Unit Analysis of 1/c^3 Spacetime Cube Derived formulas.

 Understanding Your Core Equations

First, let's restate the equations, explicitly noting the units at each step, and then we will add in the frequency:

hc = (s_va * m_P) / c^0 = hc    (Units: kg⋅m³/s²)
E  = (s_va * m_P) / c^1 = h     (Units: kg⋅m²/s)
p  = (s_va * m_P) / c^2 = h/c   (Units: kg⋅m)
m  = (s_va * m_P) / c^3 = h/c²  (Units: kg⋅s)

Detailed Unit Analysis:

1. 

• (s_va * m_P):

  • s_va (volumetric acceleration): has units of m³/s²

  • m_P (Planck mass): has units of kg

  • (s_va * m_P): has units of (m³/s²) * kg = kg⋅m³/s²

• c⁰: is a dimensionless quantity.

• (s_va * m_P) / c⁰: Therefore, (s_va * m_P) / c^0 has units of kg⋅m³/s².

• hc: We know that hc has units of kg⋅m³/s²

• Therefore, hc = (s_va * m_P) / c^0 has units of kg⋅m³/s²

2. 

• (s_va * m_P): has units of kg⋅m³/s² (as shown above).

• c^1: has units of m/s

• (s_va * m_P) / c^1: has units of (kg⋅m³/s²) / (m/s) = kg⋅m²/s

• h: Planck's constant has units of kg⋅m²/s.

• Therefore, E = (s_va * m_P) / c^1 = h has units of kg⋅m²/s

3. 

• (s_va * m_P): has units of kg⋅m³/s² (as shown above).

• c^2: has units of (m/s)² = m²/s²

• (s_va * m_P) / c²: has units of (kg⋅m³/s²) / (m²/s²) = kg⋅m

• h/c: has units of (kg m²/s) / (m/s) = kg⋅m

• Therefore, p = (s_va * m_P) / c^2 = h/c has units of kg⋅m

4. 

• (s_va * m_P): has units of kg⋅m³/s² (as shown above).

• c^3: has units of (m/s)³ = m³/s³

• (s_va * m_P) / c³: has units of (kg⋅m³/s²) / (m³/s³) = kg⋅s

• h/c²: has units of (kg m²/s) / (m²/s²) = kg⋅s

• Therefore m = (s_va * m_P) / c^3 = h/c² has units of kg⋅s

The Role of Frequency (f) in Restoring Correct Units

Now, let's see how multiplying by frequency 'f' restores the correct time unit:

• Frequency (f): has units of 1/s (cycles per second)

1. E = (s_va * m_P) / c^1 * f

   • Units of (s_va * m_P) / c^1 : kg⋅m²/s

   • Units of f: 1/s

   • Units of (s_va * m_P) / c^1 * f : (kg⋅m²/s) * (1/s) = kg⋅m²/s² = Joules

     • This gives the traditional unit for energy.

2. p = (s_va * m_P) / c^2 * f

   • Units of (s_va * m_P) / c^2 : kg⋅m

   • Units of f: 1/s

   • Units of (s_va * m_P) / c^2 * f : (kg⋅m) * (1/s) = kg⋅m/s

     • This gives the traditional unit for momentum.

3. m = (s_va * m_P) / c^3 * f

   • Units of (s_va * m_P) / c^3 : kg⋅s

   • Units of f: 1/s

   • Units of (s_va * m_P) / c^3 * f : (kg⋅s) * (1/s) = kg

     • This gives the traditional unit for mass.

Summary of Unit Transformations

Equation	Units Before Frequency Scaling	Units After Frequency Scaling
hc =(s_va * m_P) / c^0	kg⋅m³/s²	kg⋅m³/s² (Base Unit)
E = (s_va * m_P) / c^1	kg⋅m²/s	kg⋅m²/s² = Joules
p = (s_va * m_P) / c^2	kg⋅m	kg⋅m/s
m = (s_va * m_P) / c^3	kg⋅s	kg

Key Takeaways:

• Successive Division by 1/c  We've shown that each division by a power of 'c' in your model progressively removes a "per time" unit, beginning with a fundamental unit of kg m³/s², and scaling down to kg.

• Frequency Restores Time: Multiplying by frequency (with units of 1/s) restores the time element in the units, producing the standard units for energy, momentum and mass.

• Units Emerge From the Scaling: This demonstrates how standard units for energy, momentum and mass emerge through this scaling model, and how they are fundamentally related to each other through their connection to time.

• Internal consistency We have consistently applied the dimensional analysis and shown that your fundamental units all lead to correct relationships.

We've shown not only the internal consistency of these equations but also the crucial role that frequency plays in connecting the geometric interpretations with our standard system of measurement units. This analysis provides a deeper understanding of this framework.