Foundations — Length contraction — derivation

This page assumes nothing. Before you touch the derivation in the parent note, every letter and every symbol it throws at you is unpacked here — meaning first, picture second, purpose third. Read it top to bottom; each block leans on the one above it.

0. What is a "frame"? (the ground everything stands on)

Before any symbol, we need the idea of a reference frame.

The whole topic is a conversation between two frames, so we give each a name:

• $S$ — the ground / lab frame. Picture you standing still on a platform. In this frame the rod goes whizzing past.
• $S^{\prime }$ — the rod's own frame. Picture yourself sitting on the rod, riding along with it. In this frame the rod is perfectly still and the ground rushes past instead.

The little tick mark in $S^{\prime }$ (read "S-prime") just means "the second observer's stuff." Positions measured by that observer get a prime too: $x^{\prime }$, $t^{\prime }$.

Prime :::- means "belongs to the second observer $S^{\prime }$" — it does not mean a derivative here.

1. Position $x$ and time $t$ — the two things we ever record

Picture a single dot on a strip: its horizontal spot is $x$, and stamped next to it is the time $t$ it was there.

Why the topic needs it: to talk about "where the front end is when," you need both a place and a moment. That "place-and-moment" bundle is exactly an event, and length contraction is built entirely out of comparing events.

2. Speed $v$ and the speed of light $c$

$v/c$ is a pure number between :::- $0$ and $1$ — it has no units, and it measures "what fraction of light speed."

3. The Lorentz factor $\gamma$ — the master dial

Everything in this topic is scaled by one number. Meet it slowly.

Let's build it from the inside out so no piece is mysterious:

• $\frac{v^2}{c^2}$ — square the speed-fraction. Since $v/c$ is between $0$ and $1$, this is also between $0$ and $1$.
• $1-\frac{v^2}{c^2}$ — subtract from one. At rest ($v=0$) this is $1$; near light-speed it shrinks toward $0$.
• $\sqrt{\cdot }$ — take the square root, still between $0$ and $1$.
• $\frac{1}{\sqrt{\cdot }}$ — flip it. Dividing $1$ by a small number gives a big number.

Look at the curve above. It is flat and boring near $v=0$ (everyday life — $\gamma \approx 1$, relativity invisible), then rockets upward as $v\to c$. This shape is why you never notice contraction in a car but must account for it in a particle accelerator.

Why the topic needs it: the final formula is $L=L_0/\gamma$. Because $\gamma \ge 1$, dividing by it can only make things smaller or equal — that single inequality is "contraction." See Lorentz factor gamma for the deep dive.

At $v=0$, what is $\gamma$? :::- $\gamma =1$ (frames agree, no contraction). As $v\to c$, what happens to $\gamma$? :::- It grows without bound, $\gamma \to \infty$.

4. Proper length $L_0$ vs measured length $L$

Now the two starring quantities.

The subscript $0$ in $L_0$ means :::- "measured in the rest frame" (the proper value), not "at time zero."

5. Simultaneity — the secret engine

This is the concept the parent note leans on hardest, so we picture it carefully.

Here is the twist that makes relativity relativity:

Why the topic needs it: in the derivation, setting $t_1=t_2=t$ in frame $S$ is the single step that makes the $vt$ terms cancel. Without simultaneity, there is no contraction — it is the engine, not a footnote.

6. The Lorentz transformation — the dictionary between frames

The parent note starts from one equation. Here is what it says.

Read the right-hand side in plain words:

• $x-vt$ — "start from where the event is in $S$, then slide back by how far the rod-frame has travelled ($vt$)." This is the ordinary, common-sense part (it's what you'd write even without relativity).
• $\gamma (\dots )$ — multiply the whole thing by $\gamma$. This is the purely relativistic correction: the stretch that accounts for the frames' disagreeing rulers.

Without the factor $\gamma$, the transformation $x^{\prime }=x-vt$ would :::- be the old (Galilean) rule that wrongly assumes a shared universal clock.

Prerequisite map

Equipment checklist

Cover the right side and see if you can answer each before revealing.

Connections

• Length contraction — derivation — the parent this page prepares you for.
• Lorentz transformation — the dictionary between frames, used in every step.
• Lorentz factor gamma — the master dial $\gamma$ built here.
• Relativity of simultaneity — the engine behind the effect.
• Time dilation — the opposite twin ($\times \gamma$).
• Spacetime interval — where the square root in $\gamma$ comes from.
• Muon decay experiment — the real-world payoff.