Treatise of Physics

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Gravitational-Wave Theory

Contents
  1. Linearized gravity and the wave equation
  2. Transverse-traceless gauge and the two polarizations
  3. The quadrupole formula
  4. Beyond linear order

The Einstein field equations are nonlinear, but far from strong sources spacetime is very nearly flat, and the equations linearize into a wave equation. The resulting radiative solutions — gravitational waves — are the subject of this short chapter. We state the results and their physical content; the complete derivation, from the linearized field equations through the quadrupole formula, is carried out in The Quadrupole Formula for Gravitational Radiation. The experimental record, indirect and direct, occupies Experiment: Gravitational Waves.

Linearized gravity and the wave equation

Consider a metric that deviates only slightly from the Minkowski metric,

\begin{equation}\tag{46.1} g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}\ec \qquad \abs{h_{\mu\nu}} \ll 1\ep \end{equation}

Keeping only terms of first order in \(h_{\mu\nu}\), and introducing the trace-reversed perturbation \(\bar h_{\mu\nu} = h_{\mu\nu} - \tfrac{1}{2}\eta_{\mu\nu}\,h\) with \(h = \eta^{\alpha\beta}h_{\alpha\beta}\), one may always choose coordinates satisfying the Lorenz gauge condition \(\pp^{\mu}\bar h_{\mu\nu} = 0\). In this gauge the linearized Einstein equations collapse to an inhomogeneous wave equation [Einstein:1916] [Misner:1973] [Wald:1984],

\begin{equation}\tag{46.2} \Box\,\bar h_{\mu\nu} = -\frac{16\pi G}{c^{4}}\,T_{\mu\nu}\ec \end{equation}

where \(\Box = \eta^{\alpha\beta}\pp_{\alpha}\pp_{\beta}\) is the flat-space d'Alembertian. Every statement in this equation is physical: disturbances of the metric propagate at the speed of light, they are sourced by the stress–energy tensor, and the coupling \(G/c^{4} \approx 8.3\times 10^{-45}\,\mathrm{s}^{2}/\mathrm{kg}/\mathrm{m}\) is so small that only astrophysical masses moving at relativistic speeds produce perturbations of detectable size. We do not derive Equation (46.2) here; the derivation is the opening step of The Quadrupole Formula for Gravitational Radiation.

Transverse-traceless gauge and the two polarizations

In vacuum (\(T_{\mu\nu}=0\)) the residual gauge freedom within the Lorenz class can be used to set \(h_{0\mu} = 0\), \(h^{i}{}_{i} = 0\) and \(\pp^{i}h_{ij} = 0\): the transverse-traceless (TT) gauge. For a plane wave travelling along the \(z\)-axis only two independent components survive, the polarization amplitudes \(h_{+}\) and \(h_{\times}\) [Misner:1973] [Maggiore:2008]:

\begin{equation}\tag{46.3} \dd s^{2} = -c^{2}\dd t^{2} + \bigl(1 + h_{+}\bigr)\dd x^{2} + \bigl(1 - h_{+}\bigr)\dd y^{2} + 2 h_{\times}\,\dd x\,\dd y + \dd z^{2}\ec \end{equation}

with \(h_{+,\times} = h_{+,\times}(t - z/c)\). A gravitational wave therefore carries exactly two physical polarizations, as befits a massless spin-2 field.

The effect on matter is tidal. A circular ring of free test masses in the \(xy\)-plane is deformed by the \(+\) polarization into an ellipse stretched along \(x\) while squeezed along \(y\), and half a period later the reverse; the \(\times\) polarization produces the same pattern rotated by \(45^\circ\). For two free masses separated by a distance \(L\) along a principal axis of the wave, the fractional change in proper separation is

\begin{equation}\tag{46.4} \frac{\Delta L}{L} = \frac{h}{2}\ec \end{equation}

where \(h\) is the relevant polarization amplitude. This is precisely the quantity a laser interferometer measures: a differential arm-length oscillation at the wave frequency, of fractional size \(h/2\) per arm.

The quadrupole formula

Phenomenon 46.1.

Accelerated masses whose quadrupole moment varies in time radiate gravitational waves. The waves propagate at the speed of light \(c\) and carry energy away from the source; at leading order the radiated strain and luminosity are given by the quadrupole formula. This radiation has been observed indirectly, through the orbital decay of binary pulsars [Taylor:1982], and directly, by the LIGO interferometers in 2015 [Abbott:2016], exactly as predicted by the linearized theory [Einstein:1916] [Einstein:1918]. Rests on Postulate 44.3.

Full derivation in Appendix A.

Derives Phenomenon 46.1.

We state the results proved there. Let \(Q_{ij} = \int \rho\,\bigl(x_{i}x_{j} - \tfrac{1}{3}\delta_{ij}r^{2}\bigr)\dd^{3}x\) denote the traceless mass quadrupole moment of the source. A distant observer at distance \(d\) measures the strain

\begin{equation}\tag{46.5} \bar h_{ij}(t) = \frac{2G}{c^{4} d}\,\ddot{Q}_{ij}\!\left(t - \frac{d}{c}\right)\ec \end{equation}

and the total power radiated by the source, averaged over several wave periods, is

\begin{equation}\tag{46.6} P = \frac{G}{5 c^{5}}\avg{\dddot{Q}_{ij}\,\dddot{Q}^{ij}}\ep \end{equation}

The prefactor \(G/c^{5} \approx 2.8\times 10^{-53}\,\mathrm{W}^{-1}\) explains at once why gravitational radiation is negligible in the laboratory and why only compact binaries reach detectable luminosities [Maggiore:2008].

For a binary of masses \(m_{1}\) and \(m_{2}\) on a circular orbit, the gravitational-wave frequency is twice the orbital frequency, and the entire leading-order evolution is controlled by a single mass combination, the chirp mass

\begin{equation}\tag{46.7} \mathcal{M} = \frac{(m_{1} m_{2})^{3/5}}{(m_{1} + m_{2})^{1/5}}\ep \end{equation}

Energy loss through Equation (46.6) shrinks the orbit and drives the frequency upward according to

\begin{equation}\tag{46.8} \dv{f_{\mathrm{gw}}}{t} = \frac{96}{5}\,\pi^{8/3} \left(\frac{G\mathcal{M}}{c^{3}}\right)^{5/3} f_{\mathrm{gw}}^{11/3}\ec \end{equation}

so that a measurement of the frequency and its rate of change during the inspiral yields \(\mathcal{M}\) directly, with no astrophysical modelling. This accelerating sweep — the chirp — is the unmistakable signature by which compact binaries announce themselves in the data of Experiment: Gravitational Waves [Abbott:2016] [Maggiore:2008].

Beyond linear order

The linearized treatment above suffices for the propagation of the waves and for the leading-order emission from slowly moving sources. It does not by itself define the energy carried by the waves — a second-order quantity in \(h_{\mu\nu}\) — nor does it describe the late inspiral and merger of black holes, where the full nonlinearity of the field equations takes over and only post-Newtonian expansions and numerical relativity give quantitative waveforms [Misner:1973] [Wald:1984] [Maggiore:2008].

The first of these gaps is closed in this book: the second-order (Isaacson) analysis — expanding the Ricci tensor to \(O(h^{2})\), averaging over several wavelengths, and reading the averaged quadratic terms as an effective stress–energy tensor of the radiation — is carried out in Energy carried by the wave: the Isaacson tensor, culminating in Equation (A.46); it is that tensor whose flux, integrated over a sphere, yields the luminosity Equation (46.6) quoted above. The second gap is imported knowledge, and honestly so: the post-Newtonian expansion of the two-body problem has been driven to the orders required by detector phase templates [Blanchet:2014], and the merger itself yielded only to numerical solution of the full field equations [Pretorius:2005]. This book states those results where the experiments need them (Experiment: Gravitational Waves) and does not rederive them; each is a controlled approximation to, or a numerical solution of, the same field equations derived in The Einstein Field Equations, so nothing in the evidential chain rests on machinery outside the theory already stated.

The history of the subject mirrors its subtlety. Einstein's founding 1916 paper on gravitational waves [Einstein:1916] contained a celebrated error of a numerical factor in the radiation formula, which he himself corrected in the 1918 paper that first stated the quadrupole formula in its modern form [Einstein:1918]; decades of confusion about whether the waves were physical or mere coordinate ripples followed, and were settled only in the late 1950s with gauge-invariant treatments of the energy flux [Maggiore:2008] [Misner:1973]. That the formula so laboriously secured now agrees with binary-pulsar timing to a fraction of a percent, and with direct interferometry, is the subject of the next chapter: Experiment: Gravitational Waves.