Time dilation — derivation, twin paradox — Hinglish notes

2.3.29 · Physics › Modern Physics

Time dilation exist kyon karti hai bilkul bhi?

"Light clock" kya hota hai? (woh tool jisse hum derive karte hain)

Formula derive kaise karein (scratch se)

Imagine karo light clock jisme mirrors vertically height $L$ se separated hain, aur yeh horizontally speed $v$ se tumhare paas se guzar raha hai. Hum ek round-trip track karte hain (pehle upar phir neeche).

Frame S′ (clock par sawaar rider): photon seedha upar top mirror tak jaata hai aur seedha wapas neeche aata hai — total vertical distance $2 L$ . $Δ t_{0} = \frac{2 L}{c} \Rightarrow L = \frac{c Δ t _{0}}{2}$ $Δ t_{0}$ proper time hai — aur yeh legitimate proper time hai kyunki emission aur final-reception clock ke rest frame mein same jagah (bottom mirror par) hoti hain.

Yeh step kyon? Rider ke liye, clock move nahi kar rahi, isliye photon seedha gap $L$ mein upar jaata hai aur seedha wapas neeche — total path $2 L$ , dono events ek hi point par.

Frame S (tum, ise uda hua dekh rahe ho): jab photon apni round-trip karta hai, clock sideways drift karti hai. Photon do diagonals trace karta hai — ek upar-ki-taraf slanting, phir ek neeche-ki-taraf slanting, ek symmetric "Λ" (tent) shape banata hai.

Tumhare time $Δ t$ mein full round-trip ke dauran clock ne jitna horizontal distance move kiya: $v Δ t$ .
Sirf pehli half consider karo (upar jaana): tumhare time $Δ t /2$ mein clock $v Δ t /2$ drift karta hai, photon $L$ utha hai, aur uski diagonal path length $c Δ t /2$ hai.

Yeh step kyon? Postulate 2: Main bhi photon ko speed $c$ par dekhta hoon, chahe woh apne lamba diagonal route par ho. Toh woh half-path length $c \cdot (Δ t /2)$ hai, kuch bada nahi.

Pythagoras apply karo pehli half ke right triangle par (vertical $L$ , horizontal $v Δ t /2$ , hypotenuse $c Δ t /2$ ): $(\frac{c Δ t}{2})^{2} = L^{2} + (\frac{v Δ t}{2})^{2}$

Figure — Time dilation — derivation, twin paradox

$L = \frac{c Δ t _{0}}{2}$ substitute karo: $\frac{c ^{2} Δ t ^{2}}{4} = \frac{c ^{2} Δ t _{0}^{2}}{4} + \frac{v ^{2} Δ t ^{2}}{4}$ $4$ se multiply karo: $c^{2} Δ t^{2} = c^{2} Δ t_{0}^{2} + v^{2} Δ t^{2}$ $c^{2} Δ t^{2} - v^{2} Δ t^{2} = c^{2} Δ t_{0}^{2}$ $Δ t^{2} (c^{2} - v^{2}) = c^{2} Δ t_{0}^{2}$ $Δ t^{2} = \frac{c ^{2} Δ t _{0}^{2}}{c ^{2} - v ^{2}} = \frac{Δ t _{0}^{2}}{1 - v ^{2} / c ^{2}}$

Worked examples

The Twin Paradox

Common mistake Steel-manned mistakes

Mistake 1: "Time dilation symmetric hai, isliye twins same age ke hain → contradiction." Kyon sahi lagta hai: Jab dono inertially move karte hain, har ek genuinely doosre ki clock slow dekhta hai — yeh sach aur symmetric hai. Fix: Ages compare karne ke liye unhe ek jagah reunite karna hoga. Reunite karne ke liye, ek twin ko turn around karna hoga (accelerate karna), symmetry todte hue. Traveler kam age hota hai. Koi contradiction nahi.

Mistake 2: Earth time ko $Δ t_{0}$ mein plug karna. Kyon sahi lagta hai: "Earth rest par hai, toh easy variable use karo." Fix: $Δ t_{0}$ hamesha wahan measure hota hai jahan dono events space mein coincide karein (moving clock ka apna frame). Muon/ship ke liye, yeh particle/ship frame hai, Earth nahi.

Mistake 3: Yeh sochna ki fast clock "frozen" dikh ti hai lekin mechanically broken hai. Kyon sahi lagta hai: Yeh ek malfunction jaisa lagta hai. Fix: Kuch bhi broken nahi hai. Har process — chemical, biological, atomic — identically slow hota hai kyunki time khud us frame mein slow run karta hai.

Recall Feynman: ek 12-saal ke bache ko explain karo

Light sabse fast cheez hai kabhi bhi, aur weirdly yeh same speed par jaati hai chahe tum use kitna bhi chase karo. Ek aise clock imagine karo jo ek light beam ko upar mirror tak aur wapas neeche bounce karke tick karta hai. Agar main ise lekar tumhare paas se guzarta hoon, tum light ko ek slanted zig-zag mein jaate dekhte ho — ek lamba safar. Lekin light speed up nahi kar sakti, isliye har tick tumhare liye zyada time leti hai. Iska matlab meri clock — aur mera poora body — tumhare view se slow motion mein chalti hai. Ab agar main ek door star tak udata hoon aur wapas aata hoon, mujhe apna rocket turn around karna pada, aur wahi mujhe slower age karne wala banata hai. Main apne stay-at-home twin se chhota wapas aaunga. Yeh koi magic ya trick nahi hai — time sach mein stretch hota hai.

Active recall

Kaun sa Einstein postulate time dilation cause karta hai?

Sabhi inertial frames mein light ki speed

c

ki invariance.

Proper time

Δ t_{0}

define karo.

Do events ke beech time jo us frame mein measure hota hai jahan woh same jagah occur karte hain (moving clock ka apna rest frame, jaise light clock ki ek full round-trip).

Time dilation formula state karo.

Δ t = γ Δ t_{0}

with

γ = 1/ 1 - v^{2} / c^{2}

Δ t \geq Δ t_{0}

hamesha kyon?

Kyunki

γ \geq 1

; moving clocks slow chalti hain.

Light-clock derivation mein hum ek full round-trip kyon use karte hain?

Taaki emission aur final reception clock ke rest frame mein same jagah hों,

Δ t_{0}

ko true proper time banate hue.

Clock ke rest frame mein ek round-trip ke liye

Δ t_{0}

kya hai?

Δ t_{0} = 2 L / c

Derivation mein Pythagoras (pehli half) se kaun si equation nikiti hai?

(c Δ t /2)^{2} = L^{2} + (v Δ t /2)^{2}

Twin paradox resolve karo.

Traveling twin accelerate/turn around karta hai, inertial frames switch karta hai, isliye situation symmetric nahi hai; accelerating twin kam age hota hai.

Muon lifetime 2.2 µs at 0.99c — lab lifetime?

γ \approx 7.09

, isliye

Δ t \approx 15.6 μ s

0.6c par,

γ

kya hai?

1/ 1 - 0.36 = 1/0.8 = 1.25

Daily life mein time dilation kyon nahi dikhti?

Everyday speeds par

v ≪ c

γ \approx 1

Connections

Special Relativity Postulates
Length Contraction — partner effect ( $L = L_{0} / γ$ )
Lorentz Transformations — general framework jisse time dilation nikalta hai
Relativistic Velocity Addition
Muon Decay Experiment — experimental confirmation
Spacetime Interval — kyon proper time frame-invariant hai