General relativity — equivalence principle, curved spacetime (overview)

2.3.33 · Physics › Modern Physics

1. The Equivalence Principle

YEH SACH KYUN HONA CHAHIYE

Poori baat ek ajeeb experimental fact pe tiki hai:

$inertial mass m_{i} = gravitational mass m_{g}$

Inertial mass $m_{i}$ Newton ke 2nd law mein aata hai: $F = m_{i} a$ .
Gravitational mass $m_{g}$ gravity mein aata hai: $F = m_{g} g$ .

Free fall ke liye inhe equal karo: $m_{i} a = m_{g} g \Rightarrow a = \frac{m _{g}}{m _{i}} g .$

EINSTEIN NE ISSE THOUGHT EXPERIMENT MEIN KAISE BADLA

2. Pehla Consequence: Light Bend Hoti Hai & Clocks Slow Hote Hain

Light bend hoti hai (EP se derived)

Gravitational time dilation (EP se derived)

Socho light frequency $f$ ke saath field $g$ mein height $h$ chadh rahi hai. Climb ko accelerating frame samjho; receiver door ja raha hai, toh ek Doppler-jaisa shift hota hai.

Ek photon chadh ke "energy khota" hai: effective potential energy per unit mass $= g h$ , photon "mass" $= E / c^{2} = h f / c^{2}$ .

$Δ E = \frac{h f}{c ^{2}} g h \Rightarrow \frac{Δ f}{f} = - \frac{g h}{c ^{2}} .$

3. Equivalence se Curvature Tak

"Force" picture globally KYUN fail hoti hai

Elevator trick sirf locally kaam karti hai (chota box, thoda time). Earth par door-door giraaye gaye do balls Earth ke center ki taraf girate hain, toh unke raaste converge karte hain — woh bina kisi force ke ek doosre ki taraf accelerate karte hain.

Geometric reframe

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

MATHEMATICALLY YEH KAISE ENCODE HOTA HAI (overview level)

Curved spacetime mein distances ek metric $g_{μν}$ use karta hai jo flat (Minkowski) spacetime ko generalize karta hai: $d s^{2} = g_{μν} d x^{μ} d x^{ν} (flat: d s^{2} = - c^{2} d t^{2} + d x^{2} + d y^{2} + d z^{2}) .$

Master equation (tumhe sirf ise recognize karna hai):

4. Worked Numerics

5. Common Mistakes

6. Active Recall

Recall Quick self-test (answers hide karo!)

Einstein equivalence principle state karo.
Free fall mein mass cancel kyun hota hai lekin electric acceleration mein nahi?
Ek climbing photon se $Δ f / f = - g h / c^{2}$ qualitatively derive karo.
Kaunsa physical effect free-falling frame mein jaake bhi remove nahi ho sakta? (→ tidal/curvature)
Wheeler ka slogan padho aur dono halves explain karo.

Recall Feynman: ek 12-saal ke bacche ko explain karo

Socho tum ek elevator mein ho jisme koi khidki nahi hai. Agar elevator achanak tezi se upar jaaye, toh tum floor se chipak jaate ho — bilkul waise jaise gravity tumhe Earth se chipkati hai. Einstein ne kaha: yeh dono feelings exactly ek jaisi hain. Toh "gravity ka tumhe neeche kheenchna" asal mein waise hi hai jaise "floor upar push kar raha hai jabki tum float karna chahte". Aur bhaari aur halki cheezein saath girti hain kyunki gravity unke weight par nahi kheench rahi — woh raasta (space) hi bend kar rahi hai jis par sab kuch chalta hai. Ek bowling ball trampoline par ek gaddha banati hai, aur marbles uski taraf isliye roll karte hain ki ball unhe pakad rahi ho, balki isliye ki floor bent hai. Woh bent floor spacetime hai!

7. Connections

Special Relativity — GR locally / free-fall mein SR reduce ho jaati hai.
Spacetime Metric & Minkowski Diagram — flat-space starting point.
Gravitational Time Dilation & GPS Corrections — applications.
Gravitational Lensing & Black Holes — extreme curvature.
Newtonian Gravity — weak-field, slow-speed limit.
Geodesics & Curvature — math machinery.

Equivalence principle (Einstein form)

Ek freely-falling lab locally zero gravity wali lab se alag nahi pehchana ja sakta; ek uniform gravitational field locally ek accelerating frame ke barabar hai.

Saari objects same rate se kyun girti hain?

Kyunki inertial mass = gravitational mass (

m_{i} = m_{g}

), toh

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

har cheez ke liye — mass cancel ho jaata hai.

Gravity ki woh unique property kya hai jisse mass cancel hota hai?

Gravity ka coupling "charge" inertial mass hi hai, toh ratio universal hai — electromagnetism se alag jahan charge/mass vary karta hai.

EP light bending ko kaise predict karta hai?

Accelerating box mein light clearly curve karti hai (

Δ y = \frac{1}{2} g t^{2}

); EP se yahi real gravity mein bhi hota hai, toh gravity light ko bend karti hai.

Gravitational redshift formula

Δ f / f = - g h / c^{2}

— potential se bahar chadh rahi light frequency khoti hai.

Kaun se clocks slower chalte hain?

Jo clocks gravitational potential mein deeper hain (neeche ki position) woh slower chalte hain:

Δ τ / τ = + g h / c^{2}

Free-fall kya remove NAHI kar sakta?

Tidal effects — nearby free-fallers ka relative acceleration — spacetime curvature ki sachchi pehchaan.

Einstein field equations words mein

G_{μν} = \frac{8 π G}{c ^{4}} T_{μν}

: geometry/curvature = matter/energy.

Wheeler ka slogan

Matter spacetime ko batata hai kaise curve kare; spacetime matter ko batata hai kaise move kare.

Light bending par Newtonian galti

Photons ko massive maanna sirf AADHA observed deflection deta hai; poori GR ise double kar deti hai.

Elevator equivalence sirf local kyun hai?

Bade regions mein, free-fall paths converge karte hain (Earth ke center ki taraf), tidal forces reveal karte hain jo transform nahi ho sakti.

General relativity — equivalence principle, curved spacetime (overview)

1. The Equivalence Principle

YEH KEHTA KYA HAI

YEH SACH KYUN HONA CHAHIYE

EINSTEIN NE ISSE THOUGHT EXPERIMENT MEIN KAISE BADLA

2. Pehla Consequence: Light Bend Hoti Hai & Clocks Slow Hote Hain

Light bend hoti hai (EP se derived)

Gravitational time dilation (EP se derived)

3. Equivalence se Curvature Tak

"Force" picture globally KYUN fail hoti hai

Geometric reframe

MATHEMATICALLY YEH KAISE ENCODE HOTA HAI (overview level)

4. Worked Numerics

5. Common Mistakes

6. Active Recall

7. Connections

Concept Map