General Relativity

Equivalence principle

Einstein's "happiest thought" (1907) — a free-falling observer feels no gravity. Inside a free-falling elevator, objects float; if you let go of a ball, it stays put relative to you. Indistinguishable from being in deep space.

From this — accelerated frames are equivalent to gravitational fields. So gravity must be a manifestation of geometry, not a force in flat space.

Einstein field equations

G_μν + Λ·g_μν = (8πG / c⁴) · T_μν

where:

• G_μν — Einstein tensor (geometry of spacetime curvature).
• g_μν — metric tensor (defines distances/intervals).
• Λ — cosmological constant (dark energy).
• T_μν — stress-energy tensor (matter and energy distribution).
• G — gravitational constant.
• c — speed of light.

"Spacetime tells matter how to move; matter tells spacetime how to curve" (John Wheeler).

Confirmed predictions

Prediction	Observation/Test	Result
Mercury's perihelion precession	43"/century beyond Newton	GR exactly matches
Light bending	Eddington 1919 eclipse	2× Newton; matches GR
Gravitational redshift	Pound-Rebka 1959	Confirmed
Gravitational time dilation	Atomic clocks at altitude	Matches GR
Gravitational lensing	Einstein cross, etc.	Common observation
Frame dragging	Gravity Probe B (2011)	Confirmed
Gravitational waves	LIGO (2015 BH merger)	Direct detection
Black holes (M87 image)	EHT 2019	Image consistent with GR

JavaScript — GR calculations (mostly conceptual)

const G = 6.674e-11;
const c = 3e8;

// Schwarzschild radius
function schwarzschildRadius(mass_kg) {
  return 2 * G * mass_kg / (c * c);
}

console.log(`Sun (1.989e30 kg): r_s = ${schwarzschildRadius(1.989e30).toFixed(0)} m`);  // 2950 m
console.log(`Earth (5.972e24 kg): r_s = ${(schwarzschildRadius(5.972e24) * 1000).toFixed(2)} mm`); // 8.87 mm

// Sagittarius A*: 4.3 million solar masses
console.log(`Sgr A*: r_s = ${(schwarzschildRadius(4.3e6 * 1.989e30) / 1e9).toFixed(2)} Gm`);

// Gravitational time dilation (weak field)
function gravitationalTimeDilation(altitude_m, planet_mass = 5.972e24, planet_radius = 6.371e6) {
  // Δt/t ≈ -GM/(rc²) for low altitudes
  const r1 = planet_radius;
  const r2 = planet_radius + altitude_m;
  const phi1 = -G * planet_mass / r1 / (c * c);
  const phi2 = -G * planet_mass / r2 / (c * c);
  return phi2 - phi1;  // fractional time difference (positive = higher clock faster)
}

// GPS satellite at 20,200 km altitude
const gps_dilation = gravitationalTimeDilation(20.2e6);
console.log(`GPS satellite gains: ${(gps_dilation * 86400 * 1e6).toFixed(2)} µs/day from gravity`);
// ~45.8 µs/day FASTER

// Mercury perihelion precession
function mercuryPrecession() {
  // Δφ per orbit = 6πGM_sun/(a(1-e²)c²) where a = semi-major, e = eccentricity
  const M_sun = 1.989e30;
  const a = 5.79e10;  // semi-major axis
  const e = 0.206;
  const orbits_per_century = 100 * (1 / 0.241);
  const dphi_per_orbit = 6 * Math.PI * G * M_sun / (a * (1 - e*e) * c * c);  // radians
  const dphi_per_century = dphi_per_orbit * orbits_per_century;
  // Convert to arcseconds
  return dphi_per_century * 180 / Math.PI * 3600;
}

console.log(`Mercury precession: ${mercuryPrecession().toFixed(0)} arcsec/century`);  // ~43

// Schwarzschild solution: time dilation at radius r outside black hole
function timeDilationFactorBH(r, M) {
  // dτ/dt = √(1 - 2GM/rc²)
  const r_s = 2 * G * M / (c * c);
  if (r <= r_s) return 0;
  return Math.sqrt(1 - r_s / r);
}

// Just outside event horizon vs far away
const M_BH = 10 * 1.989e30;  // 10 solar mass
const r_s_BH = schwarzschildRadius(M_BH);
console.log(`At 1.001 × r_s: time runs ${timeDilationFactorBH(r_s_BH * 1.001, M_BH).toFixed(4)}× normal`);
console.log(`At 1.1 × r_s: ${timeDilationFactorBH(r_s_BH * 1.1, M_BH).toFixed(3)}×`);
console.log(`At 10 × r_s: ${timeDilationFactorBH(r_s_BH * 10, M_BH).toFixed(3)}×`);
// At infinity, factor approaches 1

Where GR matters

• GPS and satellite navigation. Without GR corrections, GPS would be off by km/day.
• Cosmology. Big Bang, expansion, dark energy, dark matter — all framed in GR.
• Black holes. Predicted by GR; observed (Cygnus X-1, Sgr A*, M87).
• Gravitational waves. LIGO, Virgo, KAGRA detect ripples in spacetime from cosmic mergers.
• Astrophysics. Neutron stars, pulsars, gravitational lensing.
• Tests of fundamental physics. Extreme-precision tests at solar system scale and gravitational wave observations.
• Future technology. Gravitational wave-based timing, cosmology probes.

Common mistakes

• Treating gravity as a force. In GR, gravity is not a force — it's spacetime geometry. Newton works for everyday cases but conceptually wrong.
• Confusing special and general relativity. SR — flat spacetime, no gravity. GR — curved spacetime, includes gravity.
• Thinking GR contradicts Newton. They agree at low gravity, low speeds. GR is a refinement that handles strong gravity, fast motion.
• Believing GR predicts wormholes/time travel. Some solutions allow these mathematically; physically realizable is debated. No experimental evidence.
• Confusing cosmological constant with dark matter. Λ accounts for dark energy (acceleration). Dark matter is a separate phenomenon (extra matter not seen via electromagnetism).
• Treating GR as complete theory of gravity. Breaks down at quantum scales (singularity, Planck scale). Quantum gravity is incomplete.