Quadrupole Radiation

Quadrupole radiation is the leading mechanism by which a system emits gravitational waves. Unlike electromagnetism, which radiates dominantly through an oscillating electric dipole, gravity has no dipole radiation: the would-be mass dipole Σ mᵢ rᵢ equals M r_cm, and the centre of mass of an isolated system cannot accelerate because total momentum is conserved. Its second time derivative vanishes identically, so the lowest radiating multipole is the quadrupole — the second mass moment Qᵢⱼ = Σ mᵢ(xᵢxⱼ − ⅓rᵢ²δᵢⱼ).

A system radiates gravitationally only when it changes shape in a way that gives a non-zero third time derivative of the quadrupole moment. A perfectly spherical pulsation radiates nothing; a rotating, asymmetric, or oscillating mass distribution does. The radiated power is given by Einstein's 1918 quadrupole formula, P = (G/5c⁵)⟨Q⃛ᵢⱼQ⃛ᵢⱼ⟩, where the prefactor G/c⁵ is so small that only violently relativistic sources — orbiting black holes and neutron stars — radiate detectably.

Because gravitational radiation starts at the quadrupole rather than the dipole, its amplitude carries an extra factor of the source velocity v/c relative to electromagnetic radiation, which is a fundamental reason gravitational waves are so weak. The same quadrupole formula governs the inspiral of binary systems, predicting the orbital decay that Hulse and Taylor confirmed in the binary pulsar to better than 1% — the first quantitative proof that the waves carry away energy.

Einstein derived the quadrupole formula in 1918, correcting an erroneous factor in his 1916 paper. Its physical validity was contested for decades — Einstein himself doubted the waves were real until the 1936 Einstein–Rosen episode and the 1957 sticky-bead argument — and was settled empirically by the 1974–2005 timing of the Hulse–Taylor binary pulsar.