Gravitational Waves

Gravitational waves basics

From general relativity, accelerating massive objects radiate gravitational waves — ripples in spacetime that propagate at c. The wave amplitude is described by "strain" h = ΔL/L (fractional length change).

For typical sources at cosmological distances, h ≈ 10⁻²¹ at Earth. Tiny — but detectable with extreme precision instruments.

How they're generated

Quadrupole formula (Einstein, 1916):

P = (G/5c⁵) · <d³Q/dt³>²

where Q is the mass quadrupole moment. Power scales as ~v⁶ for binary inspirals.

Compact binary inspiral — two compact objects orbiting close, eventually merging. Frequency grows as orbit shrinks; sound: "chirp."

Major detected events

Event	Date	Source	Significance
GW150914	14 Sep 2015	2 BHs (36 + 29 M_sun)	First direct detection
GW170817	17 Aug 2017	2 NS (1.46 + 1.27 M_sun)	First multi-messenger; kilonova
GW190521	21 May 2019	Heavy BH merger (85 + 66 M_sun)	Intermediate-mass BH
GW190425	25 Apr 2019	Heavy NS-NS	—
GW200105	5 Jan 2020	NS-BH	—

Hundreds of detections so far across LIGO/Virgo/KAGRA observation runs.

JavaScript — gravitational wave calculations

const G = 6.674e-11;
const c = 3e8;
const M_sun = 1.989e30;

// Chirp mass (from binary inspiral)
function chirpMass(m1, m2) {
  // M_chirp = (m1*m2)^(3/5) / (m1+m2)^(1/5)
  return Math.pow(m1*m2, 0.6) / Math.pow(m1+m2, 0.2);
}

// GW150914
console.log(`GW150914 chirp mass: ${(chirpMass(36 * M_sun, 29 * M_sun) / M_sun).toFixed(2)} M_sun`);
// ~28 M_sun

// Strain at Earth from binary inspiral
function strain(chirpMass_kg, distance_m, frequency_Hz) {
  // h ~ (G·M_chirp/c²)^(5/3) · ω^(2/3) / (c·d)
  // Approximate; full formula is more complex
  const omega = 2 * Math.PI * frequency_Hz;
  return Math.pow(G * chirpMass_kg / (c * c), 5/3) * Math.pow(omega, 2/3) / (c * distance_m);
}

// GW150914 at ~100 Hz, 1.3 billion ly
const ly = 9.461e15;  // m
const distance_GW150914 = 1.3e9 * ly;
const Mchirp_GW150914 = 28 * M_sun;
console.log(`GW150914 strain: ${strain(Mchirp_GW150914, distance_GW150914, 100).toExponential(2)}`);
// ~10⁻²¹

// LIGO arm length distortion
function ligoDistortion(strain, arm_length_km) {
  return strain * arm_length_km * 1000;  // in meters
}

console.log(`LIGO 4km, h=10⁻²¹: ${ligoDistortion(1e-21, 4).toExponential(2)} m`);
// ~4 × 10⁻¹⁸ m — smaller than proton

// Power radiated (rough)
function gwPower(orbitalFreq, M_chirp) {
  // For two equal masses in circular orbit
  const omega = 2 * Math.PI * orbitalFreq;
  return (32/5) * Math.pow(G, 7/3) * Math.pow(M_chirp, 10/3) * Math.pow(omega, 10/3) / Math.pow(c, 5);
}

// Earth-Sun system: extremely small (Hz ~ 10⁻⁷)
console.log(`Earth-Sun GW power: ${gwPower(3.17e-8, 1).toExponential(2)} W`);
// ~10⁻⁹⁰ W — laughably small

// Frequency at coalescence (final orbit before merge)
function isco_freq(M_total) {
  // ISCO orbital freq ~ c³/(6√6 · π · G · M)
  return Math.pow(c, 3) / (6 * Math.sqrt(6) * Math.PI * G * M_total);
}

// 30 solar mass BH: f ~ 73 Hz
console.log(`30 M_sun BH ISCO: ${isco_freq(30 * M_sun).toFixed(0)} Hz`);
// ~73 Hz

Where GWs matter

• Astrophysics. Black hole and neutron star mergers; populations and rates.
• Multi-messenger astronomy. GW + EM + neutrinos for richer event understanding.
• Cosmology. Standard sirens (GW + EM) for Hubble constant; primordial GW for inflation.
• Tests of GR. Strong-field tests; ringdown spectra match Kerr BHs.
• Heavy element creation. NS mergers produce gold, platinum (kilonova).
• Future missions. LISA (2030s), Cosmic Explorer, Einstein Telescope.
• Pulsar timing. Long-wavelength GWs from supermassive BH binaries (NANOGrav).

Common mistakes

• Confusing GWs with gravity waves (atmospheric). Different — atmospheric "gravity waves" are buoyancy oscillations in fluids. GWs in physics are spacetime ripples.
• Believing GWs travel through matter unchanged. Mostly yes, but at extreme densities (neutron stars), some interaction.
• Thinking GWs need a medium. They're spacetime distortions — no medium needed. Travel through vacuum at c.
• Underestimating energy. GW150914 emitted ~3 solar masses of energy in fraction of a second — more power than rest of observable universe combined.
• Forgetting all moving masses radiate. Earth-Sun system also emits GWs (negligibly small). Only compact, high-acceleration sources detectable.
• Believing speed of GWs ≠ c. GW170817 confirmed speed = c to 10⁻¹⁵ precision.