Treatise of Physics

Contents › Appendices › 01-quadrupole

The Quadrupole Formula for Gravitational Radiation

Contents
  1. Setup: linearized geometry
  2. Gauge freedom and the Lorenz gauge
  3. Vacuum solutions: plane waves and the transverse–traceless gauge
  4. Physical effect: geodesic deviation and the strain
  5. Energy carried by the wave: the Isaacson tensor
  6. Generation: the quadrupole formula
  7. Application: the circular compact binary

This appendix proves Phenomenon 50.1 (Chapter 50): the field equations of general relativity, linearized about flat spacetime, admit transverse wave solutions propagating at speed \(c\) with two polarizations; these waves carry energy; and a slowly moving, self-gravitating source radiates them with a power fixed by the third time derivative of its mass quadrupole moment. The chain of results proved below — wave equation, transverse–traceless gauge, geodesic-deviation observable, Isaacson energy flux, quadrupole generation formula, and the binary-inspiral application — is exactly the chain that connects Einstein's field equations to the two great experimental confirmations: the orbital decay of the binary pulsar PSR B1913+16 (Section 51.1) and the direct detection of GW150914 by the LIGO interferometers (Section 51.3).

The history deserves one sentence. Einstein obtained the linearized wave solutions in 1916 [Einstein:1916], and in 1918 published the quadrupole radiation formula in its correct form, repairing a computational slip of the earlier paper [Einstein:1918]; the modern treatments we follow — and against which every step below can be checked — are those of Misner, Thorne and Wheeler [Misner:1973], Wald [Wald:1984], and Maggiore [Maggiore:2008].

Throughout this section the metric signature is \((-,+,+,+)\), coordinates are \(x^\mu = (x^0, x^i) = (ct, \vect{x})\), Greek indices run over \(0,1,2,3\), Latin indices over the spatial values \(1,2,3\), and the flat d'Alembertian is

\begin{equation} \Box \equiv \eta^{\mu\nu}\pp_\mu\pp_\nu = -\frac{1}{c^2}\frac{\pp^2}{\pp t^2} + \nabla^2\ep \tag{A1.1} \end{equation}

Setup: linearized geometry

We consider spacetimes that deviate weakly from Minkowski space, so that global coordinates exist in which

\begin{equation} g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}\ec \qquad \abs{h_{\mu\nu}} \ll 1\ec \tag{A1.2} \end{equation}

with \(\eta_{\mu\nu} = \diag(-1,+1,+1,+1)\). All indices are raised and lowered with \(\eta\), an operation that is consistent to the linear order in \(h\) at which we work. The inverse metric is

\begin{equation} g^{\mu\nu} = \eta^{\mu\nu} - h^{\mu\nu} + O(h^2)\ec \tag{A1.3} \end{equation}

as one checks by direct multiplication: \((\eta_{\mu\rho} + h_{\mu\rho})(\eta^{\rho\nu} - h^{\rho\nu}) = \delta_\mu^{\ \nu} + h_\mu^{\ \nu} - h_\mu^{\ \nu} + O(h^2) = \delta_\mu^{\ \nu} + O(h^2)\).

Christoffel symbols.

Inserting Equations (A1.2) and (A1.3) into the definition of the Levi-Civita connection and discarding \(O(h^2)\),

\begin{align} \Gamma^{\lambda}{}_{\mu\nu} &= \frac{1}{2} g^{\lambda\rho} \left(\pp_\mu g_{\rho\nu} + \pp_\nu g_{\rho\mu} - \pp_\rho g_{\mu\nu}\right) \nonumber\\ &= \frac{1}{2}\left(\eta^{\lambda\rho} - h^{\lambda\rho}\right) \left(\pp_\mu h_{\rho\nu} + \pp_\nu h_{\rho\mu} - \pp_\rho h_{\mu\nu}\right) \nonumber\\ &= \frac{1}{2}\eta^{\lambda\rho} \left(\pp_\mu h_{\rho\nu} + \pp_\nu h_{\rho\mu} - \pp_\rho h_{\mu\nu}\right) + O(h^2)\ep \tag{A1.4} \end{align}

The connection is thus itself of first order in \(h\).

Riemann tensor.

In the curvature tensor \(R^{\rho}{}_{\sigma\mu\nu} = \pp_\mu\Gamma^{\rho}{}_{\nu\sigma} - \pp_\nu\Gamma^{\rho}{}_{\mu\sigma} + \Gamma^{\rho}{}_{\mu\lambda}\Gamma^{\lambda}{}_{\nu\sigma} - \Gamma^{\rho}{}_{\nu\lambda}\Gamma^{\lambda}{}_{\mu\sigma}\) the \(\Gamma\Gamma\) terms are \(O(h^2)\) and drop. Substituting Equation (A1.4),

\begin{align} R^{\rho}{}_{\sigma\mu\nu} &= \pp_\mu\Gamma^{\rho}{}_{\nu\sigma} - \pp_\nu\Gamma^{\rho}{}_{\mu\sigma} + O(h^2)\nonumber\\ &= \frac{1}{2}\eta^{\rho\lambda}\left( \pp_\mu\pp_\nu h_{\lambda\sigma} + \pp_\mu\pp_\sigma h_{\lambda\nu} - \pp_\mu\pp_\lambda h_{\nu\sigma} - \pp_\nu\pp_\mu h_{\lambda\sigma} - \pp_\nu\pp_\sigma h_{\lambda\mu} + \pp_\nu\pp_\lambda h_{\mu\sigma}\right)\nonumber\\ &= \frac{1}{2}\eta^{\rho\lambda}\left( \pp_\mu\pp_\sigma h_{\lambda\nu} + \pp_\nu\pp_\lambda h_{\mu\sigma} - \pp_\mu\pp_\lambda h_{\nu\sigma} - \pp_\nu\pp_\sigma h_{\lambda\mu} \right)\ec \tag{A1.5} \end{align}

where the two \(\pp_\mu\pp_\nu h_{\lambda\sigma}\) terms cancelled. Lowering the first index gives the fully covariant linearized Riemann tensor,

\begin{equation} R_{\rho\sigma\mu\nu} = \frac{1}{2}\left( \pp_\mu\pp_\sigma h_{\rho\nu} + \pp_\nu\pp_\rho h_{\mu\sigma} - \pp_\mu\pp_\rho h_{\nu\sigma} - \pp_\nu\pp_\sigma h_{\rho\mu} \right)\ep \tag{A1.6} \end{equation}

Ricci tensor, Ricci scalar, Einstein tensor.

Contracting the first and third indices of Equation (A1.5) and writing \(h \equiv \eta^{\mu\nu}h_{\mu\nu}\) for the trace,

\begin{equation} R_{\mu\nu} = R^{\rho}{}_{\mu\rho\nu} = \frac{1}{2}\left( \pp_\rho\pp_\mu h^{\rho}{}_{\nu} + \pp_\rho\pp_\nu h^{\rho}{}_{\mu} - \Box h_{\mu\nu} - \pp_\mu\pp_\nu h \right)\ec \tag{A1.7} \end{equation}

and contracting once more,

\begin{equation} R = \eta^{\mu\nu}R_{\mu\nu} = \pp_\mu\pp_\nu h^{\mu\nu} - \Box h\ep \tag{A1.8} \end{equation}

The linearized Einstein tensor is therefore

\begin{equation} G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}\eta_{\mu\nu}R = \frac{1}{2}\left( \pp_\rho\pp_\mu h^{\rho}{}_{\nu} + \pp_\rho\pp_\nu h^{\rho}{}_{\mu} - \Box h_{\mu\nu} - \pp_\mu\pp_\nu h - \eta_{\mu\nu}\pp_\rho\pp_\sigma h^{\rho\sigma} + \eta_{\mu\nu}\Box h \right)\ep \tag{A1.9} \end{equation}

Trace reversal.

The expression Equation (A1.9) simplifies decisively in terms of the trace-reversed perturbation

\begin{equation} \bar h_{\mu\nu} \equiv h_{\mu\nu} - \frac{1}{2}\eta_{\mu\nu}h\ec \qquad \bar h = \eta^{\mu\nu}\bar h_{\mu\nu} = h - 2h = -h\ec \qquad h_{\mu\nu} = \bar h_{\mu\nu} - \frac{1}{2}\eta_{\mu\nu}\bar h\ep \tag{A1.10} \end{equation}

(The map is an involution: trace-reversing twice returns \(h_{\mu\nu}\).) Substituting the third relation of Equation (A1.10) into Equation (A1.7) term by term,

\begin{align} \pp_\rho\pp_\mu h^{\rho}{}_{\nu} &= \pp_\rho\pp_\mu \bar h^{\rho}{}_{\nu} - \frac{1}{2}\pp_\mu\pp_\nu \bar h\ec & \Box h_{\mu\nu} &= \Box\bar h_{\mu\nu} - \frac{1}{2}\eta_{\mu\nu}\Box\bar h\ec & \pp_\mu\pp_\nu h &= -\pp_\mu\pp_\nu\bar h\ec \end{align}

so that

\begin{align} R_{\mu\nu} &= \frac{1}{2}\left( \pp_\rho\pp_\mu\bar h^{\rho}{}_{\nu} + \pp_\rho\pp_\nu\bar h^{\rho}{}_{\mu} - \pp_\mu\pp_\nu\bar h - \Box\bar h_{\mu\nu} + \frac{1}{2}\eta_{\mu\nu}\Box\bar h + \pp_\mu\pp_\nu\bar h \right)\nonumber\\ &= \frac{1}{2}\left( \pp_\rho\pp_\mu\bar h^{\rho}{}_{\nu} + \pp_\rho\pp_\nu\bar h^{\rho}{}_{\mu} - \Box\bar h_{\mu\nu} + \frac{1}{2}\eta_{\mu\nu}\Box\bar h \right)\ec \qquad R = \pp_\rho\pp_\sigma\bar h^{\rho\sigma} + \frac{1}{2}\Box\bar h\ec \end{align}

and the Einstein tensor assembles into

\begin{equation} G_{\mu\nu} = -\frac{1}{2}\left( \Box\bar h_{\mu\nu} + \eta_{\mu\nu}\pp^\rho\pp^\sigma\bar h_{\rho\sigma} - \pp^\rho\pp_\mu\bar h_{\rho\nu} - \pp^\rho\pp_\nu\bar h_{\rho\mu} \right) + O(h^2)\ec \tag{A1.13} \end{equation}

the two \(\frac14\eta_{\mu\nu}\Box\bar h\) contributions having cancelled between \(R_{\mu\nu}\) and \(-\frac12\eta_{\mu\nu}R\). Every term now contains \(\bar h_{\mu\nu}\) either under a d'Alembertian or under a divergence \(\pp^\rho\bar h_{\rho\nu}\) — which is the invitation to fix a gauge.

Gauge freedom and the Lorenz gauge

The split Equation (A1.2) is not unique: an infinitesimal coordinate change

\begin{equation} x'^{\mu} = x^\mu + \xi^\mu(x)\ec \qquad \abs{\pp_\mu\xi^\nu} \ll 1\ec \tag{A1.14} \end{equation}

preserves the form \(g = \eta + h\) with a different, equally small \(h\). Indeed, from the tensor transformation law and \(\pp x^\rho/\pp x'^\mu = \delta^\rho_{\ \mu} - \pp_\mu\xi^\rho + O(\xi^2)\),

\begin{align} g'_{\mu\nu} = \frac{\pp x^\rho}{\pp x'^\mu}\frac{\pp x^\sigma}{\pp x'^\nu} g_{\rho\sigma} = \eta_{\mu\nu} + h_{\mu\nu} - \pp_\mu\xi_\nu - \pp_\nu\xi_\mu + O(h\xi,\xi^2)\ec \end{align}

so the perturbation transforms as

\begin{equation} h'_{\mu\nu} = h_{\mu\nu} - \pp_\mu\xi_\nu - \pp_\nu\xi_\mu\ep \tag{A1.16} \end{equation}

This is the gauge freedom of linearized gravity, the precise analogue of \(A_\mu \to A_\mu - \pp_\mu\chi\) in electrodynamics [Jackson:1999]. The linearized Riemann tensor Equation (A1.6) is gauge invariant: substituting Equation (A1.16),

\begin{align} \delta R_{\rho\sigma\mu\nu} = -\frac{1}{2}\big( &\pp_\mu\pp_\sigma\pp_\rho\xi_\nu + \pp_\mu\pp_\sigma\pp_\nu\xi_\rho + \pp_\nu\pp_\rho\pp_\mu\xi_\sigma + \pp_\nu\pp_\rho\pp_\sigma\xi_\mu \nonumber\\ &- \pp_\mu\pp_\rho\pp_\nu\xi_\sigma - \pp_\mu\pp_\rho\pp_\sigma\xi_\nu - \pp_\nu\pp_\sigma\pp_\rho\xi_\mu - \pp_\nu\pp_\sigma\pp_\mu\xi_\rho \big) = 0\ec \end{align}

because partial derivatives commute and the eight triple-derivative terms cancel in pairs. Curvature — hence everything observable — is unaffected by the choice of \(\xi^\mu\); we may therefore spend that choice on simplifying the field equations.

Taking the trace of Equation (A1.16) gives \(h' = h - 2\pp_\rho\xi^\rho\), so the trace-reversed perturbation transforms as

\begin{equation} \bar h'_{\mu\nu} = \bar h_{\mu\nu} - \pp_\mu\xi_\nu - \pp_\nu\xi_\mu + \eta_{\mu\nu}\pp_\rho\xi^\rho\ec \tag{A1.18} \end{equation}

and its divergence as

\begin{equation} \pp^\mu\bar h'_{\mu\nu} = \pp^\mu\bar h_{\mu\nu} - \Box\xi_\nu - \pp_\nu\pp^\mu\xi_\mu + \pp_\nu\pp_\rho\xi^\rho = \pp^\mu\bar h_{\mu\nu} - \Box\xi_\nu\ep \tag{A1.19} \end{equation}

Given any \(\bar h_{\mu\nu}\), choose \(\xi_\nu\) to solve the inhomogeneous wave equation \(\Box\xi_\nu = \pp^\mu\bar h_{\mu\nu}\) — a solution always exists, for instance by the retarded Green function of \(\Box\) constructed in Generation: the quadrupole formula below. In the new coordinates the Lorenz gauge (also called harmonic or de Donder gauge) holds:

\begin{equation} \pp^\mu\bar h_{\mu\nu} = 0\ep \tag{A1.20} \end{equation}

In this gauge the last three terms of Equation (A1.13) vanish, and the full Einstein equations \(G_{\mu\nu} = (8\pi G/c^4)\,T_{\mu\nu}\), linearized, collapse to a flat-space wave equation with source:

\begin{equation} \boxed{\;\Box\bar h_{\mu\nu} = -\frac{16\pi G}{c^4}\,T_{\mu\nu}\;} \tag{A1.21} \end{equation}

Taking \(\pp^\mu\) of both sides and using Equation (A1.20) shows \(\pp^\mu T_{\mu\nu} = 0\): at linear order the source moves on the flat background, its energy–momentum conserved in the special-relativistic sense. This is the self-consistency of the linear approximation [Misner:1973] [Wald:1984].

Vacuum solutions: plane waves and the transverse–traceless gauge

In vacuum (\(T_{\mu\nu} = 0\)) equation Equation (A1.21) reads \(\Box\bar h_{\mu\nu} = 0\): each component obeys the massless wave equation, so disturbances of the metric propagate at the speed \(c\) — the first assertion of Phenomenon 50.1. Insert the plane-wave ansatz

\begin{equation} \bar h_{\mu\nu} = A_{\mu\nu}\,\ee^{\ii k_\alpha x^\alpha}\ec \qquad A_{\mu\nu} = A_{\nu\mu} \text{ constant}\ec \tag{A1.22} \end{equation}

(real part understood). Then \(\Box\bar h_{\mu\nu} = -k_\alpha k^\alpha\, \bar h_{\mu\nu} = 0\) forces the wave vector to be null,

\begin{equation} k_\alpha k^\alpha = 0 \qquad\Longleftrightarrow\qquad \omega = c\,\abs{\vect{k}}\ec \quad k^\mu = (\omega/c, \vect{k})\ec \tag{A1.23} \end{equation}

and the Lorenz condition Equation (A1.20) becomes the transversality constraint

\begin{equation} k^\mu A_{\mu\nu} = 0\ep \tag{A1.24} \end{equation}

Counting the physical degrees of freedom.

A symmetric \(A_{\mu\nu}\) has \(10\) independent components; Equation (A1.24) removes \(4\). But the gauge is not yet exhausted: any further \(\xi^\mu\) with \(\Box\xi^\mu = 0\) preserves Equation (A1.20), by Equation (A1.19). Take \(\xi^\mu = B^\mu\ee^{\ii k_\alpha x^\alpha}\) with the same null \(k\); then Equation (A1.18) shifts the amplitude by

\begin{equation} A'_{\mu\nu} = A_{\mu\nu} - \ii\left(k_\mu B_\nu + k_\nu B_\mu - \eta_{\mu\nu}k_\rho B^\rho\right)\ep \tag{A1.25} \end{equation}

The four free constants \(B_\mu\) remove four more components. Explicitly, let the wave run along \(z\), \(k^\mu = (k,0,0,k)\), \(k_\mu = (-k,0,0,k)\), and write \(\beta \equiv k_\rho B^\rho = k(B_0 + B_3)\). From Equation (A1.25),

\begin{equation} \delta A^{\mu}{}_{\mu} = 2\ii\beta\ec\qquad \delta A_{01} = \ii k B_1\ec\qquad \delta A_{02} = \ii k B_2\ec\qquad \delta A_{03} = \ii k\,(B_3 - B_0)\ec \end{equation}

and the four quantities \((B_0 + B_3,\,B_1,\,B_2,\,B_3 - B_0)\) are independent linear combinations of the \(B_\mu\): the linear system is invertible. We may therefore impose

\begin{equation} A^{\mu}{}_{\mu} = 0\ec \qquad A_{0i} = 0 \quad (i = 1,2,3)\ep \tag{A1.27} \end{equation}

The remaining component \(A_{00}\) then vanishes automatically: transversality Equation (A1.24) with \(\nu = 0\) reads \(k^0 A_{00} + k^i A_{i0} = 0\), and \(A_{i0} = 0\) with \(k^0 = \omega/c \neq 0\) forces \(A_{00} = 0\). The count of physical polarizations is thus

\begin{equation} 10\ \text{(symmetric $A_{\mu\nu}$)} \;-\; 4\ \text{(Lorenz)} \;-\; 4\ \text{(residual gauge)} \;=\; 2\ep \tag{A1.28} \end{equation}

Because the trace vanishes, \(\bar h_{\mu\nu} = h_{\mu\nu}\) in this gauge: bar and no-bar coincide. The conditions

\begin{equation} h^{\mathrm{TT}}_{0\mu} = 0\ec\qquad h^{\mathrm{TT}\,i}{}_{i} = 0\ec\qquad \pp^j h^{\mathrm{TT}}_{ij} = 0 \tag{A1.29} \end{equation}

define the transverse–traceless (TT) gauge. For the wave along \(z\), transversality further gives \(k(A_{0\nu} + A_{3\nu}) = 0\), hence \(A_{3\nu} = -A_{0\nu} = 0\): only \(A_{11} = -A_{22}\) and \(A_{12} = A_{21}\) survive. Writing \(A_{11} \equiv h_+\) and \(A_{12} \equiv h_\times\),

\begin{equation} h^{\mathrm{TT}}_{ij}(t,z) = \begin{pmatrix} h_+ & h_\times & 0\\ h_\times & -h_+ & 0\\ 0 & 0 & 0 \end{pmatrix} \cos\!\left[\omega\left(t - z/c\right)\right]\ec \tag{A1.30} \end{equation}

the two physical polarizations of Phenomenon 50.1 [Misner:1973] [Maggiore:2008]. They are spin-2 objects: a rotation by \(\psi\) about the propagation axis mixes them as \(h_+ \to h_+\cos 2\psi + h_\times\sin 2\psi\), so the pattern returns to itself after half a turn.

Physical effect: geodesic deviation and the strain

A single free test mass tells us nothing: in TT coordinates a particle initially at rest stays at fixed coordinates. Its geodesic equation with initial four-velocity \(u^\mu = (c,0,0,0)\) gives

\begin{equation} \dv{^2 x^i}{\tau^2}\bigg|_{\tau=0} = -\Gamma^{i}{}_{00}\,c^2 = -\frac{c^2}{2}\,\eta^{ij}\left(2\pp_0 h_{j0} - \pp_j h_{00}\right) = 0\ec \tag{A1.31} \end{equation}

since \(h_{0\mu} = 0\) in the TT gauge, Equation (A1.29). The coordinates simply ride along with the wave. What oscillates is the proper distance between two such masses. Let both sit on the \(x\) axis with coordinate separation \(L_{\mathrm{c}}\), the wave Equation (A1.30) passing along \(z\). The proper distance at time \(t\) is

\begin{align} s(t) &= \int_0^{L_{\mathrm{c}}}\sqrt{g_{xx}}\,\dd x = \sqrt{1 + h_+\cos\omega t}\;L_{\mathrm{c}} \nonumber\\ &= \left[1 + \frac{1}{2}h_+\cos\omega t + O(h^2)\right] L_{\mathrm{c}}\ec \tag{A1.32} \end{align}

so the fractional length change — the strain — is

\begin{equation} \boxed{\;\frac{\Delta L}{L} = \frac{1}{2}\,h\;} \tag{A1.33} \end{equation}

for a wave of amplitude \(h\) optimally oriented with respect to the separation.

The same result follows covariantly from geodesic deviation, which also exhibits the wave as a tidal force. For two neighbouring geodesics with separation vector \(\xi^\mu\) and common four-velocity \(u^\mu\) [Misner:1973] [Wald:1984],

\begin{equation} \frac{D^2\xi^\mu}{D\tau^2} = -R^{\mu}{}_{\nu\rho\sigma}\,u^\nu \xi^\rho u^\sigma\ep \tag{A1.34} \end{equation}

For slowly moving masses \(u^\mu \simeq (c,0,0,0)\), \(\tau \simeq t\), and the only components of Riemann we need follow from Equation (A1.6) with the TT conditions \(h_{0\mu} = 0\):

\begin{equation} R_{i0j0} = \frac{1}{2}\left( \pp_0\pp_j h_{i0} + \pp_i\pp_0 h_{0j} - \pp_i\pp_j h_{00} - \pp_0\pp_0 h_{ij} \right) = -\frac{1}{2c^2}\,\pp_t^2 h^{\mathrm{TT}}_{ij}\ep \tag{A1.35} \end{equation}

Then Equation (A1.34) gives, in the local proper frame of the first mass,

\begin{equation} \ddot\xi^i = -c^2 R^{i}{}_{0j0}\,\xi^j = \frac{1}{2}\,\ddot h^{\mathrm{TT}}_{ij}\,\xi^j\ec \tag{A1.36} \end{equation}

which integrates (for \(\xi\) nearly constant) to \(\delta\xi^i = \frac12 h^{\mathrm{TT}}_{ij}\xi^j\), reproducing Equation (A1.33). Note that the observable is curvature, Equation (A1.35), which we proved gauge invariant: the strain is physics, not coordinates.

Applying \(\delta\xi^i = \frac12 h^{\mathrm{TT}}_{ij}\xi^j\) to a ring of free masses of radius \(L\) in the \(xy\) plane, \(\xi^j = L(\cos\varphi, \sin\varphi, 0)\): a pure \(h_+\) wave displaces \((\delta x, \delta y) = \frac12 h_+ L\,(\cos\varphi, -\sin\varphi)\), squeezing the ring into an ellipse along \(x\) while stretching it along \(y\) and vice versa each half period; a pure \(h_\times\) wave produces the same ellipse rotated by \(45^\circ\). This quadrupolar breathing pattern is precisely what a Michelson interferometer with perpendicular arms measures: the wave lengthens one arm while it shortens the other, and the differential strain \(\Delta L_x - \Delta L_y = h_+ L\) appears as an optical phase shift at the dark port. With \(L = 4\,\mathrm{km}\) arms and strain sensitivity near \(10^{-22}\) per root hertz in the band around \(100\,\mathrm{Hz}\), the Advanced LIGO instruments [Aasi:2015] resolve arm-length changes of order \(10^{-18}\,\mathrm{m}\) — the observable exploited in Section 51.3 [Abbott:2016].

Energy carried by the wave: the Isaacson tensor

That gravitational waves carry energy was disputed into the 1950s; the resolution is that the energy is not localizable at a point (the equivalence principle forbids it) but is perfectly well defined once averaged over a few wavelengths. We follow the short-wave (Isaacson) analysis [Misner:1973] [Maggiore:2008].

Expand the Ricci tensor of \(g = \eta + h\) to second order in \(h\),

\begin{equation} R_{\mu\nu} = R^{(1)}_{\mu\nu}[h] + R^{(2)}_{\mu\nu}[h,h] + O(h^3)\ec \end{equation}

where \(R^{(1)}\) is the linear expression Equation (A1.7). In vacuum, at second order, the equation \(R_{\mu\nu} = 0\) says that the small residual background curvature generated at \(O(h^2)\) is sourced by \(-R^{(2)}_{\mu\nu}\): the wave gravitates. Averaging over a spacetime region several wavelengths across (denoted \(\avg{\cdot}\)) and defining

\begin{equation} t_{\mu\nu} \equiv -\frac{c^4}{8\pi G} \avg{R^{(2)}_{\mu\nu} - \frac{1}{2}\eta_{\mu\nu}R^{(2)}}\ec \tag{A1.38} \end{equation}

the coarse-grained field equations take the form \(G^{(1)}_{\mu\nu}[\text{background}] = (8\pi G/c^4)\,t_{\mu\nu}\): the averaged quadratic terms act as an effective stress–energy tensor of the radiation.

The straightforward if lengthy expansion of the Ricci tensor to second order gives [Maggiore:2008]

\begin{multline} R^{(2)}_{\mu\nu} = \frac{1}{2}\Big[ \frac{1}{2}\pp_\mu h_{\alpha\beta}\,\pp_\nu h^{\alpha\beta} + h^{\alpha\beta}\pp_\mu\pp_\nu h_{\alpha\beta} - h^{\alpha\beta}\pp_\nu\pp_\beta h_{\alpha\mu} - h^{\alpha\beta}\pp_\mu\pp_\beta h_{\alpha\nu} + h^{\alpha\beta}\pp_\alpha\pp_\beta h_{\mu\nu} \\ + \pp^\beta h^{\alpha}{}_{\nu}\,\pp_\beta h_{\alpha\mu} - \pp^\beta h^{\alpha}{}_{\nu}\,\pp_\alpha h_{\beta\mu} - \pp_\beta h^{\alpha\beta}\,\pp_\nu h_{\alpha\mu} - \pp_\beta h^{\alpha\beta}\,\pp_\mu h_{\alpha\nu} + \pp_\beta h^{\alpha\beta}\,\pp_\alpha h_{\mu\nu} \\ - \frac{1}{2}\pp^\alpha h\,\pp_\alpha h_{\mu\nu} + \frac{1}{2}\pp^\alpha h\,\pp_\nu h_{\alpha\mu} + \frac{1}{2}\pp^\alpha h\,\pp_\mu h_{\alpha\nu} \Big]\ep \tag{A1.39} \end{multline}

Work in the TT gauge, where \(h = 0\) and \(\pp^\beta h_{\alpha\beta} = 0\): the last six terms of Equation (A1.39) vanish identically. Under the average, total derivatives are suppressed by (wavelength)/(averaging scale) and may be dropped, which licenses integration by parts inside \(\avg{\cdot}\); and the field equations \(\Box h_{\alpha\beta} = 0\) hold. Then, term by term,

\begin{align} \avg{h^{\alpha\beta}\pp_\mu\pp_\nu h_{\alpha\beta}} &= -\avg{\pp_\mu h^{\alpha\beta}\,\pp_\nu h_{\alpha\beta}}\ec \\ \avg{h^{\alpha\beta}\pp_\nu\pp_\beta h_{\alpha\mu}} &= -\avg{\pp_\beta h^{\alpha\beta}\,\pp_\nu h_{\alpha\mu}} = 0\ec \qquad\text{(and likewise with $\mu \leftrightarrow \nu$)} \\ \avg{h^{\alpha\beta}\pp_\alpha\pp_\beta h_{\mu\nu}} &= -\avg{\pp_\alpha h^{\alpha\beta}\,\pp_\beta h_{\mu\nu}} = 0\ec \\ \avg{\pp^\beta h^{\alpha}{}_{\nu}\,\pp_\beta h_{\alpha\mu}} &= -\avg{h^{\alpha}{}_{\nu}\,\Box h_{\alpha\mu}} = 0\ec \\ \avg{\pp^\beta h^{\alpha}{}_{\nu}\,\pp_\alpha h_{\beta\mu}} &= -\avg{h^{\alpha}{}_{\nu}\,\pp_\alpha\pp^\beta h_{\beta\mu}} = 0\ep \end{align}

Only the first two terms of Equation (A1.39) survive, combining to

\begin{equation} \avg{R^{(2)}_{\mu\nu}} = \frac{1}{2}\left(\frac{1}{2} - 1\right) \avg{\pp_\mu h_{\alpha\beta}\,\pp_\nu h^{\alpha\beta}} = -\frac{1}{4}\avg{\pp_\mu h^{\mathrm{TT}}_{\alpha\beta}\, \pp_\nu h^{\mathrm{TT}\,\alpha\beta}}\ep \tag{A1.45} \end{equation}

Its trace, \(\eta^{\mu\nu}\avg{R^{(2)}_{\mu\nu}} = \frac14\avg{h_{\alpha\beta}\Box h^{\alpha\beta}} = 0\) after one more integration by parts, vanishes on shell, so Equation (A1.38) yields the Isaacson stress–energy tensor. In the TT gauge only spatial components of \(h\) are nonzero, and

\begin{equation} \boxed{\; t_{\mu\nu} = \frac{c^4}{32\pi G} \avg{\pp_\mu h^{\mathrm{TT}}_{ij}\,\pp_\nu h^{\mathrm{TT}}_{ij}} \;} \tag{A1.46} \end{equation}

(spatial indices summed; their up/down position is immaterial in the flat background). Although derived in a particular gauge, the averaged tensor is gauge invariant to the order considered [Misner:1973] [Maggiore:2008].

For the plane wave Equation (A1.30) travelling along \(z\), every component is a function of \(t - z/c\), so \(\pp_z = -c^{-1}\pp_t\) acting on \(h\), and the energy density and energy flux are

\begin{equation} t^{00} = \frac{c^2}{32\pi G}\avg{\dot h^{\mathrm{TT}}_{ij}\dot h^{\mathrm{TT}}_{ij}}\ec \qquad F = c\,t^{0z} = \frac{c^3}{32\pi G}\avg{\dot h^{\mathrm{TT}}_{ij}\dot h^{\mathrm{TT}}_{ij}} = \frac{c^3}{16\pi G}\avg{\dot h_+^2 + \dot h_\times^2}\ec \tag{A1.47} \end{equation}

using \(h^{\mathrm{TT}}_{ij}h^{\mathrm{TT}}_{ij} = 2(h_+^2 + h_\times^2)\) from Equation (A1.30). The prefactor \(c^3/16\pi G\) is enormous, \(\approx 8.0\times 10^{33}\,\mathrm{W}/\mathrm{m}^{2}\) per unit \(\avg{\dot h^2}//\mathrm{s}^{2}\): even a strain of \(10^{-21}\) oscillating at \(100\,\mathrm{Hz}\) carries a flux of order \(10^{-3}\,\mathrm{W}/\mathrm{m}^{2}\) — comparable to moonlight, delivered by a distortion of geometry a thousandth of a proton radius across a kilometre.

Generation: the quadrupole formula

We now solve the sourced equation Equation (A1.21).

Retarded Green function.

The retarded solution of \(\Box f = -S\), with \(\Box\) as in Equation (A1.1), is

\begin{equation} f(t,\vect{x}) = \frac{1}{4\pi}\int \frac{S\!\left(t - \abs{\vect{x}-\vect{x}'}/c,\ \vect{x}'\right)} {\abs{\vect{x}-\vect{x}'}}\,\dd^3 x'\ep \tag{A1.48} \end{equation}

To verify, consider a single spherical wave \(f = g(t - r/c)/r\) centred on \(\vect{x}'\), \(r = \abs{\vect{x}-\vect{x}'}\). For \(r \neq 0\), the radial Laplacian gives \(\nabla^2 f = r^{-1}\pp_r^2(rf) = g''(t-r/c)/(c^2 r) = c^{-2}\pp_t^2 f\), so the homogeneous wave equation holds; near \(r = 0\), \(f \to g(t)/r\) and \(\nabla^2(1/r) = -4\pi\delta^3(\vect{x}-\vect{x}')\) supplies a point source of strength \(4\pi g(t)\). Superposing such waves with \(g = S\,\dd^3x'/4\pi\) yields Equation (A1.48); causality selects the retarded rather than advanced root [Jackson:1999]. Applying this to Equation (A1.21) with \(S = (16\pi G/c^4)T_{\mu\nu}\),

\begin{equation} \bar h_{\mu\nu}(t,\vect{x}) = \frac{4G}{c^4}\int \frac{T_{\mu\nu}\!\left(t - \abs{\vect{x}-\vect{x}'}/c,\ \vect{x}'\right)} {\abs{\vect{x}-\vect{x}'}}\,\dd^3x'\ep \tag{A1.49} \end{equation}

Far zone and slow motion.

Let the source occupy a region of size \(d\) about the origin and let \(r = \abs{\vect{x}} \gg d\), with unit line of sight \(\vect{n} = \vect{x}/r\). Then \(\abs{\vect{x}-\vect{x}'} = r - \vect{n}\cdot\vect{x}' + O(d^2/r)\), and keeping only the leading \(1/r\),

\begin{equation} \bar h_{\mu\nu}(t,\vect{x}) = \frac{4G}{c^4 r}\int T_{\mu\nu}\!\left(t - \frac{r}{c} + \frac{\vect{n}\cdot\vect{x}'}{c},\ \vect{x}'\right)\dd^3x'\ep \tag{A1.50} \end{equation}

If moreover the internal motions are slow, \(v \ll c\) — equivalently the emitted wavelength \(\lambda \sim c\,d/v\) far exceeds the source size — the retardation spread \(\vect{n}\cdot\vect{x}'/c\) across the source may be dropped, and every component of \(T_{\mu\nu}\) is evaluated at the single retarded time \(t_r = t - r/c\).

From stress to quadrupole: the conservation identities.

The spatial integral of \(T^{ij}\) is not an independent datum: flat-space conservation \(\pp_\mu T^{\mu\nu} = 0\), guaranteed at this order by Equation (A1.21), trades it for time derivatives of the energy density. With \(x^0 = ct\), the two components of the conservation law read

\begin{equation} \frac{1}{c}\pp_t T^{00} + \pp_i T^{i0} = 0\ec \qquad \frac{1}{c}\pp_t T^{0j} + \pp_i T^{ij} = 0\ep \tag{A1.51} \end{equation}

Multiply the first by \(x^i x^j\) and integrate over all space (the source has compact support, so boundary terms vanish under integration by parts):

\begin{align} \dv{}{t}\int T^{00}x^i x^j\,\dd^3x &= -c\int \pp_k T^{k0}\,x^i x^j\,\dd^3x = c\int\left(T^{i0}x^j + T^{j0}x^i\right)\dd^3x\ep \tag{A1.52} \end{align}

Multiply the second of Equation (A1.51) by \(x^j\), integrate, and symmetrize:

\begin{align} \dv{}{t}\int\left(T^{0i}x^j + T^{0j}x^i\right)\dd^3x &= -c\int\left(\pp_k T^{ki}\,x^j + \pp_k T^{kj}\,x^i\right)\dd^3x = 2c\int T^{ij}\,\dd^3x\ep \tag{A1.53} \end{align}

Chaining Equations (A1.52) and (A1.53),

\begin{equation} \int T^{ij}\,\dd^3x = \frac{1}{2c^2}\,\dv{^2}{t^2}\int T^{00}\,x^i x^j\,\dd^3x = \frac{1}{2}\,\ddot Q_{ij}\ec \tag{A1.54} \end{equation}

where, with the mass density \(\rho \equiv T^{00}/c^2\) (the dominant part of \(T^{00}\) for slow motion), the mass quadrupole moment is

\begin{equation} Q_{ij}(t) \equiv \int \rho(t,\vect{x})\,x_i x_j\,\dd^3x\ep \tag{A1.55} \end{equation}

Substituting Equation (A1.54) into Equation (A1.50),

\begin{equation} \boxed{\; \bar h_{ij}(t,\vect{x}) = \frac{2G}{c^4 r}\,\ddot Q_{ij}(t - r/c) \;} \tag{A1.56} \end{equation}

— the quadrupole formula for the field [Einstein:1918] [Misner:1973]. Dimensionally: \(G/c^4 \sim \mathrm{s}^{2}/\mathrm{kg}/\mathrm{m}\) and \(\ddot Q \sim \mathrm{kg}\,\mathrm{m}^{2}/\mathrm{s}^{2}\), so \(h\) is a pure number divided by \(r\) — a strain falling off as \(1/r\), as a radiation field must. The time components need not be computed separately: the Lorenz condition Equation (A1.20) determines \(\bar h^{0\mu}\) from \(\bar h^{ij}\) up to stationary pieces. Physically, the monopole moment \(\int\rho\,\dd^3x\) is the conserved total mass and the dipole moment \(\int\rho\,x^i\,\dd^3x\) moves uniformly by conservation of momentum, so neither can oscillate: mass conservation kills monopole radiation, momentum conservation kills dipole radiation, and the quadrupole is the first moment free to radiate [Misner:1973]. This is the deep reason gravitational radiation is so feeble compared with the electric-dipole radiation of electromagnetism.

TT projection and the reduced quadrupole.

The physical waveform is the TT part of Equation (A1.56). Define the projector transverse to the line of sight, \(P_{ij}(\vect{n}) = \delta_{ij} - n_i n_j\) (idempotent: \(P_{ik}P_{kj} = P_{ij}\)), and the TT projector

\begin{equation} \Lambda_{ij,kl}(\vect{n}) = P_{ik}P_{jl} - \frac{1}{2}P_{ij}P_{kl}\ec \qquad \Lambda_{ij,kl}\Lambda_{kl,mn} = \Lambda_{ij,mn}\ep \tag{A1.57} \end{equation}

Because \(\Lambda\) annihilates any term proportional to \(\delta_{kl}\), we may freely replace \(Q_{ij}\) by its trace-free part. We adopt, here and in the luminosity below, the reduced quadrupole moment

\begin{equation} \mathcal{I}_{ij}(t) \equiv \int \rho(t,\vect{x}) \left(x_i x_j - \frac{1}{3}\,r'^2\,\delta_{ij}\right)\dd^3x = Q_{ij} - \frac{1}{3}\,\delta_{ij}\,Q_{kk}\ec \tag{A1.58} \end{equation}

(with \(r'^2 = x_k x_k\) the squared distance from the origin), in terms of which

\begin{equation} h^{\mathrm{TT}}_{ij}(t,\vect{x}) = \frac{2G}{c^4 r}\,\Lambda_{ij,kl}(\vect{n})\, \ddot{\mathcal{I}}_{kl}(t - r/c)\ep \tag{A1.59} \end{equation}

Radiated power.

Insert Equation (A1.59) into the flux Equation (A1.47) and integrate over a large sphere. For an outgoing wave \(h \propto r^{-1}\times(\text{function of } t - r/c)\) one has \(\pp_r h = -c^{-1}\dot h + O(1/r^2)\), so the radial energy flux is \(c\,t^{0r} = (c^3/32\pi G)\avg{\dot h^{\mathrm{TT}}_{ij}\dot h^{\mathrm{TT}}_{ij}}\) and

\begin{align} P = \dv{E}{t} &= \frac{c^3}{32\pi G}\,r^2\oint \avg{\dot h^{\mathrm{TT}}_{ij}\dot h^{\mathrm{TT}}_{ij}}\,\dd\Omega = \frac{c^3}{32\pi G}\cdot\frac{4G^2}{c^8} \oint \Lambda_{kl,mn}(\vect{n})\, \avg{\dddot{\mathcal{I}}_{kl}\,\dddot{\mathcal{I}}_{mn}}\,\dd\Omega\ec \tag{A1.60} \end{align}

where the projector identity of Equation (A1.57) collapsed the two \(\Lambda\)'s into one. The angular integral needs only the elementary moments of the unit vector,

\begin{equation} \oint n_i n_j\,\dd\Omega = \frac{4\pi}{3}\,\delta_{ij}\ec \qquad \oint n_i n_j n_k n_l\,\dd\Omega = \frac{4\pi}{15}\left(\delta_{ij}\delta_{kl} + \delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk}\right)\ec \tag{A1.61} \end{equation}

fixed by isotropy and by contracting both sides. For any constant symmetric trace-free \(A_{ij}\), expanding Equation (A1.57) gives \(\Lambda_{ij,kl}A_{ij}A_{kl} = A_{ij}A_{ij} - 2A_{ij}A_{ik}n_j n_k + \frac12(n_i n_j A_{ij})^2\), and with Equation (A1.61),

\begin{equation} \oint \Lambda_{ij,kl}\,A_{ij}A_{kl}\,\dd\Omega = 4\pi\left(1 - \frac{2}{3} + \frac{1}{15}\right)A_{ij}A_{ij} = \frac{8\pi}{5}\,A_{ij}A_{ij}\ep \tag{A1.62} \end{equation}

Substituting into Equation (A1.60),

\begin{equation} \boxed{\; P = \frac{G}{5c^5}\, \avg{\dddot{\mathcal{I}}_{ij}\,\dddot{\mathcal{I}}_{ij}} \;} \tag{A1.63} \end{equation}

— Einstein's quadrupole luminosity [Einstein:1918] [Maggiore:2008], with the reduced moment Equation (A1.58) and the third time derivative evaluated at retarded time. The prefactor \(G/c^5 \approx 2.75\times 10^{-53}\, \mathrm{W}^{-1}\) explains at a glance why laboratory sources are hopeless and only relativistic astrophysical masses radiate detectably; its inverse \(c^5/G \approx 3.6\times 10^{52}\,\mathrm{W}\) sets the natural luminosity scale of strong-field gravity.

Application: the circular compact binary

The cleanest astrophysical source is two point masses \(m_1\), \(m_2\) in a circular orbit — the configuration realized, to excellent approximation, by compact binaries in their late inspiral. Work in the centre-of-mass frame with total and reduced masses

\begin{equation} M = m_1 + m_2\ec\qquad \mu = \frac{m_1 m_2}{M}\ec \end{equation}

separation \(a\), and Kepler angular frequency

\begin{equation} \Omega^2 = \frac{GM}{a^3}\ep \tag{A1.65} \end{equation}

With the centre of mass at the origin, \(\vect{x}_1 = (m_2/M)\,\vect{x}\) and \(\vect{x}_2 = -(m_1/M)\,\vect{x}\) in terms of the relative coordinate \(\vect{x}(t) = a(\cos\Omega t, \sin\Omega t, 0)\), so the quadrupole of the two \(\delta\)-function masses reduces to that of a single mass \(\mu\) on the relative orbit:

\begin{equation} Q_{ij} = m_1 x_{1i}x_{1j} + m_2 x_{2i}x_{2j} = \left(\frac{m_1 m_2^2 + m_2 m_1^2}{M^2}\right)x_i x_j = \mu\,x_i x_j\ep \end{equation}

Using \(\cos^2\theta = \frac12(1+\cos 2\theta)\), \(\sin^2\theta = \frac12(1-\cos 2\theta)\), \(\sin\theta\cos\theta = \frac12\sin 2\theta\), the reduced moment Equation (A1.58) has time-dependent part

\begin{equation} \mathcal{I}_{xx} = \frac{\mu a^2}{2}\cos 2\Omega t + \text{const}\ec\quad \mathcal{I}_{yy} = -\frac{\mu a^2}{2}\cos 2\Omega t + \text{const}\ec\quad \mathcal{I}_{xy} = \frac{\mu a^2}{2}\sin 2\Omega t\ec \tag{A1.67} \end{equation}

with \(\mathcal{I}_{zz}\) constant. Everything oscillates at \(2\Omega\): the mass distribution returns to itself after half an orbit, so

\begin{equation} f_{\mathrm{GW}} = 2 f_{\mathrm{orb}} = \frac{\Omega}{\pi}\ep \tag{A1.68} \end{equation}

Differentiating Equation (A1.67) three times,

\begin{equation} \dddot{\mathcal{I}}_{xx} = 4\mu a^2\Omega^3\sin 2\Omega t\ec\qquad \dddot{\mathcal{I}}_{yy} = -4\mu a^2\Omega^3\sin 2\Omega t\ec\qquad \dddot{\mathcal{I}}_{xy} = -4\mu a^2\Omega^3\cos 2\Omega t\ec \tag{A1.69} \end{equation}

whence, counting \(\mathcal{I}_{xy} = \mathcal{I}_{yx}\) twice,

\begin{equation} \dddot{\mathcal{I}}_{ij}\dddot{\mathcal{I}}_{ij} = 16\mu^2 a^4\Omega^6\left(2\sin^2 2\Omega t + 2\cos^2 2\Omega t\right) = 32\,\mu^2 a^4\Omega^6\ec \end{equation}

constant in time, so the average in Equation (A1.63) is trivial:

\begin{equation} P = \frac{32}{5}\,\frac{G}{c^5}\,\mu^2 a^4\Omega^6 = \boxed{\;\frac{32}{5}\,\frac{G^4}{c^5}\, \frac{m_1^2 m_2^2\,(m_1+m_2)}{a^5}\;} \tag{A1.71} \end{equation}

after eliminating \(\Omega\) with Equation (A1.65).

Orbital decay.

The Newtonian orbital energy is, by the virial relation \(K = \frac12\mu a^2\Omega^2 = \frac12\,G m_1 m_2/a = -\frac12 U\),

\begin{equation} E = K + U = -\frac{G m_1 m_2}{2a}\ep \tag{A1.72} \end{equation}

Radiation drains it, \(\dot E = -P\); differentiating Equation (A1.72) and inserting Equation (A1.71),

\begin{equation} \frac{G m_1 m_2}{2a^2}\,\dot a = -P \qquad\Longrightarrow\qquad \dot a = -\frac{64}{5}\,\frac{G^3}{c^5}\, \frac{m_1 m_2\,(m_1+m_2)}{a^3}\ep \tag{A1.73} \end{equation}

The orbit shrinks, \(\Omega\) grows, \(P \propto a^{-5}\) grows faster: a runaway — the chirp.

Chirp mass and frequency evolution.

Express the evolution in the observable \(f \equiv f_{\mathrm{GW}}\). From Equations (A1.65) and (A1.68), \(\pi f = \Omega = (GM)^{1/2}a^{-3/2}\), i.e. \(a = (GM)^{1/3}(\pi f)^{-2/3}\). Substituting into Equations (A1.71) and (A1.72) and introducing the chirp mass

\begin{equation} \mathcal{M} \equiv \frac{(m_1 m_2)^{3/5}}{(m_1+m_2)^{1/5}} = \mu^{3/5}M^{2/5}\ec \qquad \mathcal{M}^{5/3} = \mu\,M^{2/3} = m_1 m_2\,M^{-1/3}\ec \tag{A1.74} \end{equation}

both energy and power depend on the masses only through \(\mathcal{M}\):

\begin{equation} E(f) = -\frac{1}{2}\,\mathcal{M}^{5/3} G^{2/3}(\pi f)^{2/3}\ec \qquad P(f) = \frac{32}{5}\,\frac{c^5}{G} \left(\frac{\pi G\mathcal{M} f}{c^3}\right)^{10/3}\ep \tag{A1.75} \end{equation}

Energy balance \(\dot f\,(\dd E/\dd f) = -P\) with \(\dd E/\dd f = -\frac13\mathcal{M}^{5/3}G^{2/3}\pi^{2/3}f^{-1/3}\) then gives

\begin{equation} \boxed{\; \dot f = \frac{96}{5}\,\pi^{8/3} \left(\frac{G\mathcal{M}}{c^3}\right)^{5/3} f^{11/3} \;} \tag{A1.76} \end{equation}

— the announced law \(\dot f \propto \mathcal{M}^{5/3}f^{11/3}\). Measuring \(f\) and \(\dot f\) at any moment of the inspiral therefore yields the chirp mass directly, with no knowledge of the distance; the distance then follows from the amplitude. That amplitude is read off Equations (A1.59) and (A1.67): for the optimal (face-on) orientation the two polarizations have common magnitude

\begin{equation} h = \frac{4G\mu a^2\Omega^2}{c^4 r} = \frac{4}{r}\,\frac{(G\mathcal{M})^{5/3}}{c^4}\, \left(\frac{\pi f}{c}\right)^{2/3} \cdot c^{2/3} = \frac{4}{r}\left(\frac{G\mathcal{M}}{c^2}\right)^{5/3} \left(\frac{\pi f}{c}\right)^{2/3}\ec \tag{A1.77} \end{equation}

using \(a^2\Omega^2 = GM/a\) and Equation (A1.74).

Numerical estimate (i): a GW150914-like binary.

Take two black holes of \(30\,M_\odot\) each at luminosity distance \(r = 410\,\mathrm{Mpc}\), the parameters of the first LIGO detection to within its uncertainties (the measured values were \(36\,M_\odot\) and \(29\,M_\odot\) at \(410^{+160}_{-180}\,\mathrm{Mpc}\) [Abbott:2016]). We use \(G = 6.67430(15)\times 10^{-11}\,\mathrm{m}^{3}/\mathrm{kg}/\mathrm{s}^{2}\) [Tiesinga:2021], the exact \(c = 299792458\,\mathrm{m}/\mathrm{s}\) of the SI [BIPM:2019], and \(M_\odot = 1.989\times 10^{30}\,\mathrm{kg}\). Then \(m_1 = m_2 = 30\,M_\odot = 5.97\times 10^{31}\,\mathrm{kg}\), \(M = 60\,M_\odot = 1.193\times 10^{32}\,\mathrm{kg}\), \(\mu = 15\,M_\odot\), and

\begin{equation} \mathcal{M} = \mu^{3/5}M^{2/5} = (15)^{3/5}(60)^{2/5}\,M_\odot \approx 26.1\,M_\odot = 5.19\times 10^{31}\,\mathrm{kg}\ec \end{equation}

so that

\begin{equation} \frac{G\mathcal{M}}{c^2} = \frac{6.674\times 10^{-11}\times5.19\times 10^{31}}{8.988\times 10^{16}}\,\mathrm{m} = 3.85\times 10^{4}\,\mathrm{m}\ec \qquad \frac{G\mathcal{M}}{c^3} = 1.29\times 10^{-4}\,\mathrm{s}\ep \end{equation}

Orbital geometry across the band. With \(GM = 7.97\times 10^{21}\,\mathrm{m}^{3}/\mathrm{s}^{2}\), the separation at gravitational-wave frequency \(f\) is \(a = (GM)^{1/3}(\pi f)^{-2/3}\). At the low edge of the detector band, \(f = 35\,\mathrm{Hz}\): \(a = 2.0\times 10^{7}\,\mathrm{m}/\mathrm{s}^{2/3}\times (110\,/\mathrm{s})^{-2/3} \approx 8.7\times 10^{5}\,\mathrm{m}\) — two thirty-solar-mass black holes, each of Schwarzschild radius \(2Gm/c^2 \approx 89\,\mathrm{km}\), orbiting \(870\,\mathrm{km}\) apart at an orbital frequency of \(17.5\,\mathrm{Hz}\). Near merger the separation approaches a few hundred kilometres: at \(a = 2.5\times 10^{5}\,\mathrm{m}\),

\begin{equation} \Omega = \sqrt{\frac{7.97\times 10^{21}}{(2.5\times 10^{5})^3}}\;\mathrm{rad}/\mathrm{s} = 714\,\mathrm{rad}/\mathrm{s}\ec \qquad f_{\mathrm{GW}} = \frac{\Omega}{\pi} \approx 230\,\mathrm{Hz}\ec \end{equation}

reproducing the observed sweep of GW150914 from \(35\,\mathrm{Hz}\) to about \(250\,\mathrm{Hz}\) [Abbott:2016]. Strain. At \(f = 150\,\mathrm{Hz}\) and \(r = 410\,\mathrm{Mpc} = 1.27\times 10^{25}\,\mathrm{m}\), Equation (A1.77) gives, with \((G\mathcal{M}/c^2)^{5/3} = (3.85\times 10^{4})^{5/3}\,\mathrm{m}^{5/3} = 4.4\times 10^{7}\,\mathrm{m}^{5/3}\) and \((\pi f/c)^{2/3} = (1.57\times 10^{-6})^{2/3}\,\mathrm{m}^{-2/3} = 1.35\times 10^{-4}\,\mathrm{m}^{-2/3}\),

\begin{equation} h = \frac{4\times4.4\times 10^{7}\times1.35\times 10^{-4}}{1.27\times 10^{25}} \approx 1.9\times 10^{-21}\ec \end{equation}

for optimal orientation; averaging over source inclination and detector antenna pattern reduces this by a factor of a few, in agreement with the observed peak strain of \(10^{-21}\) [Abbott:2016]. By Equation (A1.33) the LIGO arms of \(L = 4\,\mathrm{km}\) then change length by \(\Delta L = \frac12 h L \approx 2\times 10^{-18}\,\mathrm{m}\) [Aasi:2015]. Chirp rate. At \(f = 35\,\mathrm{Hz}\), Equation (A1.76) gives

\begin{equation} \dot f = \frac{96}{5}\,\pi^{8/3} \left(1.29\times 10^{-4}\right)^{5/3}\left(35\right)^{11/3} \mathrm{Hz}/\mathrm{s} \approx 61\,\mathrm{Hz}/\mathrm{s}\ec \end{equation}

so the residual time to coalescence, \(\tau \approx \frac{3}{8}f/\dot f \approx 0.2\,\mathrm{s}\), matches the observed duration of the signal in band [Abbott:2016]. All three numbers — band, amplitude, chirp — flow from Equations (A1.76) and (A1.77) with one chirp mass; that a single \(\mathcal{M}\) fits the entire waveform is the quantitative content of the statement that GW150914 was a compact binary coalescence.

Numerical estimate (ii): orbital decay of PSR B1913+16.

For a binary of period \(P_b\), Kepler's law \(P_b \propto a^{3/2}\) gives \(\dot P_b/P_b = \frac{3}{2}\,\dot a/a\), and inserting Equation (A1.73) with \(a = (GM)^{1/3}(P_b/2\pi)^{2/3}\) yields the decay of the period. The derivation above assumed a circular orbit; for an eccentric orbit the same energy-balance computation, carried through the Keplerian ellipse harmonics, multiplies the circular result by the enhancement factor \(f(e) = \left(1 + \frac{73}{24}e^2 + \frac{37}{96}e^4\right) (1-e^2)^{-7/2}\), which we quote from the standard treatment [Maggiore:2008]. Thus

\begin{equation} \dot P_b = -\frac{192\pi}{5}\,\frac{G^{5/3}}{c^5} \left(\frac{P_b}{2\pi}\right)^{-5/3} \frac{m_p m_c}{(m_p+m_c)^{1/3}}\;f(e)\ep \tag{A1.83} \end{equation}

The Hulse–Taylor pulsar PSR B1913+16, discovered in 1974 [Hulse:1975], has [Weisberg:2016]

\begin{equation} P_b = 27906.98\,\mathrm{s}\ec\quad e = 0.6171\ec\quad m_p = 1.438\,M_\odot\ec\quad m_c = 1.390\,M_\odot\ec \end{equation}

(masses themselves determined from two relativistic timing observables, so the test uses no free parameters). The pieces of Equation (A1.83) are

\begin{align} f(e) &= \frac{1 + \frac{73}{24}(0.3809) + \frac{37}{96}(0.1450)} {(0.6191)^{7/2}} = \frac{2.214}{0.1867} = 11.86\ec\\ \left(\frac{P_b}{2\pi}\right)^{5/3} &= (4441.5)^{5/3}\,\mathrm{s}^{5/3} = 1.200\times 10^{6}\,\mathrm{s}^{5/3}\ec\\ \frac{m_p m_c}{M^{1/3}} &= \frac{7.908\times 10^{60}}{1.778\times 10^{10}}\,\mathrm{kg}^{5/3} = 4.447\times 10^{50}\,\mathrm{kg}^{5/3}\ec \end{align}

and assembling them with \(G^{5/3} = 1.098\times 10^{-17}\) and \(c^5 = 2.422\times 10^{42}\) (SI),

\begin{equation} \dot P_b = -\frac{192\pi}{5}\times \frac{1.098\times 10^{-17}\times8.33\times 10^{-7}\times4.447\times 10^{50}} {2.422\times 10^{42}}\times 11.86 \approx -2.40\times 10^{-12}\ec \end{equation}

a dimensionless \(\mathrm{s}/\mathrm{s}\): the orbit of period near eight hours shortens by about \(76\,\mu\mathrm{s}\) per year. The full general-relativistic prediction with the precise timing parameters is \(\dot P_b^{\mathrm{GR}} = -2.40263(5)\times 10^{-12}\), and thirty-five years of timing give an observed intrinsic \(\dot P_b = -2.398(4)\times 10^{-12}\) — a ratio of observed to predicted of \(0.9983(16)\) [Taylor:1982] [Weisberg:2016]. The same parameters give a present gravitational-wave luminosity \(P \approx 7.8\times 10^{24}\,\mathrm{W}\) from Equation (A1.71) (times \(f(e)\)) at separation \(a \approx 1.95\times 10^{9}\,\mathrm{m}\) — about \(2\times 10^{-2}\) of the Sun's electromagnetic output, radiated by two neutron stars orbiting within what would fit inside the Sun. Integrating Equation (A1.73), the circular estimate gives coalescence after \(a^4/(4\beta) \approx 5.2\times 10^{16}\,\mathrm{s}\) (about \(1.7\) billion years, shortened to roughly \(0.3\) billion by the eccentricity); the binary pulsar is a young GW150914 caught three hundred million years before its chirp.

The two estimates close the loop of this proof: the strain, frequency sweep and chirp rate derived here are the quantities measured directly by the interferometers for GW150914, and the period decay Equation (A1.83) is the quantity measured by pulsar timing for PSR B1913+16, matching Section 51.1. The experimental record of both confrontations — apparatus, data and uncertainties — is presented in Experiment: Gravitational Waves.