Lorentz transformation — derivation — Hinglish notes

2.3.28 · Physics › Modern Physics

Hum kya transform kar rahe hain

Galilean guess (jo Newton likhta): $x^{'} = x - v t, t^{'} = t .$ Hum ise steel-man-then-fix karenge.

Derivation from first principles

Postulate 1 — Linearity

Toh sabse general linear form likhte hain ( $y, z$ trivial hain): $x^{'} = A x + B t, t^{'} = D x + E t .$ Chaar unknowns $A, B, D, E$ hain. Inhe hum physical conditions se pin down karte hain.

Condition 1 — $S^{'}$ ki origin $S$ mein $v$ speed se move karti hai

Point $x^{'} = 0$ moving origin hai; $S$ mein yeh $x = v t$ par baithti hai. $x^{'} = 0$ set karo: $0 = A x + B t \Rightarrow x = - \frac{B}{A} t$ . Yeh $v t$ ke equal honi chahiye, toh $B = - A v .$ Hence $x^{'} = A (x - v t)$ .

Condition 2 — Symmetry / Principle of Relativity

Condition 3 — Light ki speed DONO frames mein $c$ hai

Common origin se $t = t^{'} = 0$ par ek light pulse fire karo. $S$ mein: $x = c t$ . $S^{'}$ mein: $x^{'} = c t^{'}$ .

$x = c t$ ko $x^{'} = A (x - v t)$ mein substitute karo: $x^{'} = A (c t - v t) = A t (c - v) .$ $x^{'} = c t^{'}$ ko inverse $x = A (x^{'} + v t^{'})$ mein substitute karo: $c t = A (c t^{'} + v t^{'}) = A t^{'} (c + v) .$

Ab do left sides aur do right sides ko multiply karo. Pehli equation ki left side par $x^{'} = c t^{'}$ use karo: $x^{'} c t^{'} \cdot c t = A^{2} t t^{'} (c - v) (c + v) .$

Dono sides ko $t t^{'}$ se divide karo: $c^{2} = A^{2} (c^{2} - v^{2}) .$ Solve karo: $A^{2} = \frac{c ^{2}}{c ^{2} - v ^{2}} = \frac{1}{1 - v ^{2} / c ^{2}} \Rightarrow A = \frac{1}{1 - v ^{2} / c ^{2}} \equiv γ .$

Time equation nikalna

Humare paas $x^{'} = γ (x - v t)$ aur $x = γ (x^{'} + v t^{'})$ hain. Pehle ko doosre mein substitute karo $t^{'}$ solve karne ke liye: $x = γ (γ (x - v t) + v t^{'}) = γ^{2} (x - v t) + γ v t^{'} .$ $γ v t^{'}$ solve karo: $γ v t^{'} = x - γ^{2} x + γ^{2} v t = x (1 - γ^{2}) + γ^{2} v t .$ $1 - γ^{2} = 1 - \frac{1}{1 - v ^{2} / c ^{2}} = \frac{- v ^{2} / c ^{2}}{1 - v ^{2} / c ^{2}} = - γ^{2} \frac{v ^{2}}{c ^{2}}$ use karo: $γ v t^{'} = - γ^{2} \frac{v ^{2}}{c ^{2}} x + γ^{2} v t .$ $γ v$ se divide karo: $t^{'} = - γ \frac{v}{c ^{2}} x + γ t = γ (t - \frac{v x}{c ^{2}}) .$

Figure — Lorentz transformation — derivation

Woh invariant jo bach gaya

Check karo: $c^{2} t^{'2} - x^{'2} = c^{2} γ^{2} (t - \frac{v x}{c ^{2}})^{2} - γ^{2} (x - v t)^{2} .$ Expand karo aur cross terms ( $- 2 v x t$ type) cancel ho jaate hain; baaki $c^{2} t^{2} - x^{2}$ par collapse ho jaata hai (Verify dekho). Yeh invariance special relativity ka geometric heart hai.

Worked examples

Common mistakes

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho ki sabne ek rule pe agree kiya hai: light ki ek flash hamesha same fast speed se jaati dikhni chahiye, chahe tum kitni bhi fast daudo. Woh promise rakne ke liye, universe ko thoda cheat karna padta hai: jab tum bahut fast move karte ho, tera ruler thoda sa shrink ho jaata hai aur teri watch thodi si slow tick karti hai — bilkul sahi amount se taaki light abhi bhi same speed se dikhti rahe. Lorentz transformation woh exact recipe hai jisse ruler kitna shrink hoga aur watch kitni slow hogi. Normal speeds par yeh cheat itna tiny hai ki tum kabhi notice nahi karte.

Flashcards

Lorentz transformation ke neeche kaunse do postulates hain?

(1) Sabhi inertial frames mein physics ke laws identical hain; (2) sabhi inertial frames mein light ki speed

c

same hai.

Transform linear kyun hona chahiye?

Space homogeneous hai aur time uniform hai, isliye uniform-velocity motion uniform-velocity motion mein map honi chahiye — sirf linear maps yeh karte hain.

Lorentz factor batao.

γ = 1/ 1 - v^{2} / c^{2}

, hamesha

\geq 1

x^{'}

aur

t^{'}

ke liye Lorentz transformation likho.

x^{'} = γ (x - v t)

t^{'} = γ (t - v x / c^{2})

, with

y^{'} = y, z^{'} = z

A = γ

kaise milta hai?

x = c t, x^{'} = c t^{'}

c^{2} = A^{2} (c^{2} - v^{2})

milta hai.

Galilean velocity addition ki jagah kya aata hai?

u^{'} = (u - v) / (1 - uv / c^{2})

; yeh

c

ko invariant rakhta hai.

Lorentz transforms ke under kaunsi spacetime quantity invariant hai?

Interval

s^{2} = c^{2} t^{2} - x^{2}

Simultaneity relative kyun hai?

t^{'} = γ (t - v x / c^{2})

: equal

t

lekin alag

x

se alag

t^{'}

milta hai.

v ≪ c

hone par Lorentz transform ki limit?

γ \to 1

v x / c^{2} \to 0

→ Galilean

x^{'} = x - v t, t^{'} = t

Length contraction direction derive karo.

Dono ends ko same lab time par measure karo (

Δ t = 0

Δ x^{'} = γ Δ x \Rightarrow L = L_{0} / γ

Connections

Galilean transformation — woh $v ≪ c$ limit jise yeh generalize karta hai.
Time dilation — $t^{'}$ equation ka direct consequence.
Length contraction — $x^{'}$ equation ka consequence.
Relativity of simultaneity — $- v x / c^{2}$ term se janam leti hai.
Spacetime interval — woh invariant jo bach jaata hai.
Relativistic velocity addition — transform ko differentiate karke derive hoti hai.
Michelson–Morley experiment — experimental motivation.
Minkowski diagram — in transforms ki geometric picture.