Relativity ★★★ Advanced

🌌 Gravitational Wave Chirp

Two compact objects spiral inward under gravitational radiation reaction. Watch the strain waveform h(t) sweep from low to high frequency as the binary loses energy, just like GW150914 at LIGO. Set individual masses and adjustthe luminosity distance.

Binary system

Chirp parameters

Chirp mass 𝓅
Total mass M
f at merger
Peak strain h
Inspiral time
Energy radiated

Playback

Inspiral
f(t)=1/π·(5/256)³/⁸·
(G𝓅/c³)⁵/⁸·(t_c-t)⁻³/⁸
𝓅=(m₁m₂)³/⁵/(m₁+m₂)¹/⁵
h≈(4/r)(G𝓅/c²)⁵/³(πf)²/³

Gravitational Wave Physics

The Inspiral Phase

Two masses orbiting each other continuously lose energy to gravitational radiation (quadrupole formula: P = −32/5 · G&sup4;/c&sup5; · (m1m2)2(m1+m2)/r&sup5;). As the orbit shrinks, the orbital frequency rises — producing the characteristic “chirp” signal. LIGO detected GW150914 (30 + 36 M☉ black holes at 440 Mpc) on 14 September 2015.

Chirp Mass & Parameter Estimation

The chirp mass 𝓅 = (m1m2)3/5/(m1+m2)1/5 is the best-measured parameter from the inspiral phase; it controls the frequency evolution f(t). Individual masses and spin are harder to measure (require higher-order PN terms). The luminosity distance is measured from the amplitude h ∝ 𝓅5/3/r, making gravitational wave sources “standard sirens” for independent cosmology (H0 measurement).

Merger & Ringdown

Near coalescence (r → RISCO = 3rs) the post-Newtonian approximation breaks down. Numerical relativity is required. After merger, the remnant black hole rings like a bell — quasinormal modes with characteristic damped-sinusoid frequencies determined by mass and spin (Kerr). The ringdown frequency for GW150914 was ≈150 Hz with a damping time ≈4 ms.

LIGO Detection

LIGO uses 4 km Michelson interferometer arms. For h ∼ 10−²¹ (GW150914’s peak), the arm-length change is ΔL ∼ 4 × 10−¹&sup8; m — one-thousandth the diameter of a proton. The signal is extracted from detector noise (seismic, thermal, quantum shot noise) using matched filtering: cross-correlate the data with template waveforms from a bank spanning 𝓅 and η = m1m2/(m1+m2)2.

About Gravitational Wave Chirp

Gravitational waves are ripples in the fabric of spacetime produced by accelerating masses, predicted by Einstein's general relativity in 1916 and first directly detected by LIGO on September 14, 2015 (GW150914). A binary system of two compact objects (black holes or neutron stars) spirals inward as it loses energy to gravitational wave emission, gradually accelerating and emitting increasingly powerful waves—a signal called a chirp because its frequency rises rapidly just before merger, sweeping through the LIGO band (20–2000 Hz) in a characteristic pattern resembling a bird chirp.

The chirp mass M_chirp = (m₁m₂)^(3/5)/(m₁+m₂)^(1/5) is the primary parameter governing the inspiral dynamics and can be measured to better than 1% accuracy from the frequency evolution of the chirp signal alone. As the binary inspirals, gravitational wave frequency increases as f_GW = 2·f_orbital = (5/8π)^(3/8)·(c³/GM_chirp)^(5/8)·(t_merger - t)^(-3/8). In the final milliseconds before merger, frequency reaches hundreds to thousands of Hz while strain amplitude reaches a maximum, followed by the ringdown of the merged black hole oscillating in its quasinormal modes.

This simulator generates synthetic chirp waveforms for user-specified binary parameters (masses, initial frequency), visualizes time-domain strain h(t) and time-frequency spectrogram, and demonstrates matched filtering—the technique LIGO uses to detect signals buried in noise. You can observe how chirp mass affects the signal evolution, add noise to demonstrate detection sensitivity, and explore the relationship between binary parameters and observable waveform characteristics that enabled GW150914's masses (36+29 solar masses) to be measured from a signal lasting ~0.2 seconds.

Frequently Asked Questions

What is a gravitational wave and how is spacetime stretched?

A gravitational wave is a transverse oscillation of spacetime curvature propagating at the speed of light. As it passes, it alternately stretches and compresses space in perpendicular directions: a ring of free-floating masses forms an ellipse that oscillates between horizontal and vertical elongation, then back. The strain h = ΔL/L is the fractional change in distance caused by the wave. For GW150914, h ≈ 10⁻²¹—a 4 km LIGO arm changed length by ~4×10⁻¹⁸ m, less than 1/1000 the diameter of a proton. Detecting this requires laser interferometry of extraordinary precision.

Why does the gravitational wave frequency increase (chirp) as the binary inspirals?

As the two objects orbit each other, they emit gravitational waves carrying energy and angular momentum away from the system. Losing orbital energy causes the orbit to shrink (by Kepler's third law, smaller orbit means faster orbital period). Faster orbital period means higher gravitational wave frequency (f_GW = 2f_orbit). The inspiral rate accelerates as the orbit shrinks—more power radiated at smaller separations causes faster shrinkage—creating a runaway acceleration. The result is the characteristic chirp: frequency and amplitude both increase, slowly at first and then catastrophically fast in the final seconds before merger.

What happens at merger and ringdown?

As the two compact objects approach within a few Schwarzschild radii, post-Newtonian approximations break down and the full nonlinear Einstein equations govern the dynamics. In black hole mergers, the objects plunge together and merge into a single, distorted black hole oscillating in its characteristic quasinormal modes—the ringdown. The ringdown frequency and damping time are uniquely determined by the final black hole's mass and spin via the Kerr solution (no-hair theorem), allowing direct tests of general relativity. Neutron star mergers produce additional features from tidal disruption and electromagnetic counterparts from kilonova ejecta.

How does LIGO detect such tiny signals?

LIGO's Michelson interferometer splits laser light down two 4 km arms, reflects it off suspended mirrors, and recombines the beams. A gravitational wave differentially changes the two arm lengths, producing a phase shift detected as changing light intensity at the output. Key technologies enabling 10⁻²¹ strain sensitivity: Fabry-Pérot cavities that bounce light ~300 times (effectively 1200 km arm length); power recycling mirrors boosting circulating power to 100 kW; 40 kg fused silica mirrors suspended on quadruple pendulums for seismic isolation; and quantum squeezing to reduce photon shot noise below the standard quantum limit. Even with all this, signals are found by matched filtering against template waveform banks.

What have gravitational wave observations revealed about the universe?

Since GW150914, the LIGO-Virgo-KAGRA network has detected over 90 compact binary mergers. Key discoveries: black hole masses in the 5–150 solar mass range, revealing a population previously unknown from electromagnetic observations; the neutron star merger GW170817 with multi-messenger electromagnetic counterpart (kilonova), confirming neutron star mergers as sites of r-process heavy element production (gold, platinum, lanthanides), enabling an independent Hubble constant measurement; the first asymmetric mass-ratio events suggesting hierarchical mergers in dense stellar environments; and no deviations from general relativity in any observed event.

About this simulation

This simulation traces the chirp of a compact-binary inspiral: two black holes spiralling together as gravitational waves carry away orbital energy. The chirp mass, M_c = (m₁m₂)^(3/5)/(m₁+m₂)^(1/5), sets how fast the separation shrinks, while Kepler's third law gives the orbital frequency. Frequency and strain climb toward a peak at the innermost stable circular orbit (ISCO), then the remnant rings down — all recalculated live, not a canned animation.

🔬 What it shows

Separation between the two bodies shrinks under the quadrupole radiation formula. A dashed ring marks the ISCO (three Schwarzschild radii), where inspiral gives way to a merger flash and decaying ringdown. The h(t) strain plot builds up live, coloured by amplitude, with a phase label tracking Inspiral, Merger and Ringdown.

🎮 How to use

Set the masses with the m₁ and m₂ sliders (1–80 M☉, default 30/25) and distance with the Distance slider (10–5000 Mpc, default 400). Eccentricity (0–0.7) is reserved for a future orbit model — the build always runs a circular inspiral. Speed (1–200×) scales playback; Restart and Pause reset or freeze it.

💡 Did you know?

GW150914, the event this simulator echoes, merged two black holes of about 36 and 29 solar masses on 14 September 2015, radiating roughly 3 solar masses as gravitational waves in under a second — briefly the most powerful gravitational-wave source in the observable universe.

Frequently asked questions

How does the chirp mass drive the waveform?

It sets the rate constant β = (64/5)G³m₁m₂(m₁+m₂)/c⁵ in r(t)⁴ = 4β(t_merger − t) — the single combination controlling the whole frequency sweep, which is why it's the best-measured parameter from a real signal.

Why doesn't Eccentricity change the orbit's shape?

The model always evolves a circular inspiral, so the slider is a placeholder for a future update rather than an active parameter. Real binaries circularise quickly anyway, so this is realistic for the late inspiral shown here.

What sets the frequency and strain at merger?

Merger is defined at the ISCO, three Schwarzschild radii of the total mass. Merger frequency comes from Kepler's law there, and peak strain uses the same amplitude formula, h ∝ (GM_c/c²)^(5/3)(πf)^(2/3)/distance.

How is the ringdown modelled?

After the ISCO, the animation switches to a damped sinusoid for the remnant's dominant quasinormal mode: an exponentially decaying strain with a roughly 20 ms damping time, loosely matched to GW150914's own ringdown.

Why does distance shrink the waveform but not its frequency sweep?

Strain falls off as 1/distance since the wave's energy spreads over an ever-larger sphere. Frequency evolution depends only on the masses, so a distant source sounds quieter without chirping any slower or faster.