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

(b).

• (a) aur (c) weak equivalence principle hain — ye sirf girne aur masses ki baat karte hain.
• (b) Einstein version hai: ye statement ko physics ke saare experiments tak upgrade karta hai box ke andar, sirf girne tak nahi. Woh "tum kisi bhi experiment se fark nahi bata sakte" wali extra baat hai jo Einstein ko gravity ko geometry mein badalne deti hai.

(b). Ek geodesic "straight line" ka curved surface par generalization hai — ye woh path hai jo tum follow karte ho jab koi force tumhe sideways nahi dhakelta. Ek free-faller ek geodesic par travel karta hai. Dekho Geodesics & Curvature.

WHAT: Newton's 2nd law $F=m_{i}a$; gravity $F=m_{g}g$. Dono barabar karo. $m_{i}a=m_{g}g\Rightarrow a=\frac{m_g}{m_i}g.$ WHY masses cancel: equivalence principle kehta hai ki $m_g/m_i=1$ har cheez ke liye, isliye $a=g$ mass se independent hai. $a_1=a_2=9.8{\text{m/s}}^2.$ Marble aur shot-put identically accelerate karte hain. Mass kabhi answer mein nahi aata — woh cancellation GR ka seed hai.

WHY this formula: ek photon gravity well se bahar chadh ke "effective energy" khota hai, isliye uski frequency drop hoti hai — yahi Gravitational Time Dilation hai jo colour shift ke roop mein padhi jaati hai. $\frac{\Delta f}{f}=\frac{gh}{c^2}=\frac{9.8\times 22.5}{(3\times 10^8{)}^2}=\frac{220.5}{9\times 10^{16}}\approx 2.45\times 10^{-15}.$ Bahut chhota — lekin Pound aur Rebka ne measure kiya tha 1959 mein.

WHAT (figure dekho): beam space mein seedhi jaati hai, lekin box uske neeche utha, isliye andar se beam neeche curve karti dikhti hai.

• Paar karne ka time: $t=L/c=3.0/(3\times 10^8)=1.0\times 10^{-8}\text{s}$.
• Box kisi bhi accelerating cheez ki tarah rise karti hai: $\Delta y=\frac{1}{2}g{t}^2$. $\Delta y=\frac{1}{2}(9.8)(1.0\times 10^{-8}{)}^2=4.9\times 10^{-16}\text{m}.$ WHY it matters: Einstein EP se, wahi bend real gravity mein bhi honi chahiye — gravity light ko bend karti hai. Dekho Gravitational Lensing.

WHICH is faster: top wali clock — zyada gravitational potential matlab zyada fast ticking. $\frac{\Delta \tau }{\tau }=\frac{gh}{c^2}=\frac{9.8\times 100}{9\times 10^{16}}=1.089\times 10^{-14}.$ Toh top wali clock har second mein lagbhag $1.09\times 10^{-14}$ seconds gain karti hai — ek poore din ($86400$ s) mein yeh lagbhag $9.4\times 10^{-10}$ s $\approx 0.94$ ns hota hai.

WHAT (figure dekho): har ball Earth ke centre ki taraf girti hai, na ki "seedha neeche parallel." Ek hi centre ki taraf aimed do lines ko converge karna hi hai. WHY this kills the force picture: tum ek chhote box mein free-fall karke gravity cancel kar sakte ho — lekin tum dono balls ke paths ko ek saath straight nahi bana sakte. Free-fallers ka yeh irreducible convergence tidal effect hai, aur mathematicians isko exactly spacetime curvature kehte hain. Dekho Geodesics & Curvature.

WHAT: dono effects ke opposite signs hain, isliye add karo: $\Delta t_{\text{net}}=+45+(-7)=+38\mu \text{s/day}.$ WHY both: velocity (SR, Special Relativity) hamesha ek moving clock ko slow karta hai; altitude (GR) ek high clock ko speed up karta hai. GPS high bhi hai aur fast bhi, isliye dono matter karte hain. Kisi ek ko bhi ignore karo aur positions har din kilometres drift karenge. Dekho GPS Corrections.

WHAT: $\text{GR}/\text{Newton}=1.75/0.875=2.00$ exactly. WHY the factor 2: Newtonian gravity sirf spacetime ke time part ko curve karta hai (energy in a potential). GR space aur time dono ko curve karta hai, aur space-curvature ek equal second half contribute karti hai. Photon curved time mein "girta" hai aur curved space par ride karta hai — do equal contributions. The lesson: Newton se match karna kaafi nahi hai; factor of 2 (measured by Eddington, 1919) space itself ke curved hone ka direct fingerprint hai, sirf koi acting force nahi.

Step 1 — photon ki effective mass. Energy $E=hf$ (Planck), aur mass-energy $E=m{c}^2$, isliye effective mass $m_{\text{eff}}=hf/c^2$. WHY: ek photon ke gravity ke against "chadne" ki baat karne ke liye humein kuch chahiye jise gravity pakad sake — uski energy mass ka role play karti hai. Step 2 — height $h$ chadhne mein energy ka loss. Potential energy gained $=m_{\text{eff}}gh=\frac{hf}{c^2}gh$. WHY: mass $m$ ko field $g$ mein height $h$ tak uthane mein $mgh$ cost aata hai — standard mechanics. Step 3 — photon apni energy se pay karta hai. Woh slow nahi ho sakta (light hamesha $c$ hai), isliye woh frequency kho ke pay karta hai: $\Delta E=-\frac{hf}{c^2}gh,\Delta E=h\Delta f.$ $E=hf$ se divide karo: $\frac{\Delta f}{f}=\frac{\Delta E}{E}=-\frac{gh}{c^2}.$ WHAT it means: minus sign = upar chadhte waqt frequency drop hoti hai (redshift). Receiver ki clock isliye usi $+gh/c^2$ se faster tick karti hai.

WHY this formula: Newtonian potential $\Phi =-GM/r$ hai. Ek clock upar (less negative $\Phi$) faster chalti hai; fractional rate difference $\Delta \Phi /c^2$ hai. Hum (satellite − ground) chahte hain, aur kyunki ${\Phi }_{\text{sat}}-{\Phi }_{\text{ground}}=-\frac{GM}{r}+\frac{GM}{R}=GM\left(\frac{1}{R}-\frac{1}{r}\right)>0$, satellite gain karta hai. Numbers: $\frac{1}{R}-\frac{1}{r}=\frac{1}{6.37\times 10^6}-\frac{1}{2.66\times 10^7}=1.5699\times 10^{-7}-3.759\times 10^{-8}=1.194\times 10^{-7}{\text{m}}^{-1}.$ $\frac{GM}{c^2}=\frac{6.674\times 10^{-11}\times 5.97\times 10^{24}}{(3\times 10^8{)}^2}=\frac{3.984\times 10^{14}}{9\times 10^{16}}=4.427\times 10^{-3}\text{m}.$ $\frac{\Delta \tau }{\tau }=4.427\times 10^{-3}\times 1.194\times 10^{-7}=5.285\times 10^{-10}.$ Ek din mein: $\Delta \tau =5.285\times 10^{-10}\times 86400=4.57\times 10^{-5}\text{s}\approx 45.7\mu \text{s/day}.$ WHAT it means: yeh famous "$\approx 45\mu$s/day" gravitational speed-up recover karta hai — woh number jo tumhara phone har din correct karta hai. Dekho GPS Corrections.

The chain: Experiment dikhata hai ki $m_i=m_g$ lagbhag $\sim 10^{-15}$ tak, isliye sab objects identically girte hain — mass cancel ho jaata hai. Einstein ne ise equivalence principle tak promote kiya: ek freely-falling lab locally gravity-free lab se indistinguishable hai, aur ek accelerating box gravitational field mimic karta hai. Accelerating box se hum derive karte hain ki light bend hoti hai (floor beam ke neeche rise karta hai) aur neeche wali clocks slow chalti hain (chadhte photons redshift hote hain). Lekin box trick sirf locally kaam karta hai — do door ke free-fallers phir bhi Earth ke centre ki taraf converge karte hain, ek tidal effect jo koi free-fall frame nahi mita sakta. Wahi irreducible convergence hi spacetime curvature hai, isliye gravity ek force nahi balki geometry hai: matter spacetime ko curve karta hai ($G_{\mu \nu }=\frac{8\pi G}{c^4}{T}_{\mu \nu }$), aur free objects geodesics par glide karte hain — "matter spacetime ko batata hai kaise curve ho; spacetime matter ko batata hai kaise move ho."

Einstein EP ek line mein ::: Ek freely-falling lab locally ek aisi lab se indistinguishable hai jahan gravity nahi hai. Free fall mein mass cancel kyun hoti hai ::: Kyunki $m_g/m_i=1$ har cheez ke liye, isliye $a=g$ mass se independent hai. GPS net drift ka sign ::: Positive — satellite clock roughly $\approx +38\mu$s/day fast chalti hai. Free-fall kya nahi hata sakta ::: Tidal effects — nearby geodesics ka convergence = curvature. Newtonian light bending vs GR ::: Newton exactly half deta hai; GR equal space-curvature part add karta hai.