In ArXe theory, a hierarchical reduction of fundamental physical dimensions to a single temporal base is proposed.
The proposed mapping is:
$$T = T^1, \quad L = T^2, \quad M = T^3$$
In this way, every physical magnitude can be expressed as a pure power of \(T\), which unifies the traditional dimensions \((M, L, T)\) within a unique temporal hierarchical scale.
Below is the correspondence table and the consistency check.
Conversion Rule
If a magnitude \(X\) has physical dimension:
$$[X] = M^{\alpha} L^{\beta} T^{\gamma}$$
then, under the ArXe hierarchy:
$$[X]_{\text{ArXe}} = T^{3\alpha + 2\beta + \gamma}$$
Step-by-Step Dimensional Reduction
- Basic hierarchical substitution:
It is defined that each physical dimension is an exponentiation of the temporal one:
\(L = T^2\), \(M = T^3\). - Complete expansion:
Given a magnitude \(X\) with dimension \(M^{\alpha} L^{\beta} T^{\gamma}\), we substitute:$$[X] = (T^3)^{\alpha} (T^2)^{\beta} T^{\gamma}$$ - Simplification of exponents:
Adding the exponents of \(T\):$$[X] = T^{3\alpha + 2\beta + \gamma}$$ - Result:
Each physical magnitude is expressed as a unique power of hierarchical time, where the total exponent
\(n = 3\alpha + 2\beta + \gamma\) represents its ArXe exentation level.
Comparative Dimensional Table
| Magnitude | Physical Dimension | Exponents \((M, L, T)\) | ArXe Dimension \([X] = T^n\) |
|---|---|---|---|
| \(c\) | \(LT^{-1}\) | \((0, 1, -1)\) | \(T^{1}\) |
| \(t_p\) | \(T\) | \((0, 0, 1)\) | \(T^{1}\) |
| \(l_p\) | \(L\) | \((0, 1, 0)\) | \(T^{2}\) |
| \(\hbar\) | \(ML^{2}T^{-1}\) | \((1, 2, -1)\) | \(T^{6}\) |
| \(G\) | \(M^{-1}L^{3}T^{-2}\) | \((-1, 3, -2)\) | \(T^{1}\) |
| \(m_p\) | \(M\) | \((1, 0, 0)\) | \(T^{3}\) |
| \(E_p\) | \(ML^{2}T^{-2}\) | \((1, 2, -2)\) | \(T^{5}\) |
Consistency Check
1. Fundamental Relation
$$l_p = c , t_p$$
$$T^{2} = T^{1} \cdot T^{1} \quad \Rightarrow \quad \text{Consistent}$$
2. Planck Time Definition
$$t_p = \sqrt{\frac{\hbar G}{c^5}} \quad \Rightarrow \quad T^{1} = \sqrt{\frac{T^{6} \cdot T^{1}}{T^{5}}} = T^{1}$$
3. Planck Mass and Energy
$$m_p = \sqrt{\frac{\hbar c}{G}} \Rightarrow T^{3}, \qquad E_p = m_p c^2 \Rightarrow T^{5}$$
ArXe Transformation Matrix
The dimensional reduction can be expressed as a linear projection:
$$n = [3, 2, 1] \cdot \begin {bmatrix} \alpha \beta \gamma \end {bmatrix}$$
or in explicit matrix form:
$$\begin {bmatrix} n \end {bmatrix} = \begin {bmatrix} 3 & 2 & 1 \end {bmatrix} \begin {bmatrix} \alpha \beta \gamma \end {bmatrix}$$
This matrix acts as a dimensional collapser that takes any physical combination \((M, L, T)\) to a single hierarchical temporal exponent \(T^n\).
Hierarchical Interpretation
Under this assignment:
- All physical magnitudes are reduced to powers of \(T\).
- The relation \(L = T^2\) and \(M = T^3\) implies that space and mass are hierarchical exentations of time.
- The speed of light \(c = T^1\) is interpreted as the hierarchical equivalence operator between consecutive temporal levels.
- The system is dimensionally closed and self-referential, i.e., each magnitude can be expressed solely through powers of \(T\).