Worked examples — General relativity — equivalence principle, curved spacetime (overview)

2.3.33 · D3 · Physics › Modern Physics › General relativity — equivalence principle, curved spacetime

Koi bhi symbol aane se pehle, yeh raha toolbox — har woh quantity jo hum use karenge, plain words mein:

Parent se do master results, dobara likhe gaye taaki hum har piece ko point kar sakein:

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The scenario matrix

Is topic ka har problem in cells mein se ek (ya combination) hai:

Cell	Case class	Kya badalta hai	Example
A	$h>0$ (clock zyada upar)	GR: upar wali clock faster, light upar jaate redshifts	Ex 1
B	$h<0$ (clock neeche)	GR: neeche wali clock slower, neeche jaate light blueshifts	Ex 2
C	$h=0$ (same height)	Degenerate: koi GR shift nahi	Ex 3
D	Pure velocity, koi height nahi	SR only: moving clock slower	Ex 3
E	Dono effects competing (GPS)	GR $+$, SR $-$, kaun jeetega?	Ex 4
F	Strong-field / exact potential	$U=-GM/r$ use karo, $gh$ nahi	Ex 5
G	Limiting behaviour ($h\to 0$, $h\to \infty$, weak field $\to$ SR)	Formula ki edges par sanity checks	Ex 6
H	Real-world word problem	Mountain-vs-valley aging	Ex 7
I	Exam twist — light bending, clocks nahi	Free-fall / deflection reasoning	Ex 8

Neeche ke aath examples har cell ko cover karte hain.

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Example 1 — Cell A: ek clock ko tower ke UPAR le jaana ($h>0$)

Forecast: Abhi andaza lagao — nanoseconds, microseconds, ya seconds per day? Aur kya light upar jaate redder hogi ya bluer?

Steps

1. Fractional rate difference. $\frac{\Delta \tau }{\tau }=+\frac{gh}{c^2}=\frac{9.8\times 100}{(3\times 10^8{)}^2}$. Yeh step kyun? $h>0$ (roof zyada upar hai) hume seedha Cell A mein rakhta hai, isliye hum GR engine ko plus sign ke saath use karte hain — zyada upar matlab zyada fast.

2. Compute karo: $\frac{980}{9\times 10^{16}}=1.089\times 10^{-14}$ (pure number). Yeh step kyun? Yeh ratio batata hai "har second, roof kitne extra seconds gain karti hai." Hum ise real time se multiply karne se pehle dimensionless rakhte hain.

3. Ek din se multiply karo: $\Delta \tau =1.089\times 10^{-14}\times 86400=9.4\times 10^{-10}\text{s}\approx 0.94\text{ns/day}$. Yeh step kyun? Ek rate tab hi duration banti hai jab use elapsed time $\tau$ se multiply karo.

4. Light direction: light UPAR jaate energy khoती hai $\Rightarrow f$ girta hai $\Rightarrow$ redshift ($\Delta f/f=-gh/c^2<0$). Yeh step kyun? Photon "well se bahar climb" kar raha hai, upar jaate energy kharach karta hai — exactly $\Delta f$ formula ka sign.

Verify: Units: $\frac{({\text{m/s}}^2)(\text{m})}{(\text{m/s}{)}^2}=\frac{{\text{m}}^2/{\text{s}}^2}{{\text{m}}^2/{\text{s}}^2}=$ dimensionless ✓. Roughly ek nanosecond per day 100 m ke liye — wahi order jo Pound–Rebka ne 22.5 m par measure kiya tha, scaled up. ✓

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Example 2 — Cell B: ek clock ko mine mein NEECHE karna ($h<0$)

Forecast: Kya deep clock surface clock se faster chalegi ya slower?

Steps

1. Negative height plug karo: $\frac{\Delta \tau }{\tau }=+\frac{gh}{c^2}=\frac{9.8\times (-300)}{9\times 10^{16}}=-3.27\times 10^{-14}$. Yeh step kyun? Formula pehle se $h$ ka sign carry karta hai; ek lower clock ($h<0$) automatically negative aata hai = slower chalta hai. Yeh Cell B hai aur isme koi nayi physics nahi chahiye, bas minus sign ki honest bookkeeping.

2. Interpret karo: negative $\Delta \tau$ ⇒ mine clock slower chalti hai surface clock se. Yeh step kyun? "Potential well mein zyada neeche = slower" yahi gravitational time dilation ka sara content hai; negative $h$ woh tarika hai jisse formula "lower" encode karta hai.

3. Light neeche bheji: $\Delta f/f=-gh/c^2=-g(-300)/c^2=+$ positive ⇒ $f$ badhta hai ⇒ blueshift. Yeh step kyun? Ek photon jo girta hai energy gain karta hai (neechey dhaalta hai), isliye uski frequency badhti hai — Example 1 ka mirror image.

Verify: Magnitude $3.27\times 10^{-14}$ exactly $3\times$ Example 1 ka $1.089\times 10^{-14}$ hai kyunki $|h|$ $3\times$ bada hai ($300$ vs $100$) — formula $h$ mein linear hai. ✓ Sign sahi flip hua: up→redshift, down→blueshift. ✓

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Example 3 — Cells C & D: same height ($h=0$) lekin ek clock MOVING hai

Forecast: $h=0$ ke saath, kya koi time dilation hogi bhi?

Steps

1. (a) GR part: $h=0\Rightarrow \frac{\Delta \tau }{\tau }=\frac{g\cdot 0}{c^2}=0$. Yeh step kyun? Cell C degenerate case hai: $gh$ engine $h$ mein linear hai, isliye zero height difference exactly zero gravitational shift deta hai. Koi trick nahi, koi residue nahi.

2. (b) SR part (Cell D): $\frac{\Delta \tau }{\tau }=-\frac{v^2}{2c^2}=-\frac{200^2}{2(3\times 10^8{)}^2}=-2.22\times 10^{-13}$. Yeh step kyun? Motion akela clock slow kar deta hai. Hum ab velocity engine use karte hain kyunki dono clocks ke beech sirf speed ka difference hai, height ka nahi. $v^2/2c^2$ kyun aur $v^2/c^2$ kyun nahi? Slow speeds ke liye hum exact factor $\sqrt{1-v^2/c^2}\approx 1-\frac{1}{2}{v}^2/c^2$ expand karte hain; leading correction half $v^2/c^2$ hai.

3. Per day: $\Delta \tau =-2.22\times 10^{-13}\times 86400=-1.92\times 10^{-8}\text{s}\approx -19.2\text{ns/day}$. Yeh step kyun? Rate ko duration mein convert karo $\tau$ se multiply karke, pehle ki tarah.

Verify: GR shift exactly $0$ hai ✓ (sahi degenerate behaviour). SR shift negative hai ✓ (moving clock slower). Rim clock purely motion ki wajah se ~19 ns/day khoता hai. ✓

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Example 4 — Cell E: GPS, dono effects fighting

Forecast: Kya satellite clock ground clock se aage rahegi ya peechhe — aur microseconds mein ya nanoseconds mein?

Steps

1. GR (height) part, exact potential. Kyunki height bahut zyada hai, $gh$ ki jagah $U=-GM/r$ use karo: $\frac{\Delta {\tau }_{\text{GR}}}{\tau }=\frac{U_{\text{sat}}-U_{\text{ground}}}{c^2}=\frac{GM}{c^2}\left(\frac{1}{R}-\frac{1}{r}\right).$ Numerically $\frac{3.986\times 10^{14}}{9\times 10^{16}}\left(\frac{1}{6.37\times 10^6}-\frac{1}{2.66\times 10^7}\right)=5.28\times 10^{-10}$. Yeh step kyun? Cell F rule: $gh$ sirf tab kaam karta hai jab height chhoti ho aur $g$ roughly constant ho. $20,000$ km par, $g$ kam ho jaata hai, isliye hume potential integrate karna hoga — $-GM/r$ terms ka difference. Positive = satellite (zyada upar) faster chalta hai.

2. SR (velocity) part: $\frac{\Delta {\tau }_{\text{SR}}}{\tau }=-\frac{v^2}{2c^2}=-\frac{3874^2}{2(9\times 10^{16})}=-8.34\times 10^{-11}$. Yeh step kyun? Satellite move bhi karta hai, aur motion clock slow kar deta hai. Negative, GR gain ka opposition karta hai.

3. Unhe add karo (Cell E superposition hai): $(5.28-0.834)\times 10^{-10}=4.45\times 10^{-10}$. Yeh step kyun? Do chhote effects first order par simply add ho jaate hain. GR ($+$) SR ($-$) se bada hai, isliye net positive hai: satellite clock fast chalti hai.

4. Per day: $4.45\times 10^{-10}\times 86400=3.84\times 10^{-5}\text{s}\approx 38\mu \text{s/day}$. Yeh step kyun? Rate × day. Yahi famous $\approx 38\mu$s hai jo GPS ko correct karna padta hai warna positions ~11 km/day drift karti hain.

Verify: GR ($+45$-ish) SR ($-7$-ish) ko beat karta hai standard textbook split mein, net $\approx +38\mu$s/day ✓, aur yeh positive hai (satellite aage) ✓. Har piece ka sign hamari convention se match karta hai (higher=faster, moving=slower). ✓ Dekho GPS Corrections.

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Example 5 — Cell F: exact potential vs $gh$ shortcut

Forecast: 100 km par, kya simple $gh$ formula abhi bhi hold karta hai — 0.1% off? 10%? 50%?

Steps

1. Naive: $\frac{gh}{c^2}=\frac{9.8\times 10^5}{9\times 10^{16}}=1.089\times 10^{-11}$. Yeh step kyun? Yeh flat-field shortcut hai, valid tabhi jab $g$ roughly constant ho.

2. Exact: $\frac{GM}{c^2}\left(\frac{1}{R}-\frac{1}{r}\right)=\frac{3.986\times 10^{14}}{9\times 10^{16}}\left(\frac{1}{6.37\times 10^6}-\frac{1}{6.47\times 10^6}\right)=1.076\times 10^{-11}$. Yeh step kyun? Real potential difference account karta hai ki $g$ altitude ke saath thoda drop karta hai — yeh honest Cell F calculation hai.

3. Compare karo: dono mein lagbhag $1.2\%$ ka fark hai. Yeh step kyun? Yeh dikhata hai kab shortcut toot jaata hai: 100 km par error chhota par visible hai; 20,000 km par (Ex 4) yeh bahut bada hoga, exactly isliye GPS exact form use karta hai.

Verify: Exact < naive (kyunki real $g$ climb karte waqt decrease hota hai, true potential gain thoda chhota hai) ✓. Relative difference $\approx 1.2\%$ ✓.

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Example 6 — Cell G: teen limiting cases

Forecast: Kya ek accha formula in teen edges par $0$, ek known result, aur ek warning deta hai?

Steps

1. (a) $h\to 0$: $\Delta \tau /\tau =gh/c^2\to 0$. Yeh step kyun? Ek hi jagah par do clocks identically tick karni chahiye — formula vanish karna chahiye, aur karta hai ($h$ mein linear).

2. (b) Weak-field / free-fall limit: freely-falling chhote box mein, EP kehta hai gravity "off" hai, isliye sirf velocity time dilation ($-v^2/2c^2$) bachti hai — hum Special Relativity recover karte hain. Yeh step kyun? GR ko locally SR mein reduce karna chahiye; GR term ek gravitational-potential term hai, aur free-fall mein ek tiny box across potential difference $\to 0$.

3. (c) Large-$h$ warning: naive $gh/c^2$ with $h=3\times 10^{16}\text{m}$ deta hai $gh/c^2=9.8\times 3\times 10^{16}/9\times 10^{16}=3.27>1$ — ek fractional rate difference jo 1 se zyada ho woh nonsense hai. Yeh step kyun? Yeh flag karta hai ki constant-$g$ extrapolation badi distance par illegal hai; tumhe $-GM/r$ par switch karna hi hoga (Cell F). Formula tumhe batata hai kahan wo fail hota hai.

Verify: (a) exactly $0$ ✓; (c) $3.27>1$, unphysical ✓ — correctly linear approximation ka breakdown signal karta hai.

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Example 7 — Cell H: mountain vs valley (word problem)

Forecast: Poori zindagi mein milliseconds? Seconds? Minutes?

Steps

1. Rate: $\frac{\Delta \tau }{\tau }=\frac{gh}{c^2}=\frac{9.8\times 2000}{9\times 10^{16}}=2.178\times 10^{-13}$. Yeh step kyun? Cell A phir se ($h>0$): zyada upar mountain clock (aur body) faster chalti hai.

2. Total time: $\tau =80\times 3.156\times 10^7=2.525\times 10^9\text{s}$. Yeh step kyun? Ek lifetime ko seconds mein convert karo taaki fractional rate ek actual duration ban sake.

3. Extra ageing: $\Delta \tau =2.178\times 10^{-13}\times 2.525\times 10^9=5.5\times 10^{-4}\text{s}\approx 0.55\text{ms}$. Yeh step kyun? Rate × elapsed time — mountain twin genuinely aadhe millisecond zyada bada hai 80 saalon baad.

Verify: Positive (higher = faster ages) ✓, aur tiny (sub-millisecond ek lifetime mein) ✓ — Example 1 ke $\sim$ns/day-per-100m se consistent, scaled up ($20\times$ height, $\sim 2.9\times 10^4$ days).

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Example 8 — Cell I: exam twist — light bending, clocks nahi

Forecast: Micrometres, nanometres, ya kuch jo ruler se dekh sako?

Steps

1. Crossing time: $t=L/c=3/(3\times 10^8)=1.0\times 10^{-8}\text{s}$. Yeh step kyun? Light inertial frame mein seedha travel karti hai; hume jaanna hai ki flight mein kitna time lagta hai taaki dekh sakein ki floor kitna upar uthti hai.

2. Box rises: $\Delta y=\frac{1}{2}g{t}^2=\frac{1}{2}(9.8)(1.0\times 10^{-8}{)}^2=4.9\times 10^{-16}\text{m}$. Yeh step kyun? Accelerating box ke andar se, box upar move karta hai jabki light seedhi jaati hai, isliye light $\Delta y$ drop hoti dikhti hai. equivalence principle se, real gravity mein bhi wahi bend hona chahiye.

3. The trap (Cell I twist): naively photon ko Newtonian mass ki tarah girta maan kar yeh $\Delta y$ milta hai, lekin starlight ka full GR deflection twice as big hai — kyunki space khud bhi curved hai, sirf time nahi. Yeh step kyun? Yeh "photons have mass" misconception ko khatam karta hai: EP/accelerating-box argument sirf time-curvature half capture karta hai; extra factor of 2 spatial curvature se aata hai (dekho Gravitational Lensing).

Verify: $\Delta y=4.9\times 10^{-16}\text{m}$ ✓ — 3 m par bilkul tiny (isliye lab light-bending unmeasurable hai aur hume Sun ke paas se guzarti starlight chahiye). Real deflection ke liye "factor of 2" standard GR-vs-Newton result hai ✓.

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Recall Matrix ke across self-test (answers hide karo)

Ek clock ko upar uthane par $\Delta \tau$ ka sign kya hoga? ::: Positive — zyada upar wali clocks faster chalti hain. GPS ke liye kaun sa effect dominate karta hai, GR ya SR? ::: GR (higher = faster) jeetata hai; net $\approx +38\mu$s/day. $gh$ ko $-GM/r$ se kab replace karna padta hai? ::: Jab height itni badi ho ki $g$ appreciably change ho jaaye (Cell F). $h=0$ par $\Delta \tau /\tau$ kya hota hai? ::: Exactly zero (degenerate Cell C). Real light deflection accelerating-box estimate se twice kyun hai? ::: Box argument sirf time-curvature deta hai; spatial curvature doosra half add karta hai.

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Connections

• Parent topic — woh principles jo yeh examples exercise karte hain.
• Gravitational Time Dilation — Cells A/B/C/F/H ke peeche ka engine.
• GPS Corrections — Cell E puri detail mein.
• Special Relativity — $-v^2/2c^2$ velocity engine (Cells D/E, limit G(b)).
• Gravitational Lensing — Example 8 ka "factor of 2."
• Newtonian Gravity — $-GM/r$ potential ka source jo Cells E/F mein use hota hai.
• Geodesics & Curvature — free-fall weightless kyun lagta hai (Example 8 ka frame).