Linearized Einstein Equations

Einstein's field equations are famously nonlinear — the geometry that tells matter how to move is itself sourced by all the energy in spacetime, gravity included. Yet a huge fraction of what we actually use general relativity for lives in a regime where spacetime departs only slightly from flat: the solar system, gravitational lensing, binary pulsars, and the gravitational waves that LIGO now catches by the dozen. In all of these the metric is $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ with $|h_{\mu\nu}|\ll1$ , and keeping only terms linear in the perturbation collapses Einstein's equations into something dramatically simpler — a linear, wave-type field theory for a symmetric tensor $h_{\mu\nu}$ propagating on a fixed Minkowski stage. This article is the hinge of the gravitational-wave chapter. It expands $G_{\mu\nu}=8\pi G\,T_{\mu\nu}$ to first order, producing a linearized Einstein tensor built from second derivatives of $h_{\mu\nu}$ . Two devices make it usable: the trace-reversed perturbation $\bar h_{\mu\nu}$ , which absorbs the awkward trace terms, and the gauge freedom inherited from coordinate invariance, which lets us impose the Lorenz gauge and reduce everything to a d'Alembertian. What emerges — $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ — is structurally a copy of electromagnetism with one extra index, the fingerprint of gravity's spin-2 character. From this one equation flow Newton's law, the quadrupole formula, and the graviton, each developed by name in the articles that follow.

Key takeaways

Write the metric as a fixed flat background plus a small perturbation, $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ with $|h_{\mu\nu}|\ll1$ , and keep only first-order terms: curvature becomes linear in $h_{\mu\nu}$ .
The linearized field equation in the Lorenz gauge is the wave equation $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ , a near-exact copy of electromagnetism's $\Box A_\mu=-4\pi J_\mu$ with an extra index.
The trace-reversed perturbation $\bar h_{\mu\nu}=h_{\mu\nu}-\tfrac12\eta_{\mu\nu}h$ is what cancels the trace terms and turns the messy Einstein tensor into a clean d'Alembertian.
Gauge freedom $h_{\mu\nu}\to h_{\mu\nu}-\partial_\mu\xi_\nu-\partial_\nu\xi_\mu$ (infinitesimal coordinate changes) removes 4 of the 10 components; a residual harmonic freedom removes 4 more, leaving 2 physical polarizations.
The approximation is spectacular where it applies: $|h_{00}|\approx1.4\times10^{-9}$ at Earth's surface and $4\times10^{-6}$ at the Sun's, but grows to order one at a neutron star, where it breaks.
Key idea: expanding $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ and keeping only linear terms turns Einstein's nonlinear equations into a linear, wave-type field theory for $h_{\mu\nu}$ on flat spacetime, with a gauge symmetry inherited from coordinate invariance.

Where this lives today

Linearized gravity is the mathematical engine of gravitational-wave astronomy. When LIGO registered GW150914 in September 2015, the passing wave was a metric perturbation of strain $h\approx1\times10^{-21}$ — a ripple so gentle it changed each 4 km arm by about $10^{-18}\ \mathrm m$ , a thousandth of a proton's width, yet a textbook instance of $|h_{\mu\nu}|\ll1$ . The matching waveform was computed by pushing the linear theory to its multipole expansion (the quadrupole formula and its post-Newtonian corrections) for the inspiral, handing off to numerical relativity only for the nonlinear final merger where $|h|\to1$ . Every detection since is read against templates whose backbone is equation (8). Beyond radiation, the same expansion underwrites solar-system tests: the gravitomagnetic $\bar h_{0i}$ terms predict the frame-dragging Gravity Probe B measured around the Earth, and the retarded potential predicts the Shapiro time delay tracked in spacecraft ranging. Quantized, $h_{\mu\nu}$ is the massless spin-2 graviton — the starting point for every attempt at perturbative quantum gravity, all of which begin exactly where this article does: with a small ripple on flat space.

The physical picture

Take spacetime to be nearly flat: a rigid Minkowski grid, $\eta_{\mu\nu}$ , with only the gentlest of ripples, $h_{\mu\nu}$ , laid on top. The whole content of linearized gravity is the decision to treat those ripples as small and throw away everything quadratic in them. Because curvature is built from second derivatives of the metric, the leftover first-order curvature is linear in $h_{\mu\nu}$ — and linear equations superpose: double the source, double the dimple, exactly as in electrostatics. The first figure shows the split itself: a flat background, a small perturbation, and their sum, with the field equation that governs the ripple.

Figure

Linearized gravity splits the metric into a fixed flat background plus a small ripple: g = η + h with the perturbation h much less than one. The ripple obeys a wave equation □h̄ = −16πG T and travels at the speed of light c, exactly as the potential A of electromagnetism does

Why a wave equation, and not just Poisson's? Curvature carries two derivatives of the metric, and the natural second-order, Lorentz-invariant operator on flat spacetime is the d'Alembertian $\Box=-c^{-2}\partial_t^2+\nabla^2$ . In a static situation the time derivative drops and we recover Poisson's equation and Newton; when things change in time, the same operator propagates disturbances at $c$ . The catch is that the raw linearized Einstein tensor has four terms, not one; only after trace-reversing and gauge-fixing do three of them vanish and leave a pure wave. The remarkable payoff is that the approximation is not merely convenient but astonishingly accurate wherever gravity is weak — the second figure quantifies how small $h$ actually is across real bodies, and where the scheme finally fails.

Figure

How weak is weak: the size of the metric perturbation h₀₀ ≈ 2GM/Rc² at the surface of real bodies, on a log scale. Earth 1.4×10⁻⁹, the Sun 4.2×10⁻⁶, a white dwarf 2.5×10⁻⁴ — all deep in the linear regime — while a neutron star reaches 0.3 and a black-hole horizon 1, where linearization fails

The core equations

Everything begins with one choice: the metric is flat plus small. All indices are raised and lowered with $\eta_{\mu\nu}$ , not $g_{\mu\nu}$ , the difference being higher order in $h$ . We use geometrized units $c=1$ for the tensor equations and restore $c$ in the numerical examples.

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

where: $g_{\mu\nu}$ — full physical metric; $\eta_{\mu\nu}=\mathrm{diag}(-1,+1,+1,+1)$ — fixed Minkowski background; $h_{\mu\nu}$ — small symmetric perturbation, the dynamical field. Reading: the perturbation, not the metric, is now the unknown, and it lives on flat spacetime. Because $h$ is small, the inverse metric follows from $g_{\mu\nu}g^{\nu\rho}=\delta_\mu^{\ \rho}$ to first order — the sign is the one point of care.

g^{\mu\nu}=\eta^{\mu\nu}-h^{\mu\nu}+O(h^2) \qquad (2)

where: $h^{\mu\nu}=\eta^{\mu\alpha}\eta^{\nu\beta}h_{\alpha\beta}$ — perturbation with indices raised by $\eta$ . Reading: the minus sign is what keeps the product equal to the identity. The connection is built from first derivatives of the metric, so at linear order it is already $O(h)$ and covariant derivatives reduce to partials on $h$ .

\Gamma^{\lambda}{}_{\mu\nu}=\tfrac12\,\eta^{\lambda\rho}\left(\partial_\mu h_{\rho\nu}+\partial_\nu h_{\rho\mu}-\partial_\rho h_{\mu\nu}\right) \qquad (3)

where: $\Gamma^{\lambda}{}_{\mu\nu}$ — Christoffel symbols, first order in $h$ ; $\partial_\mu$ — ordinary partial derivative. Reading: since $\Gamma$ is already $O(h)$ , the $\Gamma\Gamma$ products in Riemann are $O(h^2)$ and drop. The linearized Riemann tensor is the one genuinely physical object here — gauge-invariant, so tidal forces and geodesic deviation are coordinate-independent.

R_{\mu\nu\rho\sigma}=\tfrac12\left(\partial_\rho\partial_\nu h_{\mu\sigma}+\partial_\sigma\partial_\mu h_{\nu\rho}-\partial_\sigma\partial_\nu h_{\mu\rho}-\partial_\rho\partial_\mu h_{\nu\sigma}\right) \qquad (4)

where: $R_{\mu\nu\rho\sigma}$ — linearized Riemann curvature tensor. Reading: built from second derivatives of $h$ , its gauge-invariance is what makes tidal effects physical. Contracting Riemann gives the linearized Ricci tensor, workhorse of the field equation.

R_{\mu\nu}=\tfrac12\left(\partial_\rho\partial_\mu h^{\rho}{}_{\nu}+\partial_\rho\partial_\nu h^{\rho}{}_{\mu}-\Box h_{\mu\nu}-\partial_\mu\partial_\nu h\right) \qquad (5)

where: $R_{\mu\nu}=R^{\rho}{}_{\mu\rho\nu}$ — linearized Ricci tensor; $\Box=\eta^{\rho\sigma}\partial_\rho\partial_\sigma=-\partial_t^2+\nabla^2$ — flat d'Alembertian; $h=\eta^{\mu\nu}h_{\mu\nu}$ — trace. Reading: the wanted $\Box h_{\mu\nu}$ is there, but escorted by divergence and trace terms to be tamed. The trace-reversed perturbation is the algebraic trick that makes the field equation clean; its name comes from $\bar h=-h$ , the flipped trace.

\bar h_{\mu\nu}\equiv h_{\mu\nu}-\tfrac12\,\eta_{\mu\nu}\,h \qquad (6)

where: $\bar h_{\mu\nu}$ — trace-reversed metric perturbation; $\bar h\equiv\eta^{\mu\nu}\bar h_{\mu\nu}=-h$ . Reading: inverting is symmetric, $h_{\mu\nu}=\bar h_{\mu\nu}-\tfrac12\eta_{\mu\nu}\bar h$ , so nothing is lost. Assembling $G_{\mu\nu}=R_{\mu\nu}-\tfrac12\eta_{\mu\nu}R$ through $\bar h$ gives the linearized Einstein tensor — four terms, three of them divergences.

G_{\mu\nu}=-\tfrac12\left(\Box\bar h_{\mu\nu}+\eta_{\mu\nu}\,\partial^\rho\partial^\sigma\bar h_{\rho\sigma}-\partial^\rho\partial_\nu\bar h_{\mu\rho}-\partial^\rho\partial_\mu\bar h_{\nu\rho}\right) \qquad (7)

where: $G_{\mu\nu}$ — linearized Einstein tensor. Reading: every term except $\Box\bar h_{\mu\nu}$ contains a divergence $\partial^\rho\bar h_{\rho\cdots}$ , which is precisely what the Lorenz gauge sets to zero. Imposing the Lorenz (harmonic) gauge $\partial^\mu\bar h_{\mu\nu}=0$ kills the three divergence terms and delivers the central result of the theory.

\Box\,\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu} \qquad (8)

where: $T_{\mu\nu}$ — stress-energy tensor on the flat background; $G$ — Newton's constant. Reading: this is Maxwell's $\Box A_\mu=-4\pi J_\mu$ with an extra index and coupling $16\pi G$ ; in vacuum it becomes $\Box\bar h_{\mu\nu}=0$ , a wave at $c$ . Finally, the perturbation is not unique: a coordinate change $x^\mu\to x^\mu+\xi^\mu$ shifts $h_{\mu\nu}$ without touching the geometry — the gauge symmetry.

h_{\mu\nu}\;\to\;h_{\mu\nu}-\partial_\mu\xi_\nu-\partial_\nu\xi_\mu \qquad (9)

where: $\xi^\mu$ — arbitrary infinitesimal coordinate shift. Reading: exactly like $A_\mu\to A_\mu+\partial_\mu\lambda$ in electromagnetism, this freedom lets us impose the four Lorenz conditions and later strip the field to two physical polarizations.

Derivation AThe linearized Einstein tensor from the curvature expansion

The goal is to expand the exact Einstein tensor to first order in the perturbation and arrive at the compact four-term expression in equation (7). The strategy is a chain of contractions: build the connection from the metric, feed it into the Riemann tensor while discarding the quadratic pieces, contract to Ricci and to the scalar, assemble the Einstein tensor, and finally repackage everything with the trace-reversed field so the trace terms cancel — the step that earns its keep.

Step 1. The connection $\Gamma^{\lambda}{}_{\mu\nu}=\tfrac12 g^{\lambda\rho}(\partial_\mu g_{\rho\nu}+\partial_\nu g_{\rho\mu}-\partial_\rho g_{\mu\nu})$ becomes equation (3) once we insert $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ and note $\partial_\mu\eta_{\rho\nu}=0$ . It is already $O(h)$ . Why: this is what makes the theory linear — in $R=\partial\Gamma-\partial\Gamma+\Gamma\Gamma-\Gamma\Gamma$ the $\Gamma\Gamma$ products are $O(h^2)$ and drop, leaving only $\partial\Gamma$ .

Step 2. Contracting the surviving $\partial\Gamma-\partial\Gamma$ on the first and third indices gives the linearized Ricci tensor $R_{\mu\nu}=\partial_\lambda\Gamma^{\lambda}{}_{\mu\nu}-\partial_\nu\Gamma^{\lambda}{}_{\mu\lambda}=\tfrac12(\partial_\rho\partial_\mu h^{\rho}{}_{\nu}+\partial_\rho\partial_\nu h^{\rho}{}_{\mu}-\Box h_{\mu\nu}-\partial_\mu\partial_\nu h)$ , equation (5). Why: the wanted wave operator $\Box h_{\mu\nu}$ appears, escorted by two divergence terms and one trace term that must still be tamed.

Step 3. Contract once more with $\eta^{\mu\nu}$ to get the Ricci scalar $R=\partial_\rho\partial_\sigma h^{\rho\sigma}-\Box h$ , then assemble $G_{\mu\nu}=R_{\mu\nu}-\tfrac12\eta_{\mu\nu}R$ . In $h_{\mu\nu}$ this has six terms and no obvious structure. Why: the raw $h$ variables look hopeless — exactly the motivation for changing variables.

Step 4. Substitute $h_{\mu\nu}=\bar h_{\mu\nu}-\tfrac12\eta_{\mu\nu}\bar h$ everywhere. The trace pieces $\partial_\mu\partial_\nu h$ and $\eta_{\mu\nu}\Box h$ cancel, and what survives is the four-term expression $G_{\mu\nu}=-\tfrac12(\Box\bar h_{\mu\nu}+\eta_{\mu\nu}\partial^\rho\partial^\sigma\bar h_{\rho\sigma}-\partial^\rho\partial_\nu\bar h_{\mu\rho}-\partial^\rho\partial_\mu\bar h_{\nu\rho})$ of equation (7). Why: three of the four terms are divergences of $\bar h$ , so a single gauge condition will annihilate them — the trace-reverse was chosen to expose this. *Verified: reproduces Carroll Eq. (7.6) and Wald Eqs. (4.4.5)–(4.4.6); the Ricci and Einstein tensors were recomputed by hand from the connection and match the standard four-term form.*

Go deeper · Derivation B · 1 minGauge freedom, the Lorenz gauge, and the wave equation

Derivation BGauge freedom, the Lorenz gauge, and the wave equation

The goal is to show that the four spurious terms in the linearized Einstein tensor are pure gauge and can always be removed, turning equation (7) into the clean wave equation (8). The plan is to establish that a coordinate shift acts on $h_{\mu\nu}$ exactly like a gauge transformation, count the freedom it gives, spend four of those functions imposing the Lorenz condition $\partial^\mu\bar h_{\mu\nu}=0$ , and read off what remains — the gravitational mirror of fixing the Lorenz gauge in electromagnetism.

Step 1. Under $x^\mu\to x^\mu+\xi^\mu$ the metric perturbation shifts by the Lie derivative of the background, $h_{\mu\nu}\to h_{\mu\nu}-\partial_\mu\xi_\nu-\partial_\nu\xi_\mu$ , equation (9). The linearized Riemann tensor is unchanged because its antisymmetries make every $\xi$ -term cancel in pairs. Why: physical curvature cannot depend on how we label points, so the observable — tidal deviation in Riemann — is gauge-invariant, while $h_{\mu\nu}$ itself is not.

Step 2. Count the freedom. A symmetric $4\times4$ tensor has 10 independent components; the four functions $\xi^\mu(x)$ are four knobs we may turn. Under (9) the divergence transforms as $\partial^\mu\bar h_{\mu\nu}\to\partial^\mu\bar h_{\mu\nu}-\Box\xi_\nu$ . Why: since $\Box\xi_\nu$ is freely specifiable, we can always solve $\Box\xi_\nu=\partial^\mu\bar h_{\mu\nu}$ to drive the divergence to zero — the Lorenz gauge is always reachable.

Step 3. Impose $\partial^\mu\bar h_{\mu\nu}=0$ . In equation (7) the three divergence terms $\eta_{\mu\nu}\partial^\rho\partial^\sigma\bar h_{\rho\sigma}$ , $\partial^\rho\partial_\nu\bar h_{\mu\rho}$ , and $\partial^\rho\partial_\mu\bar h_{\nu\rho}$ all vanish, leaving $G_{\mu\nu}=-\tfrac12\Box\bar h_{\mu\nu}$ . Why: this is why the Lorenz gauge is worth imposing — it is the unique linear condition that removes every term except the d'Alembertian.

Step 4. Insert Einstein's equation $G_{\mu\nu}=8\pi G\,T_{\mu\nu}$ to get $-\tfrac12\Box\bar h_{\mu\nu}=8\pi G\,T_{\mu\nu}$ , i.e. $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ . A residual freedom survives: any $\xi^\mu$ with $\Box\xi^\mu=0$ preserves the gauge, and these four harmonic functions remove four more components, $10\to6\to2$ in vacuum. Why: that leftover freedom is what reaches the transverse-traceless gauge downstream, where only the two graviton polarizations remain. *Verified: the gauge shift of the divergence, $\partial^\mu\bar h_{\mu\nu}\to\partial^\mu\bar h_{\mu\nu}-\Box\xi_\nu$ , and the collapse to $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ match Wald §4.4 and Carroll §7.1.*

Go deeper · Derivation C · 1 minRecovering Newton's law and fixing the coupling

Derivation CRecovering Newton's law and fixing the coupling

The goal is to show that the linearized field equation contains Newtonian gravity as its static limit, and that this match is what fixes the coupling to be $8\pi G$ rather than any other constant. The route is to take a static non-relativistic source whose only significant stress-energy component is its rest-energy density, solve the $00$ component, trace-reverse back to the physical $h_{00}$ , and compare with the relation between $g_{00}$ and the Newtonian potential from the slow-motion geodesic equation.

Step 1. For a static source the time derivatives vanish, so $\Box\to\nabla^2$ , and for slow non-relativistic matter the dominant stress-energy component is $T_{00}=\rho$ (the rest-mass density) with pressure and momentum negligible. Equation (8) then reads $\nabla^2\bar h_{00}=-16\pi G\,\rho$ . Why: keeping only $T_{00}$ is the defining approximation of the Newtonian limit — slow matter is almost entirely rest energy.

Step 2. Solve and trace-reverse. With only $\bar h_{00}$ sourced, the trace is $\bar h=\eta^{00}\bar h_{00}=-\bar h_{00}$ , so $h_{00}=\bar h_{00}-\tfrac12\eta_{00}\bar h=\bar h_{00}-\tfrac12(-1)(-\bar h_{00})=\tfrac12\bar h_{00}$ , and likewise $h_{ij}=-\tfrac12\bar h_{00}\,\delta_{ij}$ . Hence $\nabla^2 h_{00}=-8\pi G\,\rho$ . Why: the trace-reverse halves the coefficient and populates the spatial part, giving the standard form $ds^2=-(1+2\Phi)dt^2+(1-2\Phi)\delta_{ij}dx^i dx^j$ .

Step 3. Identify the potential. The slow-motion geodesic equation gives $\ddot x^i=-\partial_i\Phi$ provided $g_{00}=-(1+2\Phi)$ , i.e. $h_{00}=-2\Phi$ . Substituting into Step 2 yields $\nabla^2\Phi=4\pi G\,\rho$ — Poisson's equation exactly. Why: this consistency check any relativistic theory must pass, and it passes only if the coupling in equation (8) is $16\pi G$ .

Step 4. Read off the coupling. Had we written $G_{\mu\nu}=\kappa\,T_{\mu\nu}$ with unknown $\kappa$ , this derivation would fix $\kappa=8\pi G$ (or $8\pi G/c^4$ with units restored). Why: the Newtonian limit is how Einstein originally pinned the constant — the geometry is calibrated against the one gravitational law already known to work. *Verified: reproduces $\nabla^2\Phi=4\pi G\rho$ with $h_{00}=-2\Phi$ , consistent with MTW §18.4; the $10^{-9}$ -scale magnitude of $h_{00}$ at Earth's surface is checked numerically in Worked example 1.*

Limiting cases

The single equation $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ specializes cleanly into every weak-field regime.

Case	Result	Meaning
Static Newtonian limit	$\nabla^2\Phi=4\pi G\rho$ , $h_{00}=-2\Phi$	Time-independent slow source: the $00$ equation is Poisson's equation and fixes the $8\pi G$ coupling.
Vacuum radiation	$\Box\bar h_{\mu\nu}=0$	No source: source-free wave equation, gravitational waves propagating at $c$ with two polarizations.
Slowly-varying source	$\bar h_{\mu\nu}(t,\mathbf{x})=4G\!\int\dfrac{T_{\mu\nu}(t-\|\mathbf{x}-\mathbf{x}'\|,\mathbf{x}')}{\|\mathbf{x}-\mathbf{x}'\|}\,\mathrm d^3x'$	Retarded solution; its multipole expansion yields the quadrupole formula.
Stationary rotation	$\bar h_{0i}\ne0$ , gravitomagnetic field	Mass currents source a gravitomagnetic potential — frame dragging, the Lense–Thirring effect.
Strong field $\|h\|\to1$	expansion invalid	Near neutron stars ( $\|h_{00}\|\sim0.3$ ) and horizons the nonlinear terms dominate; linearization fails.

Common trapThe perturbation h is physical and coordinate-independent

It is not. The split $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ is not gauge-invariant: under a coordinate change $h_{\mu\nu}\to h_{\mu\nu}-\partial_\mu\xi_\nu-\partial_\nu\xi_\mu$ , so its individual components carry no absolute meaning. You can make a coordinate wave appear or vanish by choosing $\xi^\mu$ cleverly — exactly the trap Einstein fell into in 1936, when a bad coordinate choice made gravitational waves seem to disappear. Only gauge-invariant combinations, above all the linearized Riemann tensor and the tidal accelerations it produces, are physical. A second, technical trap: do not read a wave equation off the raw $h_{\mu\nu}$ — until you trace-reverse and impose the Lorenz gauge, the Einstein tensor is a messy four-term object, not a d'Alembertian.

Worked examples

1 · How weak is the field? (confidence builder)

Estimate the size of the metric perturbation $|h_{00}|=2GM/(Rc^2)$ at the surface of the Earth, the Sun, and a neutron star, and decide in each case whether the linearized approximation is trustworthy. Use $G=6.674\times10^{-11}\ \mathrm{m^3\,kg^{-1}\,s^{-2}}$ and $c^2=8.988\times10^{16}\ \mathrm{m^2\,s^{-2}}$ .

Step 1. The perturbation is twice the dimensionless Newtonian potential, $|h_{00}|=2|\Phi|/c^2=2GM/(Rc^2)$ , since $h_{00}=-2\Phi/c^2$ with $\Phi=-GM/R$ at a surface — the number the expansion demands be much less than one.

Step 2. Earth, with $GM_\oplus=3.986\times10^{14}\ \mathrm{m^3\,s^{-2}}$ and $R_\oplus=6.371\times10^6\ \mathrm m$ : $|h_{00}|=2(3.986\times10^{14})/[(6.371\times10^6)(8.988\times10^{16})]=\mathbf{1.4\times10^{-9}}$ . The Sun, with $GM_\odot=1.327\times10^{20}$ and $R_\odot=6.96\times10^8\ \mathrm m$ : $|h_{00}|=\mathbf{4.2\times10^{-6}}$ .

Step 3. A neutron star of $1.4\,M_\odot$ and radius $12\ \mathrm{km}$ : $GM=1.4\times1.327\times10^{20}=1.86\times10^{20}$ , $R=1.2\times10^4\ \mathrm m$ , so $|h_{00}|=2(1.86\times10^{20})/[(1.2\times10^4)(8.988\times10^{16})]=\mathbf{0.34}$ .

Step 4. Interpret. For the Earth and Sun the perturbation is one part in a billion to a million — the linearized equations are effectively exact, every higher-order term smaller still by that factor. For the neutron star $|h_{00}|\approx0.3$ is emphatically not small: the neglected $O(h^2)$ terms are only a factor of three down, so linearization has broken and one must solve the full nonlinear equations. Linear theory is superb for Earth (10⁻⁹) and the Sun (10⁻⁶), and fails at a neutron star (0.3).

2 · Counting the degrees of freedom of a gravitational wave

A plane gravitational wave in vacuum has the form $\bar h_{\mu\nu}=A_{\mu\nu}\exp(ik_\alpha x^\alpha)$ with constant amplitude $A_{\mu\nu}$ . Determine the propagation speed and count how many independent polarizations survive.

Step 1. Impose the vacuum equation $\Box\bar h_{\mu\nu}=0$ . Acting on the plane wave gives $-k_\alpha k^\alpha A_{\mu\nu}=0$ , so the wave-vector is null: $k_\alpha k^\alpha=0$ . With $k^\alpha=(\omega/c,\mathbf k)$ this is $-\omega^2/c^2+|\mathbf k|^2=0$ , so the phase speed is $\omega/|\mathbf k|=\mathbf{c}$ exactly — gravitational waves travel at light speed.

Step 2. Start the count: a symmetric $4\times4$ amplitude $A_{\mu\nu}$ has $\mathbf{10}$ independent components.

Step 3. The Lorenz gauge $\partial^\mu\bar h_{\mu\nu}=0$ becomes transversality $k^\mu A_{\mu\nu}=0$ — four equations, removing 4 components and leaving $\mathbf6$ .

Step 4. Residual harmonic freedom $\Box\xi^\mu=0$ preserves Lorenz gauge and lets us further set the trace $A^\mu{}_\mu=0$ and $A_{0\nu}=0$ (transverse-traceless gauge), removing 4 more and leaving $\mathbf2$ . For a wave along $z$ these are $A_{xx}=-A_{yy}\equiv h_+$ and $A_{xy}=A_{yx}\equiv h_\times$ — the plus and cross polarizations. The count runs 10 → 6 → 2: a massless spin-2 field has exactly two helicity states.

Physics of the portrait

Linearized gravity is Einstein's own child, and its history is a cautionary tale about mistaking coordinates for physics: within a year of the final field equations he had linearized them and found waves; two decades later he nearly convinced himself those same waves were an illusion.

Origins · 1 minAlbert Einstein and how this was discovered

Albert Einstein

1879–1955 · general relativity and gravitational waves

Albert Einstein completed general relativity in November 1915, and by June 1916 he had already taken its first weak-field limit. Linearizing $G_{\mu\nu}=8\pi G T_{\mu\nu}$ about flat space, he found the perturbation obeys a wave equation and predicted gravitational radiation — though his first paper carried an error in the energy formula, corrected in 1918 with the celebrated quadrupole formula $L=\tfrac{G}{5c^5}\langle\dddot Q_{ij}\dddot Q_{ij}\rangle$ . He remained deeply uneasy about whether the waves were real or mere coordinate artefacts. In 1936, with Nathan Rosen, he wrote a paper titled "Do gravitational waves exist?" arguing they did not — the manuscript, sent to Physical Review, was rejected after a critical referee report (now known to be Howard Percy Robertson). Einstein angrily withdrew it, but Robertson was right: the apparent singularity in their solution was a coordinate artefact of cylindrical waves, exactly the gauge trap this article warns about, and the corrected paper appeared in 1937. The reality of the waves was finally settled at the 1957 Chapel Hill conference — Feynman's "sticky bead" argument showed they carry energy — and confirmed by Hulse and Taylor's binary pulsar and by LIGO in 2015, every milestone a direct descendant of the linearization Einstein wrote down in 1916.

History

1916

— Einstein linearizes the field equations and first predicts gravitational waves

1918

— Einstein derives the quadrupole formula for radiated power, correcting his 1916 error

1936

— Einstein and Rosen argue in an unpublished draft that gravitational waves do not exist

1957

— the Chapel Hill conference and Feynman's sticky-bead argument establish that the waves carry energy

2015

— LIGO directly detects GW150914, the merger of two black holes

Going further

Texts. Misner, Thorne & Wheeler, Gravitation, §18.1–18.2 · Wald, General Relativity, §4.4 · Carroll, Spacetime and Geometry, §7.1 · Schutz, A First Course in General Relativity, ch. 8 · Hartle, Gravity, ch. 21 · Maggiore, Gravitational Waves Vol. 1, ch. 1.

Provenance. The linearized connection, Riemann, Ricci, and Einstein tensors are cross-checked against Carroll §7.1 and Wald §4.4; the collapse to $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ under the Lorenz gauge and the Newtonian-limit calibration of $8\pi G$ against MTW §18.4. A verifier should reconstruct: (1) the linearized Ricci and Einstein tensors from the connection; (2) the gauge shift $\partial^\mu\bar h_{\mu\nu}\to\partial^\mu\bar h_{\mu\nu}-\Box\xi_\nu$ and the resulting wave equation; (3) the Newtonian limit $\nabla^2\Phi=4\pi G\rho$ with $h_{00}=-2\Phi$ ; and (4) the weak-field magnitudes $|h_{00}|\approx1.4\times10^{-9}$ (Earth), $4.2\times10^{-6}$ (Sun), $0.34$ (neutron star). The degree-of-freedom count $10\to6\to2$ and the null dispersion $k_\alpha k^\alpha=0$ are standard.

Prerequisites. Einstein Field Equations · Riemann and Ricci Curvature · Minkowski Spacetime. Leads to. Metric Perturbation and Trace-Reversed Field · Gauge Freedom in Linearized Gravity · Lorenz (Harmonic) Gauge · Gravitational Wave Equation · Quadrupole Formula · Helicity and the Spin-2 Graviton.

Ask the tutor: Ask me to trace-reverse a specific $h_{\mu\nu}$ , walk the $10\to6\to2$ polarization count in detail, or show exactly how the Lorenz gauge annihilates three of the four terms in the linearized Einstein tensor.

If we already have the exact Einstein equations, why bother linearizing?

Because the exact equations are almost never solvable in closed form, and because most gravitational physics happens where the field is weak. Linearizing does three things at once: it produces analytic, superposable solutions (double the source, double the field); it exposes structure hidden by the nonlinearity — above all the wave nature of gravity and its gauge symmetry; and it delivers concrete predictions — Newton's law, light bending, frame dragging, and the gravitational-wave templates LIGO uses. The full nonlinear equations are reserved for the strong-field regime, black-hole horizons and merger, where linearization fails.

Why does the right-hand side have 16πG when Einstein's equation has 8πG?

No contradiction — the factor of two comes from the trace-reverse, not a different theory. The full equation is $G_{\mu\nu}=8\pi G\,T_{\mu\nu}$ , and in Lorenz gauge $G_{\mu\nu}=-\tfrac12\Box\bar h_{\mu\nu}$ . Setting these equal gives $-\tfrac12\Box\bar h_{\mu\nu}=8\pi G\,T_{\mu\nu}$ , and multiplying by $-2$ yields $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ . The $16\pi G$ is just $2\times8\pi G$ , an artefact of the $-\tfrac12$ in front of the d'Alembertian.

In what sense is h_μν not physical while the Riemann tensor is?

Under a coordinate change the perturbation shifts as $h_{\mu\nu}\to h_{\mu\nu}-\partial_\mu\xi_\nu-\partial_\nu\xi_\mu$ , so its components depend on your choice of coordinates — you can inflate or delete pieces of it by relabelling points. The linearized Riemann tensor is invariant under exactly this transformation, so the tidal accelerations it encodes (the stretching of a ring of test masses) are the same in every gauge. The rule of thumb: anything you call physical must be expressible through gauge-invariant quantities like Riemann, never through individual components of $h_{\mu\nu}$ .

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

The Lorenz gauge is the condition $\partial^\mu\bar h_{\mu\nu}=0$ , four equations that reduce the ten components to six and put the field equation in wave form. It does not fully fix the coordinates: a residual freedom remains in harmonic functions $\xi^\mu$ with $\Box\xi^\mu=0$ . In vacuum you can spend that leftover freedom to also demand the perturbation be traceless and purely spatial — the transverse-traceless (TT) gauge — stripping the six components to the two physical polarizations. So Lorenz is the general working gauge that gives the wave equation, and TT is the specialized vacuum gauge (reachable only where $T_{\mu\nu}=0$ ) that isolates the two graviton modes.

Does the linearized theory conserve energy, and where does gravitational-wave energy come from?

At strict linear order the matter stress-energy is conserved in the flat-space sense, $\partial^\mu T_{\mu\nu}=0$ , which is actually a limitation: it says matter moves on fixed background trajectories and the waves carry no energy of their own. The energy of a gravitational wave is a genuinely second-order effect — it lives in the $O(h^2)$ terms the linearization discards. Making it precise requires the Isaacson effective stress-energy tensor, obtained by averaging the second-order Einstein tensor over several wavelengths. This is why "does gravity gravitate?" sits just beyond linear theory, and why settling whether waves carry energy took until the 1957 sticky-bead argument.

Why is gravity a spin-2 field rather than spin-1 like electromagnetism?

The spin is read off the field that carries the force. Electromagnetism is mediated by the vector potential $A_\mu$ , a spin-1 object with one index; gravity is mediated by the metric perturbation $h_{\mu\nu}$ , a symmetric two-index tensor, which is spin-2. The degree-of-freedom count confirms it: a massless spin- $s$ field has exactly two helicity states $\pm s$ , and the plane gravitational wave has helicities $\pm2$ (its polarization pattern repeats under a $180^\circ$ rotation, versus $360^\circ$ for spin-1). Physically the spin-2, always-attractive character traces to the source being the stress-energy tensor $T_{\mu\nu}$ — itself a two-index, positive-energy object — rather than a one-index current $J_\mu$ .