Foundations — Length contraction — derivation

2.3.30 · D1 · Physics › Modern Physics › Length contraction — derivation

Yeh page kuch bhi assume nahi karta. Isse pehle ki tum parent note ki derivation ko haath lagao, usmein jo bhi letter aur symbol aaenge, unhe yahan zero se explain kiya gaya hai — pehle matlab, phir picture, phir purpose. Isse upar se neeche padho; har block uske upar waale par tikaa hua hai.

0. "Frame" kya hota hai? (woh zameen jis par sab kuch khada hai)

Kisi bhi symbol se pehle, humein reference frame ka idea chahiye.

Poora topic do frames ke beech ki ek baat-cheet hai, isliye hum dono ko ek naam dete hain:

• $S$ — ground / lab frame. Socho tum platform par bilkul still khade ho. Is frame mein rod tezi se guzarti hai.
• $S^{\prime }$ — rod ka apna frame. Socho tum rod par baith kar uske saath chal rahe ho. Is frame mein rod bilkul still hai aur zameen seedhi guzarti hai.

$S^{\prime }$ mein jo chhota tick mark hai (padho "S-prime"), iska matlab hai "doosre observer ki cheezein." Uss observer ke measure kiye gaye positions ko bhi prime milta hai: $x^{\prime }$, $t^{\prime }$.

Prime :::- ka matlab hai "doosre observer $S^{\prime }$ ka" — yahan iska matlab derivative nahi hai.

1. Position $x$ aur time $t$ — woh do cheezein jo hum kabhi bhi record karte hain

Ek strip par ek single dot imagine karo: uski horizontal position $x$ hai, aur uske paas stamped hai woh time $t$ jab woh wahan tha.

Yeh topic isko kyun use karta hai: "aage waala end kab kahan hai" ke baare mein baat karne ke liye, tumhe ek jagah aur ek moment dono chahiye. Yahi "jagah-aur-moment" ka bundle exactly ek event hai, aur length contraction poori tarah se events ko compare karne par bani hai.

2. Speed $v$ aur light ki speed $c$

$v/c$ ek pure number hai :::- $0$ aur $1$ ke beech — iska koi unit nahi, aur yeh measure karta hai "light speed ka kitna fraction."

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

Is topic mein sab kuch ek number se scale hota hai. Isse dheere dheere samjho.

Isse andar se bahar ki taraf build karte hain taaki koi bhi piece mysterious na rahe:

• $\frac{v^2}{c^2}$ — speed-fraction ko square karo. Kyunki $v/c$ $0$ aur $1$ ke beech hai, yeh bhi $0$ aur $1$ ke beech hai.
• $1-\frac{v^2}{c^2}$ — ek se ghatao. Rest par ($v=0$) yeh $1$ hai; light-speed ke paas yeh $0$ ki taraf shrink karta hai.
• $\sqrt{\cdot }$ — square root lo, abhi bhi $0$ aur $1$ ke beech.
• $\frac{1}{\sqrt{\cdot }}$ — isko flip karo. $1$ ko ek chhoti number se divide karne par ek badi number milti hai.

Upar ka curve dekho. Yeh $v=0$ ke paas flat aur boring hai (roz ki zindagi — $\gamma \approx 1$, relativity invisible), phir $v\to c$ ke saath tezi se upar jaata hai. Yahi shape wajah hai ki tum gaadi mein contraction notice nahi karte lekin particle accelerator mein account karna padta hai.

Yeh topic isko kyun use karta hai: final formula $L=L_0/\gamma$ hai. Kyunki $\gamma \ge 1$, isse divide karne par cheezein sirf chhhoti ya barabar ho sakti hain — yahi ek inequality hai "contraction." Deep dive ke liye dekho Lorentz factor gamma.

At $v=0$, what is $\gamma$? :::- $\gamma =1$ (frames agree karte hain, koi contraction nahi). As $v\to c$, what happens to $\gamma$? :::- Yeh bina kisi bound ke barta hai, $\gamma \to \infty$.

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

Ab do starring quantities.

$L_0$ mein subscript $0$ ka matlab hai :::- "rest frame mein measured" (proper value), na ki "time zero par."

5. Simultaneity — secret engine

Yeh woh concept hai jis par parent note sabse zyada tikaa hua hai, toh isko dhyan se picture karte hain.

Yahan woh twist hai jo relativity ko relativity banata hai:

Yeh topic isko kyun use karta hai: derivation mein, frame $S$ mein $t_1=t_2=t$ set karna woh ek step hai jo $vt$ terms ko cancel karata hai. Simultaneity ke bina, contraction nahi hoga — yeh engine hai, footnote nahi.

6. The Lorentz transformation — frames ke beech dictionary

Parent note ek equation se shuru hota hai. Yahan woh kya kehta hai.

Right-hand side ko plain words mein padho:

• $x-vt$ — "event jahan $S$ mein hai wahan se shuru karo, phir peeche slide karo kitna rod-frame travel kar chuka hai ($vt$) utna." Yeh ordinary, common-sense waala part hai (ise tum relativity ke bina bhi likhte).
• $\gamma (\dots )$ — poori cheez ko $\gamma$ se multiply karo. Yeh purely relativistic correction hai: woh stretch jo frames ke disagreeing rulers ko account karta hai.

Factor $\gamma$ ke bina, transformation $x^{\prime }=x-vt$ :::- purana (Galilean) rule hoga jo galat tarike se ek universal clock assume karta hai.

Prerequisite map

Equipment checklist

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

Connections

• Length contraction — derivation — woh parent jiske liye yeh page tumhe taiyaar karta hai.
• Lorentz transformation — frames ke beech ki dictionary, har step mein use hoti hai.
• Lorentz factor gamma — master dial $\gamma$ yahan build hua.
• Relativity of simultaneity — effect ke peeche ka engine.
• Time dilation — opposite twin ($\times \gamma$).
• Spacetime interval — $\gamma$ mein square root kahan se aata hai.
• Muon decay experiment — real-world payoff.