The Linearized Field Equations

The full Einstein field equations $G_{\mu\nu}=8\pi G\,T_{\mu\nu}$ are ten coupled nonlinear partial differential equations for the metric, and outside a handful of symmetric special cases they cannot be solved in closed form. Yet the situations physicists most often need to compute — the field of the Sun, the orbit of a binary, the ripple of a gravitational wave — are ones in which spacetime is barely curved, departing only slightly from flat Minkowski space. The linearized field equations harvest exactly this regime. Write the metric as $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ with the perturbation small, $|h_{\mu\nu}|\ll1$ in some global inertial frame, expand every geometric object — Christoffel symbols, Riemann, Ricci, Einstein tensor — to first order in $h$ , and throw away everything of order $h^2$ . The payoff is conceptual as much as computational. Because indices are now raised and lowered with the fixed flat metric $\eta_{\mu\nu}$ , the perturbation $h_{\mu\nu}$ becomes an ordinary symmetric tensor field living on Minkowski space rather than a deformation of the manifold. Gravity, at this order, stops being geometry and becomes a special-relativistic field theory — the classical theory of a massless spin-2 particle. It carries a gauge symmetry inherited from coordinate freedom, $h_{\mu\nu}\to h_{\mu\nu}-\partial_\mu\xi_\nu-\partial_\nu\xi_\mu$ , exactly parallel to $A_\mu\to A_\mu+\partial_\mu\lambda$ of electromagnetism. Fixing it with the Lorenz gauge collapses the ten equations into ten flat-space wave equations, $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ , whose retarded solutions describe how matter sources weak fields and radiates gravitational waves. Everything that follows — the Newtonian limit, gravitomagnetism, the quadrupole formula — is a reading of this one master equation.

Key takeaways

Write $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ with $|h_{\mu\nu}|\ll1$ and keep only first order in $h_{\mu\nu}$ ; indices are raised and lowered with the flat $\eta_{\mu\nu}$ , so gravity becomes a field theory on a rigid Minkowski background.
In the trace-reversed variable $\bar h_{\mu\nu}=h_{\mu\nu}-\tfrac12\eta_{\mu\nu}h$ the equations become $\Box\bar h_{\mu\nu}-\partial_\mu\partial^\alpha\bar h_{\nu\alpha}-\partial_\nu\partial^\alpha\bar h_{\mu\alpha}+\eta_{\mu\nu}\partial^\alpha\partial^\beta\bar h_{\alpha\beta}=-16\pi G\,T_{\mu\nu}$ .
A residual gauge symmetry $h_{\mu\nu}\to h_{\mu\nu}-\partial_\mu\xi_\nu-\partial_\nu\xi_\mu$ survives from diffeomorphism invariance; the Lorenz gauge $\partial^\mu\bar h_{\mu\nu}=0$ reduces the system to $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ .
The static limit reproduces $\nabla^2\Phi=4\pi G\rho$ with $g_{00}=-(1+2\Phi)$ ; the vacuum limit gives waves at speed $c$ with two polarizations, the count $10-4-4=2$ of a massless spin-2 field.
The linearized Bianchi identity $\partial^\mu G^{(1)}_{\mu\nu}\equiv0$ forces flat-space conservation $\partial^\mu T_{\mu\nu}=0$ , the leading term of the covariant law.
Key idea: Split $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ and keep only first order in $h$ : Einstein's equations become a Lorentz-covariant, gauge-invariant linear field theory for a symmetric tensor on flat space, reducing in Lorenz gauge to $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ .

The core insight

Split $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ and keep only first order in $h$ : Einstein's equations become a Lorentz-covariant, gauge-invariant linear field theory for a symmetric tensor on flat space, reducing in Lorenz gauge to the inhomogeneous wave equation $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ .

Where this lives today

The linearized equations are the working language of gravitational-wave astronomy. Every template in the LIGO–Virgo–KAGRA banks that matched GW150914 in 2015, and the hundreds of compact-binary events since, rests on the weak-field wave equation and its quadrupole solution; only the final merger is handed to full numerical relativity, with the retarded far-zone expansion turning a modelled orbit into a predicted strain $h(t)$ . The same physics runs at every scale: pulsar-timing arrays such as NANOGrav reported in 2023 evidence for a nanohertz stochastic background from supermassive black-hole binaries, whose angular correlation (the Hellings–Downs curve) follows directly from the two transverse-traceless polarizations. The static sector lives on too: Gravity Probe B measured the frame-dragging (Lense–Thirring) precession sourced by the $\bar h_{0i}$ of Earth's rotation, and the coming interferometer LISA will read the low-frequency band where the weak-field description is cleanest. In every case the object computed solves $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ .

The physical picture

The method rests on one act of bookkeeping: peel the physical metric into a fixed flat background plus a small correction, $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ , and treat $h_{\mu\nu}$ as a tensor field painted onto rigid Minkowski space, just as one treats $A_\mu$ in electromagnetism. The cost is real: dropping $O(h^2)$ discards all of gravity's self-interaction, and with it black holes, cosmology, and the energy the waves carry. The gain is that gravity becomes ordinary field theory, and "small" is concrete — for the Sun the deepest perturbation is a few parts in a million.

Figure

The perturbative split g = η + h. The full curved metric equals a fixed flat Minkowski background plus a small symmetric perturbation h with magnitude much less than 1. Real scales: h₀₀ ≈ 4×10⁻⁶ at the surface of the Sun, and only ≈ 10⁻²¹ for a gravitational wave passing the Earth.

Once the field is on flat space, the linearized Einstein tensor is rigid: the combination of second derivatives it contains is the unique operator (up to scale) that is symmetric, divergence-free, and invariant under $h_{\mu\nu}\to h_{\mu\nu}-\partial_\mu\xi_\nu-\partial_\nu\xi_\mu$ . That is why the trace-reversed variable $\bar h_{\mu\nu}=h_{\mu\nu}-\tfrac12\eta_{\mu\nu}h$ is the "right" field: in $\bar h$ the operator is a wave operator plus divergence pieces that vanish the instant one imposes Lorenz gauge. What survives is a spin-2 wave field, and counting its content — ten components, minus four Lorenz conditions, minus four residual-gauge functions — leaves exactly two polarizations. The second figure is that punchline: the two ways a passing wave stretches a ring of free test masses.

Figure

The physical content of the master equation. In vacuum □h̄_μν = 0 gives waves at speed c with exactly two polarizations, from the count 10 components − 4 Lorenz − 4 residual = 2. A ring of free test masses is stretched along the axes by the plus mode and along the diagonals by the cross mode.

The core equations

Everything begins with the split into background plus perturbation. The smallness condition licenses dropping quadratic terms; it is a statement about a chosen frame, not a coordinate-invariant fact, as flagged later.

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

where: $\eta_{\mu\nu}$ — Minkowski metric $\mathrm{diag}(-1,+1,+1,+1)$ ; $h_{\mu\nu}$ — symmetric metric perturbation, the dynamical field; $g_{\mu\nu}$ — full physical metric. Reading: gravity is repackaged as a small tensor field on a rigid stage, with inverse $g^{\mu\nu}=\eta^{\mu\nu}-h^{\mu\nu}+O(h^2)$ . Feeding the split through the connection, the Christoffel symbols are first order (derivatives of the constant $\eta$ vanish). Contracting the Riemann tensor and discarding the $\Gamma\Gamma$ products ( $O(h^2)$ ) gives the linearized Ricci tensor.

R_{\mu\nu}=\tfrac12\big(\partial^\alpha\partial_\mu h_{\nu\alpha}+\partial^\alpha\partial_\nu h_{\mu\alpha}-\Box h_{\mu\nu}-\partial_\mu\partial_\nu h\big)+O(h^2) \qquad (2)

where: $\Box$ — flat d'Alembertian $\eta^{\alpha\beta}\partial_\alpha\partial_\beta=-\partial_t^2+\nabla^2$ ; $h$ — trace $h\equiv\eta^{\mu\nu}h_{\mu\nu}$ ; $R_{\mu\nu}$ — once-contracted Riemann tensor. Reading: at first order covariant and partial derivatives coincide, so this is a linear operator acting on $h$ . Forming $G_{\mu\nu}=R_{\mu\nu}-\tfrac12\eta_{\mu\nu}R$ and rewriting in the trace-reversed $\bar h_{\mu\nu}$ cancels every trace term, leaving the compact linearized field equations — the centrepiece of the theory.

\Box\bar h_{\mu\nu}-\partial_\mu\partial^\alpha\bar h_{\nu\alpha}-\partial_\nu\partial^\alpha\bar h_{\mu\alpha}+\eta_{\mu\nu}\partial^\alpha\partial^\beta\bar h_{\alpha\beta}=-16\pi G\,T_{\mu\nu} \qquad (3)

where: $\bar h_{\mu\nu}\equiv h_{\mu\nu}-\tfrac12\eta_{\mu\nu}h$ — trace-reversed perturbation ( $\bar h=-h$ in 4D); $T_{\mu\nu}$ — stress-energy, to lowest order on the flat background; $G$ — Newton's constant ( $c=1$ ). Reading: the left side is the unique symmetric, divergence-free, gauge-invariant second-order operator on $\bar h$ . The equations inherit a gauge symmetry: an infinitesimal coordinate change $x^\mu\to x^\mu+\xi^\mu$ leaves the physics untouched but shifts the perturbation — the linearized remnant of diffeomorphism invariance, the analogue of $A_\mu\to A_\mu+\partial_\mu\lambda$ .

h_{\mu\nu}\to h_{\mu\nu}-\partial_\mu\xi_\nu-\partial_\nu\xi_\mu \qquad (4)

where: $\xi^\mu$ — arbitrary small vector field generating the shift, of order $h$ . Reading: two $h_{\mu\nu}$ differing by this shift describe the same spacetime, so four of the ten components are pure gauge. Spending the gauge freedom on the Lorenz (harmonic) condition annihilates every divergence term in equation (3) at once, decoupling the ten equations into ten flat-space wave equations — the master equation of the theory.

\partial^\mu\bar h_{\mu\nu}=0 \quad\Longrightarrow\quad \Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu} \qquad (5)

where: $\partial^\mu\bar h_{\mu\nu}=0$ — Lorenz gauge, attained by choosing $\xi_\nu$ solving $\Box\xi_\nu=\partial^\mu\bar h^{(\text{old})}_{\mu\nu}$ . Reading: weak-field gravity obeys the same wave equation as light, propagating at exactly $c$ . The wave equation is solved by the retarded integral: the field at an event sums contributions from source elements on its past light cone, weighted by inverse distance — the gravitational twin of the retarded potential.

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

where: $t-|\mathbf{x}-\mathbf{x}'|$ — retarded time, causal propagation at $c=1$ ; $\mathbf{x},\mathbf{x}'$ — field and source points. Reading: expanded in the far zone this yields the quadrupole formula, the workhorse of source modelling.

Derivation AFrom Einstein's equations to the linearized field equations

The goal is to descend from the exact $G_{\mu\nu}=8\pi G\,T_{\mu\nu}$ to the compact linear form of equation (3), showing why the trace-reversed variable $\bar h_{\mu\nu}$ is the natural field and why the source coefficient becomes $-16\pi G$ . We build the Ricci scalar, assemble the Einstein tensor, introduce the trace reversal, and watch every trace term cancel so only four derivative structures survive on the flat background.

Step 1. Contract the linearized Ricci tensor (2) with $\eta^{\mu\nu}$ to get the Ricci scalar $R=\partial^\alpha\partial^\beta h_{\alpha\beta}-\Box h$ : the first two terms each contract to $\partial^\alpha\partial^\beta h_{\alpha\beta}$ , the last two each give $-\Box h$ . Why: the scalar curvature is what the $-\tfrac12\eta_{\mu\nu}R$ piece of the Einstein tensor subtracts off.

Step 2. Form $G_{\mu\nu}=R_{\mu\nu}-\tfrac12\eta_{\mu\nu}R$ , using $\eta_{\mu\nu}$ in the trace term since $R$ is already first order: $G_{\mu\nu}=\tfrac12(\partial^\alpha\partial_\mu h_{\nu\alpha}+\partial^\alpha\partial_\nu h_{\mu\alpha}-\Box h_{\mu\nu}-\partial_\mu\partial_\nu h-\eta_{\mu\nu}\partial^\alpha\partial^\beta h_{\alpha\beta}+\eta_{\mu\nu}\Box h)$ . Why: this is the exact linearized Einstein tensor in the raw variable $h$ ; the two trace terms make it clumsy, motivating a change of variable.

Step 3. Define $\bar h_{\mu\nu}\equiv h_{\mu\nu}-\tfrac12\eta_{\mu\nu}h$ . Its trace is $\bar h=h-2h=-h$ (hence "trace-reversed"), and inverting gives $h_{\mu\nu}=\bar h_{\mu\nu}-\tfrac12\eta_{\mu\nu}\bar h$ . Why: this substitution is engineered to absorb the two $h$ -trace terms in $G_{\mu\nu}$ .

Step 4. Substitute into Step 2. Every trace term cancels and only four survive: $G_{\mu\nu}=\tfrac12(\partial^\alpha\partial_\mu\bar h_{\nu\alpha}+\partial^\alpha\partial_\nu\bar h_{\mu\alpha}-\Box\bar h_{\mu\nu}-\eta_{\mu\nu}\partial^\alpha\partial^\beta\bar h_{\alpha\beta})$ . Why: the trace-reversed field turns the Einstein tensor into a wave operator plus three divergence terms, nothing more.

Step 5. Set $G_{\mu\nu}=8\pi G\,T_{\mu\nu}$ and multiply by $-2$ : $\Box\bar h_{\mu\nu}-\partial_\mu\partial^\alpha\bar h_{\nu\alpha}-\partial_\nu\partial^\alpha\bar h_{\mu\alpha}+\eta_{\mu\nu}\partial^\alpha\partial^\beta\bar h_{\alpha\beta}=-16\pi G\,T_{\mu\nu}$ , equation (3). Why: clearing the one-half converts the $8\pi G$ of the exact theory into the $16\pi G$ that reproduces Newton's constant in the static limit. *Verified: reproduces Carroll §7.1 Eqs. (7.6)–(7.8) and MTW §18.1; tracing the boxed equation gives $-\Box\bar h+\dots=-16\pi G\,T$ , consistent with $R=-8\pi G\,T$ .*

Go deeper · Derivation B · 1 minFixing the Lorenz gauge to reach the wave equation

Derivation BFixing the Lorenz gauge to reach the wave equation

The aim is to prove the Lorenz condition $\partial^\mu\bar h_{\mu\nu}=0$ can always be reached by a gauge transformation, and that once reached it collapses equation (3) into the pure wave equation (5). We track how the divergence of $\bar h$ transforms under a coordinate shift, solve for the shift that kills it, and count the gauge freedom that remains — the count that fixes the physical polarizations at two.

Step 1. Under the gauge shift (4), trace-reverse to find $\bar h_{\mu\nu}\to\bar h_{\mu\nu}-\partial_\mu\xi_\nu-\partial_\nu\xi_\mu+\eta_{\mu\nu}\partial^\alpha\xi_\alpha$ , the extra piece being the footprint of the trace reversal. Why: the Lorenz condition constrains $\bar h$ , so we need its transformation law.

Step 2. Take the divergence: $\partial^\mu\bar h_{\mu\nu}\to\partial^\mu\bar h_{\mu\nu}-\Box\xi_\nu$ , three of the four contributions cancelling pairwise. Why: the change is governed by the same wave operator as the field, so the gauge condition is reachable by solving a wave equation.

Step 3. Choose $\xi_\nu$ solving $\Box\xi_\nu=\partial^\mu\bar h^{(\text{old})}_{\mu\nu}$ ; this always has a retarded solution, so afterwards $\partial^\mu\bar h_{\mu\nu}=0$ holds identically. Why: existence of the solution makes Lorenz gauge a choice available for any field, not a special assumption.

Step 4. In equation (3) every term with $\partial^\alpha\bar h_{\mu\alpha}$ or $\partial^\alpha\partial^\beta\bar h_{\alpha\beta}$ now vanishes, leaving $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ . Why: the divergence terms were the obstruction to decoupling; killing them turns ten coupled equations into ten independent wave equations.

Step 5. A residual freedom remains: any further $\xi_\mu$ with $\Box\xi_\mu=0$ preserves the gauge. Of ten components, the four Lorenz conditions remove four and these four harmonic functions remove four more, leaving $10-4-4=2$ polarizations. Why: this is the massless spin-2 count — the two modes $h_+,h_\times$ a detector measures. *Verified: matches Wald §4.4 and Schutz Ch. 8; the residual-gauge counting $10-4-4=2$ is the standard massless spin-2 result.*

Go deeper · Derivation C · 1 minConservation of the source from the linearized Bianchi identity

Derivation CConservation of the source from the linearized Bianchi identity

The purpose is to show the geometry side of the linearized equations is automatically divergence-free, and that this forces the matter source to obey flat-space conservation $\partial^\mu T_{\mu\nu}=0$ — the leading-order shadow of the exact covariant law $\nabla^\mu T_{\mu\nu}=0$ . We invoke the linearized Bianchi identity, apply it to the field equations, and read off what it means for the motion of matter.

Step 1. The four-divergence of the linearized Einstein tensor vanishes identically, $\partial^\mu G^{(1)}_{\mu\nu}=0$ : every term in the four-term operator of Step 4 above pairs off once partial derivatives are commuted on the flat background. Why: this is the linearized contracted Bianchi identity, the flat-space echo of $\nabla^\mu G_{\mu\nu}\equiv0$ , true for any $\bar h$ .

Step 2. Apply it to $G^{(1)}_{\mu\nu}=8\pi G\,T_{\mu\nu}$ : the divergence gives $8\pi G\,\partial^\mu T_{\mu\nu}=0$ , hence $\partial^\mu T_{\mu\nu}=0$ . Why: the geometry cannot source a non-conserved stress-energy, so matter conservation is a consequence, not an extra assumption.

Step 3. This is $\nabla^\mu T_{\mu\nu}=0$ at lowest order; the Christoffel corrections promoting $\partial$ to $\nabla$ are $O(h)$ times the already-first-order source, appearing only at next order. Why: at strictly linear order matter moves on straight lines of the flat background — gravity does not yet act back on its own source, which is why self-consistent orbital motion cannot come from linearized theory alone. *Verified: standard result, MTW §18.3; the identity $\partial^\mu G^{(1)}_{\mu\nu}\equiv0$ is checkable by inspection of the four-term operator.*

Limiting cases

The single master equation carries several familiar theories inside it, selected by which terms one chooses to keep.

Case	Result	Meaning
Static Newtonian limit	$\nabla^2\Phi=4\pi G\rho,\ \ g_{00}=-(1+2\Phi)$	Slow, weak, time-independent source: only $\bar h_{00}$ survives and $\Box\to\nabla^2$ , recovering Poisson's equation.
Vacuum / radiation	$\Box\bar h_{\mu\nu}=0$	With $T_{\mu\nu}=0$ the equation is homogeneous; plane waves with $k^\mu k_\mu=0$ travel at $c$ with two polarizations.
With cosmological constant	$\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}+2\Lambda\,\eta_{\mu\nu}$	A nonzero $\Lambda$ adds a constant source; linearizing about flat $\eta$ needs the curvature it induces negligible in the region.
Full theory restored	$G_{\mu\nu}=G^{(1)}_{\mu\nu}[h]+G^{(2)}_{\mu\nu}[h]+\cdots$	Reinstating the dropped $O(h^2)$ : $G^{(2)}_{\mu\nu}$ acts as effective gravitational stress-energy $t_{\mu\nu}$ — the route to wave energy flux and back to the nonlinear equations.

Common trapThinking that |h| ≪ 1 is a coordinate-invariant statement about the field

It is not. A flat spacetime in curvilinear coordinates can have large $h_{\mu\nu}$ , and — more dangerously — a strong field can be dressed up to look small in bad coordinates. The meaningful condition is that there exists a global, inertial-like frame in which both $h_{\mu\nu}$ and its derivatives over the relevant scales are small; physically the curvature invariants must be small in geometrized units throughout the region. A quieter trap lurks alongside: linearized theory is self-consistent only when the source's gravitational binding energy is negligible. To capture gravity gravitating — the energy the waves carry, the backreaction on orbits — you must go to second order, where $G^{(2)}_{\mu\nu}$ supplies the missing stress-energy.

Worked examples

1 · How weak is the Sun's field (confidence builder)

Estimate the largest metric perturbation the Sun produces, at its surface, and check that linearized theory is well justified there. Take $M_\odot=1.989\times10^{30}\ \mathrm{kg}$ , $R_\odot=6.96\times10^{8}\ \mathrm{m}$ , $G=6.674\times10^{-11}$ SI, $c=2.998\times10^{8}\ \mathrm{m/s}$ .

Step 1. In the static Newtonian limit $g_{00}=-(1+2\Phi)$ , so $h_{00}=-2\Phi$ with surface potential $\Phi=-GM_\odot/(R_\odot c^2)$ . Its magnitude has numerator $GM_\odot=6.674\times10^{-11}\times1.989\times10^{30}=1.327\times10^{20}$ and denominator $R_\odot c^2=6.96\times10^{8}\times8.988\times10^{16}=6.256\times10^{25}$ .

Step 2. Divide: $|\Phi|=1.327\times10^{20}/6.256\times10^{25}=\mathbf{2.12\times10^{-6}}$ , so $|h_{00}|=2|\Phi|=\mathbf{4.24\times10^{-6}}$ . A few parts in a million — the expansion parameter is tiny, and dropping $O(h^2)\sim10^{-11}$ is overwhelmingly justified.

Step 3. Cross-check via the Schwarzschild radius. Since $h_{00}=r_s/R_\odot$ with $r_s=2GM_\odot/c^2$ , evaluate $r_s=2\times1.327\times10^{20}/8.988\times10^{16}=\mathbf{2.95\times10^{3}\ m}\approx2.95\ \mathrm{km}$ , and $r_s/R_\odot=4.24\times10^{-6}$ matches Step 2 exactly.

Step 4. Linearized gravity would break down only where $h\to1$ , near $r\sim r_s\approx3$ km, buried deep inside where the Sun's mass sits. Everywhere a planet, spacecraft, or light ray samples the solar field the linear theory is excellent — which is why the classic Solar-System tests are computed from exactly this expansion. The Sun's largest metric perturbation is h₀₀ ≈ 4.2×10⁻⁶, so |h| ≪ 1 by six orders of magnitude.

2 · The strain of a gravitational wave at LIGO

A wave in transverse-traceless gauge stretches proper distances by $\Delta L/L=\tfrac12 h_+$ along one arm. For GW150914 the peak strain at Earth was $h\approx1.0\times10^{-21}$ . Find the length change of a LIGO arm of $L=4.0\ \mathrm{km}$ and compare it to a proton.

Step 1. In vacuum $\Box\bar h_{\mu\nu}=0$ ; a plane wave $\bar h_{\mu\nu}=A_{\mu\nu}e^{ik\cdot x}$ requires the null wavevector $k^\mu k_\mu=0$ , so it travels at $c$ . Residual gauge freedom sets it into transverse-traceless form, where $h_+,h_\times$ are the only survivors — the two polarizations counted above.

Step 2. For an arm aligned with the plus polarization a passing crest changes its length by $\Delta L=\tfrac12 h_+\,L=\tfrac12\times(1.0\times10^{-21})\times(4.0\times10^{3}\ \mathrm{m})=\mathbf{2.0\times10^{-18}\ m}$ .

Step 3. For perspective, a proton has radius $\approx0.84\times10^{-15}\ \mathrm{m}$ , so the arm moves by $\Delta L/r_p\approx\mathbf{1/420}$ of a proton radius — that a kilometre-scale instrument resolves this is why the 2015 detection was a landmark.

Step 4. The linear picture is self-consistent: with $h\sim10^{-21}$ the discarded $O(h^2)\sim10^{-42}$ terms are negligible, so the waveform template that identified the signal rests on the linearized equations — even though the merger that produced it required full numerical relativity. A LIGO arm changes by ΔL ≈ 2×10⁻¹⁸ m, about one four-hundredth of a proton radius.

Physics of the portrait

Linearized gravity is where three eras meet: Einstein's 1916 approximation that first predicted gravitational waves, the gauge and wave-equation machinery borrowed from electromagnetism, and the modern reading of $h_{\mu\nu}$ as a spin-2 field. Fittingly, the man who wrote the nonlinear equations was also the first to linearize them.

Origins · 1 minAlbert Einstein and how this was discovered

Albert Einstein

1879–1955 · the weak-field limit and gravitational waves

Having completed general relativity in November 1915, Einstein turned within months to its weak-field limit. In the June 1916 paper "Näherungsweise Integration der Feldgleichungen der Gravitation" he wrote $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ , expanded to first order, imposed what we now call the harmonic or Lorenz gauge, and obtained the flat-space wave equation for the perturbation — the first prediction that gravity propagates as waves at the speed of light. His initial paper carried an algebraic error producing a spurious monopole radiation term; he corrected it in the 1918 "Über Gravitationswellen," arriving at the quadrupole formula still used today. The subject then endured decades of doubt — Einstein himself, with Nathan Rosen, submitted a 1936 paper arguing the waves did not exist, before a referee (Howard Percy Robertson) caught the coordinate artifact behind the claim. Théophile de Donder had meanwhile systematized the harmonic gauge (1921), and in 1939 Markus Fierz and Wolfgang Pauli showed the same linear equations are the unique consistent theory of a massless spin-2 particle — recasting Einstein's approximation as the classical limit of the graviton. The waves waited until 2015, a century after the prediction, to be seen directly.

History

1915

— Einstein completes the nonlinear general field equations

1916

— Einstein linearizes the equations and predicts gravitational waves

1918

— Einstein derives the quadrupole formula, correcting his earlier factor

1921

— de Donder formalizes the harmonic (Lorenz) gauge condition

1939

— Fierz and Pauli give the massless spin-2 field theory

Going further

Texts. Misner, Thorne & Wheeler, Gravitation (1973), §18.1–18.3 · Wald, General Relativity (1984), §4.4, §7.5 · Carroll, Spacetime and Geometry (2004), §7.1–7.2 · Schutz, A First Course in General Relativity (2009), Ch. 8 · Hartle, Gravity (2003), Ch. 21.

Provenance. The linearized $R_{\mu\nu}$ and trace-reversed field equations follow MTW §18.1 and Carroll §7.1; the Lorenz-gauge reduction and degree-of-freedom count follow Wald §4.4 and Schutz Ch. 8; the retarded solution and normalization $4G$ are standard. A verifier should recompute: (1) $R_{\mu\nu}$ from $\Gamma$ dropping $O(h^2)$ ; (2) the trace-reversal cancelling all $h$ -trace terms in $G_{\mu\nu}$ ; (3) $\partial^\mu\bar h_{\mu\nu}\to\partial^\mu\bar h_{\mu\nu}-\Box\xi_\nu$ ; (4) the count $10-4-4=2$ ; (5) recovery of $\nabla^2\Phi=4\pi G\rho$ statically. The worked numbers ( $h_{00}^\odot\approx4.2\times10^{-6}$ , $r_s^\odot\approx2.95$ km, LIGO $\Delta L\approx2\times10^{-18}$ m) follow from the tabulated constants.

Prerequisites. The Einstein Field Equations · The Einstein Tensor · The Stress-Energy Tensor · The Weak-Field Approximation. Leads to. The Newtonian Limit · Local Conservation of Energy-Momentum · Variational Derivation of the Field Equations.

Ask the tutor: Want the full index gymnastics that cancel every trace term when you switch to $\bar h_{\mu\nu}$ , or to see how the retarded solution becomes the quadrupole formula? Ask and we will grind through it line by line.

Why expand around flat space rather than some curved background?

Flat Minkowski space is the metric a global inertial frame measures far from all matter, and expanding around it makes $\eta_{\mu\nu}$ constant, so its derivatives vanish and the Christoffel symbols start at first order — which lets the theory close with constant coefficients. You can linearize around a curved background (cosmological perturbation theory expands around Friedmann–Lemaître spacetimes), but then the background curvature enters the wave operator and the equations are no longer constant-coefficient. For Solar-System gravity and waves far from their source, the flat background is both accurate and maximally simple.

What is the difference between the Lorenz gauge and the transverse-traceless gauge?

The Lorenz (harmonic) gauge $\partial^\mu\bar h_{\mu\nu}=0$ is four conditions; it decouples the field equations into wave equations but does not fully fix the gauge, since harmonic $\xi$ with $\Box\xi=0$ preserve it. In vacuum one spends that residual freedom on four more conditions — setting $\bar h_{0\mu}=0$ and making the perturbation traceless — the transverse-traceless (TT) gauge. TT gauge is available only in source-free regions and only for the radiative field, but there it is maximally clean: the two survivors are exactly the polarizations $h_+,h_\times$ , to which test masses respond directly. Lorenz gauge is the general working gauge; TT is its specialization for reading off radiation.

How is this the analogue of electromagnetism?

The parallel is structural and exact at this order. The potential $A_\mu$ has gauge freedom $A_\mu\to A_\mu+\partial_\mu\lambda$ ; the perturbation has $h_{\mu\nu}\to h_{\mu\nu}-\partial_\mu\xi_\nu-\partial_\nu\xi_\mu$ . Electromagnetism's Lorenz gauge $\partial^\mu A_\mu=0$ reduces Maxwell to $\Box A_\mu=-\mu_0 J_\mu$ ; gravity's $\partial^\mu\bar h_{\mu\nu}=0$ gives $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ . Both have retarded solutions, both radiate, and both conserve their source ( $\partial^\mu J_\mu=0$ versus $\partial^\mu T_{\mu\nu}=0$ ). The difference is spin: $A_\mu$ is a spin-1 vector, $h_{\mu\nu}$ a spin-2 tensor — which is why gravity's leading radiation is quadrupole, not dipole.

Why does the coupling become 16πG and not stay 8πG?

The exact equations read $G_{\mu\nu}=8\pi G\,T_{\mu\nu}$ . Writing the linearized $G_{\mu\nu}$ in the trace-reversed variable and clearing the overall factor of one-half doubles the eight to sixteen: $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ . This is not a new constant but bookkeeping from the trace reversal. The proof it is correct is the Newtonian limit: in the static case $h_{00}=\tfrac12\bar h_{00}$ , and $\nabla^2\bar h_{00}=-16\pi G\rho$ with $g_{00}=-(1+2\Phi)$ collapses to $\nabla^2\Phi=4\pi G\rho$ , Poisson's equation with exactly Newton's constant. The factors track through so that the one number every experiment knows, $G$ , comes out right.

Can linearized theory describe the energy carried by gravitational waves?

Not at strictly linear order. At first order the source obeys $\partial^\mu T_{\mu\nu}=0$ with no gravitational contribution, so the waves appear to carry no energy and exert no backreaction. The energy shows up only at second order: restoring the dropped $O(h^2)$ terms, $G^{(2)}_{\mu\nu}$ acts as an effective stress-energy $t_{\mu\nu}$ for the field itself, and averaging over several wavelengths gives the Isaacson stress-energy tensor and the wave energy flux. So linearized theory predicts that waves exist and what their waveform is, but that they carry energy away from a binary — making it inspiral — is a second-order result, the same reason geodesic motion of a self-gravitating body cannot be derived from the linear theory alone.

Does a metric perturbation with |h| large always mean strong gravity?

No — this is the most common conceptual trap. The size of $h_{\mu\nu}$ is coordinate-dependent: flat Minkowski space in spherical or accelerating coordinates can have order-one $h$ with zero curvature, while clever coordinates can hide a genuinely strong field. What is physical is the curvature built from second derivatives — the Riemann tensor and its invariants — in geometrized units over the region. The correct weak-field condition is that there exists a global inertial-like frame in which $h_{\mu\nu}$ and its relevant derivatives are all small; when it exists, the linear expansion is valid regardless of how large $h$ looks in another chart.