Gravitational Waves

Drop a stone into a still pond and ripples spread outward. Now imagine the "pond" is space itself. When two very heavy objects whirl around each other — say, two black holes — they shake space and send ripples racing outward at the speed of light. Those ripples are gravitational waves. As one sweeps past, it gently stretches everything in one direction while squeezing it in the perpendicular direction, then swaps — but by such an unimaginably tiny amount that nobody ever feels it.

Putting a number on the stretch

We measure a wave's size by its strain $h$ — the fractional change in length, $h = \Delta L / L$. If a ruler of length $L$ momentarily grows by $\Delta L$, that ratio is the strain. For waves reaching Earth from a black-hole merger, $h \sim 10^{-21}$. Spread over one of LIGO's $L = 4$ km arms, that means a length change of

about a thousandth the width of a single proton. How "loud" the source is depends mostly on one clever combination of the two masses, the chirp mass:

Heavier, closer binaries make $h$ bigger.

The chirp and the two shapes

As the pair pours orbital energy into the waves, their orbit shrinks and they speed up — so the wave's frequency climbs and its amplitude swells in the final fraction of a second. That rising whistle is the famous "chirp," with $f_{GW}\propto(t_c-t)^{-3/8}$ as the merger time $t_c$ approaches. The wave also carries two independent patterns, the "plus" and "cross" polarizations $h_+$ and $h_\times$. A passing wave stretches a ring of free-floating test masses along one axis while squeezing the perpendicular one (the $+$ pattern), or along the diagonals (the $\times$ pattern) — never puffing in and out uniformly like a balloon. In the sim, the m₁ and m₂ sliders set the masses (and so the chirp mass and the chirp), Distance D scales the strain since $h\propto 1/D$, and Initial Freq f sets where the chirp begins.

Try this in the sim above

Push both mass sliders up and watch the strain readout $h$ and the chirp grow louder and faster. Then slide Distance D from 100 up to 1000 Mpc and watch $h$ fade away as $1/D$ — the same source, just farther off. Finally switch to the "+ and × Polarizations" mode and watch the ring of test masses deform into a $+$ and a $\times$, confirming for yourself that it never simply expands as a circle.

Gravitational waves are propagating ripples in spacetime curvature, predicted by Einstein in 1916 and detected directly by LIGO on Sept 14, 2015 (GW150914). They are produced by accelerating masses with a time-varying mass quadrupole moment.

Symbol	Meaning	SI Unit
$h_{\mu\nu}$	Metric perturbation	—
$h_+,h_\times$	Two polarizations	—
$\mathcal{M}$	Chirp mass	kg or $M_\odot$
$f_{GW}$	GW frequency = 2$f_{orb}$	Hz
$D_L$	Luminosity distance	Mpc
$P_{GW}$	GW power radiated	W

Step 1 — Quadrupole formula (Einstein 1918)

$$\bar{h}_{ij}(t,\vec{r}) = \frac{2G}{c^4 D}\,\ddot{Q}_{ij}\!\left(t - \frac{D}{c}\right)$$

Where $Q_{ij}$ is the traceless mass quadrupole. (Monopole = mass conservation; dipole = momentum conservation — neither radiates.)

Step 2 — Power radiated

$$P_{GW} = \frac{G}{5c^5}\langle\dddot{Q}_{ij}\dddot{Q}^{ij}\rangle$$

For a circular binary: $P_{GW} = \frac{32 G^4}{5 c^5}\frac{(m_1 m_2)^2 (m_1+m_2)}{r^5}$. The $c^{-5}$ makes GW emission usually negligible — but huge for compact binaries.

Step 3 — Inspiral & chirp

$$f_{GW}(t) \propto (t_c - t)^{-3/8}, \qquad \mathcal{M} = \frac{(m_1 m_2)^{3/5}}{(m_1 + m_2)^{1/5}}$$

The chirp mass $\mathcal{M}$ is directly measurable from the rate of frequency increase.

Step 4 — Strain amplitude

$$h \sim \frac{1}{D}\,\frac{G^2 m_1 m_2}{c^4 r}$$

For GW150914: $h \approx 10^{-21}$ at Earth — a fractional change of $10^{-21}$ in LIGO's 4 km arm = $4 \times 10^{-18}$ m, smaller than a proton.

Step 5 — Two polarizations

$$h_+ \propto (1+\cos^2\iota)\cos(2\Phi), \qquad h_\times \propto 2\cos\iota \sin(2\Phi)$$

A ring of test masses oscillates as a + cross or a × — never as a sphere.

Mapping to the simulation

Inspiral mode shows the orbit shrinking and chirp frequency rising. Strain mode plots $h(t)$ at the detector. Detector mode shows LIGO interferometer geometry.