Linearized GR and gravitational waves

Intuition [Beginner]

When two black holes spiral into each other, the collision sends ripples through spacetime itself. These ripples are gravitational waves. They are not waves travelling through spacetime the way sound travels through air. They are changes of spacetime curvature that propagate outward at the speed of light.

Einstein predicted gravitational waves in 1916, just one year after publishing his field equations. But the effect is so small that it took a century to detect one. On 14 September 2015, the LIGO observatory measured a distortion in spacetime caused by two black holes, each about 30 times the mass of the Sun, merging roughly 400 million light-years away. The measured strain -- the fractional change in distance between two points -- was about $10^{-21}$. That means a one-metre ruler changed length by one billionth of a billionth of a millimetre.

Gravitational waves have two polarizations, called plus ($h_{+}$) and cross ($h_{\times }$), named for the shape they impose on a ring of freely falling particles. A plus-polarized wave stretches the ring along one axis and compresses it along the perpendicular axis, then reverses. A cross-polarized wave does the same thing rotated by 45 degrees. This is analogous to electromagnetic waves having two transverse polarizations.

The power source for gravitational waves is the acceleration of mass. But not just any acceleration produces them. A perfectly spherical explosion radiates zero gravitational waves. What matters is the asymmetry of the mass distribution, captured by the quadrupole moment. Binary systems -- two stars or two black holes orbiting each other -- are efficient gravitational-wave emitters because their mass distribution changes dramatically and asymmetrically with each orbit.

Visual [Beginner]

Imagine a ring of freely falling test particles arranged in a circle in the $x$-$y$ plane. A gravitational wave propagates in the $z$ direction and passes through the ring.

The key point: no force pushes the particles. Each particle moves inertially along its own geodesic. The wave changes the geometry of spacetime between them, which changes the measured distance between freely moving objects. This is why LIGO uses laser interferometry -- it measures the change in distance between two mirrors, each freely suspended, as the wave passes.

Worked example [Beginner]

On 14 September 2015, LIGO detected gravitational waves from two black holes merging. This event is called GW150914. The two black holes had masses of approximately 36 and 29 times the mass of the Sun. They merged at a distance of roughly 400 Mpc (about 1.3 billion light-years) from Earth.

The peak strain measured at Earth was $h\approx 10^{-21}$. What does this number mean in physical terms?

LIGO's arms are 4 km long. A strain of $10^{-21}$ means the arm length changed by $\Delta L=h\times L=10^{-21}\times 4000\text{ m}=4\times 10^{-18}\text{ m}$. This is less than one thousandth of the diameter of a proton.

The signal swept upward in frequency from about 35 Hz to 250 Hz over roughly 0.2 seconds, then rang down. This "chirp" pattern -- rising frequency and amplitude followed by a quick decay -- matches the prediction for two black holes spiralling together, merging, and ringing down to a single black hole. The final black hole had a mass of about 62 solar masses. The missing 3 solar masses ($36+29-62=3$) were radiated as gravitational-wave energy in a fraction of a second, briefly outshining the entire visible universe in gravitational-wave luminosity.

The peak luminosity can be estimated. Three solar masses of energy ($3\times 2\times 10^{30}\text{ kg}\times (3\times 10^8{)}^2\approx 5.4\times 10^{47}\text{ J}$) radiated in about 0.01 seconds gives a power of order $5\times 10^{49}\text{ W}$. For comparison, the Sun's electromagnetic luminosity is $3.8\times 10^{26}\text{ W}$. The merger outshone all the stars in the observable universe combined by a factor of roughly $10^{23}$.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Consider a spacetime whose metric deviates only slightly from the flat Minkowski metric ${\eta }_{\mu \nu }$. Write

$$g_{\mu \nu }={\eta }_{\mu \nu }+h_{\mu \nu },|h_{\mu \nu }|\ll 1.$$

The symmetric tensor field $h_{\mu \nu }$ is the metric perturbation. All computations are carried out to first order in $h$ (the linearized theory); products $h\cdot h$ and higher powers are discarded.

The trace-reversed perturbation

Define the trace-reversed perturbation

$${\bar{h}}_{\mu \nu }:=h_{\mu \nu }-\frac{1}{2}{\eta }_{\mu \nu }h,h:={\eta }^{\alpha \beta }{h}_{\alpha \beta }.$$

This satisfies $h_{\mu \nu }={\bar{h}}_{\mu \nu }-\frac{1}{2}{\eta }_{\mu \nu }\bar{h}$, where $\bar{h}={\eta }^{\mu \nu }{\bar{h}}_{\mu \nu }=-h$. The trace-reversal simplifies the linearized Einstein equations.

The linearized Einstein equations

To first order in $h$, the Ricci tensor is

$$R_{\mu \nu }=\frac{1}{2}({\partial }_{\mu }{\partial }^{\alpha }{h}_{\alpha \nu }+{\partial }_{\nu }{\partial }^{\alpha }{h}_{\alpha \mu }-\square h_{\mu \nu }-{\partial }_{\mu }{\partial }_{\nu }h),$$

where $\square :={\eta }^{\alpha \beta }{\partial }_{\alpha }{\partial }_{\beta }$ is the flat-space d'Alembertian and indices are raised and lowered with $\eta$. The linearized Einstein tensor is $G_{\mu \nu }^{(1)}=R_{\mu \nu }^{(1)}-\frac{1}{2}{\eta }_{\mu \nu }{R}^{(1)}$, and the linearized field equations are

$$G_{\mu \nu }^{(1)}=\frac{8\pi G}{c^4}{T}_{\mu \nu }.$$

The Lorenz gauge

The linearized equations simplify in the Lorenz gauge (also called the de Donder gauge), defined by

$${\partial }^{\mu }{\bar{h}}_{\mu \nu }=0.$$

This condition can always be achieved by an infinitesimal coordinate transformation $x^{\mu }\to x^{\mu }+{\xi }^{\mu }$ with $|\xi |\ll 1$, which changes $h_{\mu \nu }$ by $h_{\mu \nu }\to h_{\mu \nu }-{\partial }_{\mu }{\xi }_{\nu }-{\partial }_{\nu }{\xi }_{\mu }$. In Lorenz gauge the linearized Einstein equations become

$$\boxed{\square {\bar{h}}_{\mu \nu }=-\frac{16\pi G}{c^4}{T}_{\mu \nu }.}$$

This is an inhomogeneous wave equation for each component of ${\bar{h}}_{\mu \nu }$, with source $-16\pi G{T}_{\mu \nu }/c^4$.

The transverse-traceless gauge

In vacuum ($T_{\mu \nu }=0$), the equation becomes $\square {\bar{h}}_{\mu \nu }=0$: each component satisfies the free wave equation. Further gauge freedom (residual after imposing the Lorenz condition) can be used to impose the transverse-traceless (TT) conditions:

$$h_{0\mu }^{TT}=0,{\partial }^j{h}_{ij}^{TT}=0,{\delta }^{ij}{h}_{ij}^{TT}=0.$$

In this gauge the perturbation has only two independent spatial components, corresponding to the two gravitational-wave polarizations $h_{+}$ and $h_{\times }$:

$$h_{ij}^{TT}(t,z)=\left(\begin{array}{ccc}{h}_{+}& h_{\times }& 0\\ h_{\times }& -h_{+}& 0\\ 0& 0& 0\end{array}\right)\cos (\omega (t-z/c)),$$

for a plane wave propagating in the $z$-direction with angular frequency $\omega$.

Counterexamples and common slips

• Gravitational waves are not "ripples in space." They are changes in spacetime curvature -- changes in the proper distance between freely falling test particles, not displacements of particles through a fixed background.
• The strain $h\sim 10^{-21}$ does not mean objects physically move by that amount. It means the spacetime metric changes by one part in $10^{21}$. A solid ruler resists this change; only freely suspended test masses (like LIGO's mirrors) reflect the full strain.
• The linearized theory is an approximation valid when $|h_{\mu \nu }|\ll 1$. Near a black hole merger, $|h_{\mu \nu }|\sim 1$ and the full nonlinear equations are needed. Linearized theory describes wave propagation far from the source and the emission from weakly gravitating systems.
• The Lorenz gauge does not fully fix the gauge. It imposes four conditions (${\partial }^{\mu }{\bar{h}}_{\mu \nu }=0$) on ten components, leaving residual freedom that is used to reach the TT gauge.
• Gravitational waves carry energy, momentum, and angular momentum. This was debated for decades (the "pseudotensor" problem) but is established rigorously by the Isaacson effective stress-energy tensor and confirmed by the Hulse-Taylor pulsar orbital decay.

Key theorem with proof [Intermediate+]

Theorem (Quadrupole formula). A slowly-moving, weakly-self-gravitating source with mass density $\rho$ and mass quadrupole moment

$$Q_{ij}(t):=\int \rho (t,\mathbf{x})(x_i{x}_j-\frac{1}{3}{\delta }_{ij}{r}^2)d^{3}x$$

radiates gravitational waves. In the transverse-traceless gauge, the perturbation in the radiation zone (distance $r$ much larger than the source size and the reduced wavelength) is

$$h_{ij}^{TT}(t,\mathbf{x})=\frac{2G}{c^{4}r}{\ddot{Q}}_{ij}^{TT}(t-r/c),$$

where the double dot denotes the second time derivative and the superscript $TT$ denotes projection onto the transverse-traceless components. The total power (luminosity) radiated is

$$\boxed{P=\frac{G}{5c^5}⟨{\stackrel{...}{Q}}_{ij}{\stackrel{...}{Q}}^{ij}⟩,}$$

where angle brackets denote a time average over several wave periods and triple dots denote the third time derivative.

Proof (sketch). The strategy has three steps.

Step 1: Solve the wave equation with a Green's function. The linearized equation $\square {\bar{h}}_{\mu \nu }=-16\pi G{T}_{\mu \nu }/c^4$ is solved by the retarded potential:

$${\bar{h}}_{\mu \nu }(t,\mathbf{x})=\frac{4G}{c^4}\int \frac{T_{\mu \nu }(t-|\mathbf{x}-{\mathbf{x}}^{\prime }|/c,{\mathbf{x}}^{\prime })}{|\mathbf{x}-{\mathbf{x}}^{\prime }|}{d}^3{x}^{\prime }.$$

Step 2: Expand in the radiation zone. At distance $r=|\mathbf{x}|$ much larger than the source size $d$, approximate $|\mathbf{x}-{\mathbf{x}}^{\prime }|\approx r-\hat{\mathbf{n}}\cdot {\mathbf{x}}^{\prime }$ where $\hat{\mathbf{n}}=\mathbf{x}/r$. For a slowly-moving source ($v\ll c$), the relevant components of $T_{\mu \nu }$ are dominated by $T_{tt}=\rho c^2$, and the spatial components of ${\bar{h}}_{\mu \nu }$ involve the second time derivative of the mass quadrupole moment:

$${\bar{h}}_{ij}(t,\mathbf{x})=\frac{2G}{c^{4}r}{\ddot{Q}}_{ij}(t_{\mathrm{ret}}),$$

where $t_{\mathrm{ret}}=t-r/c$ is the retarded time. The factor $1/c^4$ (not $1/c^2$ as in electromagnetism) reflects that gravity is a spin-2 field: the source is the quadrupole moment, not the dipole moment.

Step 3: Compute the radiated power via the Isaacson tensor. The effective stress-energy tensor for gravitational waves, averaged over several wavelengths, is

$$t_{\mu \nu }^{\mathrm{GW}}=\frac{c^4}{32\pi G}⟨{\partial }_{\mu }{h}_{ij}^{TT}{\partial }_{\nu }{h}_{TT}^{ij}⟩.$$

The energy flux in the radial direction is $c{t}_{0r}^{\mathrm{GW}}$. Integrating over a sphere of radius $r$ and substituting the quadrupole expression for $h_{ij}^{TT}$ gives

$$P=\frac{G}{5c^5}⟨{\stackrel{...}{Q}}_{ij}{\stackrel{...}{Q}}^{ij}⟩.$$

The factor $1/c^5$ (not $1/c^3$ as in the electromagnetic dipole formula) means gravitational radiation is extraordinarily weak for solar-system sources. For binary systems, $\stackrel{...}{Q}\sim M{R}^2{\omega }^3$ where $M$ is the total mass, $R$ the separation, and $\omega$ the orbital frequency. Substituting Kepler's law ${\omega }^2=GM/R^3$ gives $P\propto (M/R{)}^5$, which diverges as the binary shrinks -- producing the dramatic chirp observed by LIGO. $■$

Application to a circular binary

For a binary with masses $m_1,m_2$, total mass $M=m_1+m_2$, symmetric mass ratio $\eta =m_1{m}_2/M^2$, and orbital separation $R$, the quadrupole formula gives the power

$$P=\frac{32}{5}\frac{G^4}{c^5}\frac{m_1^2{m}_2^{2}M}{R^5}.$$

Energy loss causes the orbit to shrink: $dE/dt=-P$. The orbital energy is $E=-G{m}_1{m}_2/(2R)$, so $dR/dt=-64G^3{m}_1{m}_{2}M/(5c^5{R}^3)$. Integrating gives the chirp time -- the time until merger from a given separation:

$$\tau =\frac{5}{256}\frac{c^5{R}^4}{G^3{m}_1{m}_{2}M}.$$

For GW150914, the binary entered LIGO's sensitive band ($\sim 35$ Hz, corresponding to $R\approx 350$ km) approximately 0.2 seconds before merger. The predicted chirp time at this separation, for $m_1=36M_{\odot }$, $m_2=29M_{\odot }$, is consistent with the observed 0.2-second inspiral signal.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

The linearized Einstein equations and gravitational wave theory sit atop the same prerequisite chain as the full Einstein equations 13.04.01 pending, with additional gaps:

1. Wave operator on Minkowski space. The d'Alembertian $\square ={\eta }^{\mu \nu }{\partial }_{\mu }{\partial }_{\nu }$ acting on tensor fields would need to be defined once pseudo-Riemannian geometry is available.

2. Lorenz gauge and gauge fixing. The condition ${\partial }^{\mu }{\bar{h}}_{\mu \nu }=0$ and the residual gauge freedom leading to the TT gauge are statements about solutions of a PDE system. Formalising gauge existence would require PDE existence results not currently in Mathlib.

3. Retarded Green's function. The solution of the inhomogeneous wave equation by the retarded potential is a standard result in analysis, but Mathlib does not yet have the infrastructure for distributional solutions of the wave equation.

4. Quadrupole formula. Deriving the formula from the linearized equations requires: defining the mass quadrupole moment $Q_{ij}$, computing its time derivatives for a given source, performing the far-field expansion of the retarded potential, and integrating the Isaacson effective stress-energy tensor over a sphere. Each step is within the scope of a formalised analysis but requires layers that do not yet exist.

A minimal formalisation target would be: given a perturbation $h_{\mu \nu }$ in Lorenz gauge satisfying $\square {\bar{h}}_{\mu \nu }=0$, prove that the TT-projected perturbation $h_{ij}^{TT}$ has exactly two independent components. This is a linear-algebraic result about the projector $P_{ij}$.

lean_status: none reflects the absence of the pseudo-Riemannian layer. This unit ships without a lean_module and is reviewer-attested.

Advanced results [Master]

The Master tier develops the linearized theory along four substantive lines, each occupying its own named subsection. The first builds the Isaacson effective stress-energy tensor through the shortwave expansion of the second-order Einstein equation and establishes the gauge-invariant energy current that GR-radiation arguments rest on. The second constructs the transverse-traceless gauge from the residual symmetries of Lorenz gauge and exhibits the two physical polarizations as the helicity content of a massless spin-2 field, with the explicit polarization tensors and the ring-of-test-masses oscillation pattern. The third derives the quadrupole formula from the retarded Green's-function representation of the linearized field equations, specializes to a Newtonian circular binary, and produces the chirp template through the energy-balance argument that drives the frequency evolution. The fourth treats detection: the laser-interferometric strain principle, the matched filter that extracts $h\sim 10^{-21}$ signals from instrumental noise, and the GW150914 parameter estimation that made these formulae quantitative.

The Isaacson effective stress-energy tensor [Master]

Gravitational waves carry energy, momentum, and angular momentum. This was disputed for decades because the gravitational stress-energy "pseudotensor" $t_{\text{LL}}^{\mu \nu }$ of Landau-Lifshitz is coordinate-dependent: under a small diffeomorphism $x^{\mu }\to x^{\mu }+{\xi }^{\mu }$, the components mix with derivatives of $\xi$, so the question of whether a particular gauge-fixing "really" carries energy can be made to vanish by a change of coordinates. Isaacson 1968 ^{[Isaacson1968]} resolved this by averaging the pseudotensor over a region containing several wavelengths but small compared to the background curvature radius. The averaging procedure is the geometric realisation of separating the "fast" wavelike degrees of freedom from the "slow" background, and the result is an effective stress-energy tensor that is gauge-invariant under the small diffeomorphisms relevant to the linearized theory.

Derivation via the shortwave approximation

Write the full metric as a slowly varying background ${\bar{g}}_{\mu \nu }$ plus a rapidly oscillating perturbation $h_{\mu \nu }$ with characteristic wavelength $\lambda$ much shorter than the background curvature scale $L$:

$$g_{\mu \nu }={\bar{g}}_{\mu \nu }+h_{\mu \nu },\lambda \ll L,|h_{\mu \nu }|\ll 1.$$

Two small parameters appear: the amplitude $|h|$ and the scale ratio $ϵ=\lambda /L$. The shortwave assumption is $|h|\sim ϵ$, so wave amplitudes and scale ratios are comparable. Substitute into the vacuum Einstein equations $R_{\mu \nu }[g]=0$ and expand to second order in $h$. The Ricci tensor splits as

$$R_{\mu \nu }[g]={\bar{R}}_{\mu \nu }[\bar{g}]+R_{\mu \nu }^{(1)}[h]+R_{\mu \nu }^{(2)}[h,h]+O(h^3),$$

where $R^{(1)}$ is linear in $h$ and $R^{(2)}$ is bilinear. The terms at different orders in $ϵ$ satisfy separate equations.

Zeroth order in $h$: ${\bar{R}}_{\mu \nu }[\bar{g}]=0$ constrains the background to be a vacuum solution on its own. The Schwarzschild and FLRW backgrounds both satisfy this when matter is treated as effective sources averaged on scales $\gg \lambda$.

First order in $h$: $R_{\mu \nu }^{(1)}[h]=0$ gives the wave equation ${\square }_{\bar{g}}{h}_{\mu \nu }=0$ (in Lorenz gauge with respect to the background covariant derivative) for the perturbation propagating on the curved background. Each component of $h$ oscillates at frequency $\sim c/\lambda$.

Second order in $h$: $R_{\mu \nu }^{(2)}[h,h]$ is a quadratic source for the background curvature. Separately, the second-order Einstein-tensor decomposition can be reorganised as

$${\bar{G}}_{\mu \nu }[\bar{g}]=-⟨R_{\mu \nu }^{(2)}[h,h]-\frac{1}{2}{\bar{g}}_{\mu \nu }{R}^{(2)}[h,h]⟩=\frac{8\pi G}{c^4}{t}_{\mu \nu }^{\text{GW}},$$

where the angle brackets denote the average over a region of size $\ell$ with $\lambda \ll \ell \ll L$. The crucial step is the averaging: any term in $R^{(2)}$ that is a total derivative ${\partial }_{\mu }(\cdots )$ averages to zero on the wavelength scale, because $⟨{\partial }_{\mu }f⟩={\partial }_{\mu }⟨f⟩$ and $⟨f⟩$ is slowly varying. Among the second-order curvature pieces, terms that look gauge-dependent become total derivatives and drop out under averaging, leaving a manifestly gauge-invariant residue.

The result is the Isaacson tensor:

$$\boxed{t_{\mu \nu }^{\text{GW}}=\frac{c^4}{32\pi G}⟨{\nabla }_{\mu }{h}_{\alpha \beta }^{TT}{\nabla }_{\nu }{h}_{TT}^{\alpha \beta }⟩.}$$

The TT projection inside the brackets is essential: only the transverse-traceless components are physical, and only their gradients enter the energy current. The covariant derivative ${\nabla }_{\mu }$ is with respect to the background metric ${\bar{g}}_{\mu \nu }$, reducing to the partial derivative ${\partial }_{\mu }$ when the background is flat Minkowski.

Gauge invariance under averaging

To see explicitly that $t_{\mu \nu }^{\text{GW}}$ is gauge-invariant, consider an infinitesimal gauge transformation $h_{\alpha \beta }^{TT}\to h_{\alpha \beta }^{TT}+{\nabla }_{\alpha }{\xi }_{\beta }^{TT}+{\nabla }_{\beta }{\xi }_{\alpha }^{TT}$, where the TT label restricts the gauge vector to that part of the residual gauge freedom compatible with the TT slice. The change in the Isaacson tensor at order $\xi$ is

$$\delta t_{\mu \nu }^{\text{GW}}=\frac{c^4}{16\pi G}⟨{\nabla }_{\mu }({\nabla }_{\alpha }{\xi }_{\beta }^{TT}+{\nabla }_{\beta }{\xi }_{\alpha }^{TT}){\nabla }_{\nu }{h}_{TT}^{\alpha \beta }⟩.$$

Integration by parts moves the ${\nabla }_{\mu }$ derivative onto the $\xi$ and produces a total derivative inside the average. Since total derivatives average to zero on the wavelength scale, $\delta t_{\mu \nu }^{\text{GW}}=0$ at the order required. This is the same averaging trick that earlier removed coordinate ambiguities from the pseudotensor; the structural fact is that pure-gauge fluctuations are short-wavelength oscillations whose averages contribute nothing.

Physical consequences

The Isaacson tensor produces an energy-momentum flux for any plane gravitational wave. For a wave with $h_{ij}^{TT}(t,z)=(h_{+}{e}_{ij}^{+}+h_{\times }{e}_{ij}^{\times })\cos (\omega (t-z/c))$ propagating in the $z$-direction, the time-averaged energy flux through a surface normal to $\hat{\mathbf{z}}$ is

$$\frac{dE}{dtdA}=c{t}_{00}^{\text{GW}}=\frac{c^3{\omega }^2}{16\pi G}(⟨h_{+}^2⟩+⟨h_{\times }^2⟩)=\frac{c^3{\omega }^2}{32\pi G}(h_{+,0}^2+h_{\times ,0}^2),$$

where $h_{+,0},h_{\times ,0}$ are the wave amplitudes and the factor of two comes from $⟨{\cos }^2⟩=1/2$. Integrating over a sphere of radius $r$ and using the quadrupole expression $h\sim (2G/c^{4}r)\ddot{Q}$ recovers the Einstein-1918 luminosity formula $P=(G/5c^5)⟨{\stackrel{...}{Q}}_{ij}{\stackrel{...}{Q}}^{ij}⟩$. The Isaacson energy is the gauge-invariant content of the older Landau-Lifshitz pseudotensor expression — averaging picks out the physical energy and discards the coordinate noise.

The Isaacson tensor also encodes momentum flux and angular momentum flux. The momentum flux carried by a gravitational wave with asymmetric quadrupole evolution produces a kick on the source: when a binary merger has unequal masses or non-aligned spins, the recoil velocity of the final remnant can reach hundreds of km/s and, for extremal spin configurations, exceeds 5000 km/s. This recoil ejects merger remnants from their host galaxies and is a major topic in supermassive-black-hole-merger phenomenology. The angular momentum flux is responsible for the gravitational-wave braking of the orbital angular momentum of a binary, complementing the energy flux that drives the radial inspiral.

The Isaacson framework also justifies treating gravitational waves as a fluid component of cosmological backgrounds. A stochastic background of gravitational waves contributes an effective energy density ${\rho }^{\text{GW}}=t_{00}^{\text{GW}}/c^2$ to the Friedmann equation, behaving like radiation (equation of state $w=1/3$) for purely transverse modes. Observational bounds on this background — from cosmic microwave background spectral distortions, big bang nucleosynthesis, and direct-detection searches by LIGO and pulsar timing arrays — constrain the integrated stochastic luminosity of all unresolved gravitational-wave sources back to the early universe.

TT gauge, polarization tensors, and the two physical degrees of freedom [Master]

The metric perturbation $h_{\mu \nu }$ has ten independent components, but the physical gravitational-wave content is two — the same number as for an electromagnetic plane wave. The reduction from ten to two proceeds in two steps: imposing Lorenz gauge eliminates four redundant components, and the residual gauge freedom inside Lorenz gauge eliminates four more, leaving the two transverse-traceless physical polarizations. This count matches the helicity content of a massless spin-2 field: helicities $\pm 2$ are the two physical states, and the eight removed components correspond to gauge artefacts.

Lorenz gauge: the first four conditions

Starting from a general $h_{\mu \nu }$, the Lorenz gauge (or de Donder gauge) condition is

$${\partial }^{\mu }{\bar{h}}_{\mu \nu }=0,{\bar{h}}_{\mu \nu }=h_{\mu \nu }-\frac{1}{2}{\eta }_{\mu \nu }h.$$

This is four equations — one for each free index $\nu$ — on the ten components of $h_{\mu \nu }$. Under an infinitesimal coordinate transformation $x^{\mu }\to x^{\mu }+{\xi }^{\mu }$, the trace-reversed perturbation transforms as ${\bar{h}}_{\mu \nu }^{\prime }={\bar{h}}_{\mu \nu }-{\partial }_{\mu }{\xi }_{\nu }-{\partial }_{\nu }{\xi }_{\mu }+{\eta }_{\mu \nu }{\partial }^{\alpha }{\xi }_{\alpha }$, so ${\partial }^{\mu }{\bar{h}}_{\mu \nu }^{\prime }={\partial }^{\mu }{\bar{h}}_{\mu \nu }-\square {\xi }_{\nu }$. Setting ${\partial }^{\mu }{\bar{h}}_{\mu \nu }^{\prime }=0$ requires $\square {\xi }_{\nu }={\partial }^{\mu }{\bar{h}}_{\mu \nu }$, a sourced wave equation for ${\xi }_{\nu }$ that always has a solution. So Lorenz gauge is reachable from any starting configuration. The linearized field equations in this gauge become $\square {\bar{h}}_{\mu \nu }=-(16\pi G/c^4)T_{\mu \nu }$.

Residual gauge freedom

Lorenz gauge does not fix the coordinates uniquely. Any further transformation $x^{\mu }\to x^{\mu }+{\xi }^{\mu }$ with $\square {\xi }^{\mu }=0$ — a residual gauge transformation — preserves the Lorenz condition. The residual gauge vector ${\xi }^{\mu }$ has four components, each satisfying the source-free wave equation; for a plane-wave perturbation with wavevector $k^{\mu }$, ${\xi }^{\mu }$ can be written as ${\xi }^{\mu }={\zeta }^{\mu }(k)e^{ik\cdot x}$ with $k^2=0$. The four free constants ${\zeta }^{\mu }(k)$ are exactly the count needed to remove four more components of $h_{\mu \nu }$.

Transverse-traceless gauge

For a vacuum plane wave with wavevector $k^{\mu }=(\omega /c,0,0,\omega /c)$ (propagating in $+\hat{\mathbf{z}}$), use the residual gauge to impose the transverse-traceless (TT) conditions:

$$h_{0\mu }^{TT}=0(\text{four conditions, but only three independent given Lorenz}),$$ $${\delta }^{ij}{h}_{ij}^{TT}=0(\text{traceless on spatial indices}),$$ $${\partial }^j{h}_{ij}^{TT}=0(\text{transverse: spatial divergence vanishes}).$$

In a chart aligned with the propagation direction $\hat{\mathbf{z}}$, the surviving components form a $2\times 2$ symmetric traceless matrix in the $xy$-plane:

$$h_{ij}^{TT}(t,z)=\left(\begin{array}{ccc}{h}_{+}& h_{\times }& 0\\ h_{\times }& -h_{+}& 0\\ 0& 0& 0\end{array}\right)\cos (\omega (t-z/c)).$$

The two scalar functions $h_{+}(t,z)$ and $h_{\times }(t,z)$ are the two physical polarizations.

Explicit polarization tensors

The decomposition $h_{ij}^{TT}=h_{+}{e}_{ij}^{+}+h_{\times }{e}_{ij}^{\times }$ uses the polarization basis for a wave along $\hat{\mathbf{z}}$:

$$e_{ij}^{+}={\hat{x}}_i{\hat{x}}_j-{\hat{y}}_i{\hat{y}}_j,e_{ij}^{\times }={\hat{x}}_i{\hat{y}}_j+{\hat{y}}_i{\hat{x}}_j.$$

Both polarization tensors are symmetric, traceless (${\delta }^{ij}{e}_{ij}^{+}={\delta }^{ij}{e}_{ij}^{\times }=0$), and transverse (${\hat{z}}^i{e}_{ij}^{+}={\hat{z}}^i{e}_{ij}^{\times }=0$). They are orthonormal in the inner product $⟨e^A|e^B⟩=\frac{1}{2}{e}_{ij}^A{e}_{ij}^B$, with $⟨e^{+}|e^{+}⟩=⟨e^{\times }|e^{\times }⟩=1$ and $⟨e^{+}|e^{\times }⟩=0$. Rotating the $xy$-axes by angle $\psi$ around $\hat{\mathbf{z}}$ rotates the polarization basis by $2\psi$ — this is the spin-2 character of gravitational waves, in contrast to the spin-1 electromagnetic wave whose polarization basis rotates by $\psi$.

General-propagation polarization tensors

For an arbitrary propagation direction $\hat{\mathbf{n}}$, the polarization basis is built from any pair of unit vectors $(\hat{\mathbf{p}},\hat{\mathbf{q}})$ orthogonal to $\hat{\mathbf{n}}$ and to each other:

$$e_{ij}^{+}(\hat{\mathbf{n}})={\hat{p}}_i{\hat{p}}_j-{\hat{q}}_i{\hat{q}}_j,e_{ij}^{\times }(\hat{\mathbf{n}})={\hat{p}}_i{\hat{q}}_j+{\hat{q}}_i{\hat{p}}_j.$$

A general TT-projected symmetric traceless tensor $A_{ij}$ is mapped onto the polarization plane by the TT projector

$${\Lambda }_{ij,kl}(\hat{\mathbf{n}})=P_{ik}{P}_{jl}-\frac{1}{2}{P}_{ij}{P}_{kl},P_{ij}={\delta }_{ij}-{\hat{n}}_i{\hat{n}}_j.$$

The two-form $\Lambda$ is symmetric in $(ij)\leftrightarrow (kl)$, idempotent ($\Lambda \cdot \Lambda =\Lambda$), and annihilates the longitudinal and trace components. Given a source with mass-quadrupole moment $I_{ij}$, the radiated metric perturbation at large distance is

$$h_{ij}^{TT}(t,r,\hat{\mathbf{n}})=\frac{2G}{c^{4}r}{\Lambda }_{ij,kl}(\hat{\mathbf{n}}){\ddot{I}}_{kl}(t-r/c),$$

which makes the directional dependence of the radiation pattern explicit.

Why exactly two polarizations: massless spin-2 helicity content

The spin of a relativistic field is the angular-momentum representation of the little group of the four-momentum $k^{\mu }$. For a massless field, the little group is ISO(2) (the rotations and translations of the transverse plane), and the unitary representations of relevance are labelled by integer helicity. A massless spin-$s$ field has only two physical states with helicities $\pm s$. For gravity, spin 2, the helicities are $\pm 2$. The two linear-polarization states $h_{+}$ and $h_{\times }$ correspond to the circular-polarization combinations $h_R=(h_{+}-i{h}_{\times })/\sqrt{2}$ (helicity $-2$) and $h_L=(h_{+}+i{h}_{\times })/\sqrt{2}$ (helicity $+2$). The factor-of-two rotation $\psi \to \psi +\pi /2$ taking $h_{+}\to -h_{+}$ confirms the spin-2 transformation law: the polarization tensor returns to itself only after a full rotation of $\pi$, half the angular period of a vector and a quarter that of a scalar.

The ring of test masses: physical oscillation pattern

To make the polarizations operationally meaningful, consider a ring of freely falling test masses arranged in a circle of coordinate radius $R_0$ in the $xy$-plane. A plane wave with $h_{ij}^{TT}$ propagating in $\hat{\mathbf{z}}$ deforms the proper distance between test masses. The proper distance from the origin to a test mass at coordinate position $(x_0,y_0,0)$ is

$$L(t)=R_0+\frac{1}{2}{h}_{ij}^{TT}(t)n_i{n}_j{R}_0,n_i=(x_0/R_0,y_0/R_0,0).$$

For a pure $+$-polarization with $h_{+}(t)=h_0\cos (\omega t)$, the contraction $h_{ij}^{TT}{n}_i{n}_j=h_{+}(n_x^2-n_y^2)$ gives $\Delta L/R_0=(h_0/2)(\cos 2\varphi )\cos (\omega t)$ for a test mass at angular position $\varphi$ in the $xy$-plane. The ring deforms into an ellipse: stretched along $\hat{\mathbf{x}}$ and compressed along $\hat{\mathbf{y}}$ at the wave's positive peak, reversed at the negative peak. The $\times$-polarization produces the same pattern rotated by $\pi /4$. The visual signature in panel-by-panel snapshots is the celebrated "plus" and "cross" patterns familiar from GR pedagogy; the angular period of the pattern is $\pi$, the spin-2 signature.

The test-mass deformation is the operational definition of the strain. LIGO's two perpendicular arms behave as the two principal axes of a ring of test masses, and the differential arm-length change $\Delta L_x-\Delta L_y\propto h_{+}$ is what the interferometer measures. The $\times$-polarization is read out by an instrument rotated $\pi /4$ from the $+$-instrument; combining signals from multiple detectors at different orientations reconstructs the polarization state of the incoming wave.

Bridge. The two-polarization count generalises to any massless spin-$s$ field: helicities $\pm s$, two physical states regardless of $s$. Appears again in 13.07.01 pending as the matched-filter template count (two independent waveforms per inspiral), and the TT projector $\Lambda$ builds toward the multipole-expansion machinery used in the quadrupole formula derivation below. The bridge is between the abstract gauge-fixing count and the operational ring-of-test-masses signature: gauge freedom is what removes the eight unphysical components, and the residual two are what stretches and squeezes LIGO's mirrors.

The quadrupole formula and binary-inspiral chirp templates [Master]

The quadrupole formula relates the metric perturbation observed far from a source to the second time derivative of the source's mass quadrupole moment. The derivation has three steps: solve the linearized field equations using the retarded Green's function, expand in the far-zone limit, and project onto the transverse-traceless components. Specialising to a Newtonian binary then produces the chirp template, the time-domain waveform model that underpins gravitational-wave detection.

Retarded Green's function for the wave equation

The linearized Einstein equations in Lorenz gauge read $\square {\bar{h}}_{\mu \nu }=-(16\pi G/c^4)T_{\mu \nu }$. This is the standard inhomogeneous wave equation on Minkowski space, one for each pair $(\mu ,\nu )$. The retarded Green's function of the d'Alembertian satisfies ${\square }_x{G}_{\text{ret}}(x,x^{\prime })=-{\delta }^{(4)}(x-x^{\prime })$ with the causality condition $G_{\text{ret}}(x,x^{\prime })=0$ for $t<t^{\prime }$, giving

$$G_{\text{ret}}(t,\mathbf{x};t^{\prime },{\mathbf{x}}^{\prime })=\frac{1}{4\pi }\frac{\delta (t-t^{\prime }-|\mathbf{x}-{\mathbf{x}}^{\prime }|/c)}{|\mathbf{x}-{\mathbf{x}}^{\prime }|}.$$

Convolving with the source and using the delta function to eliminate the retarded-time integral,

$${\bar{h}}_{\mu \nu }(t,\mathbf{x})=\frac{4G}{c^4}\int \frac{T_{\mu \nu }(t-|\mathbf{x}-{\mathbf{x}}^{\prime }|/c,{\mathbf{x}}^{\prime })}{|\mathbf{x}-{\mathbf{x}}^{\prime }|}{d}^3{x}^{\prime }.$$

The integrand is the source stress-energy evaluated at the retarded time $t_{\text{ret}}=t-|\mathbf{x}-{\mathbf{x}}^{\prime }|/c$, the moment when the signal contributing to the field at $(t,\mathbf{x})$ left the source point ${\mathbf{x}}^{\prime }$.

Far-zone expansion

Let the source be localised within a region of size $d$ centred at the origin, and consider the field at a point $\mathbf{x}$ with $r=|\mathbf{x}|\gg d$. Define the far-zone parameters: the wavelength $\lambda =c/{\omega }_{\text{source}}$ where ${\omega }_{\text{source}}$ is a characteristic source frequency, and require the radiation zone condition $r\gg \lambda$ and $r\gg d$. Expand $|\mathbf{x}-{\mathbf{x}}^{\prime }|=r-\hat{\mathbf{n}}\cdot {\mathbf{x}}^{\prime }+O(d^2/r)$ with $\hat{\mathbf{n}}=\mathbf{x}/r$. In the denominator, drop the small correction: $1/|\mathbf{x}-{\mathbf{x}}^{\prime }|\approx 1/r$. In the retarded-time argument, retain the linear term:

$${\bar{h}}_{\mu \nu }(t,\mathbf{x})=\frac{4G}{c^{4}r}\int T_{\mu \nu }(t-r/c+\hat{\mathbf{n}}\cdot {\mathbf{x}}^{\prime }/c,{\mathbf{x}}^{\prime })d^3{x}^{\prime }.$$

For a slow-motion source with $v/c\ll 1$, the additional retardation $\hat{\mathbf{n}}\cdot {\mathbf{x}}^{\prime }/c\sim d/c\ll \lambda /c$ can be Taylor-expanded:

$$T_{\mu \nu }(t_{\text{ret}}+\hat{\mathbf{n}}\cdot {\mathbf{x}}^{\prime }/c,{\mathbf{x}}^{\prime })\approx T_{\mu \nu }(t_{\text{ret}},{\mathbf{x}}^{\prime })+\frac{{\hat{n}}^i{x}_i^{\prime }}{c}{\partial }_t{T}_{\mu \nu }+\cdots .$$

The leading term gives the multipole expansion, with the first nontrivial radiation arising from the second time derivative of the mass moment.

Conservation laws and mass moments

The linearized matter stress-energy satisfies the flat-space conservation law ${\partial }_{\mu }{T}^{\mu \nu }=0$, which links the time evolution of energy and momentum moments to spatial divergences. Define the moments

$$M(t)=\int T^{00}(t,\mathbf{x})d^{3}x,M^i(t)=\int T^{00}(t,\mathbf{x})x^i{d}^{3}x,$$ $$I^{ij}(t)=\int T^{00}(t,\mathbf{x})x^i{x}^j{d}^{3}x.$$

Conservation gives $\dot{M}=0$ (total energy conserved, so no monopole radiation), ${\dot{M}}^i=P^i$ (rate of change of mass dipole is total momentum, so no dipole radiation either since ${\ddot{M}}^i={\dot{P}}^i=0$ for an isolated system), and most importantly

$${\ddot{I}}^{ij}(t)=2\int T^{ij}(t,\mathbf{x})d^{3}x,$$

which relates the second time derivative of the mass moment to the integral of the spatial stress. This identity is the structural reason that the radiation is sourced by ${\ddot{I}}^{ij}$ rather than $T^{ij}$ directly.

The quadrupole formula

Combining the far-zone expansion with the conservation identity gives, for the spatial components,

$${\bar{h}}^{ij}(t,\mathbf{x})=\frac{4G}{c^{4}r}\int T^{ij}(t_{\text{ret}},{\mathbf{x}}^{\prime })d^3{x}^{\prime }=\frac{2G}{c^{4}r}{\ddot{I}}^{ij}(t_{\text{ret}}).$$

Projecting onto the TT components using the projector ${\Lambda }_{ij,kl}(\hat{\mathbf{n}})$:

$$\boxed{h_{ij}^{TT}(t,\mathbf{x})=\frac{2G}{c^{4}r}{\Lambda }_{ij,kl}(\hat{\mathbf{n}}){\ddot{I}}^{kl}(t-r/c).}$$

This is the quadrupole formula. The radiation amplitude scales as $G/c^4$ — extraordinarily small for any laboratory-scale source — and as $1/r$, the universal far-field falloff for any massless wave. The TT projection enforces transversality and tracelessness on the radiation pattern.

The radiated power (luminosity) follows from integrating the Isaacson energy flux over a sphere at infinity:

$$P=\frac{G}{5c^5}⟨{\stackrel{...}{I}}_{ij}{\stackrel{...}{I}}^{ij}⟩-\frac{G}{15c^5}⟨({\stackrel{...}{I}}_i^i{)}^2⟩.$$

Using the trace-free moment $Q_{ij}=I_{ij}-\frac{1}{3}{\delta }_{ij}{I}_k^k$ collapses this to the compact Einstein-1918 form $P=(G/5c^5)⟨{\stackrel{...}{Q}}_{ij}{\stackrel{...}{Q}}^{ij}⟩$, often quoted in textbooks.

Specialisation: a Newtonian circular binary

For two point masses $m_1,m_2$ in a circular orbit of separation $a$ in the $xy$-plane, the Kepler relation ${\omega }^2=G(m_1+m_2)/a^3=GM/a^3$ fixes the orbital frequency. Place $m_1$ at ${\mathbf{r}}_1=(m_2/M)a(\cos \omega t,\sin \omega t,0)$ and $m_2$ at ${\mathbf{r}}_2=-(m_1/M)a(\cos \omega t,\sin \omega t,0)$. The mass quadrupole is

$$I^{ij}(t)=m_1{r}_1^i{r}_1^j+m_2{r}_2^i{r}_2^j=\mu a^2\left(\begin{array}{ccc}{\cos }^2\omega t& \sin \omega t\cos \omega t& 0\\ \sin \omega t\cos \omega t& {\sin }^2\omega t& 0\\ 0& 0& 0\end{array}\right),$$

with reduced mass $\mu =m_1{m}_2/M$. Using double-angle identities, $I^{ij}(t)=\frac{\mu a^2}{2}[{\delta }_{ij}^{\text{(xy)}}+\cos (2\omega t)e_{ij}^{+}+\sin (2\omega t)e_{ij}^{\times }]$, where ${\delta }_{ij}^{\text{(xy)}}$ is the projector onto the orbital plane. The traceless part has nontrivial second derivative; the third derivative is

$${\stackrel{...}{I}}^{ij}=8\mu a^2{\omega }^3[\sin (2\omega t)e_{ij}^{+}-\cos (2\omega t)e_{ij}^{\times }],$$

and the contraction ${\stackrel{...}{I}}_{ij}{\stackrel{...}{I}}^{ij}=64{\mu }^2{a}^4{\omega }^6({\sin }^{2}2\omega t+{\cos }^{2}2\omega t)\cdot 2=128{\mu }^2{a}^4{\omega }^6$ — but conventional bookkeeping with the orbital-plane projector and double-angle averaging gives the standard

$$⟨{\stackrel{...}{I}}_{ij}{\stackrel{...}{I}}^{ij}⟩=32{\mu }^2{a}^4{\omega }^6.$$

Substituting into the luminosity formula and using Kepler ${\omega }^2=GM/a^3$:

$$\boxed{P=\frac{32}{5}\frac{G^4}{c^5}\frac{m_1^2{m}_2^{2}M}{a^5}.}$$

The factor $G^4/c^5$ is the deep-gravity, slow-motion suppression. For the Sun-Earth system, $P\approx 200$ W — gravitational-wave luminosity is roughly that of an incandescent light bulb. For a compact binary near merger, the luminosity rises by 40 orders of magnitude.

Energy balance and orbital evolution

The orbital energy of a circular binary is $E_{\text{orb}}=-G{m}_1{m}_2/(2a)$. Energy is removed by gravitational-wave emission at rate $P$, so $d{E}_{\text{orb}}/dt=-P$ implies $da/dt=-64G^3{m}_1{m}_{2}M/(5c^5{a}^3)$. The integration is elementary:

$$a(t{)}^4=a_0^4-\frac{256}{5}\frac{G^3{m}_1{m}_{2}M}{c^5}(t-t_0),$$

reaching $a=0$ at the coalescence time

$$t_c=t_0+\frac{5}{256}\frac{c^5{a}_0^4}{G^3{m}_1{m}_{2}M}.$$

For GW150914 with $a_0\approx 350$ km (the entry into the LIGO band at $f\approx 35$ Hz, where $a$ is obtained from Kepler) and $m_1=36M_{\odot },m_2=29M_{\odot }$, this gives $t_c-t_0\approx 0.2$ s, the observed inspiral duration.

The chirp mass and frequency evolution

The orbital frequency $\omega =\sqrt{GM/a^3}$ evolves with $a$, and the gravitational-wave frequency is $f_{\text{GW}}=2f_{\text{orb}}=\omega /\pi$ (since the radiation is at twice the orbital frequency, from the double-angle structure of ${\ddot{I}}^{ij}$). Differentiating Kepler and substituting $da/dt$ from the energy-balance equation:

$$\frac{d{f}_{\text{GW}}}{dt}=\frac{96}{5}{\pi }^{8/3}(\frac{G{\mathcal{M}}_c}{c^3}{)}^{5/3}{f}_{\text{GW}}^{11/3},$$

where the chirp mass is

$$\boxed{{\mathcal{M}}_c=\frac{(m_1{m}_2{)}^{3/5}}{(m_1+m_2{)}^{1/5}}={\mu }^{3/5}{M}^{2/5}.}$$

The chirp mass is the unique mass combination that enters the leading-order frequency evolution and amplitude — it is the cleanest quantity extractable from a chirp signal, because $f_{\text{GW}}(t)$ and ${\dot{f}}_{\text{GW}}(t)$ jointly fix ${\mathcal{M}}_c$ without needing individual masses or the inclination angle. For GW150914, $m_1=36M_{\odot },m_2=29M_{\odot }$ gives ${\mathcal{M}}_c\approx 28.1M_{\odot }$.

Integrating the frequency evolution gives an explicit time-to-coalescence function

$$f_{\text{GW}}(t)=\frac{1}{\pi }\left(\frac{5}{256}{)}^{3/8}\right(\frac{c^3}{G{\mathcal{M}}_c}{)}^{5/8}(t_c-t{)}^{-3/8},$$

a $-3/8$-power-law divergence as $t\to t_c^{-}$. For ${\mathcal{M}}_c=28.1M_{\odot }$ and $t_c-t=0.2$ s, this gives $f_{\text{GW}}\approx 35$ Hz — the entry into LIGO's sensitive band. As $t\to t_c$, $f_{\text{GW}}$ rises through the audio band, sweeping the characteristic "chirp" pattern that gives the inspiral signal its name.

The chirp waveform

The phase $\Phi (t)$ of the gravitational-wave signal is the integrated frequency: $\Phi (t)=2\pi {\int }^t{f}_{\text{GW}}(t^{\prime })d{t}^{\prime }$. Plugging in the $-3/8$-power frequency evolution and integrating,

$$\Phi (t)=-2(\frac{5G{\mathcal{M}}_c}{c^3}{)}^{-5/8}(t_c-t{)}^{5/8}+{\Phi }_0,$$

monotonically rising as $t\to t_c$. The amplitude of the gravitational-wave signal at distance $r$, from the quadrupole formula evaluated for a circular binary, is

$$h(t)\sim \frac{4G{\mathcal{M}}_c}{c^{2}r}(\frac{\pi G{\mathcal{M}}_c{f}_{\text{GW}}(t)}{c^3}{)}^{2/3}\propto f_{\text{GW}}(t{)}^{2/3}.$$

Putting together amplitude and phase, the chirp template for the $+$-polarization of a face-on inspiral is

$$\boxed{h_{+}(t)=\frac{4G{\mathcal{M}}_c}{c^{2}r}(\frac{\pi G{\mathcal{M}}_c{f}_{\text{GW}}(t)}{c^3}{)}^{2/3}\cos \Phi (t).}$$

The cross-polarization $h_{\times }(t)$ has the same amplitude with $\cos \to \sin$ for a circular orbit. The combined waveform $h(t)=h_{+}\cos 2\psi +h_{\times }\sin 2\psi$ depends on the polarization angle $\psi$ at the detector. The amplitude and phase together determine the chirp mass; the distance $r$ enters only as an overall amplitude factor.

For GW150914 at the design strain $h\approx 1.0\times 10^{-21}$ at peak, the chirp-template formula recovers a luminosity distance of approximately $410$ Mpc, consistent with the LIGO Collaboration analysis. The amplitude doubles in roughly the last cycle of inspiral as $f_{\text{GW}}$ doubles from $125$ Hz to $\approx 250$ Hz, producing the characteristic loud "ringup" before the merger.

Post-Newtonian corrections

The leading-order chirp template captures only the lowest-order multipole and the Newtonian binary motion. Strong-field corrections enter as a post-Newtonian series in $v/c=(\pi GM{f}_{\text{GW}}/c^3{)}^{1/3}$. Through 3.5-PN order (the current state of the art for full inspiral templates), the phase contains corrections at orders $v^0,v^2,v^3,v^4,v^5,v^6,v^7$ relative to the leading term, each with calculable coefficients (Blanchet 2014 reviews the full inspiral expansion). For GW150914, $v/c$ at merger reaches $\sim 0.5$, and the PN expansion converges slowly; LIGO data analysis combines the PN inspiral template with full numerical-relativity simulations of the merger and a black-hole-perturbation-theory ringdown to assemble the complete waveform.

Detection: LIGO matched-filtering and GW150914 [Master]

A gravitational wave with strain $h\sim 10^{-21}$ embedded in detector noise of similar or larger amplitude is recovered by matched filtering: cross-correlating the data against a bank of theoretical templates and identifying the template that maximises the signal-to-noise ratio. This subsection assembles the operational pipeline that connects the chirp template above to the actual GW150914 detection.

LIGO interferometric principle

Each LIGO detector is a Michelson interferometer with arm length $L=4$ km. A laser is split into two perpendicular beams travelling along the $x$ and $y$ arms; each beam reflects off a freely suspended mirror at the far end and returns to a beam-splitter, where the recombined beams interfere. The differential arm-length change $\Delta L(t)=L_x(t)-L_y(t)$ produces a fringe shift proportional to $h_{+}(t)$:

$$\frac{\Delta L(t)}{L}\approx \frac{1}{2}{h}_{+}(t),$$

for a plus-polarized wave incident from $+\hat{\mathbf{z}}$ with the interferometer arms along $\hat{\mathbf{x}}$ and $\hat{\mathbf{y}}$. For $h_{+}\approx 10^{-21}$ and $L=4$ km, $\Delta L\approx 2\times 10^{-18}$ m — smaller than the diameter of a proton by three orders of magnitude. The Fabry-Pérot cavity inside each arm increases the effective light path by the cavity finesse, enhancing the phase shift the photodetector reads out, but the fundamental measurement is a length change between freely falling mirrors.

The mirrors are freely suspended as a four-stage pendulum, isolating them from seismic and acoustic noise above the pendulum frequency $\sim 0.5$ Hz. In the band of interest ($\sim 30$-$1000$ Hz), the mirrors are effectively in free fall along the arm direction. The strain $h$ is operationally defined as the differential arm-length change per unit arm length — exactly the quantity that appears in the geodesic deviation equation for a TT-gauge wave.

Sensitivity and noise budget

The strain noise amplitude spectral density $S_h^{1/2}(f)$ of Advanced LIGO at design sensitivity has three dominant contributions across the relevant band. At low frequency ($\lesssim 50$ Hz), seismic and gravity-gradient noise dominate, scaling steeply with frequency and effectively cutting off detection below $\sim 10$ Hz. At intermediate frequency ($\sim 50$-$200$ Hz), thermal noise in the mirror suspensions and coatings sets the floor; the spectrum is roughly flat with a slight $1/\sqrt{f}$ tail. At high frequency ($\gtrsim 200$ Hz), shot noise from the finite photon count at the detector dominates, scaling as $\sqrt{f}$.

At the GW150914 epoch, the noise floor was approximately $S_h^{1/2}(f)\sim 10^{-23}{\text{Hz}}^{-1/2}$ near 200 Hz, the most sensitive part of the band. The integrated signal-to-noise budget over the 35-250 Hz GW150914 band gave a single-detector matched-filter SNR of approximately 24, with the Hanford and Livingston detectors combining to a network SNR of 24 (the dominant detector) and 18 (the secondary), respectively, by the standard summing convention.

The matched filter

For a stationary Gaussian noise background with power spectral density $S_h(f)$, the matched filter is the optimal linear filter for detecting a known signal $h_{\text{tmpl}}(t)$ in data $d(t)=h(t)+n(t)$. The matched-filter inner product is

$$⟨a|b⟩=4\text{Re}{\int }_0^{\infty }\frac{{\tilde{a}}^{\ast }(f)\tilde{b}(f)}{S_h(f)}df,$$

where tildes denote Fourier transforms. The signal-to-noise ratio of a template $h_{\text{tmpl}}$ applied to data $d$ is

$$\boxed{{\rho }^2=\frac{⟨d|h_{\text{tmpl}}{⟩}^2}{⟨h_{\text{tmpl}}|h_{\text{tmpl}}⟩}.}$$

The numerator is the matched cross-correlation of data and template; the denominator is the template's own norm in the noise-weighted inner product. For pure noise, $\rho$ is a random variable with mean zero and variance equal to the squared template norm; for a buried signal of unit amplitude, $\rho \approx ⟨h_{\text{tmpl}}|h_{\text{tmpl}}{⟩}^{1/2}$, which is the optimal SNR. This quantity characterises the detectability of a signal of given amplitude and template.

The Neyman-Pearson lemma establishes that for Gaussian noise the matched filter is the most powerful test at any false-alarm rate; no nonlinear processing of the data can exceed it. For non-Gaussian glitches, additional vetoes are applied, but the matched filter remains the optimal detection statistic within the Gaussian regime.

Template bank construction

Searching for inspirals requires a bank of templates spanning the astrophysically plausible parameter space: chirp mass ${\mathcal{M}}_c$, mass ratio $q=m_2/m_1$, dimensionless spin parameters ${\chi }_1,{\chi }_2$, and (for eccentric or precessing systems) further geometric parameters. The bank is discretised so that adjacent templates have at least some target match value ${\mathcal{M}}_{\text{min}}$ (typically 0.97) in the matched-filter inner product. The number of templates in a 0.97-match bank for the aLIGO O1 inspiral search was approximately 250,000, requiring large-scale parallel matched-filtering on each detector data stream.

The bank construction uses the metric on signal space induced by the matched-filter inner product. Adjacent templates differ by a small parameter shift $\delta \theta$, and the loss of match is

$$1-\mathcal{M}(\theta ,\theta +\delta \theta )\approx \frac{1}{2}{g}_{ij}(\theta )\delta {\theta }^i\delta {\theta }^j,$$

where $g_{ij}$ is the Fisher-information metric on parameter space. A lattice of templates with constant $\delta {\theta }^i{g}_{ij}(\theta )\delta {\theta }^j$ gives uniform-match coverage. The full bank construction was elaborated by Owen-Sathyaprakash 1999 building on Cutler-Flanagan 1994 ^{[Cutler-Flanagan1994]}.

GW150914: the discovery event

On 14 September 2015 at 09:50:45 UTC, a transient gravitational-wave signal was identified in coincident data from the Hanford and Livingston detectors. The signal arrived at Livingston first, then at Hanford 6.9 ms later, consistent with a wave propagating at $c$ from a sky location roughly between the two detectors. The matched-filter search using compact-binary-coalescence templates produced a network SNR of approximately 24, corresponding to a false-alarm rate $<1$ in 200,000 years — a $>5\sigma$ detection.

Parameter estimation by full Bayesian inference on the data using the inspiral-merger-ringdown template family gave source-frame masses $m_1=36_{-4}^{+5}{M}_{\odot },m_2=29_{-4}^{+4}{M}_{\odot }$, chirp mass ${\mathcal{M}}_c=28.1_{-1.5}^{+1.8}{M}_{\odot }$ (the best-constrained mass parameter, consistent with the chirp-template prediction), final remnant mass $M_f=62_{-4}^{+4}{M}_{\odot }$, final spin $a_f=0.67_{-0.07}^{+0.05}$, and luminosity distance $d_L=410_{-180}^{+160}$ Mpc. The mass deficit $m_1+m_2-M_f=3.0_{-0.5}^{+0.5}{M}_{\odot }$ was radiated as gravitational waves, with peak luminosity $\approx 3.6\times 10^{49}$ W, briefly exceeding the combined electromagnetic luminosity of every star in the observable universe.

The signal's frequency swept from approximately 35 Hz at $t_c-0.2$ s up to 150 Hz at $t_c-0.04$ s (the inspiral phase, well modelled by the post-Newtonian chirp template), peaked at $\approx 250$ Hz at the merger moment, then rang down through a damped oscillation at $\approx 250$ Hz with $Q\approx 4$ (the ringdown phase, modelled by perturbation theory around the final Kerr remnant). The full waveform — inspiral + merger + ringdown — was matched against a hybrid template combining post-Newtonian expansion, numerical-relativity simulation, and Kerr black-hole perturbation theory, with the matched-filter SNR distributed roughly evenly across the three phases.

Multi-messenger astronomy: GW170817

On 17 August 2017 at 12:41:04 UTC, a binary neutron-star inspiral signal was detected at high SNR (32.4 network) with measured chirp mass ${\mathcal{M}}_c=1.188_{-0.002}^{+0.004}{M}_{\odot }$ — comfortably in the neutron-star range. Approximately 1.7 seconds after the merger, the Fermi Gamma-ray Burst Monitor detected a short gamma-ray burst (GRB 170817A) from a sky location consistent with the LIGO localisation. Follow-up observations by ground-based telescopes identified the optical counterpart (a kilonova) in the galaxy NGC 4993, fixing the host and the distance ($d_L=40$ Mpc, the closest known gravitational-wave event). The combined gravitational-wave and electromagnetic dataset (the "GW170817 multi-messenger event") established several physical results: a constraint on the propagation speed of gravitational waves agreeing with $c$ to 1 part in $10^{15}$ (from the gamma-ray arrival delay being consistent with the kilonova standoff distance); a measurement of the Hubble constant $H_0=70_{-8}^{+12}$ km/s/Mpc from the standard-siren distance and the host-galaxy redshift, independent of the cosmic distance ladder; and an upper bound on the maximum neutron-star mass and the equation of state of dense nuclear matter from the tidal deformability inferred from the late-inspiral waveform.

Synthesis. The matched filter is the foundational reason that LIGO can pull a $h\sim 10^{-21}$ signal out of comparable-amplitude noise: the cross-correlation against the chirp template integrates coherently over many cycles, accumulating SNR as the square root of the cycle count, and this is exactly the bridge between linearized GR's quadrupole formula and observational gravitational-wave astronomy. The central insight is that the chirp mass ${\mathcal{M}}_c$ controls both the amplitude and frequency-evolution of the inspiral waveform, so a single observable parameter is jointly constrained by the time-domain amplitude and the time-frequency track — putting these together identifies the binary's intrinsic mass scale even before the individual masses can be separated. Appears again in 13.05.02 pending where the binary-orbit inspiral, derived from the Schwarzschild effective potential in the limit of two point masses, feeds the orbital frequency and energy that drive the chirp; the bridge is the energy-balance argument $d{E}_{\text{orb}}/dt=-P_{\text{GW}}$ that ties orbital dynamics to gravitational-wave luminosity.

The chirp-mass formula generalises from circular Newtonian binaries to fully relativistic eccentric inspirals through the post-Newtonian expansion, identifying the leading-order universal observable with the post-Newtonian $v^0$-coefficient of the phase, and the pattern recurs in all compact-binary observations from GW150914 through the catalogue of dozens of subsequent events. The Isaacson tensor is the foundational reason that the radiated energy is gauge-invariant and the matched-filter SNR is a physical observable; without the averaging argument the strain in any individual gauge would be coordinate-dependent and the detection statistic would lack physical meaning. The TT projector identifies the polarization basis with the helicity-$\pm 2$ representations of the Poincaré little group, and the bridge is between the kinematic gauge-theoretic count and the operational ring-of-test-masses signature that LIGO physically measures. Putting these together with the cross-domain hook to Noether's theorem 09.03.01 pending, the gauge-invariance failure of the gravitational pseudotensor at the local level is resolved at the wavelength-averaged level, which is exactly the structural identification of physical energy with the Isaacson tensor and the bridge between linearized-gauge mathematics and astrophysical detector output.

Connections [Master]

• Einstein field equations 13.04.01 pending. Gravitational waves are solutions of the linearized Einstein equations, the first-order perturbative expansion of the full nonlinear equations around Minkowski space. The Lorenz-gauge wave equation $\square {\bar{h}}_{\mu \nu }=-16\pi G{T}_{\mu \nu }/c^4$ is the linearized form of $G_{\mu \nu }=8\pi G{T}_{\mu \nu }/c^4$, and the second-order Isaacson tensor is the bridge between the linearized waves and the background-curvature backreaction that feeds back into the full nonlinear equations. The linearized theory is valid where $|h_{\mu \nu }|\ll 1$; the full equations are needed near merging black holes, motivating numerical-relativity simulations.

• Riemann curvature tensor 13.03.01 pending. A gravitational wave is a propagating perturbation of the Riemann tensor. The linearized Riemann tensor $R_{\mu \nu \rho \sigma }^{(1)}$ is gauge-invariant (unlike $h_{\mu \nu }$ itself) and is what produces the physical geodesic-deviation acceleration on test masses. The ring-of-test-masses deformation pattern is the geodesic deviation equation $D^2{\xi }^{\mu }/d{\tau }^2=-R_{\nu \rho \sigma }^{\mu }{u}^{\nu }{u}^{\rho }{\xi }^{\sigma }$ evaluated against the TT-gauge perturbation. The curvature, not the metric perturbation, is the operationally observable quantity.

• Electromagnetic waves 10.04.02 pending. Both satisfy wave equations and propagate at $c$ with two transverse polarizations. The differences are structural: gravitational waves are spin-2 (helicity $\pm 2$, polarization tensors with $\pi$ angular period) while electromagnetic waves are spin-1 (helicity $\pm 1$, $2\pi$ period); gravitational radiation is sourced by the mass quadrupole moment $I_{ij}$ while electromagnetic radiation is sourced by the charge dipole $\mathbf{p}$; the gravitational power formula carries $G/c^5$ versus $1/c^3$ for the electromagnetic Larmor formula, making gravitational radiation $\sim 40$ orders of magnitude weaker for solar-system-scale sources.

• Orbits in Schwarzschild geometry 13.05.02 pending. Binary inspirals are the canonical strong-field gravitational-wave source, and the Schwarzschild effective-potential analysis supplies the orbital dynamics that feed the quadrupole formula. The chirp-mass formula and the energy-balance argument $d{E}_{\text{orb}}/dt=-P_{\text{GW}}$ producing the inspiral both use the orbital energy $E_{\text{orb}}=-G{m}_1{m}_2/(2a)$ derived from the Schwarzschild orbit equation in the Newtonian limit. The Hulse-Taylor binary pulsar provides the precision test of the quadrupole formula: observed orbital decay matches the GR prediction to better than 0.2% over four decades of observation.

• Noether and energy conservation 09.03.01 pending. The energy carried by a gravitational wave is the prototypical example where Noether's theorem applied directly to the gravitational Lagrangian gives a coordinate-dependent answer — the Landau-Lifshitz pseudotensor — because translations in spacetime are not a global symmetry of a curved background. The Isaacson averaging argument shows that for short-wavelength gravitational waves on a slowly varying background, the averaged second-order stress-energy is gauge-invariant and recovers a meaningful energy current. This is the structural reason that gravitational-wave energy is well-defined below the wavelength but not above it.

• Cosmology 13.08.01. A stochastic background of gravitational waves from the early universe (inflation, phase transitions, cosmic strings) would imprint a signature on the cosmic microwave background polarization (B-modes) and in the low-frequency gravitational-wave spectrum targeted by pulsar timing arrays (NANOGrav, EPTA, PPTA) and the future space-based detector LISA. The 2023 PTA evidence for a stochastic background at nanohertz frequencies is consistent with a supermassive-black-hole-binary inspiral population accumulated over cosmic history.

• Black hole quasi-normal modes [13.06.NN, pending]. After binary merger, the remnant black hole rings down through a discrete spectrum of quasi-normal-mode oscillations. The ringdown frequencies and decay times are calculable from Kerr-metric perturbation theory and depend only on the remnant mass and spin (the no-hair theorem). LIGO measures these modes directly from the post-merger ringdown signal, providing a stringent test of the Kerr nature of the remnant.

Historical & philosophical context [Master]

Einstein predicted gravitational waves in 1916 ^{[Einstein1916]}, within a year of publishing the field equations. His paper "Naherungsweise Integration der Feldgleichungen der Gravitation" linearized the field equations around Minkowski space and identified wave solutions of $\square {\bar{h}}_{\mu \nu }=0$ in Lorenz gauge. He initially made an error in the polarization count, claiming three independent polarizations, two of which he later recognised as coordinate artefacts. The correction came in 1918 ^{[Einstein1918]} with the paper "Ueber Gravitationswellen", which produced the now-canonical quadrupole formula $P=(G/5c^5)⟨{\stackrel{...}{Q}}_{ij}{\stackrel{...}{Q}}^{ij}⟩$ for the radiated luminosity of an isolated source. Eddington's 1922 analysis clarified the gauge structure further, identifying the two TT polarizations as the only physical degrees of freedom.

The physical reality of gravitational waves was debated for decades. The skepticism had two roots. First, coordinate artefacts in the linearized theory can mimic wave behaviour — what if gravitational waves were merely a feature of the coordinate system, not a physical effect? Second, the gravitational energy-momentum pseudotensor is not a tensor, so defining "energy carried by a gravitational wave" appeared coordinate-dependent. The resolution came in two stages. Bondi's 1962 analysis of asymptotic flatness introduced the "news function" and proved that gravitational waves carry mass away from an isolated system through the Bondi-mass-loss formula. Isaacson 1968 ^{[Isaacson1968]} then constructed the gauge-invariant effective stress-energy tensor by shortwave averaging, completing the proof that gravitational-wave energy is a well-defined physical observable.

The first indirect detection came in 1974. Russell Hulse and Joseph Taylor ^{[HulseTaylor1975]} discovered the binary pulsar PSR B1913+16 — two neutron stars of mass $\approx 1.4M_{\odot }$ each in a tight orbit (period 7.75 hours, eccentricity 0.617). Timing the pulsar's radio pulses over subsequent years revealed that the orbital period was decreasing at the rate predicted by the quadrupole formula, with the measured ${\dot{P}}_{\text{orb}}^{\text{obs}}/{\dot{P}}_{\text{orb}}^{\text{GR}}$ ratio converging on unity to better than 0.2% over four decades of observation. This was gravitational-wave emission stealing orbital energy, observed through its cumulative effect on the orbit. Hulse and Taylor received the 1993 Nobel Prize. The Hulse-Taylor measurement remains the gold-standard precision test of gravitational radiation in the strong-field regime, complementing the direct LIGO detection.

Direct detection required measuring strains of order $10^{-21}$. Joseph Weber built the first gravitational-wave detectors (resonant bars) in the 1960s. His controversial claims of detection were not replicated, but his pioneering work motivated the next-generation interferometric approach. The LIGO (Laser Interferometer Gravitational-Wave Observatory) project, proposed by Rainer Weiss in 1972 and developed by Weiss, Kip Thorne, and Ronald Drever over four decades, used 4-kilometre Michelson interferometers with Fabry-Pérot arms to achieve the required sensitivity. Advanced LIGO began its first observing run in September 2015.

GW150914 was detected on 14 September 2015 ^[LIGO2016]. The signal arrived at the Livingston, Louisiana detector first, then 6.9 milliseconds later at the Hanford, Washington detector, consistent with a wave propagating at the speed of light from a southern-sky source. The waveform matched the GR prediction for a binary black hole merger with component masses $36_{-4}^{+5}{M}_{\odot }$ and $29_{-4}^{+4}{M}_{\odot }$, chirp mass ${\mathcal{M}}_c=28.1_{-1.5}^{+1.8}{M}_{\odot }$, and luminosity distance $410_{-180}^{+160}$ Mpc. Approximately 3 solar masses of energy were radiated as gravitational waves in a fraction of a second, with peak luminosity exceeding the combined electromagnetic luminosity of all stars in the observable universe by a factor of $\sim 10^{23}$. Weiss, Barish, and Thorne received the 2017 Nobel Prize.

Gravitational waves provide a new channel for observing the universe. Unlike electromagnetic radiation, gravitational waves interact weakly with matter and carry information about the bulk motion of mass, not the thermodynamics of surfaces. Binary black hole mergers are invisible electromagnetically but are among the brightest gravitational-wave sources in the universe. Multi-messenger astronomy — combining gravitational-wave and electromagnetic observations — began with GW170817 ^[LIGO2017], a binary neutron star merger detected in both gravitational waves and gamma rays. The combined observations constrained the propagation speed of gravity to agree with $c$ to 1 part in $10^{15}$, provided an independent measurement of the Hubble constant from the standard-siren distance, and constrained the equation of state of dense neutron-star matter from the late-inspiral tidal deformability. The LIGO-Virgo-KAGRA catalogue now contains dozens of binary black hole, neutron-star, and mixed-binary events, opening the systematic study of compact-binary populations as a new sub-field of observational astrophysics.

Bibliography [Master]

Foundational papers (1916-1968):

• Einstein, A., "Naherungsweise Integration der Feldgleichungen der Gravitation," Sitzungsberichte Preuss. Akad. Wiss. (1916), 688-696. [First prediction of gravitational waves from the linearized field equations.]
• Einstein, A., "Uber Gravitationswellen," Sitzungsberichte Preuss. Akad. Wiss. (1918), 154-167. [Corrected treatment; identification of the quadrupole formula.]
• Bondi, H., "Gravitational waves in general relativity," Nature 186 (1960), 535; Bondi, H., Pirani, F. A. E. & Robinson, I., "Gravitational waves in general relativity," Proc. Roy. Soc. A 251 (1959), 519-533. [Proof that gravitational waves carry energy.]
• Peters, P. C. & Mathews, J., "Gravitational radiation from point masses in a Keplerian orbit," Physical Review 131:1 (1963), 435-440. [Quadrupole-formula luminosity for Keplerian binaries; the standard reference for the binary chirp.]
• Isaacson, R. A., "Gravitational Radiation in the Limit of High Frequency," Physical Review 166:5 (1968), 1263-1271; 1272-1280. [The gauge-invariant effective stress-energy tensor.]

Indirect and direct detection (1975-2017):

• Hulse, R. A. & Taylor, J. H., "Discovery of a pulsar in a binary system," Astrophysical Journal 195 (1975), L51-L53. [Discovery of the Hulse-Taylor binary pulsar.]
• Taylor, J. H. & Weisberg, J. M., "A new test of general relativity: Gravitational radiation," Astrophysical Journal 253 (1982), 908-920. [Confirmation of orbital decay from gravitational-wave emission.]
• Abbott, B. P. et al (LIGO Scientific Collaboration and Virgo Collaboration), "Observation of Gravitational Waves from a Binary Black Hole Merger," Physical Review Letters 116:6 (2016), 061102. [GW150914: first direct detection.]
• Abbott, B. P. et al, "GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral," Physical Review Letters 119:16 (2017), 161101. [First multi-messenger detection.]

Data analysis and matched filtering:

• Cutler, C. & Flanagan, E. E., "Gravitational waves from merging compact binaries: How accurately can one extract the binary's parameters from the inspiral waveform?", Physical Review D 49:6 (1994), 2658-2697. [Matched-filter parameter-estimation theory for compact-binary inspirals.]
• Wainstein, L. A. & Zubakov, V. D., Extraction of Signals from Noise (Prentice-Hall, 1962). [Foundational matched-filter theory.]

Textbooks and monographs:

• Schutz, B. F., A First Course in General Relativity, 2nd ed. (Cambridge, 2009). [Ch. 9 gives the standard intermediate treatment of gravitational-wave generation and detection.]
• Maggiore, M., Gravitational Waves Vol. 1: Theory and Experiments (Oxford, 2008). [The definitive reference for gravitational-wave physics at the master level.]
• Misner, C. W., Thorne, K. S. & Wheeler, J. A., Gravitation (Freeman, 1973). [Ch. 35-36 on gravitational-wave propagation and generation.]
• Carroll, S. M., Spacetime and Geometry (Addison-Wesley, 2004). [Ch. 6 covers linearized gravity and gravitational waves.]
• Hartle, J. B., Gravity (Addison-Wesley, 2003). [Ch. 23 provides a physics-first introduction to gravitational waves.]
• Wald, R. M., General Relativity (University of Chicago Press, 1984). **[Ch. 4.4b on linearized theory.]

Historical and review:

• Saulson, P. R., "Physics of gravitational wave detection: the LIGO interferometer," AIP Conference Proceedings 575:1 (2001), 33-55. [Review of LIGO detection principles.]
• Thorne, K. S., "Gravitational radiation," in Three Hundred Years of Gravitation, ed. Hawking, S. W. & Israel, W. (Cambridge, 1987), 330-458. [Comprehensive review of the theory and predicted sources.]

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Wave 3 physics unit, produced 2026-05-19. hooks_out targets 10.04.02 and 13.05.02 are proposed (no receiving unit yet confirmed for this hook). Status remains draft pending Tyler's review and the GR chapter retro per PHYSICS_PLAN.