J. Rogers, SE Ohio
Materials:
Whiteboard or Projector Handout with a list of common physical quantities and their Planck unit equivalents (e.g., M_nat = M_si / m_P).
Lesson Structure & Instructor Notes
Start with a quick, powerful recap of the last class. "Last time, we proved that every dimensional 'constant' is a conversion factor—a '2.54'—that exists only to fix the dimensional imbalance in our SI unit system." Write on the board: G = l_P³ / (m_P · t_P²), c = l_P / t_P, h = m_P · l_P² / t_P. "We proved these are properties of our rulers, not properties of the universe. This is the bookkeeping ."Pose the central question for today's lesson: " So, if this is all just bookkeeping... where is the PHYSICS? What are the actual, real laws of the universe, if not these formulas? "
This opening immediately validates the student's own intellectual curiosity. It frames the lesson not as another lecture, but as the answer to the question they should all be asking. The clear separation between "bookkeeping" and "physics" is the central theme. You are now moving from the former to the latter.
"The real physics is hidden underneath our SI formulas. We're going to learn a new algorithm to strip away the unit-system artifacts and reveal the simple, invariant law underneath." Introduce "The 3-Step Algorithm for Finding the Invariant Law." Use Hawking Radiation as the first, dramatic example. Step 1: Start with the familiar SI formula. Write T = ħc³ / (8πGk_B M). (Use ħ here to show it works for that too). "This looks like a mess of constants. It looks complex. It's not."Step 2: Substitute the Planck unit definitions for every constant. Go through it systematically.ħ = h / 2π = (m_P l_P²/t_P) / 2π c³ = (l_P/t_P)³ G = l_P³ / (m_P t_P²) k_B = m_P l_P² / (t_P² T_P) Substitute them all into the formula. It will look like a horrifying mess of Planck units. This is intentional.
Step 3: Cancel everything. The algebra will reveal the truth. "Now, we just do the algebra. Watch the magic. The bookkeeping cancels itself out."Systematically cancel the Planck units. m_P cancels m_P. l_P⁵ cancels l_P⁵. t_P⁻⁴ cancels t_P⁻⁴. After the dust settles, all that will be left is: T = T_P / (8πM). (Assuming M is measured in Planck masses, or T/T_P = 1 / (8π M/m_P)).
Rewrite the result in its final, beautiful form: T_nat ~ 1 / M_nat. State the conclusion: " This is the REAL physics. The actual law of Hawking radiation is that temperature is inversely proportional to mass. That's it. The entire mess of ħc³ / Gk_B was nothing but the composite conversion factor needed to say that simple sentence in the clumsy language of kilograms and Kelvin. "
The "reveal" is crucial. Starting with the complex SI formula and reducing it to a simple proportionality through pure algebra is a powerful demonstration. The mess of canceling Planck units is theatrical and effective—it's like a magician revealing the mundane mechanics behind a grand illusion.You are teaching them to "un-compile" the code of physics. They are learning to reverse-engineer the complex machine code (SI formula) to find the simple, elegant source code (the invariant law).
" This isn't a special case. All the great laws of physics are this simple underneath. Let's un-compile a few more."Go through several key examples on the board, starting with the SI formula and reducing it to the invariant law. Einstein's Mass-Energy Equivalence: Start: E = mc² Substitute: E = m (l_P/t_P)² Normalize: E/ (m_P l_P²/t_P²) = m/m_P -> E / E_P = m / m_P Invariant Law: E_nat = m_nat . ("Energy and mass are the same thing.")
Schwarzschild Radius: Start: r_s = 2GM/c² Substitute: r_s = 2 (l_P³/m_P t_P²) M / (l_P/t_P)² Cancel & Normalize: r_s / l_P = 2 M / m_P Invariant Law: r_nat = 2 m_nat . ("The size of a black hole is proportional to its mass.")
de Broglie Wavelength: Start: λ = h/p Substitute & Normalize... Invariant Law: λ_nat = 1 / p_nat . ("Wavelength is inversely proportional to momentum.")
Build a "greatest hits" list. By revealing the simple, one-line truths behind the most famous and intimidating equations in physics, you are building an overwhelming case.Translate each invariant law into a simple English sentence. E_nat = m_nat becomes "Energy and mass are the same thing." This connects the abstract math to a clear, powerful physical concept.
Hand out a new worksheet. On one side are famous, constant-laden SI formulas. On the other side are simple, invariant proportionalities. "Your task is to match them. Use the algorithm: start with the SI formula, substitute the Planck definitions for the constants, cancel everything, and find the simple truth underneath." Let the students work in groups. This is the core activity. They are now actively doing the work of a natural philosopher. Example problems: Match F = G m₁m₂/r² to F_nat ~ m₁_nat m₂_nat / r_nat². Match the Stefan-Boltzmann law j = σT⁴ to j_nat ~ T_nat⁴.
This activity reinforces the "un-compiling" skill. It's a puzzle that requires them to apply the new algorithm. The matching format makes it less intimidating than a blank-slate derivation and provides immediate feedback. They are learning to see past the surface complexity of a formula to its essential, proportional core.
"For the last 100 years, we have trained physicists to be expert bookkeepers. They are masters of the SI formulas, the constants, the conversion factors. They are incredibly good at calculating with our human rulers." "But the real job of a physicist, the job of Newton and Einstein, is not to be a bookkeeper. It is to be a pattern-finder ."Point to the list of invariant laws on the board (E_nat = m_nat, r_nat ~ m_nat, T_nat ~ 1/m_nat). Deliver the final takeaway: " THIS is the physics. A handful of simple, elegant, geometric proportions. The universe is not complex. It is breathtakingly simple. The complexity is an illusion created by our tools. Your job is not to memorize the illusions. Your job is to find the simple patterns they are hiding. "
This conclusion redefines the identity of a physicist. It shifts the goal from computational mastery of complex formulas to the conceptual search for simple, underlying patterns.You are giving them a new, more noble, and more exciting vision of what they could be as scientists. You are not just teaching them a new method; you are giving them a new mission.
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