General relativity — equivalence principle, curved spacetime (overview): Question bank

2.3.33 · D5 · Physics › Modern Physics › General relativity — equivalence principle, curved spacetime

Shuru karne se pehle, teen words jinhe hum constantly use karte hain — yahan pin kar diye hain taaki koi bhi line inhe achanak use na kare:

Aur wo symbols jo formulas mein aate hain — har ek ko use karne se pehle samjha gaya hai:

Definition Symbol glossary (sign conventions ke saath)

$g$ = local gravitational acceleration (Earth ki surface ke paas $\approx 9.8 m/s^{2}$ ), yahan hamesha positive magnitude ke roop mein liya gaya hai.
$c$ = vacuum mein light ki speed ( $\approx 3 \times 1 0^{8} m/s$ ), har observer ke liye same.
$G$ = Newton's gravitational constant ( $\approx 6.67 \times 1 0^{- 11} N\cdotm^{2} / kg^{2}$ ), Newton's law aur Einstein equations dono mein gravity ka strength dial.
$h$ physics mein purposely overloaded hai, isliye hum disambiguate karte hain: $g h / c^{2}$ mein symbol $h$ ka matlab height difference (metres) hai; $E = h f$ (photon energy) mein symbol $h$ ka matlab Planck's constant hai. Ye dono unrelated hain — jahan bhi confusion possible ho, hum "height $h$ " ya "Planck's $h$ " likhte hain.
$Δ τ$ = compare kiye ja rahe do clocks ke beech elapsed proper time ka difference. Sign convention: $Δ τ / τ = + g h / c^{2}$ upar wale clock ko neeche wale se compare karta hai — positive value ka matlab hai upar wala clock fast chalta hai (neeche wala slow chalta hai). $Δ f / f$ bhi isi tarah received vs emitted frequency compare karta hai.

True ya false — justify karo

Neeche har "T/F" answer mein ek reason hona chahiye, aur letter se shuru hona chahiye. Sirf letter guess karna kuch nahi dilata.

T/F — Deep space mein uniformly accelerate karta rocket bilkul waise hi hai jaise Earth ki surface par baithna.

False — sirf locally true hai. Ek chhote box ke andar thode time ke liye ye indistinguishable hain (Einstein EP), lekin ek bade region mein Earth ki field uske centre ki taraf point karti hai aur height ke saath weak hoti hai, jabki rocket ki perfectly uniform hai — isliye real gravity mein tidal effects hain jo rocket mein nahi hain.

T/F — Sachche free-fall mein tum gravity ko strongly pull karte feel karte ho.

False. Free-fall mein tum weightless feel karte ho — locally gravity "switch off" ho jaati hai. Tum weight tabhi feel karte ho jab floor ya seat tumhe tumhare geodesic se rokti hai. (Dekho Gravitational Time Dilation ki free-fallers kya detect karte hain: locally kuch nahi, sirf tidal effects door se.)

T/F — Kyunki inertial mass gravitational mass ke barabar hoti hai, isliye saari cheezein ek hi acceleration se girती hain.

True.

m_{i} a = m_{g} g

set karne par

a = (m_{g} / m_{i}) g

milta hai; kyunki

m_{g} / m_{i} = 1

har cheez ke liye,

a = g

har cheez ke liye aur mass cancel ho jaata hai — yeh cancellation sirf gravity ke liye unique hai.

T/F — $m_{i} = m_{g}$ obvious hai kyunki hum dono ko ek hi symbol $m$ se likhte hain.

False. Ye logically independent laws mein enter hote hain — inertia (

F = m_{i} a

) versus gravitational coupling (

F = m_{g} g

). Unki

\sim 1 0^{- 15}

tak equality ek deep experimental fact hai, definition nahi; yahi GR ka seed hai.

T/F — Light Sun ke paas isliye bend hoti hai kyunki photons mein mass hoti hai aur gravity unhe pull karti hai.

False. Photons massless hain; light isliye bend hoti hai kyunki wo curved spacetime mein geodesics follow karti hai. Naive "photon mass" wala Newtonian guess sirf measured deflection ka aadha predict karta hai (dekho Gravitational Lensing).

T/F — Time dilation sirf is baat par depend karti hai ki clock kitni fast move karti hai.

False. Yeh special-relativistic (velocity) part hai. Potential par depend karne wala ek alag gravitational part bhi hota hai; GPS Corrections ke liye gravitational effect zyada bada hai aur velocity effect ke opposite sign ka hai.

T/F — Ek tall building ke bottom par rakha clock top par rakhe clock se slow chalta hai.

True. Jo clocks gravitational potential mein deeper hain (neeche) wo slow chalte hain; upar wale ko neeche wale se compare karne par

Δ τ / τ = + g h / c^{2} > 0

milta hai, yani top clock time gain karta hai. Pound–Rebka experiment ne exactly yahi

22.5

m par measure kiya.

T/F — General relativity special relativity ko throw away kar deti hai.

False. GR Special Relativity ko local limit ke roop mein contain karti hai: kisi bhi small free-falling patch mein, spacetime exactly flat (Minkowski) lagti hai aur SR exactly hold karta hai. GR sirf bade regions mein curvature add karti hai.

Error dhundo

Neeche har statement mein ek flawed step chupi hui hai. Use naam do.

"Equivalence principle hume free-fall karke gravity ko har jagah cancel karne deta hai, isliye gravity ek fiction hai."

Error "har jagah" mein hai. Free-fall gravity ko sirf locally cancel karta hai; nearby free-fallers ka tidal relative acceleration remove nahi kiya ja sakta — woh irreducible part real curvature hai.

"Ek girte elevator mein side by side drop ki gayi do balls drift apart ho jaati hain, jo prove karta hai ki gravity unke beech ek normal force hai."

Ye drift isliye hoti hai kyunki dono Earth ke centre ki taraf girte hain, isliye unke paths converge hote hain — yeh geodesic convergence hai, balls ke beech koi force nahi. Ise "unke beech ek force" kehna galti hai.

"Einstein Field Equations $G_{μν} = \frac{8 π G}{c ^{4}} T_{μν}$ mein, left side matter hai aur right side geometry hai."

Ulta hai.

G_{μν}

(left) geometry/curvature hai;

T_{μν}

(right) matter–energy (stress-energy) hai. Wheeler: "matter spacetime ko batata hai ki kaise curve hona hai."

"Kyunki elevator thought-experiment kaam karta hai, ek real gravitational field har tarah se acceleration ke identical hai."

"Har tarah se" galat hai. Yeh sirf ek chhote, brief region mein identical hai. Extended real fields mein curvature (tidal effects) hoti hai jise koi bhi rigid frame acceleration reproduce nahi kar sakta.

"Gravitational redshift ka matlab hai ki photon climb karte waqt slow ho jaata hai."

Light ki speed kabhi nahi badlti (

c

invariant hai). Jo badlta hai woh uski frequency (energy) hai — yeh redshift hoti hai — speed nahi. "Energy lose karna" ko "slow ho jaana" samajhna galti hai.

"Heuristic $Δ f / f = - g h / c^{2}$ jo ek photon ki potential energy $m g h$ lose karne se derive hoti hai, yeh poora exact GR result hai."

Yeh sirf ek weak-field, uniform-field heuristic hai — yeh photon ko aise treat karta hai jaise uski Newtonian potential energy

m g h

ho (effective mass

E / c^{2}

ke saath). Yeh sahi leading answer deta hai lekin exact GR derivation nahi hai, jo full metric use karta hai. Ise exact present karna galti hai.

"Trampoline par ek bowling ball dikhata hai ki marbles dent ke andar grab aur pull ho jaati hain."

Kuch bhi unhe "grab" nahi karta. Marbles bent surface (geodesics) par roll karti hain; analogy yeh illustrate karta hai ki curvature motion guide karti hai — "grabbed and pulled" ki jagah "bent floor follow karo" use karo. (Dekho Geodesics & Curvature.)

Why questions

Gravity wale free-fall mein mass kyun cancel ho jaata hai lekin electric acceleration mein nahi?

Gravity ka "charge" hi gravitational mass hai, jo inertial mass ke barabar hai, isliye

a = (m_{g} / m_{i}) g

mein

m

cancel ho jaata hai. Electric force ke liye

a = q E / m

— charge-to-mass ratio

q / m

objects ke beech vary karta hai, isliye yeh cancel nahi hota.

Einstein ne $m_{i} = m_{g}$ ko ek coincidence se ek principle mein kyun promote kiya?

Kyunki cancellation extreme precision tak test ki gayi har material ke liye perfect hai. Jo cheez itni exact ho uske liye ek structural reason chahiye — aur ise ek principle maanna seedha gravity ko geometry ke roop mein interpret karne par le gaya.

Gravity ki ek "force" picture globally kyun kaam nahi kar sakti?

Ek force objects ke beech act karti hai, lekin tum kisi bhi single free-faller ke liye gravitational acceleration vanish kar sakte ho. Jo survive karta hai woh relative acceleration (tidal effect) hai — jo geometry ki property hai, do-body force nahi, isliye geometry, force nahi, sahi global description hai.

Weak-field heuristic mein, ek climbing photon energy kyun lose karta hai, jisse redshift hoti hai?

Photon ko effective "mass"

E / c^{2} = h f / c^{2}

assign karna (yahan

h

Planck's constant hai), ise height

h

(yahan

h

height hai) tak uthane mein potential energy

(h f / c^{2}) g h

lagti hai; woh energy photon se aati hai, uski frequency

Δ f / f = - g h / c^{2}

se kam ho jaati hai. Yeh ek heuristic hai jo weak, roughly uniform field mein valid hai — exact result metric se aata hai (dekho Gravitational Time Dilation).

GPS gravitational correction positive kyun hai (satellite clock fast chalta hai)?

Satellite Earth ke potential mein upar hota hai (weak gravity), aur high-potential wale clocks fast chalte hain: satellite ko ground se compare karne par,

Δ τ / τ = + g h / c^{2} > 0

. Uski velocity (SR) effect use slow karti hai, lekin gravitational effect jeet jaata hai, isliye net satellite clock fast chalta hai.

Spacetime ko plain Euclidean distance ki jagah ek metric $g_{μν}$ kyun chahiye?

Kyunki spacetime mein "distance" time aur space ko minus sign ke saath mix karta hai aur jab curved ho toh point to point vary kar sakta hai. Metric intervals measure karne ka local rule hai; flat regions mein yeh Minkowski reduce ho jaata hai (dekho Spacetime Metric & Minkowski Diagram).

Edge cases

Equivalence-principle "elevator" ko jab tum box shrink karte ho aur experiment zero ki taraf short karte ho toh kya hota hai?

Yeh exact ho jaata hai: ek infinitesimal region ki limit mein, gravity aur acceleration truly indistinguishable hain aur spacetime exactly flat hai. Principle precisely ek local (limiting) statement hai.

Box ke bade hote hi equivalence-principle equivalence ka kya hota hai?

Yeh break down ho jaata hai: ek bade box mein field ki direction aur strength vary karti hai, tidal effects appear hote hain, aur koi single acceleration unhe mimic nahi kar sakta. Curvature measurable ho jaata hai — yahi woh boundary hai jahan "geometry" "acceleration" ki jagah leti hai.

Weak-field, slow-speed limit mein, GR kya reduce ho jaati hai?

Yeh Newtonian Gravity reproduce karta hai — Einstein's equations

F = G m_{1} m_{2} / r^{2}

aur

g = GM / r^{2}

recover karte hain jab curvature tiny ho aur speeds

≪ c

hon. GR mein woh classical answer hona chahiye jise yeh generalise karne ki koshish karta hai.

$g$ ya $h \to 0$ hone par gravitational time dilation ka kya hota hai?

Δ τ / τ = g h / c^{2} \to 0

: koi field nahi ya koi height difference nahi toh koi relative slowing nahi — clocks saath tick karte hain, jaise required hai, flat-spacetime SR recover karta hai.

Jab curvature itni extreme ho jaaye ki light bhi escape na kar sake toh limiting case kya hai?

Ek black hole: geometry itni sharply bend ho jaati hai ki saare future-pointing geodesics event horizon ke paas andar ki taraf jaate hain. Yeh usi curvature ka extreme end hai jo Gravitational Lensing mein mildly starlight bend karta hai.

Equivalence principle ek charged particle ke baare mein kya kehta hai jo table par still baitha hai versus ek jo free space mein accelerate kar raha hai — kya yeh radiate karta hai?

Yeh genuinely subtle hai: Earth par statically rakha ek charge accelerated hai (table use uske geodesic se dhakelta hai), jabki ek free-falling charge geometric sense mein accelerating nahi hai. EP strictly local hai, isliye naive "accelerated charges radiate" arguments ko local free-falling frame mein carefully apply karna padta hai.

Do sabse mushkil ideas ke liye ek picture

Text geometry hide kar sakta hai. Do sketches concept traps ko visual banate hain — pehla dikhata hai ki free-fall tidal effects kyun cancel nahi kar sakta (geodesic deviation), doosra dikhata hai ki "curved spacetime" ka matlab kya hai ek bent surface ke roop mein jo ek geodesic guide karta hai.

Figure — General relativity — equivalence principle, curved spacetime (overview)

Recall Lock karne ke liye one-line summary

Locally, gravity = acceleration hai aur free-fall se cancel kiya ja sakta hai; jo nahi cancel ho sakta — tidal effects — woh curvature hai, aur curvature gravity ka poora essence hai.

Connections

Parent overview — woh concepts jinhe ye traps test karti hain.
Special Relativity — woh local flat-space limit jahan ye edge cases return karte hain.
Gravitational Time Dilation · GPS Corrections — sign aur magnitude traps.
Gravitational Lensing · Black Holes — extreme-curvature edge cases.
Newtonian Gravity · Geodesics & Curvature · Spacetime Metric & Minkowski Diagram — limits aur machinery.