2.3.28 · D1Lorentz transformation — derivation
Poori Lorentz transformation ek dictionary hai jo translate karti hai ki koi cheez kahan aur kab hui — ek moving observer ke notebook se doosre ke notebook mein — aur saath mein ek hi ek vaada nibhaati hai: light hamesha same fixed speed pe travel karti hai, har kisi ke liye. Neeche har ek symbol usi dictionary ka ek word hai, aur jab aap sab jaante hain, derivation ek sentence ki tarah padhti hai.
Pehle aapko parent note dekhna hai ki Newton ki clock ko kaise theek kiya, lekin uske liye pehle har ek letter samajhna zaroori hai. Hum sab kuch zero se banate hain, ek doosre pe depend karne ke order mein. Skip mat karo — pehli line assume karti hai zero prior notation.
Ek event ek akela happening hai — ek jagah pe, ek instant pe — ek patakha phatna, ek clock ka tick karna, do origins ka milna. Ye koi object nahi hai; ye space mein ek point aur time mein ek point hai, ek saath bundle hua.
Intuition Ise ek stretched-out map pe ek dot ki tarah sochoEk horizontal line banao space ke liye (track ke saath kitni door) aur ek vertical line time ke liye (kitne seconds beet gaye). Ek patakha jo 4 metre right mein phataa, 3 seconds pe, woh usi map pe ek akela dot hai. Woh dot hi event hai. Poori relativity is baare mein hai ki alag-alag observers us same dot ko alag-alag coordinates pe rakhte hain .
Is topic ko iske kyun zaroorat hai: Lorentz transformation ek machine hai jiske inputs aur outputs hain. Inputs hain ek observer ke coordinates ek dot ke liye; outputs hain doosre observer ke coordinates us same dot ke liye. Event ke idea ke bina, transform karne ke liye kuch bhi nahi hoga.
Definition Coordinates aur reference frameEk reference frame ek observer ka poora grid hai — rulers aur synchronized clocks se bhara hua poore space mein. Frame S
x y , z t
x t ( t , x , y , z )
Intuition Char numbers kyun, aury , z x y z poori duniya mein rehta hai, sirf ek line pe nahi.
Ek prime, likha jaata hai ′ tag hai jiska matlab hai "jaise doosre observer ne measure kiya." Frame S ′ S S ′ ( t ′ , x ′ , y ′ , z ′ )
Common mistake "Prime ka matlab derivative hai."Kyun sahi lagta hai: calculus mein f ′ Fix: yahan x ′ derivative nahi hai — ye simply woh x S ′
Is topic ko iske kyun zaroorat hai: transform karne ka matlab hai unprimed se primed jaana. Bina ek symbol ke jo dono notebooks ko alag kare, hum sawaal bhi nahi likh sakte.
Definition Relative speedv v jisse S ′ S , metres per second mein, shared x S S ′ v
Intuition Do grids sliding ki picture banaoSocho S S ′ S ′ v v
v backwards − v v → − v inverse transform ko forward jaisi dikhata hai — aap "symmetry" step parent derivation mein milenge.
c c ≈ 3 × 1 0 8 Michelson–Morley experiment ) ki har observer yahi same c c
c v u u + v c c + v
Definition Light-pulse constraintsx = c t x ′ = c t ′ t = t ′ = 0 S x = c t = × S ′ wahi pulse obey karni chahiye ==x ′ = c t ′ c dono frames mein c
Is topic ko iske kyun zaroorat hai: c x = c t x ′ = c t ′
Intuition Kyun yeh obviously sahi lagta haiAgar aapka friend aapke right mein v t x x − v t Ek hi flaw hai woh hidden claim t ′ = t
Is topic ko iske kyun zaroorat hai: parent note pehle is ko "steel-man karta hai phir fix karta hai." Guess jaanna zaroori hai taaki repair ki appreciation ho sake. Yeh woh low-speed limit bhi hai jisme final answer reduce ho jaana chahiye.
v 2 / c 2 v 2 v c 2 square ki light ki speed ke square se comparison karte ho. Kyunki koi bhi c 0 1
v 2 / c 2 0 1
ek car (v = 30 1 0 − 14 zero , Newton theek hai.
v = 0.6 c 0.36 v → c 1
Is topic ko iske kyun zaroorat hai: yeh fraction Lorentz factor γ 1
Definition Square root aurγ q kaunsa number apne aap se multiply hoke q " Lorentz factor hai
γ = 1 − v 2 / c 2 1 . v 2 / c 2 1
γ ≥ 1 0 1 1 − v 2 / c 2 1 se kam hai. Uski square root bhi 1 se kam hai. 1 ko 1 se kam kisi number se divide karo toh 1 se bada aata hai. Toh γ ≥ 1
γ Kyun sahi lagta hai: root ke andar minus sign lagta hai jaise overshoot ho sakta hai. Fix: 0 ≤ v < c 0 ≤ v 2 / c 2 < 1 < 1 > 1 kabhi 1
Is topic ko iske kyun zaroorat hai: γ woh number hai jis pe poori derivation converge karti hai. Har final formula ek γ
Definition Linear relationEk relation linear hai agar naya coordinate ek constant times old coordinate, plus ek aur constant times doosra old coordinate ho — koi squares nahi, koi curves nahi. Jaise x ′ = A x + B t
A B x ′ = A x + B t A B abhi-tak-unknown fixed numbers hain — placeholders jinki values hum shuroo mein nahi jaante. Derivation ka kaam hai unhe solve karna using physical conditions (moving origin, dono frames ke beech symmetry, aur light-speed constraints x = c t x ′ = c t ′ γ v
Intuition Picture: ek ruler stretch aur slide kar sakta hai, lekin bend nahiEk linear map grid ko stretch kar sakta hai, shear kar sakta hai (squares ko tilt karna), ya slide kar sakta hai — lekin woh kabhi ek seedhi line ko curve mein bend nahi kar sakta. Constant velocity pe drift karta hua ek particle event-map pe ek seedhi line banaata hai. Dono observers agree karte hain ki woh freely drift kar raha hai (koi phantom forces nahi). Sirf bending-free — yani linear — dictionaries hi yeh guarantee kar sakti hain.
Is topic ko iske kyun zaroorat hai: derivation shuru karta hai sabse general linear guess x ′ = A x + B t A , B
s s akela number hai jo ek event se attached hai (origin se measure kiya gaya), aur hum usually uske square, s 2 s 2 = c 2 t 2 − x 2 − y 2 − z 2 . y z x t 1+1 dimensions kehte hain: ek space axis, ek time axis) aur likhte hain
s 2 = c 2 t 2 − x 2 . Spacetime interval .
Intuition Minus sign kyun, ordinary distance ke plus ki jagahOrdinary flat-map distance use karta hai x 2 + y 2 opposite signs ke saath enter karte hain. Yeh akela minus wahi hai jo unchanged rehta hai jab aap notebooks switch karte ho, chahe t x Minkowski diagram aisa kyun dikhta hai.
Is topic ko iske kyun zaroorat hai: saari mixing ke baad, hum jaanna chahte hain ki kya bachta hai. s 2 x t y ′ = y z ′ = z
Event = dot in space and time
Coordinates t x y z in frame S
Primed frame S prime and prime tag
Relative speed v between frames
Speed of light c is the same for all
Galilean guess x minus vt
Linearity x prime equals A x plus B t
Ratio v squared over c squared
Spacetime interval invariant
Har arrow ka matlab hai "left wala idea pehle chahiye tab right wala sense dega." Notice karo ki c linearity do upstream springs hain jisse baaki sab kuch peeta hai.
Right side cover karo, aloud jawaab do, phir reveal karo.
Event kya hota hai? Ek jagah aur ek instant pe ek akela happening — space-time map pe ek dot.
Yahan prime mark ′ "Jaise doosre observer S ′
v Woh speed jisse frame S ′ S x
Platform + v − v Motion relative hai; har frame doosre ko opposite direction mein move karte dekhta hai, jo symmetry step drive karta hai.
c Har inertial observer same light speed c
Do light-pulse constraints likho. S x = c t S ′ x ′ = c t ′ c
Galilean transformation likho (charon equations). x ′ = x − v t t ′ = t y ′ = y z ′ = z
Galileo ne kaunsa hidden assumption liya jo relativity discard karta hai? Ki time universal hai, t ′ = t
x ′ = A x + B t A B Unknown constants jo physical conditions se solve kiye jaane hain.
v 2 / c 2 0 1 c
γ ≥ 1 ≤ 1 γ = 1/ 1 − v 2 / c 2 , hamesha
≥ 1 .
Transform linear kyun hona chahiye? Taaki straight (constant-velocity) motion har frame mein straight rahe — linear maps stretch aur slide kar sakte hain lekin kabhi bend nahi karte.
Full spacetime interval aur uska 1+1D form likho. s 2 = c 2 t 2 − x 2 − y 2 − z 2 s 2 = c 2 t 2 − x 2
Hum interval comparison se y z Kyunki y ′ = y z ′ = z
Jab har reveal fast aur clean aaye, tab aap parent derivation line by line walk karne ke liye ready ho.
