Gravitational Wave Equation

Einstein's field equations are famously nonlinear, and their full content — black holes, cosmology, frame dragging — lives in that nonlinearity. But strip the geometry to a faint tremor on flat spacetime, keep only the terms linear in the perturbation, and the equations shed their machinery and reduce to the plainest wave equation in physics, the same d'Alembert operator that carries light. That reduction is the gravitational wave equation: small ripples in the metric propagate at $c$ , obey superposition, admit retarded Green functions, and carry two polarizations — everything you know about electromagnetic radiation, transposed one rung up the spin ladder. Two ingredients do the work. The first is a change of variable: instead of the raw perturbation $h_{\mu\nu}$ one uses its trace-reversed form $\bar h_{\mu\nu}=h_{\mu\nu}-\tfrac12\eta_{\mu\nu}h$ . The second is a choice of coordinates: the Lorenz (harmonic) gauge $\partial^\mu\bar h_{\mu\nu}=0$ . Together they collapse the linearized Einstein tensor into a single term, and the field equation becomes $\Box\bar h_{\mu\nu}=-(16\pi G/c^4)\,T_{\mu\nu}$ . In vacuum the right side vanishes and gravity radiates freely at $c$ . This one line is the hinge between the geometric theory of gravity and the working physics of gravitational-wave astronomy: it underlies the quadrupole formula for emission, the transverse-traceless modes a detector measures, and the spin-2 character that sets gravity apart from the spin-1 photon. When LIGO recorded GW150914 in 2015, it was measuring a solution of exactly this equation.

Key takeaways

In the weak field the metric is $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ with $|h_{\mu\nu}|\ll1$ ; the trace-reversed variable $\bar h_{\mu\nu}$ plus the Lorenz gauge collapse the linearized Einstein equations to a flat-space wave equation.
The master equation is $\Box\bar h_{\mu\nu}=-(16\pi G/c^4)\,T_{\mu\nu}$ , with $\Box=-c^{-2}\partial_t^2+\nabla^2$ ; in vacuum $\Box\bar h_{\mu\nu}=0$ , so disturbances travel at the speed of light.
The null wavevector $k_\alpha k^\alpha=0$ that follows means gravitational waves are massless; sourced solutions are retarded integrals over $T_{\mu\nu}$ , exactly as in electromagnetism.
Of the ten components of $\bar h_{\mu\nu}$ , Lorenz gauge removes four and the residual gauge removes four more, leaving two physical polarizations — the plus and cross modes of a helicity-±2 field.
The equation's specific form is gauge-dependent, but its physical content — two transverse-traceless radiative degrees of freedom propagating at $c$ — is invariant and encoded in the linearized Riemann tensor.
Key idea: in Lorenz gauge with the trace-reversed variable the linearized Einstein equations become the flat-space d'Alembert equation $\Box\bar h_{\mu\nu}=-(16\pi G/c^4)\,T_{\mu\nu}$ — gravity radiates like a massless relativistic field moving at $c$ .

The core insight

In Lorenz gauge with the trace-reversed variable, the linearized Einstein equations become the flat-space d'Alembert equation $\Box\bar h_{\mu\nu}=-(16\pi G/c^4)\,T_{\mu\nu}$ — gravity radiates exactly like a massless relativistic field propagating at $c$ .

Where this lives today

The wave equation is no longer theory awaiting confirmation; it is the working model of an observational science. Ground-based interferometers — LIGO, Virgo, and KAGRA — routinely detect the retarded, transverse-traceless solutions of $\Box\bar h_{\mu\nu}=0$ radiated by merging compact objects, in a catalogue now past ninety events. GW170817, seen in both gravitational waves and light, turned the equation's null character into a precision measurement of the speed of gravity and an independent value for the Hubble constant. Looking ahead, LISA will open the millihertz band of supermassive black-hole binaries, while pulsar timing arrays — NANOGrav and its partners, which in 2023 reported a nanohertz stochastic background — read the same equation at wavelengths of light-years. Its would-be modifications — a graviton mass, extra polarizations, dispersion — are among the sharpest tests of general relativity we have.

The physical picture

A gravitational wave is not a wave in anything; it is a wave of the geometry itself — no medium, no aether, nothing that vibrates. What oscillates is the rule for measuring distances. When a wave passes through a cloud of freely falling test masses, they feel no force — each follows its own geodesic — yet the proper distances between them swell and shrink in a rhythmic, quadrupolar pattern. A stretch along one transverse axis is always paired with an equal squeeze along the perpendicular one, so the ring's area is preserved to first order. That paired stretch-and-squeeze, and its 45°-rotated twin, are the two polarizations — the fingerprint of a spin-2 field.

Figure

A ring of freely falling test masses distorted by a passing wave. The plus mode stretches one transverse axis while squeezing the perpendicular one; the cross mode does the same rotated by 45°. The fractional stretch is ΔL/L = ½h, and for a LIGO source h ≈ 10⁻²¹ — a change of about 10⁻¹⁸ m across a 4 km arm

Why does the geometry ripple at all, and why at the speed of light? Once the field equations are linear, the metric perturbation is just another massless field on flat spacetime, and no massless field can stand still — the d'Alembertian $\Box$ has null characteristics, so its disturbances propagate along light cones. The same operator that makes the electromagnetic potential radiate makes $\bar h_{\mu\nu}$ radiate, with the identical dispersion $\omega=c\,|\mathbf k|$ , a straight line through the origin. A nonzero graviton mass would lift that line off the origin; the absence of any such lift, tested to exquisite precision, is a direct probe of the vacuum wave equation.

Figure

The vacuum wave equation forces a null wavevector k_α k^α = 0, i.e. the linear dispersion ω = ck straight through the origin — gravitational waves are massless and move at c. A graviton mass would raise the curve to a threshold ω₀ = mc²/ℏ (dashed). LIGO-Virgo bound the graviton mass below 1.3×10⁻²³ eV/c², and GW170817 fixed the speed of gravity to within a part in 10¹⁵

The catch is that not every wiggle of $h_{\mu\nu}$ is a wave. Most of its ten components are coordinate artifacts — ripples removable by relabeling points — or Newtonian-like potentials slaved to the source. Only after the gauge freedom is spent do the two genuine radiative modes stand exposed. Sorting physics from bookkeeping is the whole game, and the wave equation is where it is played.

The core equations

Everything begins by declaring the field weak: split the metric into a fixed Minkowski background and a small perturbation, and drop everything beyond first order. This is an expansion, not a convenience — it is exact to the order kept, with errors quadratic in a quantity that is $10^{-21}$ at a detector.

g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu},\qquad |h_{\mu\nu}|\ll 1 \qquad (1)

where: $g_{\mu\nu}$ — full spacetime metric; $\eta_{\mu\nu}$ — Minkowski background, signature $(-,+,+,+)$ ; $h_{\mu\nu}$ — small symmetric perturbation. Reading: the background provides a fixed flat arena, so derivatives of $h_{\mu\nu}$ are ordinary flat-space derivatives and indices are raised with $\eta$ . The linearized Einstein tensor built from $h_{\mu\nu}$ is a clumsy sum of four second-derivative terms. It becomes tractable in the trace-reversed variable, defined by subtracting half the trace — an operation that flips the sign of the trace, $\bar h=-h$ , and undoes itself when applied twice.

\bar h_{\mu\nu}\equiv h_{\mu\nu}-\tfrac12\eta_{\mu\nu}h,\qquad h\equiv\eta^{\mu\nu}h_{\mu\nu} \qquad (2)

where: $\bar h_{\mu\nu}$ — trace-reversed perturbation; $h$ — trace of $h_{\mu\nu}$ . Reading: this substitution is engineered so the "wave" piece of the Einstein tensor stands alone and every leftover term is a divergence — which the next equation kills. The leftover clutter is a set of terms proportional to $\partial^\mu\bar h_{\mu\nu}$ , the four-divergence of the field. The Lorenz (harmonic) gauge sets that divergence to zero; it is always reachable, the exact analogue of $\partial_\mu A^\mu=0$ that cleans up Maxwell's potential.

\partial^\mu\bar h_{\mu\nu}=0 \qquad (3)

where: $\partial^\mu=\eta^{\mu\alpha}\partial_\alpha$ — flat-space derivative. Reading: four conditions, one per free index $\nu$ ; they do not fully fix the coordinates, and the leftover freedom later trims ten components to two. With (2) and (3) in hand the field equation collapses — the master result: the linearized Einstein equations, in Lorenz gauge and trace-reversed variables, are nothing but the sourced flat-space wave equation.

\Box\bar h_{\mu\nu}=-\frac{16\pi G}{c^4}\,T_{\mu\nu} \qquad (4)

where: $\Box=\eta^{\alpha\beta}\partial_\alpha\partial_\beta=-c^{-2}\partial_t^2+\nabla^2$ — the d'Alembertian; $T_{\mu\nu}$ — stress-energy of matter; $G$ — Newton's constant. Reading: the source is the full stress-energy tensor, not a scalar or current — that tensor character makes the radiation spin-2. The equation holds only after both (2) and (3); skip either and the raw Einstein tensor is not a simple d'Alembertian. Away from matter the source vanishes and the field obeys the homogeneous wave equation, whose plane-wave solutions ride null wavevectors — where "gravity travels at $c$ " enters as a theorem, not an assumption.

\Box\bar h_{\mu\nu}=0\quad\Longrightarrow\quad \bar h_{\mu\nu}=A_{\mu\nu}\,e^{ik_\alpha x^\alpha},\quad k_\alpha k^\alpha=0,\quad k^\mu A_{\mu\nu}=0 \qquad (5)

where: $A_{\mu\nu}$ — constant symmetric polarization amplitude; $k^\alpha=(\omega/c,\mathbf k)$ — wavevector. Reading: nullity $k_\alpha k^\alpha=0$ gives $\omega=c\,|\mathbf k|$ (propagation at $c$ , massless), and the Lorenz condition becomes transversality $k^\mu A_{\mu\nu}=0$ of the amplitude to the travel direction. With sources, (4) is inverted by the retarded Green function of $\Box$ — identical in form to electromagnetism, differing only in the prefactor and the tensor source. Each field point samples the source on its past light cone.

\bar h_{\mu\nu}(t,\mathbf x)=\frac{4G}{c^4}\int\frac{T_{\mu\nu}\!\left(t-|\mathbf x-\mathbf x'|/c,\ \mathbf x'\right)}{|\mathbf x-\mathbf x'|}\,\mathrm d^3x' \qquad (6)

where: the retarded time $t-|\mathbf x-\mathbf x'|/c$ enforces causality at speed $c$ . Reading: the $16\pi$ of (4) becomes a $4$ because the Green function of $\nabla^2$ carries a $1/4\pi$ ; the structure is Coulomb-like near the source, radiative far away. Expanding (6) in the far zone and using conservation of $T_{\mu\nu}$ leaves, at leading order, the second time derivative of the mass quadrupole moment — the quadrupole formula. Monopole and dipole radiation are forbidden by conservation of energy and momentum.

\bar h^{\,TT}_{ij}(t,r)=\frac{2G}{c^4\,r}\,\ddot I_{ij}\!\left(t-r/c\right),\qquad I_{ij}=\int\rho\left(x_i x_j-\tfrac13\delta_{ij}r^2\right)\mathrm d^3x \qquad (7)

where: $I_{ij}$ — (reduced) mass quadrupole moment; $r$ — distance to the source; $\ddot I_{ij}$ — evaluated at retarded time. Reading: gravity's lowest radiating multipole is the quadrupole, which is why gravitational-wave sources must be aspherical and time-varying — a spherically pulsating star radiates nothing.

The core derivations

Derivation AReduction to the wave equation in Lorenz gauge

The linearized Einstein tensor is a tangle of four second-derivative terms with no wave operator obviously in sight. The goal is to show that one change of variable — the trace reversal — together with one gauge choice — the Lorenz condition — collapses that tangle into a lone d'Alembertian, turning Einstein's geometric field equation into the same wave equation that governs light. We track every term so it is clear which pieces are physics and which are coordinate bookkeeping the gauge may discard.

Step 1. Expand the Christoffel symbols to first order in $h_{\mu\nu}$ and contract to build the linearized Ricci tensor, $R_{\mu\nu}=\tfrac12\!\left(\partial^\alpha\partial_\mu h_{\nu\alpha}+\partial^\alpha\partial_\nu h_{\mu\alpha}-\Box h_{\mu\nu}-\partial_\mu\partial_\nu h\right)$ . Why: of these four terms only $\Box h_{\mu\nu}$ has the shape of a wave operator; the other three are the coordinate clutter we intend to remove.

Step 2. Form $G_{\mu\nu}=R_{\mu\nu}-\tfrac12\eta_{\mu\nu}R$ and re-express it in the trace-reversed field. A short computation gives $G_{\mu\nu}=-\tfrac12\Box\bar h_{\mu\nu}-\tfrac12\eta_{\mu\nu}\,\partial^\alpha\partial^\beta\bar h_{\alpha\beta}+\partial^\alpha\partial_{(\mu}\bar h_{\nu)\alpha}$ . Why: the trace reversal is chosen precisely so the pure-wave term stands alone and every surviving extra term is a divergence of $\bar h_{\mu\nu}$ — which the gauge can annihilate.

Step 3. Impose the Lorenz gauge $\partial^\mu\bar h_{\mu\nu}=0$ ; the last two terms of Step 2 vanish identically, leaving $G_{\mu\nu}=-\tfrac12\Box\bar h_{\mu\nu}$ . Why: this gauge is always reachable — a shift $x^\mu\to x^\mu+\xi^\mu$ with $\Box\xi^\mu=\partial^\nu\bar h_{\mu\nu}$ sets the divergence to zero, exactly as $\partial_\mu A^\mu=0$ cleans up the electromagnetic potential.

Step 4. Insert Einstein's equation $G_{\mu\nu}=(8\pi G/c^4)\,T_{\mu\nu}$ and multiply through by $-2$ to reach $\Box\bar h_{\mu\nu}=-(16\pi G/c^4)\,T_{\mu\nu}$ . Why: the factor of two produced by the trace reversal is exactly what turns Einstein's $8\pi$ into the wave equation's $16\pi$ — the "16" is not an independent constant but a bookkeeping consequence of the variable change. *Verified: reproduces MTW §18.1 eqs. (18.5)–(18.9), Wald eqs. (4.4.5)–(4.4.6), and Carroll eq. (7.10). Trace check: contracting (4) gives $\Box\bar h=+ (16\pi G/c^4)T$ , consistent with $\bar h=-h$ .*

Go deeper · Derivation B · 1 minCounting the two physical polarizations

Derivation BCounting the two physical polarizations

A symmetric $4\times4$ amplitude has ten independent components, yet a gravitational wave carries only two. The goal is to account for the missing eight honestly, showing which are killed by the gauge condition, which by the leftover gauge freedom, and which are non-radiative. We work with a vacuum plane wave $\bar h_{\mu\nu}=A_{\mu\nu}e^{ik\cdot x}$ travelling along the $z$ -axis, so $k^\mu=(\omega/c)(1,0,0,1)$ , and track the count at each stage.

Step 1. Start from symmetry: $A_{\mu\nu}=A_{\nu\mu}$ gives ten independent components. Why: this is the raw field content before any physics is imposed — ten functions, most of them destined to be unphysical.

Step 2. Apply the Lorenz condition $k^\mu A_{\mu\nu}=0$ ; these are four linear constraints, one per index $\nu$ , cutting ten down to six. Why: transversality to $k^\mu$ removes exactly the longitudinal components that a divergence-free field cannot support.

Step 3. Use the residual gauge. Lorenz gauge is preserved by any $\xi^\mu$ with $\Box\xi^\mu=0$ ; writing $\xi^\mu=i\varepsilon^\mu e^{ik\cdot x}$ gives four free parameters $\varepsilon^\mu$ , each shifting $A_{\mu\nu}$ without changing the physics, removing four more: six down to two. Why: the gauge was never fully fixed — Lorenz constrains a coordinate freedom with more slack, and spending it exposes the true degrees of freedom.

Step 4. Fix the residual freedom by the transverse-traceless choice $A^\mu{}_\mu=0$ and $A_{0\mu}=0$ , so only spatial components transverse to $\mathbf k$ survive, of the form $A_{ij}$ with $A_{iz}=0$ and $A_{xx}=-A_{yy}$ . Why: these conditions use up exactly the freedom counted in Step 3 and leave a manifestly physical, source-independent field.

Step 5. Read off the survivors: $A_{xx}=-A_{yy}\equiv h_+$ and $A_{xy}=A_{yx}\equiv h_\times$ , giving $h^{TT}_{ij}=\begin{pmatrix}h_+&h_\times&0\\ h_\times&-h_+&0\\ 0&0&0\end{pmatrix}$ . Why: two independent amplitudes, rotated into one another by $45°$ rather than $90°$ , are the hallmark of a massless helicity-±2 field — the plus and cross polarizations. *Verified: counting $10\to6\to2$ matches Carroll §7.2, MTW §35.4, and Maggiore §1.2; the $45°$ rotation between $h_+$ and $h_\times$ identifies spin 2.*

Go deeper · Derivation C · 1 minFrom the retarded integral to the quadrupole formula

Derivation CFrom the retarded integral to the quadrupole formula

The sourced equation (4) is inverted by the retarded Green function, but the raw integral over $T_{\mu\nu}$ is not yet illuminating. The goal is to extract its leading far-zone radiation, show it is governed by the second time derivative of the mass quadrupole, and see why gravity carries no monopole or dipole radiation. We work in the far zone and use conservation of the source, $\partial^\mu T_{\mu\nu}=0$ , as the crucial lever.

Step 1. Invert (4) with the retarded Green function of $\Box$ to obtain $\bar h_{\mu\nu}=(4G/c^4)\int T_{\mu\nu}(t-|\mathbf x-\mathbf x'|/c,\mathbf x')/|\mathbf x-\mathbf x'|\,\mathrm d^3x'$ , then take $|\mathbf x-\mathbf x'|\approx r$ outside the slowly varying source. Why: in the far zone every source element radiates along nearly parallel rays, so the denominator becomes a common $1/r$ and the retardation a common $t-r/c$ .

Step 2. Focus on the spatial components $\bar h_{ij}=(4G/c^4r)\int T^{ij}\,\mathrm d^3x'$ . Why: the time-time and time-space components are fixed by the gauge and conservation once $\bar h_{ij}$ is known, so the spatial stress integral carries all the radiative information.

Step 3. Use conservation $\partial_\mu T^{\mu\nu}=0$ twice to prove the tensor virial identity $\int T^{ij}\,\mathrm d^3x=\tfrac12\,\dfrac{\mathrm d^2}{\mathrm dt^2}\!\int T^{00}x^i x^j\,\mathrm d^3x$ . Why: this converts the hard-to-measure internal stresses into the second time derivative of the mass distribution's second moment — an observable of the source's shape.

Step 4. Identify $\int T^{00}x^ix^j\,\mathrm d^3x/c^2$ with the mass quadrupole $I_{ij}$ (its trace-free part radiates) and assemble $\bar h^{TT}_{ij}=(2G/c^4r)\,\ddot I_{ij}(t-r/c)$ . Why: the factor $\tfrac12$ from Step 3 halves the $4$ into a $2$ , giving the standard quadrupole prefactor $2G/c^4r$ .

Step 5. Note what is absent: conservation of mass forbids a changing monopole, and conservation of momentum forbids a changing dipole, so the quadrupole is the first radiating multipole. Why: sources must therefore be non-spherical and rapidly varying — a pulsating spherical star, or a spinning axisymmetric one, is silent. *Verified: prefactor $2G/c^4r$ and the virial identity match Maggiore §3.3 and MTW §36.1; monopole/dipole suppression follows directly from $\partial_\mu T^{\mu\nu}=0$ .*

Limiting cases

Case	Result	Meaning
Vacuum, far from sources	$\Box\bar h_{\mu\nu}=0$	Homogeneous massless wave equation; freely propagating radiation at $c$ with null wavevectors $k^\mu k_\mu=0$ .
With a matter source	$\Box\bar h_{\mu\nu}=-\tfrac{16\pi G}{c^4}T_{\mu\nu}$	Inhomogeneous equation solved by retarded integrals; the leading far-zone radiation is the mass quadrupole.
Static / Newtonian limit	$\nabla^2\bar h_{00}=-\tfrac{16\pi G}{c^2}\rho,\ \ \bar h_{00}=-4\phi/c^2$	Time derivatives drop and $\Box\to\nabla^2$ ; the wave equation degenerates to Poisson's equation, recovering Newtonian gravity with potential $\phi$ .
Electromagnetic analogue	$\Box A^\mu=-\mu_0 J^\mu$	Identical operator and gauge structure to the photon; gravity differs by a tensor source $T_{\mu\nu}$ (spin 2) rather than a vector current $J^\mu$ (spin 1).

Common trapIf a passing wave stretches space, it stretches the laser light with it, so an interferometer can measure nothing

This sounds airtight and is false. The trap is to imagine the wave uniformly rescaling everything — ruler, arm, and wavelength alike — so that ratios never change. Two things break it. First, the observable is not a coordinate distance but the proper light travel time, and a TT wave modulates the time a photon takes to cross an arm and return; the two perpendicular arms are affected oppositely, so their round-trip times differ and the beams return out of phase. Second, the mirrors are freely falling test masses, not points painted on a stretching sheet: the gauge-invariant statement is geodesic deviation driven by the Riemann tensor, $\ddot\xi^i=-R^i{}_{0j0}\,c^2\xi^j$ , a genuine tidal acceleration. The worry was settled by the sticky-bead argument at Chapel Hill in 1957: a bead sliding on a rod is heated by a passing wave, so the wave does real work — it carries energy and is detectable. What you must not do is read amplitudes off $h_{\mu\nu}$ before reaching TT gauge, since coordinate ripples can mimic a wave that isn't there.

Worked examples

1 · Frequency, wavelength, and speed of a gravitational wave (confidence builder)

A compact binary emits gravitational waves at $f=100\ \text{Hz}$ , in the middle of LIGO's sensitive band. Find the period, the wavelength, and the propagation speed, and confirm the wavevector is null.

Step 1. The period is the reciprocal of the frequency: $T=1/f=1/100=\mathbf{10\ ms}$ . The angular frequency is $\omega=2\pi f=628\ \text{rad/s}$ . (For a binary the wave frequency is twice the orbital frequency — the mass distribution repeats every half orbit, a quadrupole effect.)

Step 2. From the null condition $k_\alpha k^\alpha=0$ the dispersion is $\omega=c|\mathbf k|$ , so $|\mathbf k|=\omega/c$ and the wavelength is $\lambda=c/f=(2.998\times10^8)/100=\mathbf{3.0\times10^6\ m}\approx3000\ \text{km}$ . The wave is enormous compared with the 4 km detector, which is why the arm-length response is quasi-static.

Step 3. The propagation speed is $v=\lambda f=(3.0\times10^6)(100)=\mathbf{3.0\times10^8\ m/s}=c$ . Nullity is automatic: $k_\alpha k^\alpha=-(\omega/c)^2+|\mathbf k|^2=-(\omega/c)^2+(\omega/c)^2=0$ . The wave is null: it travels at exactly $c$ , with $T=10$ ms and $\lambda\approx3000$ km.

2 · The speed of gravity from GW170817 (fully worked)

On 17 August 2017 the merger of two neutron stars produced both a gravitational-wave signal (GW170817) and a gamma-ray burst (GRB 170817A) from a source $\sim40\ \text{Mpc}$ away. The gamma rays arrived $1.74\ \text{s}$ after the gravitational-wave merger. Bound the fractional difference between the speed of gravity and the speed of light.

Step 1. Convert the distance to a light travel time. With $40\ \text{Mpc}=40\times3.086\times10^{22}\ \text{m}=1.23\times10^{24}\ \text{m}$ , the photon travel time is $T=D/c=(1.23\times10^{24})/(2.998\times10^8)=4.1\times10^{15}\ \text{s}$ , about $\mathbf{1.3\times10^{8}\ \text{yr}}$ . The two signals raced side by side for 130 million years.

Step 2. If gravity travelled at $v=c(1+\delta)$ , the two arrival times would differ by roughly $\Delta t\approx T\,|\delta|$ (small allowance for emission-time offsets aside). Solving for the fractional speed difference, $|\delta|=|v-c|/c\approx \Delta t/T=1.74/(4.1\times10^{15})=\mathbf{4.2\times10^{-16}}$ .

Step 3. Allowing for the astrophysical uncertainty that the gamma rays may have been emitted seconds after the merger, the published two-sided bound is $-3\times10^{-15}\le(v_{\rm gw}-c)/c\le+7\times10^{-16}$ . Gravity and light travel at the same speed to about one part in $10^{15}$ — a direct confirmation of the null character of $\Box\bar h_{\mu\nu}=0$ , and it killed a large class of dispersive modified-gravity theories.

3 · Why linearization is safe: the perturbation near the Sun

Estimate the metric perturbation at the Earth's orbit due to the Sun, confirming that $|h_{\mu\nu}|\ll1$ holds overwhelmingly.

Step 1. In the Newtonian limit $h_{00}=-2\phi/c^2$ with $\phi=-GM/r$ , so $|h_{00}|=2GM_\odot/(c^2r)$ ; here $GM_\odot/c^2=1.48\ \text{km}$ , half the Sun's Schwarzschild radius. This length is the yardstick for how curved the Sun makes spacetime.

Step 2. At one astronomical unit, $r=1.496\times10^{11}\ \text{m}$ , so $|h_{00}|=2(1476)/(1.496\times10^{11})=\mathbf{2.0\times10^{-8}}$ , with $|\bar h_{00}|=4GM_\odot/(c^2r)\approx4.0\times10^{-8}$ .

Step 3. Since $2\times10^{-8}\ll1$ , the neglected second-order terms are $\sim(10^{-8})^2\sim10^{-16}$ — negligible. The Solar System is a textbook weak field, and linearized gravity is valid there to one part in $10^{8}$ . By contrast a merging black-hole horizon has $|h|\sim1$ , where linear theory fails and numerical relativity takes over.

Physics of the portrait

The gravitational wave equation carries one name above all others. Einstein wrote it down in 1916, doubted it, rederived it, doubted it again, and did not live to see it vindicated — a century-long arc from a line of algebra to a Nobel Prize.

Origins · 1 minEinstein and how this was discovered

Einstein

1879–1955 · linearized general relativity and the prediction of gravitational waves

Within months of completing general relativity, Einstein turned to its weak-field limit and, in a 1916 paper and corrected 1918 treatment, showed the linearized equations admit wavelike solutions propagating at $c$ and derived the quadrupole formula for the power radiated. He was never fully comfortable with it: the waves seemed to depend on the choice of coordinates, and it was unclear whether they carried energy or were mere "coordinate ripples." In 1936 he and Nathan Rosen submitted a paper titled Do gravitational waves exist? arguing that they do not — only for the referee to find an error; Einstein reversed course and the published version affirmed them, though he stayed wary. The fog lasted decades until, at the 1957 Chapel Hill conference, Bondi, Pirani, Feynman, and others settled that the waves are physical and carry energy via Feynman's "sticky bead" argument. Indirect proof came in 1974 from the Hulse–Taylor binary pulsar, whose orbit decays at exactly the quadrupole rate; direct proof came in September 2015, when LIGO caught two black holes merging — a solution of Einstein's own 1916 equation, ninety-nine years late, winning the 2017 Nobel Prize.

History

1916

— Einstein predicts gravitational waves from the linearized field equations

1936

— Einstein and Rosen submit "Do gravitational waves exist?", doubting them, then reverse after a referee finds an error

1957

— the Chapel Hill sticky-bead argument establishes that the waves carry energy and are detectable

1974

— Hulse and Taylor discover the binary pulsar PSR B1913+16, whose orbital decay confirms quadrupole emission

2015

— LIGO records GW150914, the first direct detection of a gravitational wave

Going further

Texts. Misner, Thorne & Wheeler, Gravitation, ch. 18 (weak fields) and chs. 35–36 (radiation) · Wald, General Relativity, §4.4 and §4.4b · Carroll, Spacetime and Geometry, §7.1–7.3 · Schutz, A First Course in General Relativity, ch. 9 · Hartle, Gravity, ch. 16 · Maggiore, Gravitational Waves Vol. 1, chs. 1 and 3.

Provenance. A verifier should recompute the linearized Ricci and Einstein tensors and confirm the collapse to $\Box\bar h_{\mu\nu}=-(16\pi G/c^4)T_{\mu\nu}$ in Lorenz gauge (MTW §18.1, Wald §4.4, Carroll §7.1); reproduce the count $10\to6\to2$ (Carroll §7.2); and derive the prefactor $2G/c^4r$ from the retarded Green function and the tensor virial identity (Maggiore §3.3). The worked numbers are exact given their inputs: $\lambda=c/f=3000\ \text{km}$ at $100\ \text{Hz}$ ; the GW170817 bound $\Delta t/T\approx4\times10^{-16}$ ; and $|h_{00}|\approx2\times10^{-8}$ at $1\ \text{AU}$ . Observational values (strain $\sim10^{-21}$ , speed bound $\lesssim10^{-15}$ , graviton-mass bound $m_g<1.3\times10^{-23}\ \text{eV}/c^2$ ) are standard published results.

Prerequisites. Linearized Einstein Equations · Metric Perturbation and the Trace-Reversed Field · Lorenz (Harmonic) Gauge · Gauge Freedom in Linearized Gravity. Leads to. Transverse-Traceless Gauge · Plus and Cross Polarizations · The Quadrupole Formula · Helicity and the Spin-2 Graviton.

Ask the tutor: Ask me to derive the quadrupole formula's radiated power, work through the residual-gauge transformation into TT gauge explicitly, or estimate the strain from a specific binary — and I'll walk it through step by step.

Why call it the "wave equation" when the linearized Einstein equations look nothing like one?

Because after two moves they are one. The raw linearized Einstein tensor is a sum of four second-derivative terms, only one of which is the wave operator $\Box h_{\mu\nu}$ . The trace-reversed variable $\bar h_{\mu\nu}$ regroups the terms so the three unwanted ones become divergences $\partial^\mu\bar h_{\mu\nu}$ , and the Lorenz gauge sets those to zero. What remains is literally $\Box\bar h_{\mu\nu}=-(16\pi G/c^4)T_{\mu\nu}$ . The name is earned — the same retarded Green functions, superposition, and null propagation that govern light now govern gravity.

Is the wave equation gauge-invariant, or does it depend on my coordinates?

Its form depends on the Lorenz gauge; its content does not. Change coordinates and the tidy $\Box\bar h_{\mu\nu}=0$ can turn messier, and individual components of $h_{\mu\nu}$ can be created or destroyed by relabeling points. But the physical output — two transverse-traceless modes propagating at $c$ — is invariant, encoded in the linearized Riemann tensor, which is gauge-invariant to first order. That is why the honest observable is geodesic deviation, $\ddot\xi^i=-R^i{}_{0j0}c^2\xi^j$ , driven by Riemann rather than by any components of $h_{\mu\nu}$ .

Where does the factor of 16 in the source term come from?

From the trace reversal, not new physics. Einstein's equation is $G_{\mu\nu}=(8\pi G/c^4)T_{\mu\nu}$ , and in Lorenz gauge the linearized Einstein tensor is exactly $G_{\mu\nu}=-\tfrac12\Box\bar h_{\mu\nu}$ . Equating and multiplying by $-2$ to isolate $\Box\bar h_{\mu\nu}$ doubles the $8\pi$ into $16\pi$ . The retarded solution then wears a $4G/c^4$ instead, because inverting $\Box$ brings a $1/4\pi$ from the Green function of the Laplacian — the same $1/4\pi$ that turns Gauss's law into the Coulomb potential.

Why only two polarizations, when a symmetric tensor has ten components?

Because eight of the ten are not radiation. The four Lorenz conditions $k^\mu A_{\mu\nu}=0$ cut ten to six; the residual gauge freedom — shifts $\xi^\mu$ with $\Box\xi^\mu=0$ , four functions — cuts six to two. The eight that vanished were pure gauge or non-propagating potentials tied to the source. The surviving two are the plus and cross modes, and the number is fixed by gravity being a massless spin-2 field: a massless particle of spin $s$ has just two helicity states, $\pm s$ , here $\pm2$ .

Does the linear wave equation say gravitational waves carry energy?

Not directly — the source of decades of confusion. At strict first order the vacuum equation $\Box\bar h_{\mu\nu}=0$ has no obvious energy density, and energy in general relativity is not localizable in a coordinate-invariant way. The resolution lives at second order: the quadratic-in- $h$ terms act as an effective stress-energy, the Isaacson tensor, built from averaged squared gradients of $h^{TT}_{ij}$ . That averaged flux drives a binary's inspiral and heats a sticky bead. So the linear equation describes propagation, while the energy is a genuine but second-order effect — measured in the Hulse–Taylor pulsar's orbital decay.

How does this connect to Newtonian gravity?

Through the static limit. Drop all time derivatives and $\Box=-c^{-2}\partial_t^2+\nabla^2$ becomes the plain Laplacian $\nabla^2$ . The $00$ component then reads $\nabla^2\bar h_{00}=-(16\pi G/c^2)\rho$ , Poisson's equation in disguise: identifying $\bar h_{00}=-4\phi/c^2$ recovers $\nabla^2\phi=4\pi G\rho$ exactly. So Newtonian gravity is the zero-frequency corner of the same equation whose high-frequency corner is radiation — the static Sun and a merging binary are two limits of one operator, as Worked Example 3 shows numerically.