining part of the space;
- a single scale factor \(\beta\);
- physical identification \(\beta=v/c\).
The mere presence of idempotent splittings does not select the Lorentz factor among all possible functions. It is uniquely selected by the combination of splitting, orthogonality, and scaling self-similarity.
It is the combination of these properties that defines the model of space under consideration.
Conclusion
In the paper on the transformation of a scalar into a vector, the Lorentz factor was represented as the length of an infinite-dimensional vector:
\[ \boldsymbol{\Gamma}_\beta = \mathbf j_0+ \beta\mathbf j_1+ \beta^2\mathbf j_2+\cdots. \] This work demonstrates a possible internal origin for this vector.
Successive multi-level idempotent splitting creates a chain of mutually orthogonal states:
\[ E_0,E_1,E_2,\ldots. \] After introducing the metric, they correspond to an orthonormal basis:
\[ \mathbf j_0,\mathbf j_1,\mathbf j_2,\ldots. \] The part of the space remaining after the first direction has the same structure as the entire space. If the transition to the next level is accompanied by a single scaling factor \(\beta\), the natural state must satisfy the equation
\[ \boldsymbol{\Gamma}_\beta = \mathbf j_0+\beta S\boldsymbol{\Gamma}_\beta. \] It has been proven that for \(|\beta|<1\), this equation has a unique solution of finite norm:
\[ \boldsymbol{\Gamma}_\beta = \sum_{n=0}^{\infty}\beta^n\mathbf j_n. \] Its length is
\[ \boxed{ \left\|\boldsymbol{\Gamma}_\beta\right\| = \frac{1}{\sqrt{1-\beta^2}}. } \] After identifying \(\beta=v/c\), this length coincides with the Lorentz factor.
Thus, the Lorentz factor can be viewed not as an external correction imposed on space, but as the length of its own infinitely deep state. The larger \(|\beta|\), the slower the higher-level components decay and the more orthogonal states participate in forming the total geometric length.
The main result of the work can be expressed as follows:
\[ \boxed{ \begin{gathered} \text{infinite splittability} + \text{orthogonality} + \text{scale self-similarity}\\ \Downarrow\\ \boldsymbol{\Gamma}_\beta = \{1,\beta,\beta^2,\beta^3,\ldots\}\\ \Downarrow\\ \gamma = \left\|\boldsymbol{\Gamma}_\beta\right\| = \frac{1}{\sqrt{1-\beta^2}}. \end{gathered} } \] Therefore, the Lorentz factor is a natural metric characteristic of an infinitely split self-similar space.