Simultaneity — relativity of simultaneity: Question bank

2.3.27 · D5 · Physics › Modern Physics › Simultaneity — relativity of simultaneity

True or false — justify

TF1. "Agar do events ek frame mein simultaneous hain, toh woh har frame mein simultaneous hain."

False — simultaneity sabhi frames mein sirf tab hold karti hai jab

Δ x = 0

ho; kisi bhi nonzero separation ke liye ek moving observer

Δ t^{'} = - γ v Δ x / c^{2} \neq = 0

measure karta hai.

TF2. "Relativity of simultaneity isliye hoti hai kyunki light ko tumhari aankhon tak pahunchne mein finite time lagti hai."

False — dono observers pehle se hi usi

c

se woh travel time subtract kar lete hain; yeh disagreement uss correction ke baad bhi bachti hai kyunki

c

dono ke liye identical hai, isliye yeh spacetime ki ek real feature hai, koi viewing delay nahi.

TF3. "Do events jo space mein exact same point par hain woh har kisi ke liye simultaneous hain."

True —

Δ x = 0

ke saath formula

Δ t^{'} = 0

deta hai chahe

v

γ

kuch bhi ho; coincident events ko koi bhi observer reorder nahi kar sakta.

TF4. "Agar Bob (train par) kehta hai front bolt pehle strike ki, toh Alice zaroor galat hai jo unhe simultaneous bata rahi hai."

False — koi bhi galat nahi hai; dono ne apne frame mein same speed of light sahi se apply ki. Koi privileged "true now" nahi hai jo unke beech decide kare.

TF5. "Ek fixed separation ke liye faster relative speed hamesha simultaneity gap badhati hai."

True par subtly — jaise

v

badhti hai,

v

aur

γ

dono badhte hain, isliye

γ v

monotonically infinity ki taraf

v \to c

hone par badhta hai, disagreement ko unboundedly bada karta hai (jab tak

v = c

na ho, jo unreachable hai).

TF6. "Relativity of simultaneity kisi effect ko uske cause se pehle dikha sakti hai."

False — sirf spacelike-separated events (itni door ki light connect nahi kar sakti) order swap kar sakti hain, aur woh pairs kabhi cause aur effect nahi ho sakte; timelike (causally linked) pairs har frame mein apna order maintain karte hain. Dekho Causality and the Light Cone.

TF7. "

Δ t^{'} = - γ v Δ x / c^{2}

mein minus sign sirf ek convention hai aur drop kiya ja sakta hai."

False — sign encode karta hai ki kaun sa event ek diya gaya moving observer pehle dekhta hai (leading clock lags). Isse drop karne se effect ki physical direction chali jaati hai.

TF8. "Agar tum swap kar do ki kaun sa 'event 1' hai aur kaun sa 'event 2', toh physics change ho jaati hai."

False — unhe swap karne se

Δ x

ka sign aur isliye

Δ t^{'}

ka sign flip hota hai, lekin physical statement ("front bolt train frame mein early hai") unchanged rehti hai; sirf label bookkeeping flip hoti hai.

Spot the error

SE1. "

Δ t^{'} = - γ v Δ x / c^{2}

, aur main

Δ x

= train ki length train ke measure ke anusaar (uski rest length) use karunga."

Error —

Δ x

frame

S

mein separation honi chahiye jahan events simultaneous hain (platform), naa ki

S^{'}

mein rest length; frames ko mix karne se galat number milta hai. Related idea: Length Contraction.

SE2. "Bob front bolt pehle dekhta hai kyunki woh train ke front ke zyada paas hai."

Error — Bob apne frame mein bilkul middle mein baitha hai, dono ends se equidistant. Woh front ki light pehle dekhta hai kyunki woh light travel karte waqt uss mein move karta hai, naa ki kisi distance imbalance ki wajah se.

SE3. "Kyunki platform strikes ko simultaneous dekhta hai, aur train length-contracted hai, toh train obviously unhe simultaneous bhi dekhti hai, bas squeezed."

Error — length contraction lengths change karta hai, timing nahi; timing disagreement Lorentz Transformation mein

- v x / c^{2}

term se aati hai, jo ek alag effect hai jo contraction ignore karne par bhi bachi rehti hai.

SE4. "Formula mein maine

v = 0.6

(plain number ke roop mein) plug kiya aur nonsense units mile."

Error — ya toh

v = 0.6 c

symbolically rakhna ta ki

c

's,

c^{2}

ke saath cancel ho jayein, ya

v = 1.8 \times 1 0^{8} m/s

use karo; ek bare "0.6" yahan dimensionally meaningless hai.

SE5. "Alice aur Bob disagree karte hain, toh kam se kam unki ek clock galat chal rahi hai ya toot gayi hai."

Error — dono sets of clocks ideal hain aur apne frame ke andar sahi se synchronized hain. Point yeh hai ki synchronization khud frame-dependent hai; koi bhi clock faulty nahi hai. Related clock-rate effect ke liye dekho Time Dilation.

SE6. "Effect tiny hai, toh yeh zaroor ek rounding artifact hai jo careful measurement se disappear ho jaata hai."

Error — gap

γ v Δ x / c^{2}

ke proportional hai, ek exact prediction, noise nahi. Yeh sirf isliye chhota hai kyunki everyday speeds par

Δ x / c^{2}

chhota hota hai; relativistic

v

par yeh microseconds aur bada ho jaata hai.

SE7. "Simultaneity relative hai, toh dono events ke baare mein sab kuch (kaun 'pehle' hai) pure opinion hai bina kisi rules ke."

Error — ordering constrained hai: timelike-separated events ke liye har frame order par agree karta hai (light cone isse fix karta hai); sirf spacelike separation ke liye order frame-dependent hai.

Why questions

WHY1. Disagreement spatial separation

Δ x

par depend kyun karta hai, time-gap par nahi?

Kyunki events simultaneous start karti hain (

Δ t = 0

), isliye poora

S^{'}

time difference position-dependent term

- γ v x / c^{2}

se aata hai; equal times ke saath, sirf differing positions hi differing

t^{'}

produce kar sakti hain.

WHY2.

c

ki constancy kyun force karti hai ki simultaneity relative ho?

Agar har koi light ko

c

par measure kare, toh ek moving observer jo ek flash ki taraf travel kar raha hai aur doosre se door, unequal arrival ko unequal emission times se attribute karna hoga, kyunki woh unequal light speeds ko blame nahi kar sakta.

WHY3. Hamesha leading clock (front, direction of motion mein) kyun lag karti hai?

- v x / c^{2}

term bade

x

(front) ke liye zyada negative hai, isliye front clock ki

t^{'}

reading aur peeche push ho jaati hai — motion ki direction front ko "strike karne mein jaldi, clock mein peeche" end ke roop mein pick out karti hai.

WHY4. Hum ek universal master clock kyun define karke saare disputes settle nahi kar sakte?

Distant clocks ko synchronize karne ke liye signal bhejna padta hai (fastest hai light), aur "same instant" jo synchronization define karta hai woh tumhari motion par depend karta hai; koi frame-independent procedure nahi hai, isliye koi universal clock exist nahi karta.

WHY5. Relativity of simultaneity cause and effect ko kyun nahi todti?

Cause-effect pairs timelike-separated hote hain (ek signal

\leq c

par unhe link kar sakta hai), aur timelike order invariant hai; reorderable pairs spacelike hain, precisely woh jinhe koi signal connect nahi kar sakta, isliye koi cause kabhi apne effect ke baad nahi dekha jaata.

WHY6. Effect

v = 0

par kyun vanish ho jaata hai jab bhi

Δ x

huge ho?

Koi relative motion nahi hone par observer kisi bhi light beam mein move nahi kar raha, isliye arrivals mein distinguish karne ke liye kuch nahi hai; algebraically

v = 0

poore

- γ v Δ x / c^{2}

term ko khatam kar deta hai.

WHY7. Hum kyun kehte hain ki simultaneity "light postulate ki pehli casualty" hai, naa ki time dilation ya length contraction?

Kyunki dilation aur contraction dono shared "now" ke loss se derive kiye ja sakte hain; jab alag jagahon par clocks synchronization par disagree karein, mismatched clock rates aur shrunken lengths follow karte hain. Yeh logically upstream hai.

Edge cases

EC1. Ek fixed nonzero

Δ x

ke liye

v \to c

hone par

Δ t^{'}

ka kya hota hai?

γ \to \infty

isliye

Δ t^{'} \to \infty

— disagreement unboundedly badhti hai, lekin

v = c

khud massive observers ke liye forbidden hai, isliye divergence ek aise limit hai jo kabhi reach nahi hoti.

EC2. Jab

Δ x = 0

ho lekin

v

huge ho (maan lo

0.999 c

) toh

Δ t^{'}

kya hai?

Exactly zero — ek bada

γ v

Δ x = 0

se multiply karke phir bhi zero hai. Coincident events absolutely simultaneous hain chahe observer kitni bhi fast move kare.

EC3. Agar do events motion ki direction ke perpendicular separated hain (

y

ke along, equal

x

ke saath), toh ek moving observer kya dekhta hai?

Woh simultaneous rehte hain — formula motion ke along

Δ x

use karta hai; purely transverse separation mein

Δ x = 0

hai, isliye

Δ t^{'} = 0

EC4. Do events

S

mein simultaneous hain; ek frame

S^{'}

+ x

direction mein move karta hai aur doosra frame

S^{''}

- x

direction mein, dono speed

v

par. Kya differ karta hai?

Gap ka magnitude equal hai lekin sign flip hota hai:

S^{'}

aur

S^{''}

disagree karte hain kaun sa event pehle hai, kyunki

v

reverse karne se

Δ t^{'}

ka sign reverse hota hai.

EC5. Maano events

Δ x \neq = 0

ke saath

S

mein simultaneous hain. Kya koi aisa frame hai jahan woh same

S^{'}

-time par lekin alag

S^{'}

-place par occur karein?

Sirf

S

(aur frames jinka relative

v = 0

hai) unhe simultaneous rakhta hai;

x

ke along koi bhi

v \neq = 0

Δ t^{'} \neq = 0

deta hai, isliye koi bhi distinct moving frame unki simultaneity preserve nahi karta. Ise spacetime diagram par draw karo: axes ko tilt karna hamesha equal-time events ko split karta hai.

EC6. Do events

S

mein simultaneous aur co-located hain. Kya woh actually ek hi event hain?

Effectively haan — same time aur same place matlab same spacetime point; har observer agree karta hai ki woh coincide karte hain, aur koi reordering possible nahi hai.

EC7. Agar

Δ x

spacelike-large enough hai ki koi light events ko connect nahi kar sakti, toh kya koi frame unka order reverse kar sakta hai?

Haan — spacelike-separated events ke liye aise frames hain jahan ya toh koi bhi pehle ho, ya woh simultaneous hon; yeh exactly reorderable regime hai, aur yeh safe hai kyunki koi causal signal unhe link nahi karta.

Recall Jaane se pehle ek-line self-test

$Δ t^{'} = - γ v Δ x / c^{2}$ mein kaun si single quantity saari disagreement ka trigger hai, aur ise kis frame mein measure kiya jaata hai? Trigger aur uska frame ::: Spatial separation $Δ x$ , frame $S$ mein measure ki gayi jahan events simultaneous hain; agar yeh zero hai, toh sab agree karte hain.