Relativity That ClicksBonus / Dimensionless: one line. Put the units in: a nightmare.

The one line that runs through the whole series — "what matters is the dimensionless ratio; units are just stage scenery" — here is its clearest demonstration, in two formulas

Dimensionless: one line. Put the units in: a nightmare. Velocity addition, seen through rapidity, is just addition. The ground-state energy of hydrogen, dimensionless, is α²/2.
── The $c^2$ and $\varepsilon_0^2$ that bubble up the moment you put the units back in were never physics — they were the cost of carrying the ruler around.

Tools you'll need: division, tanh (a hyperbolic function — a rough feel is fine), and β = v/c from Episode 1 The ratios this time: the composition of β, and α²/2

Throughout this series we've said the same thing over and over ── physics is decided by dimensionless ratios alone. $c$, $\hbar$, and $\varepsilon_0$ are stage scenery, answering only "which ruler are we measuring with?". But when we say "adding the units makes the formula complicated," just how complicated, concretely? This bonus episode makes that gap visible with two examples. One is velocity addition ── viewed dimensionlessly, it astonishingly collapses back into "just addition." The other is the energy of the hydrogen atom ── dimensionless it's the one line $\alpha^2/2$, yet put the units in and the constants bubble up like a snowball. At the end we'll reveal exactly where that gap comes from.

01Recap ── why does "putting the units in" look complicated?

In Episode 1 we saw that $c$ is not a "fast vehicle" but an exchange rate between time and space. It's only because we measure with mismatched units ── seconds and metres ── that the big number $c=3\times10^8$ shows up; measure both in the same unit and $c=1$, vanishing from the formula. The physical content is held by the dimensionless ratio $\beta=v/c$ ── that was the starting point of the series.

If so, the reverse must happen too. Make the units explicit, and you're forced to write the $c$ and $\hbar$ you could have cancelled all over the formula. The content is the same, yet the appearance alone becomes complicated. Let's lay two of these side by side and actually see it.

02Example 1 ── velocity addition collapses back into "addition" when dimensionless

On a train (speed $v_1$) you throw a ball (speed $v_2$) in the direction of travel. How fast is the ball seen from the ground? For Newton the answer is simple ── just add them, $v_1+v_2$. But as you approach the speed of light, this naive addition breaks down (left as is, it would exceed the speed of light). The correct relativistic composition rule, written with units in, looks like this.

Composition rule with units in (v in m/s, c ≈ 3×10⁸ m/s)

$$v=\frac{v_1+v_2}{\,1+\dfrac{v_1 v_2}{c^{2}}\,}$$

A $c^2$ (= $9\times10^{16}$) squats in the denominator. In everyday life $v_1v_2/c^2\approx0$, so $v\approx v_1+v_2$ (back to Newton). But near the speed of light this $c^2$ kicks in, and the sum can never exceed $c$. The shape of the formula, honestly, isn't very pretty.

But divide everything by $c$ to make it dimensionless ($\beta=v/c$), and $c$ vanishes, tidying things up a little.

Dimensionless composition rule ── c vanishes

$$\beta=\frac{\beta_1+\beta_2}{1+\beta_1\beta_2}$$

The $c^2$ is gone. But it still doesn't look like "addition" ── the $1+\beta_1\beta_2$ in the denominator remains. Take one more step ── swap the ruler itself ── and this changes dramatically.

That ruler is rapidity $\theta$ (the velocity parameter). Instead of the speed $\beta$, use the $\theta$ fixed by $\beta=\tanh\theta$. $\theta$ is dimensionless, and in fact it is precisely the "rotation angle" of spacetime we saw in the spacetime interval of Episode 3 (the same way angles add under spatial rotation). Write velocity addition with this $\theta$ and ──

The ultimate form ── with rapidity, just addition

$$\boxed{\;\theta=\theta_1+\theta_2\;}\qquad(\beta=\tanh\theta)$$

That complicated relativistic composition rule has become just addition. The same shape as Galileo's $v=v_1+v_2$. Indeed, undo it with the addition formula for the hyperbolic tangent, $\tanh(\theta_1+\theta_2)=\dfrac{\tanh\theta_1+\tanh\theta_2}{1+\tanh\theta_1\tanh\theta_2}$, and it matches the $\beta$ formula above exactly. The "difficulty" of relativity was only that we were looking through the wrong ruler ($v$ or $\beta$).

Try it ── add half the speed of light to itself

Naive addition (Newton)

$$\beta_1+\beta_2=0.5+0.5=1.0\quad(\text{= the speed of light. impossible})$$

Relativistic sum (dimensionless)

$$\beta=\frac{0.5+0.5}{1+0.5\times0.5}=\frac{1.0}{1.25}=0.8$$

Rapidity (just add)

$$\theta=\operatorname{artanh}(0.5)+\operatorname{artanh}(0.5)=0.549+0.549=1.098,\quad \tanh(1.098)=0.8\ \checkmark$$

Three roads reach the same 0.8. Naive addition breaks the speed of light, but the addition of rapidity never exceeds $\beta=1$ (no matter how much $\theta$ you add, $\tanh$ stays below 1). The "wall you can't cross" shows up naturally, in the world of addition, as infinity.

03Play with it ── the β ruler vs the rapidity ruler

Set the two velocities $\beta_1,\beta_2$ with the sliders. The top row is the β ruler (velocity itself, 0–1). The naive sum blasts through the light-speed wall (✗), while the relativistic sum stops just short of it (✓). The bottom row is the rapidity ruler. Just join $\theta_1$ and $\theta_2$ head to tail and the composition is done ── and bringing it back to the top row via $\tanh$ matches the relativistic sum exactly.

Figure: top = the β ruler (velocity 0–1, right end is the light-speed wall). The naive sum β₁+β₂ crosses the wall (✗), but the relativistic sum ✓ stays short of it. Bottom = the rapidity θ ruler. Just join θ₁ and θ₂ (= addition). The dashed line connects "bring it back with tanh and it's the same speed"

β₁ = v₁/c 0.50 β₂ = v₂/c 0.50

β₁ / θ₁ β₂ / θ₂ relativistic sum ✓ naive sum ✗

04Example 2 ── the snowball of constants (the energy of the hydrogen atom)

Now for "complexity" wearing a different face: the energy levels held by the electron in a hydrogen atom. Written dimensionlessly, taking the electron's rest energy $m_e c^2$ as the unit, it's only this much ──

Dimensionless (with m_e c² as the unit, written with α = 1/137)

$$E_n=-\frac{1}{2}\,\frac{\alpha^{2}}{n^{2}}$$

The ground state ($n=1$) is essentially "half of $\alpha^2$". Only the fine-structure constant $\alpha$ appears — nothing else. It's the very $\alpha$ from Episode 2, where the electron's orbital speed was $v/c=\alpha$.

But write this out "in full" in SI units, and the constants that were hidden all bubble up at once.

Put the units in (write it all out in SI)

$$E_n=-\frac{m_e\,e^{4}}{8\,\varepsilon_0^{2}\,h^{2}\,n^{2}}$$

$m_e$, $e^4$, $\varepsilon_0^2$, $h^2$ ── four kinds of constant. The content is the same 13.6 eV, yet what was folded inside $\alpha^2$ in the dimensionless version spills out into the open the instant you put the units in. Incidentally, $\alpha=\dfrac{e^2}{4\pi\varepsilon_0\hbar c}$, so these four are all the contents of $\alpha$.

Bonus ── the same thing happens with the Bohr radius

Dimensionless (with the Compton wavelength ℏ/m_e c as the unit)

$$a_0=\frac{1}{\alpha}\,\frac{\hbar}{m_e c}$$

Written out in full in SI

$$a_0=\frac{4\pi\varepsilon_0\hbar^{2}}{m_e e^{2}}$$

Dimensionless, it's a one-liner: "$1/\alpha$ (= 137 times) the Compton wavelength." Put the units in and $\varepsilon_0$ and $e^2$ show their faces again. Same physics, different appearance ── the gap always comes from the same place.

05The reveal ── complexity is "the cost of carrying the ruler around"

The true nature of the gap common to both examples is this. What decides the physical content is dimensionless ratios alone ── $\beta$ for velocity addition (or, folded further, $\theta$), and $\alpha$ for hydrogen. The dimensioned constants like $c$, $\hbar$, and $\varepsilon_0$ are nothing but luggage tags for carrying, within the formula, the "seconds, metres, kilograms, coulombs — rulers humans arbitrarily chose."

The true nature of the gap

• Speak in ratios only → the tags ($c^2,\varepsilon_0^2,h^2$…) cancel and vanish, and the structure (addition, $\alpha^2$) is laid bare.
• Make the units explicit → you're forced to write the whole ruler set into the formula, and the appearance grows complicated.
Complexity is not the depth of the physics but the cost of carrying the ruler around. That's why physicists set $c=\hbar=1$ (natural units) ── put down the luggage tags and the formula returns to the one line it was always meant to be.

This is the very viewpoint that runs through this series and the whole collection. Quantities with units are stage scenery; what matters is the dimensionless ratio. Relativity looked hard, too, because we kept thinking while gripping the stage scenery ── $v$, seconds, metres. The moment we swapped the ruler for $\beta$, and then for $\theta$, that composition rule collapsed back into "just addition" ── in the same way, choosing the right dimensionless ruler is understanding itself.

◇ ◇ ◇

The honest line ── how much of this is "the units' fault"

That $v=(v_1+v_2)/(1+v_1v_2/c^2)$ becomes $\beta=(\beta_1+\beta_2)/(1+\beta_1\beta_2)$ under dimensionlessing and turns additive as $\theta=\theta_1+\theta_2$ in terms of the rapidity $\theta=\operatorname{artanh}\beta$; that $\theta$ is the "angle" of the Lorentz transformation (the hyperbolic rotation of spacetime); the hydrogen $E_n=-\tfrac12\alpha^2 m_e c^2/n^2=-m_e e^4/(8\varepsilon_0^2 h^2 n^2)$; and the Bohr radius $a_0=\hbar/(\alpha m_e c)=4\pi\varepsilon_0\hbar^2/(m_e e^2)$ ── all of these are established standard physics.

But. ① The additivity of rapidity $\theta=\theta_1+\theta_2$ is a matter of velocity composition along one and the same line (one dimension). Compose velocities in different directions and an extra rotation (Thomas precession) appears, and simple addition no longer suffices. ② "Complexity is the units' fault" refers to apparent complexity. The content ── that velocities don't simply add and cap out at the speed of light ── is a physical fact, and changing rulers doesn't make it disappear (the same caveat as Episode 1's "the universality of $c$ is not a matter of units"). Rapidity "folds the difficulty into addition so you can see it"; it doesn't remove it. ③ Even though $\alpha\approx1/137$ is dimensionless, why its value is 1/137 is unsolved ── a genuine mystery that remains even when you erase the units (see the sister series "Cosmology That Clicks").

Practice problems (solvable with just this episode's formulas)

Add $\beta_1=0.8,\ \beta_2=0.8$ with the relativistic composition rule. Compare with the naive sum, 1.6.
See the answer
$\beta=(0.8+0.8)/(1+0.64)=1.6/1.64\approx0.976$. The naive sum 1.6 is 1.6 times the speed of light — impossible. The relativistic sum is 0.976, safely short of the wall (1).
In rapidity, when $\theta_1=\theta_2=1$, what is the composed $\beta$? ($\tanh1\approx0.762$, $\tanh2\approx0.964$)
See the answer
$\theta=\theta_1+\theta_2=2$, so $\beta=\tanh2\approx0.964$. Each one is $\tanh1\approx0.762$. It's done in a single move of addition, 1+1=2.
Estimate hydrogen's ground-state energy in eV from the dimensionless formula $-\tfrac12\alpha^2$ (unit $m_ec^2$). Take $m_ec^2=511{,}000$ eV, $\alpha=1/137$.
See the answer
$-\tfrac12(1/137)^2\times511000\approx-\tfrac12\times5.3\times10^{-5}\times511000\approx-13.6$ eV. You reach the same value as SI's $m_e e^4/(8\varepsilon_0^2h^2)$ without writing a single constant.
Why does "putting the units in make the formula complicated"? Explain in one sentence in this episode's language.
See the answer
Physics is decided by dimensionless ratios alone, and dimensioned constants (c, ℏ, ε₀…) are nothing but luggage tags for carrying the rulers around inside the formula. Make the units explicit and you're forced to write all those tags, so the appearance alone grows complicated (= the cost of carrying the ruler around).

Bonus summaryDimensionless: one line. Put the units in: a nightmare.

We demonstrated "putting the units in makes it complicated" with two formulas. Velocity addition, with units, drags around a $c^2$ as $v=(v_1+v_2)/(1+v_1v_2/c^2)$; make it dimensionless and $c$ vanishes; swap in the rapidity $\theta=\operatorname{artanh}\beta$ and it becomes $\theta=\theta_1+\theta_2$ ── back to just addition, the same as Galileo. The energy of hydrogen is the one line $-\tfrac12\alpha^2/n^2$ dimensionless, yet write it in SI and the constants snowball into $m_e e^4/(8\varepsilon_0^2h^2n^2)$.

The true nature of the gap is the cost of carrying the ruler around. What holds the physics is the dimensionless ratios ($\beta,\theta,\alpha$) alone, and $c$, $\hbar$, $\varepsilon_0$ are the ruler's luggage tags. Speak in ratios only and the tags cancel and vanish, laying the structure bare. Choosing the right dimensionless ruler is understanding itself ── this is the most straightforward demonstration of the series' guiding line, "what matters is the dimensionless ratio."

This bonus episode is an application of Episode 1 "β = v/c" and Episode 3 "the spacetime interval." The rapidity $\theta$ is precisely the angle of the "hyperbolic rotation" that preserves the spacetime invariant $s^2=(ct)^2-x^2$ we saw in Episode 3 ── for the same reason angles add straightforwardly under spatial rotation, velocities too can be added in $\theta$. Back to the main series → Table of contents. In the sister series "Cosmology That Clicks," we chase the mystery that never vanishes ── $\alpha=1/137$ itself.

This document is a bonus episode of the "Relativity That Clicks" series, reading for physics-loving high-school and university students. The relativistic velocity composition $v=(v_1+v_2)/(1+v_1v_2/c^2)$, its dimensionless form $\beta=(\beta_1+\beta_2)/(1+\beta_1\beta_2)$, the additivity $\theta=\theta_1+\theta_2$ via the rapidity $\theta=\operatorname{artanh}\beta$ ($\theta$ being the hyperbolic rotation angle of a Lorentz boost), the hydrogen energy levels $E_n=-\tfrac12\alpha^2 m_ec^2/n^2=-m_e e^4/(8\varepsilon_0^2h^2n^2)$, and the Bohr radius $a_0=\hbar/(\alpha m_ec)=4\pi\varepsilon_0\hbar^2/(m_e e^2)$ are all established standard physics. That the additivity of rapidity is limited to composition along one and the same line (1+1 dimensions), with Thomas precession (Wigner rotation) added for non-collinear composition; that what dimensionlessing removes is the apparent complexity, while the physical content ── "velocities don't simply add and cap out at c" ── is unchanged; and that the origin of $\alpha$'s numerical value 1/137 is an unsolved problem that remains even when the units are erased ── all are stated explicitly in the "The honest line" section. ── To print, use your browser's "Print" and "Save as PDF" (in the print version the sliders and answers are static and hidden). Related: Table of contents / Episode 1: β / Episode 3: The spacetime interval / sister series Cosmology That Clicks.

Print / save as PDF: ⌘+P (Ctrl+P on Windows). On screen, the β₁ and β₂ sliders link the top row (the velocity ruler) — "naive sum ✗ / relativistic sum ✓" — with the bottom row (the rapidity ruler) — "the addition that's just joining them." "See the answer" opens each solution.