Visual walkthrough — Relativistic momentum p = γmv

2.3.31 · D2 · Physics › Modern Physics › Relativistic momentum p = γmv

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Step 1 — Pehle, momentum hota kya hai?

KYA HAI. Newton ka jawab seedha hai "inhe ek saath multiply karo": $\vec{p}=\underset{\text{kitna stuff}}{\underbrace{m}}\times \underset{\text{kitna tez, aur kis taraf}}{\underbrace{\vec{v}}}$

KYUN. Double mass, toh rokna do guna mushkil. Double speed, toh rokna do guna mushkil. Ek plain product dono ko capture karta hai. Yahi hai Newtonian momentum p = mv.

PICTURE. Neeche, mass $m$ ki ek ball ek track par right slide kar rahi hai. Green arrow uska momentum $\vec{p}=m\vec{v}$ hai — yoh usi taraf point karta hai jis taraf ball move kar rahi hai aur agar $m$ ya $v$ badhta hai toh uski length bhi badhti hai.

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Step 2 — Velocity secretly kiske clock pe depend karti hai

KYA HAI. Hum dono clocks ko alag kar dete hain aur unhe naam dete hain:

• $dt$ = lab clock ki ek tiny tick (woh observer jo still khada hai).
• $d\tau$ = proper-time clock ki ek tiny tick, woh jo ball ke saath travel kar raha hai. Greek letter $\tau$ ("tau") sirf uska naam hai.

KYUN. $v=dx/dt$ mein "$dt$" lab ki clock hai — ek aisi quantity jis par har alag observer disagreement karta hai. Agar humara momentum ek aisi number se bana hai jis par observers disagree karte hain, toh woh frames ke beech pass hone par survive nahi karega. Hume ek aisi clock chahiye jis par sab agree karte hain. Woh clock hai $d\tau$. Dekho Lorentz factor and time dilation.

PICTURE. Ek hi journey par do clocks: lab clock (coral) ne ball ki apni clock (mint) se zyada time sweep kiya hai usi identical trip ke liye. Ball ki clock woh honest, shared wali hai.

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Step 3 — Dono clocks kitna disagree karte hain? Milo $\gamma$ se

KYA HAI. $\gamma$ ki exact size time dilation se aati hai: $\gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\text{jahan }c=\text{speed of light}.$ Pieces padho: $v^2/c^2$ hai "light speed ke kitna close, squared" — $0$ aur $1$ ke beech ka number. Use $1$ se subtract karo, square root lo, phir flip karo (yehi reason hai ki woh ek $1$ ke neeche baitha hai). Flip karne se $\gamma$ badhta hai jab $v$ badhta hai.

YEH SHAPE KYUN AUR KOI AUR NAHI? Kyunki hume kuch chahiye jo ho

• bilkul $1$ jab ball slow ho ($v=0\Rightarrow \sqrt{1-0}=1\Rightarrow \gamma =1$: clocks agree),
• aur $\infty$ tak blow up ho jab $v\to c$ ($1-1=0$ root ke neeche $\Rightarrow$ zero se divide karna).

Yahi exactly woh behaviour hai jo experiments demand karte hain. Note karo $\gamma$ $v^2$ par depend karta hai, toh woh same hai chahe ball right jaaye ($v>0$) ya left ($v<0$).

RECIPROCAL KYUN (bare root nahi)? Ek bahut common trap yeh hai ki $\gamma =\sqrt{1-v^2/c^2}$ likho, jo 1 se neeche shrink karta hai. Lekin lab clock ko aage run karna chahiye, toh $\gamma$ $\ge 1$ hona chahiye. $1/(\dots )$ yeh guarantee karta hai.

PICTURE. $\gamma$ ka curve $v/c$ ke against: slow speeds ke liye flat aur 1 ke barabar, phir $v=c$ par ek vertical wall ban jaata hai.

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Step 4 — Momentum ko honest clock se rebuild karo

KYA HAI. Repaired momentum define karo: $\vec{p}\equiv m\frac{d\vec{x}}{d\tau }$ Term by term: $d\vec{x}$ = woh tiny step jo ball leta hai (ek arrow, direction hai); $d\tau$ = ball ki apni clock-tick (sab se agreed, Step 2 se); $m$ = invariant rest mass, har frame mein same label.

KYUN. $d\vec{x}$ aur $d\tau$ dono frames ke beech cleanly transform karte hain (woh Energy-momentum four-vector ke pieces hain). Toh unka ratio bhi karta hai — matlab agar yeh $\vec{p}$ ek frame mein conserved hai, toh woh automatically sab frames mein conserved hai. Yehi poora goal tha.

PICTURE. Same displacement arrow $d\vec{x}$, lekin ab hum divisor ko mint proper-clock tick label karte hain coral lab tick ki jagah.

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Step 5 — Woh speed mein translate karo jo hum actually measure karte hain

KYA HAI. Chain rule use karo — ek aise clock ko insert karne ka tarika jo hum jaante hain woh mein jo hum nahi jaante: $\vec{p}=m\frac{d\vec{x}}{d\tau }=m\underset{=\vec{v}}{\underbrace{\frac{d\vec{x}}{dt}}}\cdot \underset{=\gamma }{\underbrace{\frac{dt}{d\tau }}}=\gamma m\vec{v}$

Do factors appear hote dekho:

• $\frac{d\vec{x}}{dt}=\vec{v}$ exactly woh everyday velocity hai Step 1 se.
• $\frac{dt}{d\tau }=\gamma$ exactly woh clock-mismatch hai Step 3 se ($dt=\gamma d\tau$ flip karo).

$\boxed{\vec{p}=\underset{\text{clock boost}}{\underbrace{\gamma }}\underset{\text{rest mass}}{\underbrace{m}}\underset{\text{lab velocity}}{\underbrace{\vec{v}}}}$

Kyunki $\vec{v}$ direction carry karta hai, $\vec{p}$ automatically usi taraf point karta hai jis taraf ball move karti hai — $v>0$ ke liye right, $v<0$ ke liye left — jabki $\gamma$ sirf uski length scale karta hai.

CHAIN RULE SPECIFICALLY KYUN? Yeh woh tool hai jo kehta hai "rate with respect to $\tau$ equals rate with respect to $t$, times $t$ $\tau$ per kitna fast change karta hai". Yeh hume woh clock borrow karne deta hai jo hum samajhte hain ($t$) us clock mein rate compute karne ke liye jo hume chahiye ($\tau$). Dono clocks ko itne cleanly kuch aur bridge nahi karta.

PICTURE. Newton ka chhota green arrow $mv$, aur uske upar $\gamma$ ki extra stretch, jo longer relativistic arrow deta hai.

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Step 6 — Edge case: slow ball (kya yeh Newton par reduce hota hai?)

KYA HAI. Everyday speeds par naya formula purane wale par collapse kar jaata hai.

YEH HONA CHAHIYE KYUN. Newton ke laws cars aur cricket balls ke liye behtareen kaam karte hain. Kisi bhi correct upgrade ko unse agree karna hoga jahan unhe test kiya gaya tha. Yeh hai correspondence principle — ek physics safety check, koi coincidence nahi.

PICTURE. $\sim 0.1c$ se neeche relativistic curve aur straight Newtonian line ek doosre ke upar hain; woh sirf high speed par alag hote hain.

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Step 7 — Edge case: ball light speed ke paas aa rahi hai

KYA HAI. Infinite momentum ko infinite push ki zaroorat hogi. Toh koi bhi finite force ek massive particle ko $c$ tak drive nahi kar sakti.

YEH KYUN MAAYINE RAKHTA HAI. Yeh Why nothing exceeds the speed of light ki direct root hai — momentum, koi mystical barrier nahi, woh hai jo ise forbid karta hai.

PICTURE. $p$ versus $v/c$: gentle straight-ish start, phir ek runaway vertical asymptote $v=c$ se chipka hua.

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Step 8 — Degenerate case: massless particle ($m=0$)

KYA HAI. Massless particles ke liye hum energy–momentum relation se momentum read karte hain (from Energy-momentum four-vector), jahan $E$ total energy hai jaise abhi define ki: $E^2=(pc{)}^2+(m{c}^2{)}^2\stackrel{m=0}{\to }E=pc\Rightarrow p=\frac{E}{c}.$

KYUN. $p=\gamma mv$ massive particles ke liye build kiya gaya tha (unke paas proper-time clock hoti hai). Light ka koi rest frame nahi hai aur koi proper clock nahi, toh derivation ka pehla ingredient hi missing hai. Dekho Photon momentum and radiation pressure.

PICTURE. Energy–momentum right triangle: legs $pc$ aur $m{c}^2$, hypotenuse $E$. $m{c}^2$ leg ko zero shrink karo aur hypotenuse doosri leg ban jaati hai: $E=pc$.

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Ek-picture summary

Ek frame mein poora safar: $mv$ se shuru karo, notice karo ki lab clock ball ki clock nahi hai, mismatch $\gamma$ measure karo, honest clock se divide karo, wapas convert karo, aur arrow exactly $\gamma$ se grow karta hai.

Recall Feynman retelling (plain words)

Socho aap ek fair score chahte ho "yeh cheez rokna kitna mushkil hai" ke liye. Aap lete ho woh kitna door gaya aur time se divide karte ho. Lekin kiska watch? Zameen par aapka watch, ya woh watch jo flying cheez carry karti hai? Woh dono watches disagree karte hain — flying wali slower tick karti hai, aur woh jitni peeche rehti hai woh ek number hai jise hum gamma kehte hain. Poori universe mein sab flying cheez ki apni watch par agree karte hain, kyunki woh cheez se hi chipki hai aur usi ek light speed se jo sab same measure karte hain. Einstein kehte hain: woh honest watch use karo. Jab aap us se division karte ho aur phir jawab ko un speeds mein translate karte ho jo aapka ground-radar read kar sake, toh ek spare gamma pop out karta hai aur purane $m$ times $v$ ke upar ride karta hai. Toh momentum hai $\gamma mv$. Ise jis taraf bhi cheez jaaye point karo — right plus hai, left minus hai. Slow cheezein ke liye gamma sirf 1 hai aur aap Newton par wapas ho. Super-fast cheezein ke liye gamma explode karta hai, toh momentum explode karta hai, toh aap kabhi kisi cheez ko light speed tak push nahi kar sakte. Aur light ke liye — jiske paas apni koi watch nahi — yeh recipe toot jaati hai, aur hum $p=E/c$ use karte hain, jahan $E$ uski total energy hai.

Recall

Proper time $d\tau$ se kyun divide karte hain lab time $dt$ ki jagah? ::: $d\tau$ har observer ke liye same hai, toh us se bana momentum sab frames mein conserved hai. $d\tau$ sab observers ke liye same kyun hai? ::: Yeh invariant interval $c^{2}d{\tau }^2=c^{2}d{t}^2-d{x}^2$ se aata hai, sirf $c$ se bana (sabke liye same). Chain rule mein extra $\gamma$ kahan se aata hai? ::: $dt/d\tau =\gamma$ se, woh time-dilation clock mismatch. Photon ke liye $p=\gamma mv$ ka kya hota hai? ::: Yeh $\infty \times 0$ deta hai (undefined); use $p=E/c$ instead. Ek massive particle $c$ kyun nahi reach kar sakta? ::: Jab $v\to c$, $\gamma \to \infty$ toh $p\to \infty$, jisme infinite push chahiye. $E^2=(pc{)}^2+(m{c}^2{)}^2$ mein $E$ kya hai? ::: Particle ki total energy (massive particle ke liye $=\gamma m{c}^2$).

Connections

• Lorentz factor and time dilation — Step 2 aur Step 3 supply karta hai
• Newtonian momentum p = mv — Step 1 starting point aur Step 6 limit
• Energy-momentum four-vector — kyun $d\vec{x}/d\tau$ cleanly transform karta hai (Step 4)
• Why nothing exceeds the speed of light — Step 7 consequence
• Photon momentum and radiation pressure — Step 8 massless case
• Mass-energy equivalence E = γmc² — energy partner Step 8 mein use hoti hai