Gravitational Waves: Ripples in Spacetime
Just as accelerating charges produce electromagnetic waves, accelerating masses produce gravitational waves — ripples in the fabric of spacetime that propagate at the speed of light. Einstein predicted them in 1916; LIGO detected them directly a century later.
Linearised gravity
In weak gravitational fields, write the metric as a small perturbation of flat spacetime:
$$g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu} \qquad |h_{\mu\nu}| \ll 1$$
Substitute into the Einstein field equations, keep only terms linear in $h$, and choose a convenient gauge (the Lorenz gauge: $\partial_\mu \bar{h}^{\mu\nu} = 0$, where $\bar{h}{\mu\nu} = h{\mu\nu} - \frac{1}{2}\eta_{\mu\nu}h$). The result:
$$\Box \bar{h}{\mu\nu} = -\frac{16\pi G}{c^4} T{\mu\nu}$$
This is a wave equation — the same mathematical structure as electromagnetic waves. In vacuum ($T_{\mu\nu} = 0$): $\Box \bar{h}_{\mu\nu} = 0$, and the solutions are plane waves propagating at $c$.
Polarisations
Gravitational waves are transverse and have two polarisations: “+” and “×”. They distort space perpendicular to the direction of travel:
- “+” polarisation: alternately stretches the $x$-direction and compresses the $y$-direction (and vice versa)
- “×” polarisation: the same pattern, rotated 45°
A ring of test particles distorts as a gravitational wave passes through. The wave travels into the screen. Toggle the polarisation to compare “+” and “×”.
Unlike electromagnetic waves (spin 1, dipole radiation), gravitational waves have spin 2 and are produced by quadrupole oscillations — mass distributions whose second moment changes in time. A spinning perfect sphere produces no gravitational waves (its quadrupole moment is constant). Two orbiting masses do.
Sources
| Source | Frequency | Strain $h$ | Detector |
|---|---|---|---|
| Binary neutron stars (inspiral) | 10–1000 Hz | $\sim 10^{-22}$ | LIGO/Virgo |
| Binary black holes (merger) | 30–300 Hz | $\sim 10^{-21}$ | LIGO/Virgo |
| Supermassive BH binaries | nHz–μHz | $\sim 10^{-15}$ | Pulsar timing arrays |
| Primordial (Big Bang) | μHz–nHz | $\sim 10^{-16}$? | LISA (planned) |
The quadrupole formula
For a slowly moving source, the gravitational wave strain at distance $r$ is:
$$h_{ij} \sim \frac{2G}{c^4 r} \ddot{I}_{ij}$$
where $\ddot{I}_{ij}$ is the second time derivative of the mass quadrupole moment. The prefactor $G/c^4$ is tiny — this is why gravitational waves are so weak and why you need colliding black holes (masses of tens of solar masses, speeds approaching $c$) to produce detectable signals.
Detection: LIGO
LIGO (Laser Interferometer Gravitational-Wave Observatory) uses two 4-km laser interferometer arms arranged in an L. A passing gravitational wave stretches one arm and compresses the other, producing a measurable phase shift in the laser light.
The first detection, GW150914 (September 14, 2015), matched the predicted waveform of two black holes (36 + 29 $M_\odot$) spiralling together, merging, and ringing down — in 0.2 seconds, converting 3 $M_\odot$ into gravitational radiation. The peak luminosity briefly exceeded the combined light output of all stars in the observable universe.
Indirect detection: the Hulse-Taylor pulsar
Before LIGO, the binary pulsar PSR B1913+16 (discovered 1974) provided indirect evidence. Its orbital period decreases by 76.5 μs/year — exactly matching the energy loss predicted by GR due to gravitational wave emission. Hulse and Taylor received the 1993 Nobel Prize for this measurement.