Foundations — Time dilation — derivation, twin paradox

2.3.29 · D1 · Physics › Modern Physics › Time dilation — derivation, twin paradox

Pehle se time dilation derivation padhne se pehle tumhe har letter khud se samajhna hoga. Yeh page har ek ko zero se build karta hai: plain words → picture → yeh topic uske bina kyun kaam nahi kar sakta. Upar se neeche padho; har item upar wale par lean karta hai.

0. "Reference frame" — kiska viewpoint iske liye word

Is poore page mein sirf do viewpoints hain, aur hum unhe ek baar yahan name karte hain taaki baad mein kabhi surprise na karein:

• Tum (the watcher): tum still khade ho aur clock ko zoom karte hue dekhte ho. Yeh hai "your frame."
• The rider: ek imagined person jo clock par baitha hai, uske saath move kar raha hai, toh unke liye clock bilkul move nahi kar raha. Yeh hai "the clock's own frame" (rider's frame).

Neeche mention har "frame" in donon mein se ek hai. Picture raho: do log, har ek ke paas ek ruler aur ek stopwatch, us baat par disagree karte hue ki woh stopwatches kya read karte hain.

1. Speed — woh idea jo sab kuch start karta hai

Yeh kyun matter karta hai? Agar tum speed fix karo lekin distance stretch karo, toh time force hokar grow karta hai. Wahi trade-off literally wahan se aata hai jahan se time dilation aata hai. Is fraction ko pakde raho — neeche sab kuch iske upar aur neeche ke liye ek battle hai.

2. $c$ — speed of light (woh ek cheez jo kabhi nahi badlti)

Figure dekho. Ek person jo still khada hai aur ek person fast train par dono same light beam ko $c$ par clock karte hain. Normal life mein speeds add up hoti hain (train par ball aage fenko toh ground us ball ko faster jaate dekhti hai). Light yeh game khelne se mana kar deti hai — woh stubborn hai. Woh stubbornness Einstein's second postulate hai, aur yeh cause hai har ek weird result ka jo follow karta hai.

3. $v$ — moving thing ki relative speed

Ek clock ko left-to-right speed $v$ par tumhare paas se zoom karte hue imagine karo. Agar $v=0$ toh kuch move nahi karta aur bilkul koi time-dilation effect nahi hai. Jaise $v$, $c$ ki taraf creep karta hai, effect enormous ho jaata hai. Toh $v$ woh dial hai jo control karta hai ki poora phenomenon kitna strong hai.

4. $L$ — mirror gap (ek fixed length)

Figure mein do mirrors height $L$ apart baithte hain, aur ek photon (light ka ek single particle) unke beech upar-neeche bounce karta hai. Aisa strange clock kyun banao? Kyunki iska ticking light se bana hai, isliye iska tick-rate directly $c$ se chain hai. Yeh use testing ke liye perfect probe banata hai ki constant $c$ time ke saath kya karta hai.

5. Right triangle & Pythagoras — geometry tool

Clocks ke baare mein ek physics problem mein triangle kyun aata hai? Kyunki jab clock sideways drift karta hai jabki photon upar jaata hai, photon ka actual path (tumhare frame mein) ek slanted diagonal hai — aur woh diagonal ek right triangle ka hypotenuse hai jiska vertical side $L$ hai aur horizontal side yeh hai ki clock kitna drift kiya. Pythagoras woh tool hai jo answer deta hai "woh diagonal kitna lamba hai?" — aur iska length, $c$ se divide hoke, moving clock ki tick hai.

6. $\Delta t$ aur $\Delta t_0$ — do alag times

Greek letter $\Delta$ (delta) ka matlab hai "ki matra" ya "mein change." Toh $\Delta t$ literally padhta hai "time ki ek matra." Topic in donon mein se do use karta hai, aur unhe confuse karna #1 error hai.

Poori derivation sirf ek equation hai jo in donon ko connect karti hai: $\Delta t$ hamesha $\Delta t_0$ se bada hota hai. Kitna bada yeh next symbol answer karta hai.

7. $\gamma$ — stretch factor (Lorentz factor)

Ab jab hum $c$, $v$, $L$, $\Delta t$ aur $\Delta t_0$ ke malik hain, hum $\gamma$ actually build kar sakte hain sirf quote karne ki jagah. Dekho yeh triangle se kaise appear karta hai. Poore time, "tum" watcher rahte ho aur "the rider" clock par rahta hai — woh do frames jo humne section 0 mein name kiye the.

Hum sirf half tick kyun analyse karte hain. Ek full tick upar phir neeche hai. Picture ki up-down symmetry se, downward diagonal sirf upward wale ka mirror-image hai — same length, same drift, same time. Toh jo bhi hum pehle half ke liye prove karte hain woh identically doosre half par bhi apply hota hai, aur ek half study karna poori tick capture karta hai.

$c\Delta t/2$ hypotenuse kyun hai. Tumhare frame mein (watcher's) full tick time $\Delta t$ leta hai, toh har half $\Delta t/2$ rehta hai. Us half ke dauran photon poore time speed $c$ par move karta hai (postulate 2). Distance = speed × time, toh iska slanted path $c\Delta t/2$ lamba hai — wahi hypotenuse hai. Usi half ke dauran clock sideways $v\Delta t/2$ drift karta hai (distance = $v\times \Delta t/2$), aur photon abhi bhi fixed gap $L$ climb karta hai. Toh right triangle mein hai:

• vertical side $L$,
• horizontal side $v\Delta t/2$,
• hypotenuse $c\Delta t/2$.

Pythagoras apply karo (hypotenuse² = side² + side²): ${\left(\frac{c\Delta t}{2}\right)}^2=L^2+{\left(\frac{v\Delta t}{2}\right)}^2$

Rider ki clock laao. Rider ke frame se photon seedha upar aur wapas jaata hai, distance $2L$ speed $c$ par, toh $\Delta t_0=2L/c$, yaani $L=\frac{c\Delta t_0}{2}$. Substitute karo: $\frac{c^2\Delta t^2}{4}=\frac{c^2\Delta t_0^2}{4}+\frac{v^2\Delta t^2}{4}$

Clean it up. Har term ko $4$ se multiply karo, phir $\Delta t$ terms ek side gather karo: $c^2\Delta t^2-v^2\Delta t^2=c^2\Delta t_0^2\Rightarrow \Delta t^2(c^2-v^2)=c^2\Delta t_0^2$

$\Delta t$ ke liye solve karo — carefully. Pehle dono sides ko $(c^2-v^2)$ se divide karo. Yeh allowed hai kyunki $v<c$ hai, toh $c^2-v^2>0$ (kabhi zero se divide nahi): $\Delta t^2=\frac{c^2\Delta t_0^2}{c^2-v^2}$ Ab fraction ke upar aur neeche $c^2$ se divide karo taaki use $c$ ke fractions mein tidy karo: $\Delta t^2=\frac{\Delta t_0^2}{1-v^2/c^2}$ Finally dono sides ka positive square root lo. Hum sirf positive root rakhte hain kyunki $\Delta t$ ek duration hai — ek real elapsed time — aur durations kabhi negative nahi hote: $\Delta t=\frac{\Delta t_0}{\sqrt{1-v^2/c^2}}$

$\Delta t_0$ ko multiply kar raha messy factor ab apna naam earn kar chuka hai:

Figure ko rising cliff ke graph ki tarah padhо:

• Jab $v=0$: $v^2/c^2=0$, toh $\gamma =1/\sqrt{1}=1$. Koi stretch nahi — everyday life se match karta hai.
• Jab $v$ small ho (maan lo $0.01c$): $\gamma \approx 1.00005$. Barely koi stretch — yahi reason hai ki Newton "worked" kiya.
• Jab $v\to c$: $v^2/c^2\to 1$, square root $\to 0$, aur near-zero se divide karna $\gamma \to \infty$ banata hai. Time bina limit ke stretch hota hai.
• $v$ kabhi $c$ ke equal nahi ho sakta ek clock ke liye, warna hum exactly zero se divide karenge (infinite time) — ek signpost ki massive objects light speed reach nahi kar sakti.

Wahi $\gamma$ Lorentz Transformations, Length Contraction, aur Spacetime Interval mein dubara aata hai.

Prerequisite map

Sab kuch boxed formula mein downward flow karta hai, jo phir twin paradox aur (baad mein) Muon Decay Experiment ko power karta hai.

Equipment checklist

Right side cover karo aur khud ko test karo. Agar koi bhi answer surprise kare, toh woh section dobara padho.