Math 2.0

Relativity and gravitational theory Gravitational waves Proposition

Quadrupole formula

Builds on Gravitational waveLinearized gravity

Statement

Let an isolated source of mass density ρ(t,x)\rho(t, x) move slowly, with internal speeds v≪cv \ll c, and let its gravitational field be weak. Define its reduced quadrupole moment Qij(t)=∫ρ(t,x)(xixj−13δij∣x∣2)d3x,i,j∈{1,2,3}.Q_{ij}(t) = \int \rho(t, x) \left(x_i x_j - \tfrac{1}{3}\delta_{ij} \abs{x}^2\right) d^3x, \qquad i, j \in \set{1, 2, 3}. Let rr be the distance to the source, much larger than the wavelength. Then, to leading order in v/cv/c, the emitted gravitational wave in transverse-traceless gauge is hijTT(t,x)=2Gc4r(Q¨ij)TT ⁣(t−rc).h^{TT}_{ij}(t, x) = \frac{2G}{c^4 r} \left(\ddot Q_{ij}\right)^{TT}\!\left(t - \frac{r}{c}\right). The radiated power is P=G5c5⟨Q...ij Q...ij⟩.P = \frac{G}{5c^5} \left\langle \dddot Q_{ij}\, \dddot Q_{ij} \right\rangle. Here dots denote time derivatives, repeated indices are summed, and ⟨⋅⟩\langle \cdot \rangle is an average over several periods.

Remarks

The formula comes from linearized gravity: in Lorenz gauge the perturbation solves a wave equation with the stress-energy tensor as source, and the retarded solution is expanded in v/cv/c. Conservation of the stress-energy tensor turns the leading spatial stresses into the second time derivative of the quadrupole moment. The monopole and dipole parts do not radiate, because mass and momentum are conserved. Einstein derived the formula in 1918, with a corrected factor relative to his 1916 work. The formula also holds for self-gravitating sources such as binary stars, as shown by post-Newtonian matching.

Example: two point masses m1,m2m_1, m_2 on a Newtonian circular orbit of separation aa radiate the power P=325G4m12m22(m1+m2)c5a5.P = \frac{32}{5} \frac{G^4 m_1^2 m_2^2 (m_1 + m_2)}{c^5 a^5}. The observed slow decay of the orbital period of the Hulse–Taylor binary pulsar agrees with this prediction.

Going deeper

For a student

It plays the role for gravity that the dipole formula plays for electromagnetism; there is no gravitational dipole radiation because momentum is conserved. With energy balance it gives the inspiral of binaries through gravitational radiation reaction. Higher post-Newtonian corrections to the formula are needed to model the signals of compact binaries seen by detectors.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Quadrupole_formula
About this page
TagF541Z4R4ZX
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Kindstatement, proposition
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MSC83C35, 83C25
StatusThis copy of the site has no build data. Run m2 status F541Z4R4ZX.
AboutGravitational wave
UsesGravitational wave, Linearized gravity
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Changed3 November 2026

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