The Weak-Field Approximation

General relativity is a nonlinear theory, and that nonlinearity is its reputation for difficulty: the metric $g_{\mu\nu}$ is at once the ruler that measures spacetime and the field whose curvature the equations determine, so gravity gravitates and the equations feed back on themselves. Yet almost everywhere we test the theory — the solar system, the laboratory, a gravitational-wave detector — gravity is weak: the metric departs from flat Minkowski spacetime by only a whisper. The weak-field approximation makes that whisper precise by writing $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ and keeping only terms linear in the small perturbation $h_{\mu\nu}$ . The theory then collapses into something tractable: a Lorentz-covariant field theory of a symmetric rank-two field $h_{\mu\nu}$ on a fixed flat background — linearized gravity, the spin-2 cousin of electromagnetism, with gravity's self-interaction dropped to return only at second order. Two ideas carry the whole framework. First, gauge freedom: because the split is not unique, an infinitesimal change of coordinates $x^\mu\to x^\mu+\xi^\mu$ reshuffles $h_{\mu\nu}$ without touching any measurement, exactly as a gauge transformation does in electromagnetism. Choosing the harmonic (Lorenz) gauge $\partial^\mu\bar h_{\mu\nu}=0$ decouples the equations into a single flat-space wave equation $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ — the master equation of the subject. Second, one small parameter: $\Phi\sim GM/(rc^2)$ , about $10^{-9}$ at Earth's surface and only order unity at a black hole. From this one framework flow the Newtonian limit, the solar-system tests, gravitational waves, and frame dragging — the bridge back to Newton and the engine of precision gravity at once.

Key takeaways

The weak-field approximation writes $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ with $|h_{\mu\nu}|\ll1$ and keeps only terms linear in $h_{\mu\nu}$ , turning Einstein's nonlinear theory into a field theory on flat spacetime.
In the harmonic gauge $\partial^\mu\bar h_{\mu\nu}=0$ the field equations reduce to the flat-space wave equation $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ for the trace-reversed $\bar h_{\mu\nu}=h_{\mu\nu}-\tfrac12\eta_{\mu\nu}h$ .
Coordinate changes act as a gauge symmetry $h_{\mu\nu}\to h_{\mu\nu}-\partial_\mu\xi_\nu-\partial_\nu\xi_\mu$ ; only quantities built from the gauge-invariant linearized Riemann tensor are physical.
The single small parameter is $\Phi\sim GM/(rc^2)$ : about $7\times10^{-10}$ at Earth's surface, $2\times10^{-6}$ at the Sun's, $\sim0.2$ on a neutron star — superb wherever it stays below $\sim10^{-3}$ .
Adding slow motion recovers Newton via $h_{00}=-2\Phi$ and $\nabla^2\Phi=4\pi G\rho$ ; the same linear theory predicts gravitational waves, light bending, and frame dragging.
Key idea: split the metric as $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ , keep only linear terms in $h_{\mu\nu}$ , use the residual coordinate (gauge) freedom to impose $\partial^\mu\bar h_{\mu\nu}=0$ , and the Einstein equations collapse to the flat-space wave equation $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ .

The core insight

Split the metric as $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ , keep only linear terms in $h_{\mu\nu}$ , exploit the residual coordinate (gauge) freedom to impose the harmonic condition $\partial^\mu\bar h_{\mu\nu}=0$ , and the Einstein equations collapse to the flat-space wave equation $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ for the trace-reversed perturbation $\bar h_{\mu\nu}=h_{\mu\nu}-\tfrac12\eta_{\mu\nu}h$ .

Where this lives today

Linearized gravity is not a historical stepping stone but the daily working theory of precision gravitation. Every gravitational-wave detection rests on it: LIGO, Virgo, and KAGRA measure a passing transverse-traceless wave as a strain $h\sim10^{-21}$ and match it against templates whose leading behaviour is the quadrupole formula. Since GW150914 in 2015 the catalogue has passed a hundred compact-binary mergers, and in 2023 pulsar-timing arrays (NANOGrav and partners) reported a nanohertz gravitational-wave background — the same $h_{\mu\nu}$ imprinted on the arrival times of millisecond pulsars. Frame dragging, the gravitomagnetic $\bar h_{0i}$ of a rotating body, has been measured directly: Gravity Probe B clocked Earth's dragging of a gyroscope at about 37 milliarcseconds per year, and LAGEOS and LARES confirmed the Lense–Thirring orbital precession. The identification $h_{00}=-2\Phi$ is corrected for continuously in GPS, whose clocks would drift tens of microseconds per day without it. And LISA will push the expansion into the gravitational self-force and extreme-mass-ratio regime — a frontier where linearized gravity is being extended, not retired.

The physical picture

Picture Minkowski spacetime as a perfectly flat floor and $h_{\mu\nu}$ as a faint ripple painted onto it. Because the ripple is tiny, you may compute how it moves and bends paths as if the floor stayed flat — that is linearization, the entire trick. The price is one physical effect: you ignore the ripple's own weight pressing on the floor, the gravitational field gravitating, which is genuinely second order in $h$ and so costs nothing at leading order. What survives is a symmetric tensor field $h_{\mu\nu}$ on the fixed background — and, being spin-2, it radiates with two transverse-traceless polarizations rather than the single one of a vector field.

Figure

The metric split g = η + h: a small perturbation h_μν, much less than 1, on flat Minkowski spacetime η_μν gives the physical metric g_μν. The spin-2 field carries two transverse-traceless polarizations, the plus and cross modes at right

The whole edifice is governed by one dimensionless number, the Newtonian potential in units of $c^2$ : $\Phi\sim GM/(rc^2)$ . Where it is small the perturbation is small ( $|h_{\mu\nu}|\sim2|\Phi|$ ) and every relativistic effect is a clean correction to flat space; where it approaches unity — a black-hole horizon, the early universe — the linear theory breaks. The second figure places the real systems on this single axis, telling you at a glance whether you may reach for linearized gravity or must solve Einstein's equations in full.

Figure

The single small parameter Φ = GM/(rc²) across real systems: Earth's surface ≈ 7×10⁻¹⁰, the Sun's ≈ 2×10⁻⁶, a white dwarf ≈ 10⁻⁴, a neutron star ≈ 0.2, a black-hole horizon ≈ 0.5. Linearization is excellent where Φ ≲ 10⁻³ and fails as Φ approaches 1, since h is of order 2Φ

The parallel with electromagnetism is not a loose analogy but a structural identity: $\bar h_{\mu\nu}$ plays the role of the four-potential $A_\mu$ , the harmonic gauge mirrors the Lorenz gauge $\partial^\mu A_\mu=0$ , and $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ is the twin of $\Box A_\mu=-\mu_0 J_\mu$ . The gauge freedom is just that you may re-draw the coordinate grid by an infinitesimal $\xi^\mu$ with no observable consequence: the four $\xi^\mu$ strip four of the ten components of $h_{\mu\nu}$ , and four residual transformations strip four more, leaving two physical degrees of freedom — the two polarizations of the spin-2 graviton.

The core equations

Everything begins with the split of the metric into a fixed flat background and a small perturbation — a choice of bookkeeping, not yet an approximation; that enters only when we discard powers of $h$ .

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

where: $g_{\mu\nu}$ — full spacetime metric; $\eta_{\mu\nu}$ — flat Minkowski metric $\mathrm{diag}(-1,+1,+1,+1)$ ; $h_{\mu\nu}$ — small symmetric perturbation, the dynamical variable. The single assumption $|h_{\mu\nu}|\ll1$ licenses everything that follows. The inverse metric is a geometric series truncated at first order, carrying a crucial convention: indices move with $\eta$ , never $g$ .

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

where: $g^{\mu\nu}$ — inverse full metric to first order; $h^{\mu\nu}$ — perturbation with indices raised by $\eta^{\mu\nu}$ . Moving indices with $g^{\mu\nu}$ would smuggle in $O(h^2)$ terms. The connection is built from first derivatives of the metric, so the linearized Christoffel symbols are first derivatives of $h$ .

\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)+O(h^2) \qquad (3)

where: $\Gamma^\lambda_{\mu\nu}$ — connection coefficients, first order in $h$ ; $\partial_\mu$ — ordinary partial derivative. There is no $\Gamma\Gamma$ term, since each $\Gamma$ is first order. Contracting the linearized Riemann tensor gives the Ricci tensor; each term is a second derivative of $h$ .

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 (4)

where: $R_{\mu\nu}$ — Ricci tensor to first order; $\Box=\eta^{\rho\sigma}\partial_\rho\partial_\sigma=-\partial_t^2+\nabla^2$ — flat-space d'Alembertian; $h=\eta^{\mu\nu}h_{\mu\nu}$ — trace. The lone $\Box h_{\mu\nu}$ becomes the wave operator. One change of variable makes the field equations tidy: the trace-reversed perturbation flips the trace sign and absorbs the awkward $\eta_{\mu\nu}$ pieces.

\bar h_{\mu\nu}=h_{\mu\nu}-\tfrac12\,\eta_{\mu\nu}\,h,\qquad \bar h=-h \qquad (5)

where: $\bar h_{\mu\nu}$ — trace-reversed perturbation; $\bar h=\eta^{\mu\nu}\bar h_{\mu\nu}$ — its trace. The map is its own inverse: $h_{\mu\nu}=\bar h_{\mu\nu}-\tfrac12\eta_{\mu\nu}\bar h$ . The split is not unique, and that is a genuine symmetry: an infinitesimal coordinate change acts on the perturbation exactly like a gauge transformation acts on $A_\mu$ .

h_{\mu\nu}\to h_{\mu\nu}-\partial_\mu\xi_\nu-\partial_\nu\xi_\mu \qquad (6)

where: $\xi^\mu(x)$ — infinitesimal coordinate shift, $x^\mu\to x^\mu+\xi^\mu$ . Physical quantities are invariant under (6), so a nonzero $h_{\mu\nu}$ alone proves nothing about real curvature. Fixing the harmonic (Lorenz) gauge decouples the equations: Einstein's ten coupled nonlinear equations become one flat-space wave equation, the master equation.

\Box\,\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu},\qquad \partial^\mu\bar h_{\mu\nu}=0 \qquad (7)

where: $T_{\mu\nu}$ — stress-energy tensor (to lowest order); $G$ — Newton's constant; $\partial^\mu\bar h_{\mu\nu}=0$ — harmonic gauge. Matter tells the ripple how to wave, and it propagates causally at $c$ . The sourced wave equation has the standard retarded solution — gravity's Liénard–Wiechert analogue, the starting point for radiation calculations.

\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 (8)

where: the source is evaluated at the retarded time $t-|\mathbf x-\mathbf x'|$ (with $c=1$ ), so a change reaches the field point only after light travels there.

Derivation AFrom the full Einstein equations to the linearized wave equation

The goal is to start from the exact field equations $G_{\mu\nu}=8\pi G\,T_{\mu\nu}$ , insert the split $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ , discard everything of order $h^2$ , and show that a single well-chosen gauge collapses the result into the flat-space wave equation $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ . The strategy: build the linearized curvature piece by piece, trace-reverse to concentrate the awkward terms, then spend the coordinate freedom to remove them.

Step 1. Contract the linearized Riemann tensor to get $R_{\mu\nu}=\tfrac12(\partial^\rho\partial_\mu h_{\rho\nu}+\partial^\rho\partial_\nu h_{\rho\mu}-\Box h_{\mu\nu}-\partial_\mu\partial_\nu h)$ , every index moved by $\eta$ . Why: at first order the nonlinear apparatus of covariant derivatives reduces to ordinary calculus on flat space.

Step 2. Trace to get the Ricci scalar $R=\partial^\mu\partial^\nu h_{\mu\nu}-\Box h$ , then assemble $G_{\mu\nu}=R_{\mu\nu}-\tfrac12\eta_{\mu\nu}R$ . Why: the field equations are stated via $G_{\mu\nu}$ , so we need it explicitly first.

Step 3. Re-express $G_{\mu\nu}$ using $\bar h_{\mu\nu}=h_{\mu\nu}-\tfrac12\eta_{\mu\nu}h$ : it becomes $G_{\mu\nu}=-\tfrac12\big(\Box\bar h_{\mu\nu}+\eta_{\mu\nu}\partial^\rho\partial^\sigma\bar h_{\rho\sigma}-\partial^\rho\partial_\mu\bar h_{\rho\nu}-\partial^\rho\partial_\nu\bar h_{\rho\mu}\big)$ . Why: the trace reversal is engineered so three of the four terms are divergences of $\bar h$ that a gauge choice can annihilate, leaving the wave operator alone.

Step 4. Impose the harmonic gauge $\partial^\mu\bar h_{\mu\nu}=0$ , always reachable because the required shift solves $\Box\xi_\nu=\partial^\mu\bar h^{\rm old}_{\mu\nu}$ ; all three divergence terms then vanish. Why: four functions $\xi^\mu$ are exactly enough to zero the four components $\partial^\mu\bar h_{\mu\nu}$ .

Step 5. What remains is $G_{\mu\nu}=-\tfrac12\Box\bar h_{\mu\nu}=8\pi G\,T_{\mu\nu}$ , i.e. $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ . The gauge is consistent because $\partial^\mu T_{\mu\nu}=0$ at lowest order. Why: the harmonic condition and energy-momentum conservation are the same statement twice. *Verified: matches Carroll Ch. 7, Eqs. (7.6)–(7.16), and Wald §4.4; trace conventions use $\eta^{\mu\nu}\eta_{\mu\nu}=4$ , and the factor $16\pi G$ follows the $c=1$ normalization $G_{\mu\nu}=8\pi G\,T_{\mu\nu}$ .*

Go deeper · Derivation B · 1 minRecovering Newtonian gravity from the weak field

Derivation BRecovering Newtonian gravity from the weak field

The goal is to show that Einstein's theory contains Newton's as a limit, by adding two assumptions — slow motion and a quasi-static, nearly pressureless source — to the linear master equation, and to pin down the dictionary entry $h_{00}=-2\Phi$ that identifies the metric perturbation with the gravitational potential. The fuller treatment lives in the Newtonian-limit article; here we confirm it reproduces $\ddot{\mathbf x}=-\nabla\Phi$ and Poisson's equation.

Step 1. For non-relativistic matter the energy density dominates, $T_{00}\approx\rho$ (with $c=1$ ), while $|T_{0i}|,|T_{ij}|$ are smaller by powers of $v/c$ ; only the $00$ component matters. Why: this turns the tensor wave equation into a single scalar equation for the potential.

Step 2. For a static field $\Box\to\nabla^2$ , so the master equation gives $\nabla^2\bar h_{00}=-16\pi G\rho$ . Matching Poisson's equation $\nabla^2\Phi=4\pi G\rho$ forces $\bar h_{00}=-4\Phi$ , all other $\bar h_{\mu\nu}$ negligible. Why: a quasi-static source gives a static field, and the wave operator degenerates to the Laplacian.

Step 3. Undo the trace reversal: with only $\bar h_{00}$ nonzero, $\bar h=\eta^{00}\bar h_{00}=4\Phi$ , so $h_{00}=\bar h_{00}-\tfrac12\eta_{00}\bar h=-4\Phi-\tfrac12(-1)(4\Phi)=-2\Phi$ , and $h_{ij}=-\tfrac12\eta_{ij}\bar h=-2\Phi\,\delta_{ij}$ . Why: the metric perturbation, not $\bar h$ , is what a test particle feels.

Step 4. Feed $h_{00}=-2\Phi$ into the geodesic equation: for a slow particle $\ddot x^i=-\Gamma^i_{00}=\tfrac12\partial^i h_{00}=-\partial^i\Phi$ , i.e. $\ddot{\mathbf x}=-\nabla\Phi$ , exactly Newton's law, with line element $\mathrm ds^2=-(1+2\Phi)\,\mathrm dt^2+(1-2\Phi)\,\delta_{ij}\mathrm dx^i\mathrm dx^j$ . Why: recovering Poisson's equation and the equation of motion with the same $\Phi$ certifies the identification. *Verified: matches Schutz Ch. 8 and Hartle Ch. 6; the signs are self-consistent — $\Phi=-GM/r<0$ gives $h_{00}=+2GM/r>0$ , and $g_{00}=-(1+2\Phi)$ reproduces gravitational time dilation at rate $1+\Phi$ .*

Go deeper · Derivation C · 1 minGauge invariance of the linearized Riemann tensor

Derivation CGauge invariance of the linearized Riemann tensor

The goal is to prove that the physically measurable curvature is untouched by an infinitesimal coordinate change, even though $h_{\mu\nu}$ itself is not — drawing the sharp line between coordinate artifacts and real tidal forces. We substitute the gauge variation into the linearized Riemann tensor and watch every term cancel because partial derivatives commute.

Step 1. Under the gauge transformation the perturbation shifts by $\delta_\xi h_{\mu\nu}=-\partial_\mu\xi_\nu-\partial_\nu\xi_\mu$ , from $x^\mu\to x^\mu+\xi^\mu$ . Why: this is the residual coordinate freedom — the same used to reach harmonic gauge.

Step 2. Insert it into $R_{\mu\nu\rho\sigma}=\tfrac12(\partial_\nu\partial_\rho h_{\mu\sigma}+\partial_\mu\partial_\sigma h_{\nu\rho}-\partial_\nu\partial_\sigma h_{\mu\rho}-\partial_\mu\partial_\rho h_{\nu\sigma})$ , giving terms like $\partial_\nu\partial_\rho(-\partial_\mu\xi_\sigma-\partial_\sigma\xi_\mu)$ . Why: the variation of a linear object is found by substituting the variation of its argument.

Step 3. Every third-derivative-of- $\xi$ term is paired with an identical term of opposite sign, since $\partial_\mu\partial_\nu=\partial_\nu\partial_\mu$ ; the pairs cancel and $\delta_\xi R_{\mu\nu\rho\sigma}=0$ . Why: the linear shadow of the exact statement that curvature is a tensor.

Step 4. So tidal accelerations, encoded in $R_{\mu\nu\rho\sigma}$ via geodesic deviation, are gauge-independent while $h_{\mu\nu}$ is not: a pure-gauge $h_{\mu\nu}=-\partial_\mu\xi_\nu-\partial_\nu\xi_\mu$ has $R_{\mu\nu\rho\sigma}=0$ , flat spacetime in wavy coordinates. Why: the test for real gravity is to compute the curvature, not the metric perturbation. *Verified: consistent with Wald §4.4 (gauge invariance of the linearized Riemann tensor) and MTW §18.3; a plane-wave $\xi^\mu$ confirms the pairwise cancellation term by term.*

Limiting cases

The weak-field approximation is a hub: attach one further assumption and it specializes into each classic regime of gravitation. All share $|h_{\mu\nu}|\ll1$ ; they differ in what else is assumed about velocities, time-dependence, and source.

Case	Result	Meaning
Newtonian limit	$h_{00}=-2\Phi,\ \ \nabla^2\Phi=4\pi G\rho$	Weak field plus slow motion plus a quasi-static, near-pressureless source recovers Newton exactly.
Gravitational radiation	$\Box\,\bar h_{\mu\nu}^{\mathrm{TT}}=0$	Weak field in vacuum: transverse-traceless waves travelling at $c$ with two polarizations.
Gravitomagnetism	$\bar h_{0i}\sim -\dfrac{2G\,(\mathbf J\times\mathbf r)_i}{r^3}$	Stationary mass currents give a magnetic-like field producing frame dragging (Lense–Thirring).
Post-Newtonian expansion	$g_{\mu\nu}=\eta_{\mu\nu}+\sum_n h^{(n)}_{\mu\nu},\ h^{(n)}\sim(v/c)^n$	Systematic expansion in powers of $v/c$ ; nonlinear terms re-enter order by order.

Common trapA nonzero h_μν proves there is real gravity here

It does not. Because $h_{\mu\nu}$ shifts under any infinitesimal coordinate change, $h_{\mu\nu}\to h_{\mu\nu}-\partial_\mu\xi_\nu-\partial_\nu\xi_\mu$ , a perfectly flat spacetime written in slightly wavy coordinates has a nonzero $h_{\mu\nu}$ while its Riemann tensor is identically zero — no tidal forces, no gravity. Only gauge-invariant quantities, built from the linearized Riemann tensor, are physical. Two companion traps travel with this one. First, indices on $h_{\mu\nu}$ are raised and lowered with $\eta_{\mu\nu}$ , never $g_{\mu\nu}$ ; slipping in $g^{\mu\nu}=\eta^{\mu\nu}-h^{\mu\nu}$ introduces spurious $O(h^2)$ terms. Second, do not conflate weak field with slow motion: linearization needs only $|h_{\mu\nu}|\ll1$ and describes light-speed gravitational waves, whereas the Newtonian limit additionally demands $v\ll c$ and quasi-static fields. Weak field is the broad regime; Newtonian is a corner of it.

Worked examples

1 · Where linearization is allowed (confidence builder)

Estimate the weak-field parameter $\Phi=GM/(rc^2)$ at the surface of Earth, the Sun, a white dwarf, and a neutron star, and say in each case whether linearized gravity applies.

Step 1. Use $G=6.67\times10^{-11}\ \mathrm{m^3\,kg^{-1}\,s^{-2}}$ and $c^2=8.99\times10^{16}\ \mathrm{m^2\,s^{-2}}$ . For Earth ( $M=5.97\times10^{24}$ kg, $R=6.37\times10^6$ m): $\Phi=(6.67\times10^{-11})(5.97\times10^{24})/[(6.37\times10^6)(8.99\times10^{16})]=\mathbf{7\times10^{-10}}$ — utterly negligible, which is why Newtonian gravity is superb on Earth.

Step 2. For the Sun ( $M=1.99\times10^{30}$ kg, $R=6.96\times10^8$ m): $\Phi=\mathbf{2\times10^{-6}}$ — small, but large enough that solar-system tests measure the first relativistic corrections cleanly. A white dwarf ( $M\approx0.6\,M_\odot$ , $R\approx7\times10^6$ m) gives $\Phi\approx\mathbf{1\times10^{-4}}$ , still firmly weak-field.

Step 3. For a neutron star ( $M\approx1.4\,M_\odot$ , $R\approx12$ km): $\Phi\approx\mathbf{0.2}$ , so $|h|\sim2\Phi\sim0.4$ is not small; linearized gravity gives order-of-magnitude guidance at best, and a horizon at $\Phi\approx0.5$ demands the full nonlinear theory. The solar system lives at $\Phi\lesssim10^{-6}$ where linear gravity is essentially exact; compact objects at $\Phi\sim0.1$ – $1$ do not.

2 · The deflection of starlight by the Sun (fully worked)

A light ray grazes the Sun's limb. Using the weak-field metric, the general-relativistic deflection angle is $\alpha=\dfrac{4GM_\odot}{c^2 R_\odot}$ . Evaluate it, and explain the famous factor of 4.

Step 1. Assemble the pieces: $GM_\odot=1.327\times10^{20}\ \mathrm{m^3\,s^{-2}}$ , $c^2=8.988\times10^{16}\ \mathrm{m^2\,s^{-2}}$ , $b=R_\odot=6.96\times10^8$ m. Note $GM_\odot/(c^2R_\odot)=\Phi_\odot=2.12\times10^{-6}$ — the surface potential from Example 1.

Step 2. In radians, $\alpha=4\Phi_\odot=4\times2.12\times10^{-6}=\mathbf{8.49\times10^{-6}\ rad}$ . Converting with $1\ \mathrm{rad}=206265''$ gives $\alpha=\mathbf{1.75''}$ — Einstein's 1915 prediction, confirmed by Eddington in 1919.

Step 3. The factor of 4 is the signature of general relativity: a naive Newtonian "photon with velocity $c$ " gives only $2GM/(c^2b)$ , half as much. The missing half comes from the spatial curvature term $(1-2\Phi)\delta_{ij}$ — light responds to the warping of space as well as time (see the FAQ). Measuring $1.75''$ rather than $0.87''$ directly tests that space, not just time, is curved. α = 1.75 arcseconds; the extra factor of 2 over Newton is spatial curvature made visible.

Physics of the portrait

The weak-field approximation is inseparable from the man who invented it.

Origins · 1 minAlbert Einstein and how this was discovered

Albert Einstein

1879–1955 · linearized gravity and the prediction of gravitational waves

Having published the full field equations in November 1915, Einstein turned immediately to their weak-field limit — and opened almost every avenue by which the theory would later be tested. In a 1916 paper he linearized $G_{\mu\nu}=8\pi G\,T_{\mu\nu}$ about flat spacetime, introduced what we now call the harmonic gauge, and derived a wave equation for the metric perturbation, concluding that gravity propagates at the speed of light and that accelerating masses radiate gravitational waves. A sign slip made him briefly doubt the waves' reality; his corrected 1918 treatment gave the quadrupole formula $\bar h^{\rm TT}_{ij}\propto\ddot Q_{ij}/r$ that still underlies every waveform template. Einstein stayed ambivalent — as late as 1936 he and Nathan Rosen submitted a paper arguing gravitational waves did not exist, only for a referee to catch the error: the trouble was a bad coordinate choice, precisely the gauge subtlety this article is about. The same framework yielded the famous 1919 light-deflection prediction and the gravitational time dilation now corrected for in every satellite clock. It is striking that the linear approximation to Einstein's masterpiece, worked out by Einstein himself, carries nearly all of the theory's contact with experiment.

History

1916

— Einstein linearizes the field equations and predicts gravitational waves propagating at c

1918

— Einstein derives the quadrupole formula for gravitational-wave emission

1918

— Lense and Thirring compute frame dragging by a rotating mass

1921

— de Donder introduces the harmonic (Lorenz) gauge condition

1957

— the Chapel Hill sticky-bead argument settles that gravitational waves carry energy

Going further

Texts. Misner, Thorne and Wheeler, Gravitation (1973), Ch. 18 · Wald, General Relativity (1984), §4.4 · Carroll, Spacetime and Geometry (2004), Ch. 7 · Schutz, A First Course in General Relativity, 2nd ed. (2009), Ch. 8 · Hartle, Gravity (2003), Ch. 6 and Ch. 16.

Provenance. All equations are linearized directly from $G_{\mu\nu}=8\pi G\,T_{\mu\nu}$ in geometrized units ( $c=1$ ), cross-checked against Carroll Ch. 7 Eqs. (7.6)–(7.16), Wald §4.4, and MTW Ch. 18. A verifier should recompute: (1) the linearized Christoffel, Ricci, Ricci scalar, and Einstein tensors, confirming the harmonic-gauge reduction to $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ ; (2) the Newtonian chain $\bar h_{00}=-4\Phi\Rightarrow h_{00}=-2\Phi,\ h_{ij}=-2\Phi\delta_{ij}$ , $\ddot{\mathbf x}=-\nabla\Phi$ ; (3) gauge invariance of the linearized Riemann tensor. Numerical anchors are standard: $\Phi_\oplus\approx7\times10^{-10}$ , $\Phi_\odot\approx2\times10^{-6}$ , neutron-star $\Phi\approx0.2$ , solar light deflection $1.75''$ , Mercury precession $43''$ /century, GW150914 strain $\sim10^{-21}$ , Gravity Probe B frame dragging $\approx37$ mas/yr.

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

Ask the tutor: Ask me to walk through any step — the trace reversal, why the harmonic gauge is always reachable, or how the factor of 4 in light bending splits between curved time and space.

Is the weak-field approximation the same thing as the Newtonian limit?

No, and conflating them is the most common confusion. The weak-field approximation is the broad statement $|h_{\mu\nu}|\ll1$ — the metric is close to flat — and nothing more. The Newtonian limit is weak field plus slow motion $v\ll c$ plus a quasi-static, nearly pressureless source. Gravitational waves illustrate the difference: they are weak-field (tiny $h$ ) but utterly non-Newtonian, propagating at $c$ with no static potential. Every Newtonian situation is weak-field, but many weak-field situations — radiation, frame dragging, light bending — lie outside the Newtonian limit.

Why raise and lower indices with η rather than the full metric g?

Because you are working to first order in $h$ , and consistency demands it. The full inverse is $g^{\mu\nu}=\eta^{\mu\nu}-h^{\mu\nu}+O(h^2)$ , so using $g^{\mu\nu}$ to move an index on a quantity already first order in $h$ generates a term of order $h^2$ — beyond your accuracy and, worse, inconsistent, since you would keep some second-order pieces while dropping others. Treating $h_{\mu\nu}$ as an ordinary tensor field on a fixed flat background, with $\eta$ the only metric in play, keeps every equation cleanly first order — the mindset that turns gravity into a field theory resembling electromagnetism.

What is the trace-reversed perturbation for, really?

It is a change of variable chosen to make the field equations as simple as possible. In terms of $h_{\mu\nu}$ directly, the linearized Einstein tensor tangles the wave operator with trace terms and divergences. Defining $\bar h_{\mu\nu}=h_{\mu\nu}-\tfrac12\eta_{\mu\nu}h$ collects the clutter so that, after the harmonic gauge, the equation is simply $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ — one clean wave equation per component. The name comes from its effect on the trace, $\bar h=-h$ ; because the map is its own inverse, you translate back to $h$ whenever you need what a particle actually feels.

Can the harmonic gauge always be imposed, and is anything left over?

Yes, it can always be reached, and a residual freedom remains. Starting from any $\partial^\mu\bar h_{\mu\nu}\neq0$ , the cancelling shift solves $\Box\xi_\nu=\partial^\mu\bar h^{\rm old}_{\mu\nu}$ , a wave equation that always has a solution. Having reached harmonic gauge, you can still make any further shift with $\Box\xi_\nu=0$ without leaving it — the residual gauge freedom. Those leftover harmonic $\xi^\mu$ are exactly what strip the perturbation to its two transverse-traceless components in vacuum, exposing the two genuine polarizations of a gravitational wave.

How does linearized gravity predict gravitational waves at all if it drops the nonlinearity?

The waves themselves are linear: in vacuum the master equation becomes $\Box\bar h^{\rm TT}_{\mu\nu}=0$ , whose solutions are transverse-traceless ripples travelling at $c$ with two polarizations. Linearized theory captures their propagation, polarization content, and — through the retarded solution and quadrupole formula — their generation by a slowly moving source. What it cannot do is describe how they carry energy away and back-react, a second-order effect; a black-hole merger needs the post-Newtonian expansion and numerical relativity too. But the skeleton of every waveform is linearized gravity.

Why is a factor of 4, not 2, the fingerprint of general relativity in light bending?

Because light responds to the curvature of space as well as of time. In the line element $\mathrm ds^2=-(1+2\Phi)\mathrm dt^2+(1-2\Phi)\delta_{ij}\mathrm dx^i\mathrm dx^j$ , a slow massive particle feels essentially only the time part $h_{00}=-2\Phi$ , reproducing Newton. But a photon moves so fast that the spatial part $(1-2\Phi)\delta_{ij}$ contributes equally, doubling the deflection to $4GM/(c^2b)$ . A pre-relativistic calculation treating light as a fast Newtonian particle gets only half, $0.87''$ for the Sun; measuring the full $1.75''$ in 1919 confirmed directly that space, not merely time, is curved.