Foundations — Special relativity — Michelson-Morley experiment

Before you can trust a single line of the parent note, you need to own every symbol it throws at you. Below we build them one at a time, from nothing. Each gets: what it means in plain words → the picture it lives in → why the topic can't live without it. Read top to bottom; each rung of the ladder stands on the one below.

1. Speed — distance over time

Why the topic needs it: the entire derivation is nothing but adding up little "distance ÷ speed" times for light on each leg of its journey. If you're solid on $t=d/\text{speed}$, half the algebra is already yours.

2. $c$ — the speed of light

Why the topic needs it: the whole experiment asks "is $c$ really the same in every direction, or does the Earth's motion tilt it?" You can't ask that question without a name for the thing being measured. See Maxwell's Equations and the Speed of Light for where the value of $c$ comes from.

3. $v$ — the Earth's speed and the "aether wind"

Why the topic needs it: without a wind there is no speed difference to detect. $v$ is the villain the experiment hunts. The whole prediction lives or dies on whether this $v$ has any effect.

4. Adding and subtracting speeds (Galilean intuition)

Why the topic needs it: the parallel-arm times $\frac{L}{c-v}$ and $\frac{L}{c+v}$ are built entirely from this rule. When the experiment fails, it is this rule (for light) that must be abandoned — see Galilean Relativity & Velocity Addition.

5. $L$ — the arm length

Why the topic needs it: $L$ sets the scale of the effect. Longer arms mean more time for the tiny wind effect to accumulate — that's why the real apparatus folded the beam with many mirrors to make $L$ effectively $\sim 11$ m.

6. The right triangle and the Pythagorean theorem

This is the one piece of geometry the perpendicular arm demands, so we build it fully.

Why the topic needs it: this is exactly where the perpendicular arm's speed $c_{\perp }=\sqrt{c^2-v^2}$ comes from. Without Pythagoras there is no cross-arm formula, and no comparison between the two arms.

7. $\beta$ — the speed ratio $v/c$

Why the topic needs it: the whole prediction shrinks to $\propto {\beta }^2$. Naming the ratio lets us say "the effect is second order in $v/c$" — small, but not zero. $\beta$ reappears everywhere in Lorentz Transformation, Length Contraction and Time Dilation.

8. The two arm times $t_{\parallel }$ and $t_{\perp }$

Now that we own $c$, $v$, $L$ and $c_{\perp }$, we can build the two round-trip times the experiment compares.

Why the topic needs it: these two formulas are the heart of the whole prediction. Every symbol in them — $L$, $c$, $v$, $c_{\perp }$ — was built above, so you can now see exactly how they combine. The experiment measures the difference between them.

9. $\beta$ and the approximation $\frac{1}{1-x}\approx 1+x$

Why the topic needs it: the exact bracket is unreadable. The tool answers the question "what does this messy formula look like when $v$ is tiny?" — and tiny is exactly our case, melting it to the clean $\Delta t\approx \frac{L{v}^2}{c^3}$.

10. Waves: wavelength $\lambda$, and interference fringes

Why the topic needs it: the time difference $\Delta t$ is far too small to time with a clock. But turned into a fringe shift — a visible stripe movement measured in units of $\lambda$ — it becomes something the eye and the apparatus can catch. This is the engine of the whole measurement; deepen it in Interference of Light & Fringes.

11. $\Delta$ (delta), $\Delta t$, and the fringe-shift formula $\Delta N$

Why the topic needs it: this is the bridge from theory (a time difference) to observation (a visible stripe shift). Everything the experiment reports is a statement about this $\Delta N$ — and the headline result is that it came out zero.

The prerequisite ladder (text)

Each rung below stands on the one before it — this is the same chain the parent note walks, now with every symbol owned:

1. Speed = distance ÷ time — the one arithmetic rule behind every arm.
2. $c$ (speed of light) and $v$ (Earth's speed) — the two speeds in the race.
3. Adding speeds ($c\pm v$) — gives the parallel-arm effective speeds.
4. $L$ — sets how far the light travels, hence the size of the effect.
5. Right triangle + Pythagoras — produces the cross speed $c_{\perp }=\sqrt{c^2-v^2}$.
6. $t_{\parallel }$ and $t_{\perp }$ — the two round-trip times, built from steps 1–5.
7. $\beta =v/c$ and the small-$x$ approximation — melt the difference $\Delta t$ to $\frac{L{v}^2}{c^3}$.
8. $\lambda$, fringes, and $\Delta N=\frac{c(2\Delta t)}{\lambda }$ — turn the time difference into a visible stripe shift.
9. The null result — $\Delta N$ measured as zero, killing the aether.

Equipment checklist

Self-test: cover the right side and see if you can say each answer out loud.

Once every line above comes back instantly, you are ready for the Michelson–Morley derivation and its consequences in Special Relativity — Einstein's Postulates.