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2.3.33Modern Physics

General relativity — equivalence principle, curved spacetime (overview)

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🇮🇳 Yeh note Hinglish mein padho →

1. The Equivalence Principle

WHAT it says

WHY it must be true

The whole thing rests on a curious experimental fact:

inertial mass mi=gravitational mass mg\text{inertial mass } m_i \quad=\quad \text{gravitational mass } m_g

  • Inertial mass mim_i appears in Newton's 2nd law: F=miaF = m_i a.
  • Gravitational mass mgm_g appears in gravity: F=mggF = m_g g.

Set them equal for free fall: mia=mgg  ⇒  a=mgmi g.m_i a = m_g g \;\Rightarrow\; a = \frac{m_g}{m_i}\,g.

HOW Einstein turned it into a thought experiment

2. First Consequence: Light Bends & Clocks Slow

Light bends (derived from EP)

Gravitational time dilation (derived from EP)

Consider light of frequency ff climbing a height hh in a field gg. Treat the climb as an accelerating frame; the receiver recedes, so a Doppler-like shift occurs.

A photon "loses energy" climbing: effective potential energy per unit mass =gh= gh, photon "mass" =E/c2=hf/c2= E/c^2 = hf/c^2.

ΔE=hfc2 gh  ⇒  Δff=−ghc2.\Delta E = \frac{hf}{c^2}\,gh \;\Rightarrow\; \frac{\Delta f}{f} = -\frac{gh}{c^2}.

3. From Equivalence to Curvature

WHY a "force" picture fails globally

The elevator trick only works locally (small box, short time). Two balls dropped far apart on Earth fall toward Earth's center, so their paths converge — they accelerate toward each other with no force between them.

The geometric reframe

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

HOW it's encoded mathematically (overview level)

Distances in curved spacetime use a metric gμνg_{\mu\nu} generalizing flat (Minkowski) spacetime: ds2=gμν dxμdxν(flat: ds2=−c2dt2+dx2+dy2+dz2).ds^2 = g_{\mu\nu}\,dx^\mu dx^\nu \quad(\text{flat: } ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2).

The master equation (you only need to recognize it):

4. Worked Numerics

5. Common Mistakes

6. Active Recall

Recall Quick self-test (hide answers!)
  • State the Einstein equivalence principle.
  • Why does mass cancel in free fall but not in electric acceleration?
  • Derive Δf/f=−gh/c2\Delta f/f = -gh/c^2 qualitatively from a climbing photon.
  • What physical effect cannot be removed by going to a free-falling frame? (→ tidal/curvature)
  • Read Wheeler's slogan and explain each half.
Recall Feynman: explain to a 12-year-old

Imagine you're in an elevator with no windows. If the elevator suddenly shoots upward really fast, you get squished to the floor — just like gravity squishes you to Earth. Einstein said: these two feelings are exactly the same thing. So "gravity pulling you down" is really just like "the floor pushing up while you'd rather float." And heavy things and light things fall together because gravity isn't pulling on their weight — it's bending the road (space) that everything rolls along. A bowling ball on a trampoline makes a dip, and marbles roll toward it not because the ball grabs them, but because the floor is bent. That bent floor is spacetime!

7. Connections

  • Special Relativity — GR reduces to SR locally / in free-fall.
  • Spacetime Metric & Minkowski Diagram — flat-space starting point.
  • Gravitational Time Dilation & GPS Corrections — applications.
  • Gravitational Lensing & Black Holes — extreme curvature.
  • Newtonian Gravity — the weak-field, slow-speed limit.
  • Geodesics & Curvature — the math machinery.
Equivalence principle (Einstein form)
A freely-falling lab is locally indistinguishable from one in zero gravity; a uniform gravitational field is locally equivalent to an accelerating frame.
Why do all objects fall at the same rate?
Because inertial mass = gravitational mass (mi=mgm_i=m_g), so a=(mg/mi)g=ga=(m_g/m_i)g=g for everything — mass cancels.
What unique property of gravity lets mass cancel?
The coupling "charge" of gravity is the inertial mass, so the ratio is universal — unlike electromagnetism where charge/mass varies.
How does EP predict light bending?
In an accelerating box light visibly curves (Δy=12gt2\Delta y=\tfrac12 g t^2); by EP the same occurs in real gravity, so gravity bends light.
Gravitational redshift formula
Δf/f=−gh/c2\Delta f/f = -gh/c^2 — light climbing out of a potential loses frequency.
Which clocks run slower?
Clocks deeper in a gravitational potential (lower position) run slower: Δτ/τ=+gh/c2\Delta\tau/\tau=+gh/c^2.
What can free-fall NOT remove?
Tidal effects — relative acceleration of nearby free-fallers — the true signature of spacetime curvature.
Einstein field equations in words
Gμν=8πGc4TμνG_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}: geometry/curvature = matter/energy.
Wheeler's slogan
Matter tells spacetime how to curve; spacetime tells matter how to move.
Newtonian mistake on light bending
Treating photons as massive gives only HALF the observed deflection; full GR doubles it.
Why is the elevator equivalence only local?
Over large regions, free-fall paths converge (toward Earth's center), revealing tidal forces that can't be transformed away.

Concept Map

tested to 10^-15

weak form

Einstein form

accel indistinguishable from gravity

straightest paths

light falls in accel frame

confirmed 1919

Doppler on climbing photon

df/f = -gh/c^2

lower clocks run slow

gravity not a force

Mass equality mi = mg

Equivalence Principle

All objects fall at same a

Elevator thought experiment

Gravity is geometry

Geodesics in curved spacetime

Light bends in gravity

Eddington eclipse

Gravitational redshift

Time dilation

Potential slows clocks

Free-fall follows geometry

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, General Relativity ka core idea bahut elegant hai: gravity koi "force" nahi hai, balki spacetime ka geometry (curvature) hai. Equivalence principle isko shuru karta hai. Socho tum ek band lift mein ho. Agar lift deep space mein gg se upar accelerate ho rahi hai, tumhe lagega jaise floor tumhe push kar raha hai aur dropped ball neeche girti hai. Bilkul wahi feeling Earth pe khadi lift mein hogi. Einstein ne kaha — inn dono ko andar se distinguish karna impossible hai. Matlab acceleration aur gravity locally same cheez hain.

Iska sabse pyara result: saari cheezein ek hi rate se girti hain kyunki inertial mass = gravitational mass (mi=mgm_i = m_g), to free fall mein mass cancel ho jaata hai aur a=ga=g sabke liye. Yahi se light bhi bend hoti hai — accelerating box mein light curve dikhti hai, to real gravity mein bhi light curve karegi. Aur clocks: jo clock neeche (deeper potential) hai wo slowly chalti hai, formula Δf/f=−gh/c2\Delta f/f = -gh/c^2. Yeh sab abstract nahi hai — tumhare phone ka GPS rozana lagभग 45 microsecond/day ka GR correction lagata hai, warna 11 km error aa jaata.

Lekin lift wala trick sirf local hai. Agar do balls door-door se Earth pe girein, dono Earth ke center ki taraf jaate hain, to unke paths aapas mein converge karte hain — yeh tidal effect hai jise free-fall se hata nahi sakte. Yahi irreducible cheez hi curvature kehlaati hai. Isiliye picture badal jaati hai: massive object spacetime ko bend karta hai, aur free object us bent geometry mein geodesic (straightest path) follow karta hai. Wheeler ne perfect bola — "Matter tells spacetime how to curve; spacetime tells matter how to move." Bas yahi GR ka dil hai.

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Connections

Special relativity — Michelson-Morley experimentPhysics · 2.3.25Time dilation — derivation, twin paradoxPhysics · 2.3.29Newton's law of gravitation — universal, action at distancePhysics · 1.2.19