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

2.3.33 · D1 · Physics › Modern Physics › General relativity — equivalence principle, curved spacetime

Parent note on general relativity padhne se pehle, aapko uske har symbol ko dekhkar ek picture dikhni chahiye. Yeh page unhe ek ek karke leta hai — sabse ordinary (mass, acceleration) se lekar darawne ($g_{\mu \nu }$, $G_{\mu \nu }$) tak. Yahan kuch bhi pehle se jaanna zaroori nahin. Agar aap add, multiply kar sakte hain, aur ek ball ko floor par imagine kar sakte hain, toh aap har line follow kar sakte hain.

Yeh parent overview ka foundations page hai.

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1. Mass ke do prakar — sab kuch ki shuruat

Sab log ek word "mass" jaante hain. Physics isse secretly do alag kamon ke liye use karti hai, aur general relativity ki poori kahani tab shuru hoti hai jab hum notice karte hain ki yeh do kaam ek hi number dete hain.

Ise define karne wala rule hai Newton's second law:

Yahan do naye symbols aaye, toh inhe samajhte hain:

• $F$ = force = ek push ya pull, newtons mein maapa jaata hai. Ek arrow imagine karo: lamba arrow = zyada push.
• $a$ = acceleration = speed khud kitni tezi se badal rahi hai (tez hona, slow hona, ya morna). Socho ek car ka speedometer needle ghoomta hai — us ghumne ki speed hai $a$.

Ab mass ka doosra kaam:

Ise define karne wala gravity rule:

$F=m_{g}g$

jahan lowercase $g$ = gravitational field ki strength (Earth ki surface par, lagbhag $9.8{\text{m/s}}^2$). $g$ ko socho "har second gravity aapko kitne metres-per-second ki downward speed deti hai."

Ek freely falling object ke liye dono definitions equal set karo (sirf ek force gravity hai, $F=m_{g}g$, aur Newton kehte hain $F=m_{i}a$):

$m_{i}a=m_{g}g⟹a=\frac{m_g}{m_i}g.$

Kyunki $m_g/m_i=1$ sab cheez ke liye, $a=g$ sab ke liye — mass cancel ho jaata hai. Ek feather aur ek hammer saath girte hain. Woh cancellation hi GR ka darwaaza hai. Neeche ki figure mein dono objects hain, alag alag size mein drawn, identical falls trace karte hue:

Dono downward arrows dekho: woh ek hi waqt floor tak pahunchte hain chahe amber "hammer" cyan "feather" se kahin zyada massive ho. Beech ka chhota box batata hai kyun — ratio $m_g/m_i$ hai $1$, toh mass answer mein kabhi aata hi nahin.

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2. "Change" symbols padhna: $\Delta$, ratios, aur squares

Parent note mein $\frac{1}{2}g{t}^2$ jaise expressions bhari hain. Aao inke har piece ko pehle hi samjho.

• $t$ = time, seconds. Ek stopwatch.
• $t^2$ = $t$ times $t$. Girne mein time kyun square hota hai? Kyunki free fall mein distance tezi se badhta jaata hai — do gune time ke baad aap chaar guna zyada gire ho. Woh "do ke liye chaar" wala pattern exactly squaring karta hai.

$\Delta y=\frac{1}{2}g{t}^2$ mein $\frac{1}{2}$ kahan se aata hai

Yeh half koi fudge factor nahin hai — aap ise dekh sakte ho. Rest se giraya hua object ki speed shuruat mein $0$ hai aur time $t$ par speed $gt$ hai (kyunki har second gravity $g$ speed mein jodhti hai). Speed ko time ke against plot karo aur ek seedhi line milti hai jo $0$ se $gt$ tak jaati hai.

Girne ki distance = us speed line ke neeche ka area (speed $\times$ time, jod ke). Lekin graph ek triangle hai, rectangle nahin: base $t$, height $gt$. Triangle ka area hai $\frac{1}{2}\times \text{base}\times \text{height}=\frac{1}{2}t(gt)=\frac{1}{2}g{t}^2$. Woh triangle exactly wahi hai jahan se half janam leta hai — yeh "area of a triangle" mein jo "half" hai.

Figure mein shaded triangle ko dashed rectangle se compare karo: agar object apni final speed par poore waqt chalta toh rectangle ka area cover karta; kyunki yeh dheere shuru hua, yeh sirf aadha cover karta hai — triangle.

Yahan parent formulas mein ek naya symbol $h$ aata hai. Ise define karte hain:

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3. Frequency $f$ — kyun light ek "clock" carry karti hai

Figure mein ek wave neeche emit hoti hai (tight crests, zyada $f$) aur wahi wave oopar receive hoti hai (stretched crests, kam $f$) gravity ke khilaf chadh ke:

Safed arrow ko upar follow karo: oopar wali cyan wave mein neeche wali amber wave se kam crests per stretch hain — woh visible stretching hi redshift hai, aur utni hi "clock" ka slow ticking.

Ab $f$ exist karta hai, toh Section 2 ka symbol $\Delta f$ apna matlab paata hai: $\Delta f$ = emission aur reception ke beech frequency mein change. Fraction $\Delta f/f$ (poore se compare mein change) woh hai jo parent ka redshift formula predict karta hai — lekin woh formula tab tak nahin likh sakte jab tak ek aur symbol, $c$, table par na aa jaaye.

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4. Proper time $\tau$ — woh clock jo traveller actually padhta hai

Toh ab dono deferred symbols earn ho gaye: $\Delta f$ = frequency mein change, $\Delta \tau$ = ek clock ki reading mein change. Redshift statement inhe saath bandhega: ek stretched-out wave ($\Delta f<0$) aur ek slow clock ($\Delta \tau$ peechhe chal raha) ek hi fact hain.

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5. Light ki speed $c$, aur kyun yeh har jagah squared hai

$c^2$ (ek behad bada number) $gh/c^2$ jaise expressions ke denominator mein kyun baitha hai? Kyunki relativity ke effects maape jaate hain relative to cheezein $c$ ke kitne kareeb move kar rahi hain. Uss bade $c^2$ se divide karna hi wajah hai ki gravitational time shifts aksar tiny hote hain — lekin "tiny" "zero" nahin hai, isiliye GPS ko abhi bhi inki zaroorat hai.

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6. Redshift formula $\Delta f/f=-gh/c^2$ ko pieces se banana

Ab parent ke key formula mein har symbol defined hai: $g$ (field strength, §1), $h$ (height, §2), $f$ (frequency, §3), $c$ (light-speed, §5), $\Delta$ aur ratio idea (§2). Aao formula ko banate hain taaki yeh ek story ho, koi spell nahin.

Step 1 — photon ko ek effective mass do (KYA & KYUN). Ek photon ki energy hai $E=hf$. $E=m{c}^2$ se, woh energy ek mass ki tarah behave karti hai $m=E/c^2=hf/c^2$. Hum yeh isliye karte hain kyunki gravity kisi bhi mass-energy wali cheez par kaam karta hai, toh photon par ek "gravitational handle" chahiye.

Step 2 — ise chadhao aur gravitational toll do (KYA & KYUN). Ek mass $m$ ko field $g$ mein height $h$ tak uthane mein energy $mgh$ lagti hai (yeh everyday "oopar uthana = kaam karna" wala rule hai). Photon ke paas pay karne ke liye koi pocket nahin hai siwaay apni energy ke, toh woh utni khota hai: $\Delta E=-mgh=-\frac{hf}{c^2}gh.$ Minus sign kehta hai "energy upar jaate waqt neeche gayi."

Step 3 — khoi energy ko khoi frequency mein badlo (KYA & KYUN). Kyunki $E=hf$, energy mein change frequency mein change hai: $\Delta E=h\Delta f$. Hum yeh isliye use karte hain kyunki $f$ woh hai jo hum actually colour/redshift ke roop mein observe karte hain. Substitute karo: $h\Delta f=-\frac{hf}{c^2}gh.$

Step 4 — cancel karo aur fraction padho (KAISA DIKHTA HAI). Planck's $h$ dono sides cancel ho jaata hai; $f$ se divide karo:

Exactly yeh size kyun? Do ingredients ise set karte hain: energy per unit ka toll hai $gh/c^2$ (gravity ka pull $g$ times climb $h$, light ki speed-squared ke against maapa), aur frequency energy ke proportional hai, toh fractional energy loss aur fractional frequency loss ek hi number hain. §3 ki figure is formula ki picture hai: crests exactly is fraction se stretch hoti hain.

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7. Reference frames, free-fall, aur "weightless"

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8. Tidal effects, geodesics, aur curvature — geometry ke words

Ek chhote box ke andar aap free-fall karke gravity mita sakte ho. Lekin do balls Earth ke upar dur dur drop karo: dono Earth ke centre ki taraf aim karte hain, toh unke paths dheere dheere ek doosre ki taraf jhukenge.

Figure mein do free-fallers Earth ke kaafi upar side by side shuru hote hain; unke amber paths planet ke centre ki taraf andar jhukate hue dekho:

Dotted white lines Earth ke centre ki taraf seedhe point karti hain — kyunki dono fallers wahan aim karte hain, unhe zaroor converge karna hoga. "Still room" ki koi bhi re-choice us lean-together ko nahi hata sakti; woh ziddi convergence curvature ko visible banata hai.

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9. Metric $g_{\mu \nu }$ aur field equations $G_{\mu \nu }=T_{\mu \nu }$

Yeh parent note ke sabse daraware symbols hain. Tame karo, aur yeh sirf bookkeeping tables hain.

Pehle: indices $\mu$ aur $\nu$ ka matlab kya hai?

Doosra: shorthand jo ek sum chhupata hai

Teesra: metric khud

Iske saath likha line element, $d{s}^2=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }(\text{sum over }\mu ,\nu =0,1,2,3),$ sirf ek accountant ka total hai jo summation convention ne unpack kiya: $ds$ sachchi chhoti distance hai, $dx$'s chhote coordinate steps hain, aur har $g_{\mu \nu }$ matching pair ko weight karta hai sab solah jodte waqt. Flat spacetime mein yeh uss jaani-pehchani Pythagoras-with-a-time-term mein collapse ho jaati hai jo aap Spacetime Metric & Minkowski Diagram mein milte ho.

Chautha: field equations

Aapse is page par inhe compute karne ki expect nahin ki jaati — sirf inhe "curvature table = matter table" ke roop mein dekhna, taaki parent note ka boxed equation ek sentence ho, wall nahin. Machinery Geodesics & Curvature mein hai.

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Prerequisite map

Neeche ka diagram dikhata hai ki har foundation agla kaise feed karta hai, ending at parent overview. Ise top-to-bottom padho: do masses equivalence principle mein milte hain; light-and-frequency plus woh principle time dilation aur bending dete hain; free-fall plus tidal effects curvature dete hain; curvature metric aur field equations deta hai — aur sab kuch GR mein flow karta hai.

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Equipment checklist

Daayaan side cover karo aur khud test karo. Agar koi bhi answer surprise kare, toh parent note kholne se pehle woh section dobara padho.