The Quadrupole Formula

The previous lesson solved the vacuum wave equation and read off the two polarizations. Turning on the source, $□ \overset{ˉ}{h}_{μν} = - 16 π G c^{- 4} T_{μν}$ , gives the field a slowly moving mass distribution radiates. The result is that the leading radiation comes from the second time derivative of the mass quadrupole moment, the monopole and dipole being killed by conservation laws. This lesson derives the quadrupole formula for the strain and the luminosity, applies it to a compact binary to produce the inspiral chirp, and compares it with the orbital decay of the Hulse-Taylor binary pulsar.

The retarded field of a source

The sourced wave equation is solved by the retarded integral, the same Green's function that governs the electromagnetic potential,

\overset{ˉ}{h}_{μν} (t, x) = \frac{4 G}{c ^{4}} \int \frac{T _{μν} ( t - ∣ x - x ^{'} ∣/ c , x ^{'} )}{∣ x - x ^{'} ∣} d^{3} x^{'} .

The field at an event is the sum of source contributions on the past light cone, each delayed by the light-travel time from $x^{'}$ to $x$ . Two approximations reduce this to a usable form.

Far zone: the observer is at distance $r$ much larger than the source size, so $∣ x - x^{'} ∣ \approx r$ in the denominator, which comes out of the integral.
Slow motion: the source moves slowly compared with $c$ , so its characteristic size is small compared with the wavelength, and the retardation can be evaluated at the single retarded time $t_{r} = t - r / c$ .

With both, the spatial components are

\overset{ˉ}{h}_{ij} (t, x) \approx \frac{4 G}{c ^{4} r} \int T_{ij} (t_{r}, x^{'}) d^{3} x^{'} .

The stress–energy identity

The integral of $T_{ij}$ is rewritten as a second time derivative of a mass integral using conservation of the source. Local conservation $\partial_{μ} T^{μν} = 0$ splits into $\partial_{t} T^{00} + \partial_{k} T^{k 0} = 0$ and $\partial_{t} T^{0 i} + \partial_{k} T^{k i} = 0$ . Multiplying the first by $x^{i} x^{j}$ and integrating twice by parts over a volume enclosing the source (so surface terms vanish) gives the tensor virial identity,

\int T^{ij} d^{3} x = \frac{1}{2} \frac{d ^{2}}{d t ^{2}} \int T^{00} x^{i} x^{j} d^{3} x .

In the slow-motion limit $T^{00} \approx ρ c^{2}$ is the rest-mass energy density. Defining the mass quadrupole moment

Q^{ij} (t) = \int ρ (t, x^{'}) x^{' i} x^{' j} d^{3} x^{'},

the field of the source is

\overset{ˉ}{h}_{ij} (t, x) = \frac{2 G}{c ^{4} r} \ddot{Q}^{ij} (t_{r}) .

The radiation is governed by the second time derivative of the mass quadrupole, evaluated at the retarded time.

Why there is no monopole or dipole radiation

Two lower multipoles are absent, each removed by a conservation law.

Monopole: the analogue of the electric monopole is the total mass $M = \int ρ d^{3} x$ . Mass–energy is conserved, $\dot{M} = 0$ , so it cannot radiate. In electromagnetism the total charge is likewise conserved and monopole radiation is likewise absent.
Mass dipole: the mass dipole is $d = \int ρ x d^{3} x = M x_{cm}$ . Its first derivative is the total momentum, $\dot{d} = P$ , and its second derivative $\ddot{d} = \dot{P} = 0$ by momentum conservation for an isolated system. The dipole term in the radiation therefore vanishes identically.
Magnetic-type dipole: the current analogue involves the total angular momentum $J$ , conserved for an isolated system, $\dot{J} = 0$ , so it too does not radiate.

Electromagnetic radiation is dominated by the electric dipole because positive and negative charges can separate and oscillate. Mass has one sign only, so the mass dipole is rigidly tied to the centre of mass and the first radiating multipole is the quadrupole. This is why gravitational waves are intrinsically weak: the leading term is one multipole order higher than in electromagnetism.

Conservation laws remove the two lowest multipoles: constant total mass kills the monopole, constant total momentum kills the mass dipole, and constant angular momentum kills the current dipole, leaving the quadrupole as the first radiating moment.

The luminosity

The energy carried away per unit time follows from the effective stress–energy of the wave, averaged over a wavelength. Using the traceless reduced quadrupole

I^{ij} = \int ρ (x^{i} x^{j} - \frac{1}{3} δ^{ij} r^{'2}) d^{3} x^{'} = Q^{ij} - \frac{1}{3} δ^{ij} Q^{k}_{k},

the total radiated power is the Einstein quadrupole formula,

L_{GW} = \frac{G}{5 c ^{5}} ⟨ I ..._{ij} I ...^{ij} ⟩,

where angle brackets denote an average over several periods. The prefactor $G / c^{5}$ carries the units: $c^{5} / G \approx 3.6 \times 1 0^{52} W$ is the natural luminosity scale of general relativity, so any realistic source with a dimensionless $I ...$ factor far below unity radiates a tiny fraction of it, yet a source approaching that scale would be a merger of compact objects moving near $c$ .

The compact binary

The cleanest application is two masses $m_{1}$ and $m_{2}$ in a circular orbit. Work in the centre-of-mass frame, where the problem reduces to a single body of reduced mass $μ = m_{1} m_{2} / M$ at separation $a$ , with $M = m_{1} + m_{2}$ . Place the orbit in the $x$ – $y$ plane with angular frequency $ω$ ,

x^{'} (t) = a cos ω t, y^{'} (t) = a sin ω t .

The quadrupole components, using double-angle identities, are

Q_{xx} = \frac{1}{2} μ a^{2} (1 + cos 2 ω t), Q_{y y} = \frac{1}{2} μ a^{2} (1 - cos 2 ω t), Q_{x y} = \frac{1}{2} μ a^{2} sin 2 ω t .

Every component oscillates at $2 ω$ : the gravitational-wave frequency is twice the orbital frequency, because the mass distribution returns to itself after half an orbit. Differentiating three times and inserting into the luminosity formula gives

L_{GW} = \frac{32}{5} \frac{G}{c ^{5}} μ^{2} a^{4} ω^{6} .

Eliminating $ω$ with Kepler's third law $ω^{2} = GM / a^{3}$ expresses the luminosity in terms of the masses and separation,

L_{GW} = \frac{32}{5} \frac{G ^{4}}{c ^{5}} \frac{m _{1}^{2} m _{2}^{2} ( m _{1} + m _{2} )}{a ^{5}} .

The steep $a^{- 5}$ dependence is decisive: as the orbit shrinks the luminosity climbs sharply, driving a runaway.

The radiated luminosity of a circular binary rises as the inverse fifth power of the separation and as the square of each mass, so emission is negligible at wide separation and diverges as the bodies approach contact.

The orbiting binary carries a rotating mass quadrupole; because the configuration repeats every half orbit, the emitted wave oscillates at twice the orbital frequency, and the luminosity rises as the inverse fifth power of the separation.

The inspiral chirp

The orbit is not closed: radiated energy comes from the orbital energy, which for a bound circular orbit is $E = - G m_{1} m_{2} / (2 a)$ . Energy balance $\dot{E} = - L_{GW}$ gives

\frac{G m _{1} m _{2}}{2 a ^{2}} \overset{a}{˙} = - \frac{32}{5} \frac{G ^{4}}{c ^{5}} \frac{m _{1}^{2} m _{2}^{2} M}{a ^{5}} ⟹ \overset{a}{˙} = - \frac{64}{5} \frac{G ^{3}}{c ^{5}} \frac{m _{1} m _{2} M}{a ^{3}} .

The separation shrinks, $ω$ rises, the luminosity rises, and the shrinkage accelerates: an inspiral. The gravitational-wave frequency $f = ω / π$ and amplitude both sweep upward toward merger, a chirp. Introducing the chirp mass

M = \frac{( m _{1} m _{2} ) ^{3/5}}{( m _{1} + m _{2} ) ^{1/5}} = μ^{3/5} M^{2/5},

the frequency evolution takes a form depending on the masses only through $M$ ,

\dot{f} = \frac{96}{5} π^{8/3} (\frac{G M}{c ^{3}})^{5/3} f^{11/3} .

Measuring $f (t)$ and $\dot{f} (t)$ from the waveform therefore delivers the chirp mass directly, the single best-determined parameter of an inspiral. The $f^{11/3}$ dependence makes the sweep slow at low frequency and explosively fast near merger, the audible-analogue rise that named the chirp.

As the binary loses energy the separation shrinks, the frequency and amplitude climb, and the waveform sweeps upward into the merger; the rate of frequency increase fixes the chirp mass.

The Hulse-Taylor binary pulsar

The first evidence came not from a detector but from timing a pulsar. PSR B1913+16 is a neutron star in an $7.75$ -hour eccentric orbit with a companion neutron star; the pulsar's radio pulses act as a precise clock, and their arrival times map the orbit. General relativity predicts the orbit loses energy to gravitational radiation, so the orbital period should shrink at a computed rate. The quadrupole formula, generalized to the eccentric orbit, predicts a period derivative

\dot{P}_{b} \approx - 2.40 \times 1 0^{- 12},

a decrease of about $76.5$ microseconds per year. The measured cumulative shift of the time of periastron follows a parabola that matches the general-relativity prediction to better than $0.2%$ over decades of timing, the first confirmation that gravitational waves carry energy exactly as the quadrupole formula says.

The cumulative shift in the time of periastron of the Hulse-Taylor pulsar accumulates as a parabola in time; the data points track the general relativity prediction from the quadrupole formula, confirming orbital energy loss to gravitational radiation.

The luminosity scaling makes the physical picture concrete: the power radiated grows as the fifth inverse power of separation and the square of the reduced mass, so the last seconds of a compact-object inspiral outshine, in gravitational-wave luminosity, the electromagnetic output of every star in the observable universe combined. Detecting that final burst directly is the subject of the next lesson.