Natural Units and Projection to SI: 50-Minute Lesson Plan

J. Rogers, SE Ohio

Understanding Physics from First Principles

Learning Objectives

By the end of this lesson, students will:

1. Understand that the universe operates on dimensionless proportional relationships

2. Recognize that these natural proportions are the actual physics, true in any measurement system

3. See that physical constants only appear when we project natural ratios into arbitrary unit systems

4. Be able to derive familiar laws by projecting from natural proportions to SI units

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Part 1: The Universe's Natural Proportions (15 minutes)

The Fundamental Truth

The universe doesn't know what a "meter" is. It doesn't know what a "second" is. It doesn't know what a "kilogram" is.

What the universe does have are proportional relationships between different aspects of reality:

    Temperature = Frequency = Mass = 1/Length = Energy = Momentum = Force
  

Or written more clearly:

    T = f = m = 1/l = E = p = F
  

This is the actual real physics. These ratios are invariant - they're true regardless of how you choose to measure them.

What Does This Mean?

At the natural scale:

• Energy IS mass (not "converts to" - IS)

• Mass IS frequency (they're the same thing)

• Momentum IS inverse wavelength (direct proportionality)

• Energy IS temperature (no conversion needed)

• Frequency IS inverse length (fundamental wave relationship)

All of these quantities are different aspects or projections of the same underlying thing.

Why Is This True?

For the same reason π has its value. π = circumference/diameter ≈ 3.14159... isn't something we "discovered through experiment." It's a structural necessity of how circles work in flat space.

The proportionality E = m = f = 1/l is the same kind of thing. It's a structural relationship that's geometrically necessary given how these measurement axes relate to each other.

Just as π is true whether you measure in inches or centimeters, these proportions are true whether you measure in SI units, Imperial units, or any other consistent system.

Student Exercise (3 minutes)

"If energy and mass are the same thing at the natural scale (E = m), why does E = mc² have that huge c² factor? Where does it come from?"

Take responses. Guide toward: "Maybe it's about our choice of arbitrary units..."

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Part 2: Introducing Arbitrary Human Units (12 minutes)

Humans Invented Units

We created measurement systems for practical purposes:

• Meter: Originally one ten-millionth of the distance from equator to North Pole

• Second: 1/86,400 of Earth's rotation period

• Kilogram: Mass of a specific platinum-iridium cylinder in Paris

These choices were:

• Convenient for human-scale activities

• Completely arbitrary relative to fundamental physics

• Not coordinated with each other

The Problem This Creates

The universe says: E = m (at natural scale, ratio 1:1)

But we measure:

• Energy in joules (1 joule = work to lift 102 grams up 1 meter)

• Mass in kilograms (mass of a metal cylinder)

These two units have nothing to do with each other in their construction. They're scaled completely differently.

So when we try to express the natural relationship E = m using these mismatched units, we need a correction factor to account for how differently they're scaled.

The Planck Scale: Finding Natural Proportions

To find the natural proportions, we can construct units from how far off our units of measure are from the geometry of spacetime itself:

• Planck mass: m_P = √(hc/G) ≈ 2.2 × 10⁻⁸ kg

• Planck time: t_P = √(hG/c⁵) ≈ 5.4 × 10⁻⁴⁴ s

• Planck length: l_P = √(hG/c³) ≈ 1.6 × 10⁻³⁵ m

• Planck energy: E_P = m_P c² ≈ 2.0 × 10⁹ J

• Planck temperature: T_P = E_P/k_B ≈ 1.4 × 10³² K

At this scale, all the natural proportions equal 1:

    T/T_P = f·t_P = m/m_P = l_P/l = E/E_P = p/p_P = F/F_P
  

This reveals the actual physics: the dimensionless ratios.

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Part 3: Projecting Natural Physics into SI Units (18 minutes)

The Projection Process

Now we'll see how the natural proportions project into SI, and where constants come from.

Example 1: Energy = Mass

Natural physics:

    E = m
  

In terms of our units:

At the natural scale: E/E_P = m/m_P

Since we're not working at the Planck scale, we need to express E and m in our SI units:

    E/E_P = m/m_P
  

Solve for E:

    E = m · (E_P/m_P)
  

Calculate the ratio E_P/m_P:

    E_P/m_P = (m_P c²)/m_P = c²
  

Therefore in SI:

    E = mc²
  

Where did c² come from?

It's the Jacobian - the scaling factor that converts between our arbitrary mass units (kilograms) and our arbitrary energy units (joules). c = l_P/t_P, this just converts between length and time in our units of measurement. 

The number c² ≈ 9×10¹⁶ isn't telling us something profound about nature. It's telling us: "Your kilogram and your joule are scaled very differently from each other compared to natural proportions."

It's like being amazed there are 63,360 inches in a mile. Big number, but it's just unit mismatch.

Example 2: Energy = Frequency

Natural physics:

    E = f
  

Projection to SI:

At natural scale: E/E_P = f·t_P

Solve for E:

    E = f · (E_P · t_P)
  

Calculate the Jacobian E_P · t_P:

    E_P · t_P = √(hc⁵/G) · √(hG/c⁵) = h
  

Therefore in SI:

    E = hf
  

Or using ν = f and λ = c/ν, we get the familiar: E = hf

The constant h is not fundamental physics. It's the scaling factor (E_P · t_P) that exists because we chose to measure energy in joules and frequency in hertz, and these units are mismatched relative to natural proportions.

Example 3: Momentum = Inverse Wavelength

Natural physics:

    p = 1/l
  

Or: p·l = 1

Projection to SI:

At natural scale: (p/p_P)·(l/l_P) = 1

Rearranging: p·l = p_P·l_P

Calculate p_P·l_P:

       p_P·l_P = (m_P·c)·(l_P) = m_P·c·√(hG/c³) = √(m_P²·c²·hG/c³)

         = √(hc/G)·√(hG/c) = h
  

Therefore in SI:

    p·λ = h  →  p = h/λ
  

This is de Broglie's relation! But it's not something mysterious about quantum mechanics. It's just the natural proportion p = 1/l projected into SI units.

Example 4: Simple Wave Relation

Natural physics:

    f = 1/l
  

This is just: things that oscillate faster have shorter wavelengths.

Projection to SI:

At natural scale: f·t_P = l_P/l

Rearranging: f·l = l_P/t_P

Calculate l_P/t_P:

    l_P/t_P = √(hG/c³) / √(hG/c⁵) = √(c²) = c
  

Therefore in SI:

     λ = c/f
  

The most basic wave relationship. It falls straight out of natural proportions.

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Part 4: The Big Picture (5 minutes)

What We've Learned

1. The actual physics is dimensionless proportional relationships:

   • E = m = f = 1/l (and so on)

   • These are true in any measurement system

   • They're structural necessities, like π

2. Physical constants (c, h, G, k_B) are not fundamental:

   • They're Jacobian scaling factors

   • They exist because we chose arbitrary, mismatched units

   • They're products of Planck units: h = t_P·E_P, c = l_P/t_P, etc.

3. Every physical law can be derived by:

   • Starting with natural proportions

   • Projecting into your chosen unit system

   • The constants appear automatically as Jacobians

Why This Matters

Old way: Memorize E = mc², memorize E = hf, memorize p = h/λ... each formula seems separate, each constant seems mysterious.

New way: Understand E = m = f = 1/l at the natural scale. All formulas are just projections. Constants are just unit conversion.

The physics is simple. The complexity comes from our arbitrary measurement choices.

Looking Forward

In future courses, you'll work more at the natural scale where the physics is transparent:

• Set c = h = k_B = 1

• Do your calculations with simple dimensionless ratios

• Project back to SI at the end when you need numbers

This is how theoretical physics actually works. Now you know why.

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Student Exercise / Homework

Derive the following in SI units starting from natural proportions:

1. Stefan-Boltzmann relationship between energy density and temperature

   • Start with: E = T at natural scale

2. Compton wavelength relationship

   • Start with: m = 1/l at natural scale

3. Newton's gravitational constant relationship

   • Start with: F = m²/l² at natural scale

Show all steps: write the natural proportion, express in Planck units, solve for the SI quantity, calculate the Jacobian.