quadrupole radiation formulaequation
The quadrupole radiation formula follows from Einstein's equations in the weak-field approximation. Gravity, like electromagnetism, can create waves, but with an important difference: an electric dipole (a pair of + and - charges) radiates dipole radiation, but in gravity, dipole radiation is forbidden because of the conservation of momentum. The first allowed term is quadrupole. It's like sound: a uniformly pulsating sphere is silent, but one that flattens and stretches makes noise. Einstein realized this in 1916, but only in the 1970s was the formula refined for astrophysical calculations.
How it works
This law explains why a single spherical star does not emit gravitational waves—its shape remains unchanged. Binary pulsars, on the contrary, constantly lose energy, and their orbits shrink. This is how astronomers have confirmed the existence of gravitational waves for decades, even before direct detection by LIGO.
💡 The famous binary pulsar PSR B1913+16, discovered by Hulse and Taylor, decays exactly according to the quadrupole formula, which earned them the Nobel Prize.
L = \frac{G}{5c^5} \left\langle \frac{d^3 Q_{ij}}{dt^3} \frac{d^3 Q_{ij}}{dt^3} \right\rangle
L — power of gravitational waves carried away by the system; G — gravitational constant, G ≈ 6.674×10⁻¹¹ m³/(kg·s²); c — speed of light in vacuum, c ≈ 2.998×10⁸ m/s; Q_{ij} — reduced quadrupole moment of the system (tensor); d^3/dt^3 — third derivative with respect to time; ⟨...⟩ — averaging over several periods. The formula shows that radiation is generated only by the change in the quadrupole moment with acceleration.