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2.3.30Modern Physics

Length contraction — derivation

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🇮🇳 Yeh note Hinglish mein padho →

WHAT are we even measuring?

WHY does measurement need simultaneity? If a rod moves past you, to get its length you must note where its front end is and where its back end is at the same instant. If you record the front at time t1t_1 and the back at a later time t2t_2, the rod will have moved, and you'd get nonsense. So length = (positions of both ends recorded simultaneously). Since simultaneity differs between frames, length does too.

HOW to derive it — from the Lorentz transformation

We use two frames:

  • SS = ground (lab) frame, where the rod moves with speed vv.
  • S′S' = rod's rest frame, moving at +v+v relative to SS.

The Lorentz transformation from SS to S′S': x′=γ (x−vt),γ=11−v2/c2x' = \gamma\,(x - v t), \qquad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}}

Step 1 — Define the proper length in S′S'

In the rod's own frame S′S' the ends sit at fixed coordinates x1′x'_1 (back) and x2′x'_2 (front), so L0=x2′−x1′.L_0 = x'_2 - x'_1. Why this step? The rod is at rest in S′S', so its ends don't move — we can read them off whenever we like.

Step 2 — Measure in SS at one instant

In SS the observer records both ends at the same time tt: t1=t2=t.t_1 = t_2 = t. Why this step? This is the whole physics — a length measurement of a moving rod demands simultaneous endpoint readings in the measuring frame.

Step 3 — Apply the transformation to each end

x1′=γ (x1−vt),x2′=γ (x2−vt).x'_1 = \gamma\,(x_1 - v t), \qquad x'_2 = \gamma\,(x_2 - v t).

Step 4 — Subtract

x2′−x1′=γ[(x2−vt)−(x1−vt)]=γ (x2−x1).x'_2 - x'_1 = \gamma\big[(x_2 - v t) - (x_1 - v t)\big] = \gamma\,(x_2 - x_1). The vtvt terms cancel only because t1=t2t_1 = t_2 — this is where simultaneity does the work.

Now x2′−x1′=L0x'_2 - x'_1 = L_0 and x2−x1=Lx_2 - x_1 = L (the length in SS). So: L0=γ L.L_0 = \gamma\, L.

Step 5 — Solve for the measured length

Figure — Length contraction — derivation

Worked examples

Common mistakes

Recall Feynman: explain it to a 12-year-old

Imagine a train zooming past you super fast. To measure how long the train is, you have to snap a photo catching the front and back at the exact same moment. But here's the weird thing Einstein found: when something moves near light speed, "the exact same moment" means different things to you and to the people on the train. Because of that mismatch, your measurement comes out shorter than what the train passengers measure for their own train. Nothing is being crushed — it's just that fast motion scrambles what "now" means, and that scrambling makes moving things look squished in the direction they travel.

Active recall

What is proper length L0L_0?
The length measured in the frame where the object is at rest; it is the maximum measured length.
State the length contraction formula.
L=L01−v2/c2=L0/γL = L_0\sqrt{1 - v^2/c^2} = L_0/\gamma.
Why does a moving rod measure shorter?
Because measuring its length requires recording both ends simultaneously, and simultaneity is frame-dependent (relativity of simultaneity).
In the derivation, why do the vtvt terms cancel?
Because the two ends are recorded at the same time t1=t2t_1=t_2 in SS, so the vtvt terms are identical and subtract away.
Does length contract perpendicular to the motion?
No — only the dimension along the direction of motion contracts.
Compare contraction and dilation directions.
Length divides by γ\gamma (gets smaller); time multiplies by γ\gamma (gets larger); proper length is max, proper time is min.
At v=0.8cv=0.8c, by what factor is a rod contracted?
γ=5/3\gamma=5/3, so L=0.6 L0L=0.6\,L_0 (60% of rest length).
Is length contraction a real physical squeezing force?
No — it's a geometric consequence of spacetime/simultaneity, not a mechanical compression.

Connections

  • Lorentz transformation — the master equations this is derived from.
  • Time dilation — the "opposite" twin effect (×γ\times\gamma).
  • Relativity of simultaneity — the root cause of contraction.
  • Lorentz factor gamma — the universal γ=1/1−v2/c2\gamma=1/\sqrt{1-v^2/c^2}.
  • Muon decay experiment — real-world evidence.
  • Spacetime interval — the invariant both effects preserve.

Concept Map

forces

frame-dependent

longest length

shorter

applied to both ends

t1=t2 so vt cancels

gives L0=gamma L

since gamma>=1

opposite scaling

Relativity of simultaneity

Length needs simultaneous endpoint marks

Length contraction

Proper length L0 in rest frame S'

Measured length L in frame S

Lorentz transformation x'=gamma x-vt

Subtract endpoint equations

L = L0 / gamma

Time dilation dt=gamma dt0

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, length contraction ka matlab hai: jo cheez tezi se move kar rahi hai, woh apni motion ki direction mein chhoti measure hoti hai. Yeh koi physical squeezing nahi hai — koi force usse daba nahi raha. Asli reason hai relativity of simultaneity: kisi moving rod ki length naapne ke liye tumhe uske dono ends ek hi waqt (same instant) par mark karne padte hain, aur "same instant" har frame mein alag hota hai. Isi mismatch ki wajah se length chhoti aati hai.

Derivation simple hai. Rod ke apne rest frame S′S' mein length L0L_0 hai (yeh proper length, sabse badi). Ground frame SS mein observer dono ends ko same time tt par measure karta hai. Lorentz transformation x′=γ(x−vt)x' = \gamma(x - vt) dono ends pe lagao aur subtract karo — kyunki tt same hai, vtvt wala term cancel ho jaata hai aur milta hai L0=γLL_0 = \gamma L, yaani L=L0/γL = L_0/\gamma. Bas, γ\gamma hamesha ≥1\ge 1 hota hai, isliye LL hamesha chhoti.

Yaad rakhne ki cheez: length mein divide by gamma (chhoti hoti hai), lekin time dilation mein multiply by gamma (badi hoti hai) — dono opposite hain. Proper length maximum hai, proper time minimum hai. Aur ek important baat — sirf motion ki direction mein contraction hota hai, perpendicular (height/width) mein nahi.

Yeh real hai aur prove ho chuka hai — muons jab atmosphere se aate hain, unke frame mein atmosphere patli (contracted) dikhti hai, isliye woh ground tak pohonch jaate hain before decay. Toh agle baar koi bole "yeh sirf theory hai", bol dena ki experiments isse roz confirm karte hain!

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Connections

Lorentz transformation — derivationPhysics · 2.3.28Time dilation — derivation, twin paradoxPhysics · 2.3.29Special relativity — Michelson-Morley experimentPhysics · 2.3.25Postulates of SRPhysics · 2.3.26Simultaneity — relativity of simultaneityPhysics · 2.3.27Lorentz transformation — derivationPhysics · 2.3.28Time dilation — derivation, twin paradoxPhysics · 2.3.29