The Algebraic Derivation of SI Unit Variables from Unit Dependent Constants

J. Rogers, SE Ohio

Abstract

This paper treats the base units of the International System (SI) not as fundamental ontological categories, but as dependent variables. By treating

$kg$

,

$s$

,

$m$

, and

$T$

as algebraic unknowns, we demonstrate that these jacobians are simply the scalar solution to the system of equations defined by the constants

$G$

,

$c$

,

$h$

, and

$k_B$

. The constants act as the fixed definitions; the units are the derived quantities.  This is operationally true since the redefinition of constants by committee vote in 2019. 

All we are doing here is showing that the units are not abstractions, they are concrete jacobins inside a unit chart, whose values define that unit chart against a single dimensionless invariant scale of the universe, not features of the universe. 

1. Solving for the Mass Variable (kg)

We begin by isolating the jacobian of Mass. From the system of constants, the mass dimension is isolated by the ratio of action and light speed against gravity.

$kg=\sqrt{\frac{hc}{G}}$

Expanding the units of the constants:

• $h$
    
  $\to [kg\cdot m^2/s]$
   
  
  
       
• $c$
    
  $\to [m/s]$
• $G$
   
  $\to [m^3/(kg\cdot s^2)]$
Substituting these unit dimensions into the equation:

$kg=\sqrt{\frac{[kg\cdot m^2/s]\cdot [m/s]}{[m^3/(kg\cdot s^2)]}}$

 

Simplifying the numerator and inverting the denominator:

$kg=\sqrt{\frac{kg\cdot m^3/s^2}{m^3}\cdot (kg\cdot s^2)}$

 

Canceling the length (

$m^3$

) and time (

$s^2$

) units:

$kg=\sqrt{k{g}^2}$

$kg=kg$

 

The Mass variable is consistent. The constant

$G$

defines the scaling factor for mass relative to Length and Time.

2. Solving for the Time Variable (s)

Having established the Mass variable, we solve for the Time variable using the ratio of Planck's constant to the square of the speed of light.

We identify the composite unit

$kg\cdot s$

:

$\frac{h}{c^2}=\frac{[kg\cdot m^2/s]}{[m^2/s^2]}$

 

Simplifying the right side:

$\frac{h}{c^2}=kg\cdot s$

 

To isolate the second (

$s$

), we divide this quantity by the Mass variable (

$kg$

) established in Step 1:

$s=\frac{h/c^2}{kg}$

 

This shows that the jacobian of Time is a derived jacobian, dependent on the scaling of Length, and Mass.

3. Solving for the Length Variable (m)

Finally, we solve for the unit of Length. The speed of light c defines the relationship between length and time:

$c=\frac{m}{s}$

 

Rearranging to solve for the meter (

$m$

):

$m=c\cdot s$

 

We substitute the Time scaling derived in Step 2 into this equation:

$m=c\cdot \left(\frac{h/c^2}{kg}\right)$

 

We simplify the velocity term (

$c^1/c^2=1/c$

):

$m=\frac{h/c}{kg}$

 

Thus, the jacobian of Length is derived trivially by canceling the second scaling from the speed of light constant, normalizing against the Mass variable.

4. Solving for the Temperature Variable (T)

Temperature is defined by the energy ratio of the Joule (

$J$

) to the Boltzmann constant (

$k_B$

). We solve for the Temperature variable (

$T$

) to show its dimensional consistency.

$T=\frac{J}{k_B}$

Expanding the units:

• $J$
  (Joules)
  $\to [kg\cdot m^2/s^2]$
    
  
  
• $k_B$
  (Boltzmann)
  $\to [J/T]\to [kg\cdot m^2/(s^2\cdot T)]$
Substituting these expansions into the equation:

$T=\frac{[kg\cdot m^2/s^2]}{[kg\cdot m^2/(s^2\cdot T)]}$

 

To solve for

$T$

, we multiply the numerator by the inverse of the denominator:

$T=[kg\cdot m^2/s^2]\cdot \left(\frac{s^2\cdot T}{kg\cdot m^2}\right)$

 

Canceling the Mass (

$kg$

), Length (

$m^2$

), and Time (

$s^2$

) units:

$T=T$

 

Conclusion 

The units

$kg$

,

$s$

,

$m$

, and

$T$

are not independent axioms. They are algebraic solutions to the system of constraints

$G,c,h,k_B$

. The "Planck Scale" is simply the definition of our si unit chart relative to a single invariant scale in the universe.