Technopedia

The technologist’s encyclopedia

Gravitational waves

Gravitational waves are propagating ripples of spacetime curvature: the radiative solutions of the linearized Einstein equations, traveling at the speed of light. They are generated by time-varying quadrupole mass distributions — binaries, supernovae, spinning deformed neutron stars — and they stretch and squeeze distances transversely as they pass.

Authoritative sources

The quadrupole formula

Definition. In the weak-field, slow-motion limit, the radiated power of a source with time-varying mass quadrupole moment QijQ_{ij} is

P=G5c5 ⟨\dddotQij\dddotQij⟩,P = \frac{G}{5c^5}\,\left\langle \dddot{Q}_{ij}\dddot{Q}_{ij} \right\rangle,

the famous G/c5G/c^5 prefactor explaining why gravitational radiation is fantastically weak: only astrophysical sources radiate measurably.

Details
  1. Why quadrupole? A spherically symmetric collapse emits nothing (Birkhoff’s theorem), and a spinning axisymmetric body emits nothing either — the monopole (mass) and dipole (center of mass) are conserved, so the first radiating multipole is the quadrupole. This is why a spinning perfect neutron star is silent while a lumpy one is not.
  2. Polarizations. A wave has two transverse polarizations, ++ and ×\times, which stretch one transverse axis while squeezing the other, swapping every half-period. A ring of test particles is alternately stretched into ellipses at ±45°.
  3. Effect on detectors. The strain h=ΔL/Lh = \Delta L/L from astrophysical sources is ∼10−21\sim 10^{-21}: a 4 km interferometer arm changes length by ∼4×10−18\sim 4\times10^{-18} m, a thousandth of a proton radius. LIGO detects it with laser interferometry, and pulsar timing arrays extend the technique to nanohertz frequencies.

Example. GW150914 (the first detection): two black holes of 36 and 29 solar masses merged 1.3 billion light-years away; the peak strain at Earth was ∼10−21\sim 10^{-21}, and the peak luminosity exceeded the light output of the observable universe by a factor of ~50 — but only in gravitational energy.

Chirp signals and binary inspiral

Definition. As a binary loses energy to gravitational radiation, its orbit shrinks and its orbital frequency rises, producing a characteristic accelerating “chirp” whose frequency–time profile measures the system’s chirp mass M=(m1m2)3/5/(m1+m2)1/5\mathcal{M} = (m_1 m_2)^{3/5}/(m_1+m_2)^{1/5}.

Details
  1. Inspiral. The quadrupole formula gives the orbital decay: f˙\dot{f} grows as f11/3f^{11/3}. Measuring ff and f˙\dot{f} in a detected signal fixes the chirp mass without any distance information — the basis of “standard siren” cosmology.
  2. Merger and ringdown. As the bodies approach, the signal becomes nonlinear; after merger, the remnant black hole rings down at its quasinormal frequencies, like a struck bell, settling into a stationary Kerr state.
  3. Indirect evidence first. The Hulse–Taylor binary pulsar’s orbit decays at exactly the quadrupole-formula rate (Nobel Prize 1993) — gravitational waves were confirmed before they were directly detected.

Example problem. Two neutron stars of 1.4 and 1.3 solar masses merge. Why does the detected chirp constrain their masses so precisely?

Solution

The chirp rate f˙\dot{f} at a given frequency ff depends only on the chirp mass: f˙∝M5/3f11/3\dot{f} \propto \mathcal{M}^{5/3} f^{11/3}. Since the signal is tracked over thousands of cycles with millisecond timing, M\mathcal{M} is measured to sub-percent precision; combined with the amplitude (which encodes distance) and the merger frequency (which encodes the total mass via the last stable orbit), the individual masses are recovered. The same physics that makes the orbit shrink — energy carried away by the waves — is what encodes the masses in the waveform.

Relation to other classes

Gravitational waves are the radiative sector of General relativity: linearized solutions carrying energy away from sources. They are central to Compact objects (binary black-hole and neutron-star mergers) and to Cosmological gravity (a stochastic background from the early universe would probe epochs the electromagnetic cosmic microwave background cannot reach).