Foundations — Postulates of SR

2.3.26 · D1 · Physics › Modern Physics › Postulates of SR

Is page par ye assume kiya gaya hai ki tumne parent note ki notation mein se kuch bhi nahi dekha. Hum har symbol ko ground up se build karte hain, us order mein jis order mein ek dusre par depend karte hain. Koi bhi cheez use se pehle nahi aati jab tak draw na ho.

0. Do postulates — wo rules jinpar sab kuch tika hai

Kyunki poora page baar baar unhi ki taraf point karta hai, chaliye Special Relativity ke do rules seedhe plain words mein bata dete hain.

Hum khaas taur par Postulate 2 par depend karenge: yahi wajah hai ki hamare light-clock (§6) mein photon exactly $c$ par chalna forced hai, tab bhi jab clock sideway slide kar raha ho.

1. "Event" aur "frame of reference" — wo ruler-and-clock jo tum saath rakhte ho

Kisi bhi physics se pehle, hume do sabse basic words chahiye.

Figure dekho: black grid ek khade insaan ka hai; red grid ek dayi taraf glide karne wale ka. Same event (dot) dono ke liye alag grid coordinates par baithta hai — yahi poora drama hai relativity ka ek tasveer mein.

2. "Inertial" — special, well-behaved frames

Har frame fair game nahi hota. Postulate 1 sirf inertial frames ki baat karta hai.

Picture-test se pehle, hume "mysterious tug" ka naam dena hoga.

Picture-test: ek frictionless table par ball rakho.

• Smoothly chalte train par → ball wahi rahi. Inertial. ✓
• Achanak brake lagate train par → ball ko koi touch kiye bina aage roll kar gai. Wo phantom dhakka ek pseudo-force hai; braking frame non-inertial hai. ✗
• Spinning merry-go-round par → ball khud sideway curve karti hai (Coriolis pseudo-force), aur bahar flung feel hoti hai. Isliye ek rotating frame bhi non-inertial hota hai, chahe spin rate constant hi kyun na ho — kyunki turning ek tarah ka acceleration hai. ✗

Topic ko ye kyun chahiye: Postulate 1 sirf inertial frames ke liye claim ki gayi hai. Accelerating aur rotating frames ko General Relativity chahiye — ek alag, baad ki kahani.

3. Velocity $v$ — number "kitna tez, kis taraf"

Red arrow moving frame ki velocity hai. Parent note mein, $v$ hamesha do observers ke beech relative speed hai — kabhi light ki speed nahi, jiske liye apna letter hai (next).

4. Light ki speed $c$ — universe ka speed limit

Sabse important baat — jo relativity ko relativity banati hai — ye hai ki har inertial observer yahi same $c$ measure karta hai (exactly Postulate 2 yehi hai), chahe wo light beam ke peeche bhaag raha ho. Contrast karo $v$ se, jo alag observers ke liye alag hota hai.

5. $v^2/c^2$ — apni speed ko light se compare karna

Ab hum do symbols combine karte hain jo hamare paas hain. Parent note baar baar $v^2/c^2$ likhta hai. Chaliye har piece earn karte hain.

• $v/c$ = "main light-speed ka kitna fraction ja raha hoon?" $v=0.6c$ par ek rocket ka $v/c=0.6$ hai, yaani light-speed ka 60%.
• Squaring ($v^2/c^2$) sign hata deta hai: chahe tum left jao ya right, $v^2$ positive hai. Direction ko ye nahi change karna chahiye ki time kitna slow hota hai, to squaring exactly sahi tool hai — ye jawaab deta hai "kitna tez" jabki "kis taraf" ignore karta hai.

6. Square root $\sqrt{1-v^2/c^2}$ — ek triangle se step by step bana

Parent ka key expression hai $\sqrt{1-v^2/c^2}$. Ye exact shape kahan se aata hai? Pythagoras se, jo light-clock naam ki ek device par apply hota hai. Pehle, wo device kya hai?

Ab same clock ko ek aisi frame se dekho jisme ye $v$ speed se right glide karta hai. Ek half-tick consider karo (light neeche se upar wale mirror tak ja rahi hai). Do frames ise dekhte hain:

• Clock ke apne rest frame mein light seedha upar $L_0$ distance jaati hai, ek proper half-tick time mein jo hum $\frac{1}{2}\Delta t_0$ label karenge.
• Us frame mein jahan clock move karta hai, upar wala mirror aage slide karta hai jab light charhti hai, to light ek diagonal trace karti hai. Is observer ka half-tick time $\frac{1}{2}\Delta t$ kahein (ye moving-frame interval hai — exactly wahi $\Delta t$ jo hum §8 mein define karte hain).

Diagonal (red) ek right triangle ki hypotenuse hai jiske teen sides hain, sab moving frame mein measured:

• Vertical leg $=L_0$ (mirror gap — unchanged, kyunki motion sideways hai, upar-neeche nahi),
• Horizontal leg $=v\cdot \frac{1}{2}\Delta t$ (is half-tick mein mirror kitna slide hua),
• Hypotenuse $=c\cdot \frac{1}{2}\Delta t$ (light ka actual path — ise $c$ par jaana hi padega, Postulate 2 se, chahe path lamba kyun na ho).

Step 1 — Pythagoras lagao (leg² + leg² = hypotenuse²). Algebra clean rakhne ke liye $\tau \equiv \frac{1}{2}\Delta t$ likhte hain: $L_0^2+(v\tau {)}^2=(c\tau {)}^2$ Humne kya kiya: teen sides ko relate kiya. Kyun: right triangle exactly wahi hai jo "moving light-clock" draw karta hai, aur Pythagoras woh ek tool hai jo legs ko hypotenuse se jodta hai.

Step 2 — ${\tau }^2$ terms ek side gather karo: $c^2{\tau }^2-v^2{\tau }^2=L_0^2\Rightarrow {\tau }^2(c^2-v^2)=L_0^2$

Step 3 — $\tau$ ke liye solve karo divide karke phir square root leke: $\tau =\frac{L_0}{\sqrt{c^2-v^2}}$

Step 4 — root mein se $c$ bahar nikalo taaki fraction $v^2/c^2$ appear ho. Likho $c^2-v^2=c^2\left(1-\frac{v^2}{c^2}\right)$, aur $\sqrt{c^2(\cdots )}=c\sqrt{\cdots }$: $\tau =\frac{L_0}{c\sqrt{1-v^2/c^2}}$

Ye raha: exact shape $\sqrt{1-v^2/c^2}$ triangle se emerge hota hai, decree se nahi. Clock ke apne rest frame mein half-tick sirf $L_0/c$ hai (koi sideways slide nahi), to moving observer ka half-tick $\tau$ factor $1/\sqrt{1-v^2/c^2}$ se lamba hai. Dono ko double karke full ticks milate hain aur ye parent note ka $\Delta t=\gamma \Delta t_0$ ban jaata hai jab hum us factor ko naam dein (§7).

Note karo $1-v^2/c^2$ hamesha $0$ aur $1$ ke beech hota hai (kyunki $v^2/c^2$ $0$ aur $1$ ke beech baithta hai), to uska square root bhi $0$ aur $1$ ke beech hota hai.

7. Lorentz factor $\gamma$ — master stretch number

Kyunki neeche (denominator) $0$ aur $1$ ke beech ka number hai, $1$ ko usse divide karne par $\ge 1$ milta hai. To $\gamma$ hamesha kam se kam $1$ hota hai, aur $v\to c$ par bina bound ke badhta hai.

Curve padho: kam $v$ par ye $1$ ke paas rehta hai (Newton ki duniya), phir $v=c$ ke paas sky high uda jaata hai. Har relativistic effect — slow clocks, shorter rulers — is baat se measure hota hai ki $\gamma$ $1$ se kitna upar chadha hai. Kuch landmark values:

• $v=0$ par: $v^2/c^2=0$, to $\gamma =1$ (kuch nahi badalta).
• $v=0.6c$ par: $v^2/c^2=0.36$, to $\gamma =1/\sqrt{0.64}=1.25$.
• $v=0.9c$ par: $v^2/c^2=0.81$, to $\gamma \approx 2.29$.
• jab $v\to c$: $v^2/c^2\to 1$, to $\gamma \to \infty$ (uda jaata hai).

8. Times $\Delta t_0$ aur $\Delta t$ — "kiska clock tika?"

Symbol $\Delta$ (Greek "delta") ka matlab hai "mein change" ya "interval of". To $\Delta t$ = "elapsed time ki ek amount" — do events ke timestamps ke beech ka gap.

Parent ka headline result hai $\Delta t=\gamma \Delta t_0$. Kyunki $\gamma \ge 1$, moving observer hamesha lamba interval measure karta hai — moving clock slow tick karta dikha hai. Wahi hai Time Dilation.

9. Length $L_0$ aur $L$ — proper length vs. contracted length

Length distance hai metres mein — jaise light-clock mein mirror gap. Lekin time ki tarah, length bhi is par depend karti hai ki kaun measure kar raha hai, to hum do symbols rakhte hain, exactly wahi $L_0$ match karte hue jo hum §6 mein use kar chuke hain.

Wahi $\gamma$ jo time stretch karta hai ab length shrink karta hai.

Ye foundations topic ko kaise feed karte hain

Chain ko top-down padho. Frames (events se bane) batate hain kaun measure karta hai. Do speeds $c$ aur $v$ ratio $v^2/c^2$ banate hain. Moving light-clock par Pythagoras us ratio ko $\sqrt{1-v^2/c^2}$ ki shape deta hai, jiska reciprocal Lorentz factor $\gamma$ hai. Finally $\gamma$, do postulates ke saath ("same laws" force karta hai "$c$ is constant"), produce karta hai time dilation $\Delta t=\gamma \Delta t_0$ aur length contraction $L=L_0/\gamma$.

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