Just as Einstein showed that time stretches when you move fast, he also showed that space itself compresses. This is called length contraction and it follows almost directly from the same thought experiment we used to derive time dilation in the book. In fact, once you have the time dilation equation in hand, length contraction is only a few steps away.

The key insight: if time can vary between two observers, then distance must vary too. Otherwise, the speed of light would be different for each observer—which we already know it can't be.

Here's how to derive it, step by step.

1. 1. Set Up the Same Two Observers—Go back to the train. Observer R' is riding on the train, and observer R is standing on the platform watching it pass. The train is moving at speed v.
2. 2. Measure the Train's Length—Now ask: how long is the train? Each observer will measure it differently.

• From inside the train (R'), the train has a "proper length"—the length when measured at rest. Call it L'.
• From the platform (R), the train flies by. To measure its length, R times how long the train takes to pass a fixed point on the platform.
• If the train passes at speed v in time t, then the length R measures is simply L = v · t.

3. Now connect the two frames of reference. From R's perspective, the train passes at speed v and takes time t to go by. But which t are we talking about?
   
   R is watching the front of the train pass a fixed point, and then the back of the train pass that same point. Both events happen at the same place in R's frame—that fixed spot on the platform. So the time R measures is the "proper time": let's call it t.
   
   Meanwhile, from inside the train (frame R'), that same process looks different. The platform is moving, and the fixed platform point sweeps from the front of the train to the back. The time R' measures for this is t'.
   
   And from our time dilation equation, we already know how these two times are related:

4. Write the length in each frame. In both frames, length = speed × time. The speed v is the same for both (it's just the relative speed between train and platform). So:

• In R (platform): L = v · t
• In R' (train): L' = v · t'
• Or rewritten: t = L/v and t' = L'/v

5. Substitute into the time dilation equation. Plug t = L/v and t' = L'/v into what we already know:

The v's cancel on both sides—and you're left with the length contraction equation:

If you're standing on the platform watching a fast-moving train, the length you measure (L) is shorter than the train's actual rest length (L'). The factor √(1 − v²/c²) is always less than or equal to 1, which means L ≤ L'. The faster the object moves, the more it contracts along its direction of motion.

At everyday speeds (say, a car or even a rocket) the contraction is so tiny it's completely unmeasurable. But as v approaches c, the factor approaches zero. In the (physically impossible) limit of moving at the speed of light, length would contract to nothing.

This isn't just theoretical strangeness, it's deeply connected to why the speed of light stays constant for every observer. Time stretches and space compresses in precisely the right amounts to keep c the same, no matter how you're moving.

And that, derived on the back of a napkin from a thought experiment about a train, is one of the most mind-bending conclusions in the history of science. Round of applause to you for working through it.

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^* Note that L' here is the "proper length"—the length measured at rest. This is the length the person inside the train would measure with a ruler. L is what the outside observer measures as the train flies by. The contraction only happens along the direction of motion—width and height are unaffected.