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Physics · Document 6

General Relativity

Curved spacetime, Einstein's equations, and gravity as geometry

1. The Equivalence Principle

Einstein's key insight: gravity is indistinguishable from acceleration. In a falling elevator, you feel weightless. In an accelerating rocket, you feel weight.

Weak Equivalence Principle:
All objects fall at the same rate, regardless of mass or composition.
Strong Equivalence Principle:
Physics in a freely falling frame is locally the same as in flat spacetime.
The equivalence principle implies that gravity curves spacetime. Free-falling objects follow straight lines (geodesics) in curved spacetime.

2. Curved Spacetime

Mass and energy curve the fabric of spacetime. Objects then follow the natural paths (geodesics) through this curved geometry.

🎮 Interactive: Spacetime Curvature from Mass
Spacetime curvature from mass
Einstein's insight: Mass tells spacetime how to curve. Objects follow the straightest possible paths (geodesics) in curved spacetime.

The Metric Tensor

Spacetime geometry is encoded in the metric tensor g_uv, which defines distances and angles:

ds2 = g_uv dx^u dx^v
Spacetime interval
Flat spacetime (Minkowski):
ds2 = -c2dt2 + dx2 + dy2 + dz2
Schwarzschild (black hole):
ds2 = -(1-r_s/r)c2dt2 + dr2/(1-r_s/r) + ...

3. Geodesics

A geodesic is the straightest possible path through curved spacetime. Massive objects follow timelike geodesics; light follows null geodesics.

🎮 Interactive: Light Bending Around Mass
MLight bends near massive objects
Geodesic: The shortest path through curved spacetime. Light always follows null geodesics (ds2 = 0).
Gravitational lensing: Light from distant stars bends around the Sun, confirmed in the 1919 eclipse.
d2x^u/dt2 + G^u_ab (dx^a/dt)(dx^b/dt) = 0
Geodesic equation

The Christoffel symbols G^u_ab encode how coordinates curve, derived from the metric: G^u_ab = (1/2)g^um(g_ma,b + g_mb,a - g_ab,m)

4. Einstein Field Equations

Einstein's equations relate spacetime curvature to matter and energy content:

G_uv + Lambda g_uv = (8 pi G / c4) T_uv
Einstein field equations
G_uv
Einstein tensor (curvature)
Lambda
Cosmological constant
T_uv
Stress-energy tensor (matter)

The Einstein Tensor

G_uv = R_uv - (1/2)R g_uv

Where R_uv is the Ricci tensor (contraction of Riemann curvature) and R is the Ricci scalar.

5. Real-World Application: GPS

General relativity is not just abstract theory - GPS satellites must account for relativistic time dilation to maintain accuracy.

🎮 Interactive: GPS Time Dilation
Low Earth OrbitGPS (~20,200 km)Geostationary
Satellite Clock+144.9 us/day (gravity)-7.2 us/day (velocity)Net Effect+137.7 us/day
Without corrections: GPS would accumulate ~10 km/day of error!
General relativity is not just theory - it is essential technology.
GPS satellites experience two competing effects:
1. Gravitational time dilation: Clocks run faster at higher altitude (+45 us/day)
2. Velocity time dilation: Moving clocks run slower (-7 us/day)
Net effect: satellite clocks gain ~38 microseconds per day.

6. Key Solutions

SolutionDescribesKey Feature
SchwarzschildNon-rotating black holeEvent horizon at r_s = 2GM/c2
KerrRotating black holeFrame dragging, ergosphere
FLRWExpanding universeScale factor a(t), Hubble flow
de SitterPure dark energyExponential expansion

7. Connection to Z2 Framework

Z²
G_uv + Lambda g_uv = 8piG T_uv
Einstein equations must be satisfied by any physical spacetime, including the Z2 compactification
Z²
Lambda = 13/19 * rho_crit
The cosmological constant fraction emerges from moduli stabilization on T3/Z2
Z²
AdS_5 x S5 / Z2
The near-horizon geometry of D3-branes includes a Z2 orbifold

Why General Relativity Matters for Z2

  • * Compactification requires solving Einstein equations on compact spaces
  • * Moduli stabilization fixes the size of extra dimensions
  • * ADM formalism separates time from space for canonical quantization
  • * Cosmological constant arises from vacuum energy of compactified dimensions

Exercises

  1. Calculate the Schwarzschild radius for the Sun (M = 2 x 10^30 kg). Is the Sun a black hole?
  2. At what altitude does gravitational time dilation equal velocity time dilation?
  3. Show that the Einstein tensor is divergence-free: nabla_u G^uv = 0.
  4. Derive the Newtonian limit: show g_00 approx -(1 + 2Phi/c2) where Phi is gravitational potential.
  5. Why does the cosmological constant have units of (length)^-2?