Sources and the Quadrupole Formula
How accelerating masses generate gravitational radiation, and the remarkable confirmation from binary pulsars
Having established that the linearized Einstein equations admit wave solutions, we now turn to the question of how gravitational waves are generated. Unlike electromagnetic radiation, which can be produced by oscillating dipoles, gravitational radiation requires at least a time-varying quadrupole moment. This fundamental difference arises because there is no gravitational dipole radiation -- conservation of momentum forbids it.
Why No Gravitational Dipole Radiation?
In electromagnetism, the leading-order radiation comes from the oscillating electric dipole moment. For gravity, the mass monopole moment is simply the total mass-energy, which is conserved. The mass dipole moment is \(\sum m_a \mathbf{x}_a\), whose time derivative is the total momentum -- also conserved. Therefore:
$$\dot{M} = 0 \quad \text{(mass conservation } \Rightarrow \text{ no monopole radiation)}$$
$$\ddot{D}^i = \dot{P}^i = 0 \quad \text{(momentum conservation } \Rightarrow \text{ no dipole radiation)}$$
The lowest-order gravitational radiation is quadrupolar
This is why gravitational waves are so much weaker than electromagnetic waves: the leading radiation mechanism is suppressed by an additional power of \(v/c\) compared to dipole radiation.
The Mass Quadrupole Moment Tensor
The source of gravitational radiation is characterized by the second mass moment (quadrupole moment tensor):
$$I_{ij} = \int \rho(\mathbf{x}, t)\, x^i x^j \, d^3x$$
The second moment of the mass distribution -- analogous to the moment of inertia tensor
Often it is more convenient to work with the trace-free quadrupole moment:
$$Q_{ij} = I_{ij} - \frac{1}{3}\delta_{ij} I_{kk} = \int \rho\left(x^i x^j - \frac{1}{3}\delta_{ij}|\mathbf{x}|^2\right)d^3x$$
The trace-free part is what matters because the trace contributes only to the non-radiative parts of the field (it is pure gauge in TT gauge).
The Quadrupole Radiation Formula
The gravitational wave field at a distance r from the source, in the far-field (radiation) zone where \(r \gg \lambda_{\text{GW}}\), is given by the celebrated quadrupole formula:
$$h_{ij}^{TT}(t, \mathbf{x}) = \frac{2G}{c^4 r}\,\ddot{Q}_{ij}^{TT}\!\left(t - \frac{r}{c}\right)$$
The TT projection of the second time derivative of the quadrupole moment, evaluated at the retarded time
The derivation proceeds by solving the sourced wave equation\(\Box\bar{h}_{\mu\nu} = -16\pi G T_{\mu\nu}/c^4\) using the retarded Green's function, then expanding in powers of \(v/c\) (the slow-motion approximation). The leading term in this multipole expansion is the quadrupole contribution above.
The superscript TT indicates that we must project the result into the transverse-traceless gauge. For a wave propagating in the direction \(\hat{n}\), the TT projection operator is:
$$\Lambda_{ij,kl}(\hat{n}) = P_{ik}P_{jl} - \frac{1}{2}P_{ij}P_{kl}, \quad P_{ij} = \delta_{ij} - n_i n_j$$
\(P_{ij}\) projects onto the plane transverse to the propagation direction
Gravitational Wave Luminosity
The total power (luminosity) radiated in gravitational waves is obtained by integrating the energy flux over a sphere surrounding the source. The result is the Einstein quadrupole formula for radiated power:
$$P_{\text{GW}} = \frac{G}{5c^5}\left\langle\dddot{Q}_{ij}\,\dddot{Q}^{ij}\right\rangle$$
The angle brackets denote time-averaging over several wave periods
The prefactor \(G/c^5 \approx 2.6 \times 10^{-53}\) W\(^{-1}\) s\(^5\) m\(^{-2}\) kg\(^{-2}\) is extraordinarily small, which is why only the most violent astrophysical events produce detectable gravitational waves. The natural luminosity scale is:
$$L_0 = \frac{c^5}{G} \approx 3.6 \times 10^{52} \text{ W}$$
The "Planck luminosity" -- the maximum luminosity any source can radiate
Worked Example: Circular Binary System
Consider two masses \(m_1\) and \(m_2\) in a circular orbit of radius R and angular frequency \(\omega\). This is the most important source for gravitational wave astronomy. Place the orbit in the x-y plane with the center of mass at the origin.
The positions of the two masses are:
$$x_1(t) = \frac{m_2}{M}R\cos(\omega t), \quad y_1(t) = \frac{m_2}{M}R\sin(\omega t)$$
$$x_2(t) = -\frac{m_1}{M}R\cos(\omega t), \quad y_2(t) = -\frac{m_1}{M}R\sin(\omega t)$$
where \(M = m_1 + m_2\) is the total mass
Computing the quadrupole moment tensor and its time derivatives, we find:
$$Q_{xx} = \mu R^2\cos(2\omega t), \quad Q_{yy} = -\mu R^2\cos(2\omega t)$$
$$Q_{xy} = Q_{yx} = \mu R^2\sin(2\omega t)$$
where \(\mu = m_1 m_2 / M\) is the reduced mass. Note the frequency doubling: GW frequency = 2 × orbital frequency
The gravitational wave strain for an observer along the z-axis (face-on binary) is:
$$h_+ = \frac{4G\mu\omega^2 R^2}{c^4 r}\cos(2\omega t_{\text{ret}})$$
$$h_\times = \frac{4G\mu\omega^2 R^2}{c^4 r}\sin(2\omega t_{\text{ret}})$$
Circularly polarized waves at twice the orbital frequency
Using Kepler's third law \(\omega^2 R^3 = GM\) to eliminate R, we can express the strain in terms of the chirp mass \(\mathcal{M} = \mu^{3/5}M^{2/5}\):
$$h \sim \frac{4}{r}\left(\frac{G\mathcal{M}}{c^2}\right)^{5/3}\left(\frac{\pi f_{\text{GW}}}{c}\right)^{2/3}$$
The chirp mass is the single combination of masses that determines the inspiral waveform at leading order
The radiated power for the circular binary is:
$$P_{\text{GW}} = \frac{32G^4}{5c^5}\frac{m_1^2 m_2^2(m_1 + m_2)}{R^5}$$
Extremely strong R-dependence: the power grows dramatically as the orbit shrinks
Peters Formula: Orbital Decay
As the binary radiates gravitational waves, it loses energy, causing the orbit to shrink and the frequency to increase. This is the characteristic "chirp" signal. Peters and Mathews (1963) derived the rate of orbital decay for circular orbits:
$$\frac{dR}{dt} = -\frac{64G^3}{5c^5}\frac{m_1 m_2(m_1 + m_2)}{R^3}$$
The orbit shrinks faster as R decreases, leading to a runaway inspiral
Integrating, the time remaining until coalescence from an initial separation \(R_0\) is:
$$t_{\text{merge}} = \frac{5c^5}{256G^3}\frac{R_0^4}{m_1 m_2(m_1 + m_2)}$$
The frequency evolution (the "chirp") follows:
$$\dot{f}_{\text{GW}} = \frac{96}{5}\pi^{8/3}\left(\frac{G\mathcal{M}}{c^3}\right)^{5/3}f_{\text{GW}}^{11/3}$$
Measuring \(f\) and \(\dot{f}\) directly determines the chirp mass
The Hulse-Taylor Binary Pulsar: Indirect Detection
The binary pulsar PSR B1913+16, discovered by Russell Hulse and Joseph Taylor in 1974, provided the first indirect evidence for gravitational waves. This system consists of two neutron stars in a tight 7.75-hour orbit, one of which is a radio pulsar that serves as an exquisite clock.
System Parameters
- Pulsar mass: \(m_1 \approx 1.4408\,M_\odot\)
- Companion mass: \(m_2 \approx 1.3886\,M_\odot\)
- Orbital period: \(P_b \approx 7.752\) hours
- Eccentricity: \(e \approx 0.617\)
- Semi-major axis: \(a \approx 1.95 \times 10^9\) m
Observed Period Decay
- Predicted: \(\dot{P}_b = -2.40263 \times 10^{-12}\) s/s
- Observed: \(\dot{P}_b = -2.4056 \times 10^{-12}\) s/s
- Agreement: better than 0.2%
- Cumulative shift: ~40 s over 30+ years
- This earned Hulse & Taylor the 1993 Nobel Prize
$$\dot{P}_b = -\frac{192\pi}{5}\left(\frac{2\pi G\mathcal{M}}{c^3 P_b}\right)^{5/3}\frac{1 + \frac{73}{24}e^2 + \frac{37}{96}e^4}{(1-e^2)^{7/2}}$$
Peters formula generalized to eccentric orbits -- predicts the orbital decay with exquisite precision
The extraordinary agreement between the observed orbital decay and the prediction from gravitational wave emission was the first strong evidence that gravitational waves carry energy exactly as general relativity predicts. Over decades of observation, the cumulative shift in the time of periastron passage has followed the theoretical curve with remarkable precision.
Physical Insight
The Hulse-Taylor binary loses energy at a rate of about \(7.35 \times 10^{24}\) watts -- roughly 2% of the Sun's luminosity. Yet this enormous power output changes the orbital period by only about 76 microseconds per year. The system will merge in approximately 300 million years.
Other Binary Pulsar Systems
Since the discovery of PSR B1913+16, several other binary pulsar systems have been found that confirm gravitational wave predictions:
The Double Pulsar (J0737-3039)
Discovered in 2003, this is the only known system where both neutron stars are observed as pulsars. It has an orbital period of only 2.4 hours and provides the most stringent tests of general relativity in the strong-field regime. The orbital decay agrees with GR to better than 0.05%.
PSR J1738+0333
A millisecond pulsar with a white dwarf companion. This system provides tight constraints on alternative theories of gravity that predict dipole gravitational radiation. The absence of excess orbital decay confirms the quadrupole-only nature of gravitational radiation as predicted by GR.