Weak-Field Limit and Linearized Gravity

Almost everywhere in the universe, gravity is weak. The Sun bends starlight by less than two arcseconds; the Earth warps the flow of time by barely one part in a billion; even a neutron star, the most compact object short of a black hole, sits at the edge of where "strong" begins. Linearized gravity is the theory built for this vast weak regime. Its move is disarmingly simple: write the full spacetime metric as the flat Minkowski background plus a small correction, $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ with $|h_{\mu\nu}|\ll1$ , and throw away everything beyond first order in $h$ . What survives is no longer Einstein's forbidding nonlinear geometry but a special-relativistic field theory — a symmetric rank-2 tensor field $h_{\mu\nu}$ propagating on flat spacetime, sourced by the stress-energy tensor exactly as the electromagnetic potential is sourced by charge. The payoff is that the whole apparatus of flat-space physics — Fourier transforms, Green's functions, retarded potentials — comes back into play, and the intractable Einstein tensor collapses into a single linear wave operator. The price is a subtlety that turns out to be a feature: the split is not unique. Slide the coordinates infinitesimally and $h_{\mu\nu}$ changes while the physics does not — gauge freedom, the exact analog of $A_\mu\to A_\mu+\partial_\mu\lambda$ . Once it is fixed with the Lorenz gauge the equations reduce to one luminous statement, $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ , which in different limits contains Newtonian gravity, frame dragging, light bending, the Shapiro delay, and gravitational waves — the launchpad for every precision test of general relativity in the Solar System.

Key takeaways

Linearized gravity writes the metric as $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ with $|h_{\mu\nu}|\ll1$ and keeps only terms linear in $h_{\mu\nu}$ , turning general relativity into a field theory on flat spacetime.
The split carries a gauge symmetry $h_{\mu\nu}\to h_{\mu\nu}-\partial_\mu\xi_\nu-\partial_\nu\xi_\mu$ — an infinitesimal coordinate change mirroring $A_\mu\to A_\mu+\partial_\mu\lambda$ — that must be fixed before solving.
With $\bar h_{\mu\nu}=h_{\mu\nu}-\tfrac12\eta_{\mu\nu}h$ and the Lorenz gauge $\partial^\mu\bar h_{\mu\nu}=0$ , the field equation becomes a decoupled flat-space wave equation $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ .
One equation, many regimes: the static limit gives Newton's $\nabla^2\Phi=4\pi G\rho$ , mass currents give gravitomagnetism and frame dragging, the transverse-traceless part gives gravitational waves at $c$ .
The size of $h$ is set by $\varepsilon=GM/(rc^2)$ : $\approx7\times10^{-10}$ at Earth's surface, $\approx2\times10^{-6}$ at the Sun's — weak-field territory to parts per million.
Key idea: Write $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ with $|h|\ll1$ , treat $h_{\mu\nu}$ as a tensor field on flat spacetime, fix the gauge freedom $h_{\mu\nu}\to h_{\mu\nu}-\partial_\mu\xi_\nu-\partial_\nu\xi_\mu$ with the Lorenz condition, and the Einstein equations become $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ — a flat-space wave equation that encodes all weak-field gravity.

Where this lives today

Linearized gravity is the quiet workhorse behind nearly every precision test of general relativity. The weak-field metric is the starting point for the parametrized post-Newtonian (PPN) formalism, whose leading coefficients become parameters $\beta,\gamma$ ; general relativity predicts $\beta=\gamma=1$ , and the 2003 Cassini experiment pinned $\gamma-1$ to about $2\times10^{-5}$ from the Shapiro delay of signals grazing the Sun. In the vector sector, the gravitomagnetic field of the spinning Earth drags inertial frames: Gravity Probe B measured this Lense–Thirring precession at $37\pm7$ mas/yr against a predicted $39.2$ . And in the radiative sector, all of gravitational-wave astronomy — LIGO, Virgo, KAGRA, and the future LISA — rests on the linearized wave equation: the matched-filter templates that pull a binary black-hole chirp from noise descend from the quadrupole formula, a child of $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ .

The physical picture

Think of gravity, almost everywhere, as a faint ripple painted on perfectly flat spacetime. The background is Minkowski, $\eta_{\mu\nu}=\mathrm{diag}(-1,+1,+1,+1)$ , rigid and featureless; the ripple $h_{\mu\nu}$ is the field of the Sun, the Earth, a passing wave. As long as it stays shallow — and the first figure shows just how shallow — you can use ordinary flat-space tools and treat $h_{\mu\nu}$ as just another field, like $A_\mu$ , that happens to carry two indices instead of one.

Figure

The dimensionless field strength ε = GM/(rc²) sets the size of h. At Earth's surface ε ≈ 7×10⁻¹⁰ and at the Sun's ε ≈ 2×10⁻⁶ — deep in the regime where linearized gravity is exact to parts per million — while a neutron-star surface (ε ≈ 0.17) and a black-hole horizon (ε = 0.5) demand the full nonlinear theory

The analogy with electromagnetism is exact in structure — the single most useful mental model in the subject. The gauge shift $h_{\mu\nu}\to h_{\mu\nu}-\partial_\mu\xi_\nu-\partial_\nu\xi_\mu$ mirrors $A_\mu\to A_\mu+\partial_\mu\lambda$ ; the Lorenz gauge $\partial^\mu\bar h_{\mu\nu}=0$ mirrors $\partial^\mu A_\mu=0$ ; and $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ mirrors $\Box A_\mu=-\mu_0 j_\mu$ . Under this dictionary $\bar h_{00}$ is the scalar potential (Newtonian gravity), $\bar h_{0i}$ the vector potential (gravitomagnetism), and the transverse-traceless remainder the radiation field — one tensor equation, sorted by index structure, is the entire weak-field zoo.

Figure

The single wave equation □h̄_μν = −16πG T_μν read in four limits gives all of weak-field gravity: the static 00-part is Newton's Poisson equation, the 0i mass-current part gives gravitomagnetism and frame dragging, the transverse-traceless part gives gravitational waves at speed c, and when ε approaches 1 linearization fails and full GR takes over

Curvature is the gauge-invariant content all four branches share. Just as $F_{\mu\nu}$ is invariant under $A_\mu\to A_\mu+\partial_\mu\lambda$ , the linearized Riemann tensor is invariant under the gravitational gauge shift. Any feature of $h_{\mu\nu}$ that a coordinate change can erase carries no tidal force and no physics; only the curvature is real — the deepest lesson of the weak-field theory.

The core equations

Everything begins with the split of the metric into a fixed flat background and a small dynamical correction — not an approximation to be justified after the fact but the definition of the framework: $h_{\mu\nu}$ is the object we solve for, and $|h_{\mu\nu}|\ll1$ is what lets us drop every higher power of it.

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. Reading: geometry is Minkowski plus a ripple that carries the field. Demanding $g_{\mu\nu}g^{\nu\rho}=\delta_\mu{}^\rho$ at first order gives the inverse $g^{\mu\nu}=\eta^{\mu\nu}-h^{\mu\nu}+\mathcal{O}(h^2)$ — the sign simply flips. From here on all indices are raised and lowered with $\eta$ (using $g$ would add only terms of order $h^2$ ), so $h^{\mu\nu}=\eta^{\mu\alpha}\eta^{\nu\beta}h_{\alpha\beta}$ . Feeding the split into the connection at first order gives the linearized Christoffel symbols, built from flat-space derivatives of $h_{\mu\nu}$ .

\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 (2)

where: $\Gamma^{\lambda}{}_{\mu\nu}$ — connection coefficients, first order in $h$ ; $\partial_\mu$ — flat-space derivative. Reading: the gravitational "force" is a first derivative of the field, as $\mathbf{E}$ is of $A_\mu$ . The perturbation is not physically unique: an infinitesimal coordinate change $x^\mu\to x^\mu+\xi^\mu$ shifts $h_{\mu\nu}$ while leaving all curvature untouched. This gauge freedom must be fixed before solving.

h_{\mu\nu}\to h_{\mu\nu}-\partial_\mu\xi_\nu-\partial_\nu\xi_\mu \qquad (3)

where: $\xi^\mu$ — infinitesimal coordinate displacement (the gauge generator). Reading: four free functions $\xi^\mu$ , matching the four conditions we impose to simplify the field equation. The field equation is cleanest in the trace-reversed variable, which absorbs the trace terms that clutter the Einstein tensor; its trace is the negative of the original in four dimensions.

\bar h_{\mu\nu}\equiv h_{\mu\nu}-\tfrac12\eta_{\mu\nu}h,\qquad h=\eta^{\mu\nu}h_{\mu\nu},\qquad \bar h=-h \qquad (4)

where: $\bar h_{\mu\nu}$ — trace-reversed perturbation, the natural field variable; $\bar h$ — its trace. Reading: an invertible repackaging of $h_{\mu\nu}$ — no information lost, the algebra tidier. Imposing the Lorenz (harmonic) gauge $\partial^\mu\bar h_{\mu\nu}=0$ spends the freedom of (3) and annihilates three of the four derivative clusters in the Einstein tensor, leaving one flat-space wave operator per component.

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

where: $\Box=\eta^{\mu\nu}\partial_\mu\partial_\nu=-\partial_t^2+\nabla^2$ ( $c=1$ ) — the flat d'Alembertian; $T_{\mu\nu}$ — source stress-energy; $G$ — Newton's constant. Reading: the master equation — ten decoupled wave equations, identical in form to electromagnetism. The master equation is solved by the retarded Green's function of the d'Alembertian, giving a potential that lags its source by the light-travel time — the seed of radiation.

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

where: $t-|\mathbf{x}-\mathbf{x}'|$ — retarded time, the light-travel delay ( $c=1$ ; restore it as $t-|\mathbf{x}-\mathbf{x}'|/c$ , and the prefactor as $4G/c^4$ , if you work in SI). Reading: gravity propagates, not instantaneously; drop the retardation and this is the Newtonian potential integral. Feeding the static solution back through (4) and (1) yields the weak-field line element every Solar-System test is computed from, with $\Phi=-GM/r$ the Newtonian potential — dimensionless here because $c=1$ .

\mathrm{d}s^2=-(1+2\Phi)\,\mathrm{d}t^2+(1-2\Phi)\,\delta_{ij}\,\mathrm{d}x^i\mathrm{d}x^j \qquad (7)

where: $\Phi$ — Newtonian potential, dimensionless in these $c=1$ units; restoring $c$ sends $\Phi\to\Phi/c^2$ , which is the form used in the worked examples below ( $\mathrm{d}s^2=-(1+2\Phi/c^2)c^2\mathrm{d}t^2+(1-2\Phi/c^2)\delta_{ij}\,\mathrm{d}x^i\mathrm{d}x^j$ ). Reading: both time and space parts are warped by $\Phi$ ; that spatial warp doubles the Newtonian light-bending estimate.

Derivation AFrom the Einstein equations to the linearized wave equation

The goal is to take Einstein's exact nonlinear field equation $G_{\mu\nu}=8\pi G\,T_{\mu\nu}$ and show that, once the metric is split and every term of order $h^2$ is discarded, it collapses into a single flat-space wave equation. The engine is a change of variable (the trace-reversed perturbation) and a gauge choice (Lorenz) that together clear away all but one derivative term.

Step 1. Insert the split into $R^\lambda{}_{\mu\nu\sigma}=\partial_\nu\Gamma^\lambda{}_{\mu\sigma}-\partial_\sigma\Gamma^\lambda{}_{\mu\nu}+\Gamma\Gamma-\Gamma\Gamma$ . The two $\Gamma\Gamma$ products are order $h^2$ and drop; contracting gives $R_{\mu\nu}=\tfrac12\big(\partial^\lambda\partial_\mu h_{\lambda\nu}+\partial^\lambda\partial_\nu h_{\lambda\mu}-\Box h_{\mu\nu}-\partial_\mu\partial_\nu h\big)$ . Why: discarding the quadratic connection terms is what turns curvature into a linear operator on $h$ .

Step 2. Trace with $\eta^{\mu\nu}$ for $R=\partial^\mu\partial^\nu h_{\mu\nu}-\Box h$ , and assemble $G_{\mu\nu}=R_{\mu\nu}-\tfrac12\eta_{\mu\nu}R$ — correct but a tangle of four terms. Why: the trace terms in $R$ and $G$ knot together; the next step untangles them.

Step 3. Switch to $\bar h_{\mu\nu}=h_{\mu\nu}-\tfrac12\eta_{\mu\nu}h$ . The Einstein tensor becomes $G_{\mu\nu}=-\tfrac12\big(\Box\bar h_{\mu\nu}+\eta_{\mu\nu}\partial^\alpha\partial^\beta\bar h_{\alpha\beta}-\partial^\lambda\partial_\nu\bar h_{\mu\lambda}-\partial^\lambda\partial_\mu\bar h_{\nu\lambda}\big)$ . Why: this variable isolates a clean $\Box\bar h_{\mu\nu}$ ; the other three terms carry a divergence $\partial^\lambda\bar h_{\lambda\cdot}$ that the gauge kills.

Step 4. Impose $\partial^\mu\bar h_{\mu\nu}=0$ . Each of the last three terms is such a divergence and vanishes, leaving $G_{\mu\nu}=-\tfrac12\Box\bar h_{\mu\nu}$ . The gauge is always reachable: the divergence shifts by $-\Box\xi_\nu$ , so solving $\Box\xi_\nu=\partial^\mu\bar h_{\mu\nu}$ enforces it. Why: the four $\xi^\mu$ are exactly what is needed to zero the four components of the divergence.

Step 5. Equate to the source, $-\tfrac12\Box\bar h_{\mu\nu}=8\pi G\,T_{\mu\nu}$ , and multiply by $-2$ : $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ , ten decoupled wave equations mirroring $\Box A_\mu=-\mu_0 j_\mu$ . Why: the $16\pi$ (not $8\pi$ ) is the fingerprint of the trace reversal, and it makes the retarded solution carry the coefficient $4G$ .

Go deeper · Derivation B · 1 minRecovery of the Newtonian limit and Newton's law of gravity

Derivation BRecovery of the Newtonian limit and Newton's law of gravity

The goal is to prove that the master equation contains Newtonian gravity exactly, fixing the coefficient $h_{00}=-2\Phi$ and returning $\ddot{\mathbf{x}}=-\nabla\Phi$ from the geodesic equation. The setting is a static, non-relativistic source, where the mass-energy density dwarfs every pressure and momentum flux, so $T_{00}=\rho$ dominates ( $=\rho c^2$ with $c$ restored).

Step 1. For a static source all time derivatives vanish, so $\Box\to\nabla^2$ , and with only $T_{00}=\rho$ surviving ( $c=1$ ) the master equation becomes $\nabla^2\bar h_{00}=-16\pi G\rho$ . Comparing with Poisson's $\nabla^2\Phi=4\pi G\rho$ gives $\bar h_{00}=-4\Phi$ . Why: the wave operator degenerating to the Laplacian is the content of "slow and static" — an instantaneous potential, no radiation.

Step 2. Undo the reversal with $h_{\mu\nu}=\bar h_{\mu\nu}-\tfrac12\eta_{\mu\nu}\bar h$ . With only $\bar h_{00}$ nonzero, $\bar h=\eta^{00}\bar h_{00}=4\Phi$ , so $h_{00}=-4\Phi+2\Phi=-2\Phi$ and $h_{ij}=-2\Phi\,\delta_{ij}$ . Why: the reversal spreads the single source component into both the time and space parts of $h_{\mu\nu}$ — the origin of the spatial curvature that later doubles light bending.

Step 3. Read off $g_{00}=-(1+2\Phi)$ and $g_{ij}=(1-2\Phi)\delta_{ij}$ — the weak-field line element (7). Why: both coefficients carrying the same $\Phi$ is a genuine prediction; Newtonian gravity alone would warp only $g_{00}$ .

Step 4. Drop a slow test particle in. The static connection is $\Gamma^i{}_{00}=-\tfrac12\partial_i h_{00}=\partial_i\Phi$ , so the geodesic equation gives $\ddot{\mathbf{x}}=-\nabla\Phi$ — Newton's law. Why: the defining check; a weak-field theory that failed to return $\mathbf{g}=-\nabla\Phi$ would be dead on arrival.

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

Derivation CGauge invariance of the linearized Riemann tensor

The goal is to separate what is physical from what is mere coordinate choice, by showing that the linearized Riemann tensor is unchanged by the gauge transformation while $h_{\mu\nu}$ is not. This is the counterpart of the invariance of $F_{\mu\nu}$ under $A_\mu\to A_\mu+\partial_\mu\lambda$ , and it licenses the whole apparatus: only curvature carries tidal forces and observable content.

Step 1. From the first-order connection, $R_{\mu\nu\rho\sigma}=\tfrac12\big(\partial_\nu\partial_\rho h_{\mu\sigma}+\partial_\mu\partial_\sigma h_{\nu\rho}-\partial_\mu\partial_\rho h_{\nu\sigma}-\partial_\nu\partial_\sigma h_{\mu\rho}\big)$ , with $\Gamma\Gamma$ terms dropped as second order. Why: this is a pure second derivative of $h$ , so its response to a gauge shift is governed entirely by second derivatives of the gauge terms.

Step 2. Apply $\delta h_{\mu\sigma}=-\partial_\mu\xi_\sigma-\partial_\sigma\xi_\mu$ to each of the four terms. A representative contribution, from $\partial_\nu\partial_\rho h_{\mu\sigma}$ , is $-\partial_\nu\partial_\rho\partial_\mu\xi_\sigma-\partial_\nu\partial_\rho\partial_\sigma\xi_\mu$ , with similar triple-derivative terms from the rest. Why: each gauge term is a third derivative of some $\xi$ , and third derivatives commute freely — the key to cancellation.

Step 3. Collect. Every third-derivative term is matched by an identical one of opposite sign from another Riemann term, because $\partial_a\partial_b=\partial_b\partial_a$ , so $\delta R_{\mu\nu\rho\sigma}=0$ identically. Why: the antisymmetric pairing of the four Riemann terms is engineered so symmetric-derivative gauge junk cancels in pairs.

Step 4. Interpret. The linearized Riemann tensor — and the tidal acceleration $a^\mu=-R^\mu{}_{0\nu0}\,\xi^\nu$ — is gauge invariant, while $h_{\mu\nu}$ is not. A perturbation whose Riemann tensor vanishes is pure gauge: a coordinate ripple with no gravity in it. Why: this criterion separates a real gravitational wave from a coordinate artifact, and is why observers report curvature, never $h_{\mu\nu}$ .

Limiting cases

The single equation $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ specializes to distinct physical regimes depending on which source components dominate and whether the field is static or time-dependent.

Case	Result	Meaning
Static Newtonian	$\nabla^2\Phi=4\pi G\rho,\ \ h_{00}=-2\Phi$	Slow, time-independent sources: only $T_{00}=\rho c^2$ matters and $\Box\to\nabla^2$ , recovering Poisson's equation and Newtonian orbits.
Gravitomagnetic (stationary)	$\nabla^2\bar h_{0i}=-16\pi G\,T_{0i}$	A steady mass current $T_{0i}=-\rho v_i$ sources a vector potential — frame dragging and Lense–Thirring, in exact analogy with magnetostatics.
Radiative (vacuum, time-dependent)	$\Box\,h^{TT}_{ij}=0$	Far from the source the transverse-traceless part obeys the homogeneous wave equation: gravitational waves at $c$ with two polarizations.
Strong-field breakdown	$\lvert h_{\mu\nu}\rvert\sim1\ \Leftrightarrow\ GM/(rc^2)\sim1$	Near horizons or in neutron-star interiors $h$ is no longer small; linearization fails and the full nonlinear equations are required.

Common trapA nonzero h_μν always means real gravity is present

Not so — this is the single most common trap in linearized gravity. Because the split is not unique, a nonzero $h_{\mu\nu}$ can be pure gauge: flat spacetime in wobbly coordinates, with no gravity whatsoever. Writing Minkowski space in slightly accelerated coordinates produces a nonzero $h_{\mu\nu}$ carrying not one gram of mass and not one tidal force. The only way to tell whether a given $h_{\mu\nu}$ is physical is to compute a gauge-invariant quantity — above all the linearized Riemann tensor: if it vanishes, the field is a coordinate mirage. A related slip is raising and lowering indices with the full $g_{\mu\nu}$ instead of $\eta_{\mu\nu}$ ; the difference is order $h^2$ , so a first-order theory must always use the flat metric.

Worked examples

1 · How weak is weak, and does it give back g = 9.8 m/s²? (fully worked)

Estimate $h_{00}$ at the Earth's surface, confirm linearization is wildly justified, and check that the theory returns the familiar surface gravity.

Step 1. The relevant number is $\varepsilon=GM/(Rc^2)=-\Phi/c^2$ . For the Earth, $GM_\oplus=3.986\times10^{14}\ \mathrm{m^3\,s^{-2}}$ , $R_\oplus=6.371\times10^{6}\ \mathrm{m}$ , $c^2=8.988\times10^{16}\ \mathrm{m^2\,s^{-2}}$ , so

\varepsilon=\frac{3.986\times10^{14}}{(6.371\times10^{6})(8.988\times10^{16})}=6.96\times10^{-10},

nine orders below the linearization threshold $|h|\sim1$ .

Step 2. The metric coefficient is $h_{00}=-2\Phi/c^2=2\varepsilon$ , so $h_{00}=1.39\times10^{-9}$ . Thus $g_{00}$ differs from the flat value $-1$ by one part in a billion — exactly the fractional rate at which ground clocks run slow relative to distant ones, the Earth's gravitational redshift.

Step 3. Check Newton. The static limit gives $\ddot{\mathbf{x}}=-\nabla\Phi$ with $\Phi=-GM/r$ , so

g=\frac{GM_\oplus}{R_\oplus^2}=\frac{3.986\times10^{14}}{(6.371\times10^{6})^2}=\mathbf{9.82\ m\,s^{-2}}.

The lesson: $h$ is minuscule, one part in $10^9$ , yet its gradient is all of everyday gravity — weak field and strong force are not in conflict.

2 · Light bending at the solar limb and the factor of two (fully worked)

A light ray grazes the Sun. Using the weak-field metric, compute the deflection angle and show why it is twice the naive "photon as a falling stone" value.

Step 1. The deflection of a null ray with impact parameter $b$ in the metric (7) is $\Delta\phi=4GM/(bc^2)$ . For a grazing solar ray $b=R_\odot$ , with $GM_\odot=1.327\times10^{20}\ \mathrm{m^3\,s^{-2}}$ and $R_\odot=6.96\times10^{8}\ \mathrm{m}$ :

\Delta\phi=\frac{4(1.327\times10^{20})}{(6.96\times10^{8})(8.988\times10^{16})}=8.49\times10^{-6}\ \mathrm{rad}.

Step 2. Convert with $1\ \mathrm{rad}=206\,265''$ : $\Delta\phi=8.49\times10^{-6}\times206\,265=\mathbf{1.75''}$ . This is what Eddington's 1919 eclipse expedition set out to test, and it agrees with modern radio measurements to better than one part in $10^4$ .

Step 3. Where the factor of two hides. A photon treated as a Newtonian particle moving at $c$ , deflected by $g_{00}$ alone, would bend by only $2GM/(bc^2)=\mathbf{0.875''}$ — exactly half. The missing half comes from the spatial metric $g_{ij}=(1-2\Phi)\delta_{ij}$ : light, moving fast, samples the curvature of space as strongly as that of time. Both contribute equally, summing to $1.75''$ — a clean discriminator between general relativity and any theory that curves only time.

Physics of the portrait

Linearized gravity is Einstein's own creation: the linear approximation, the gauge idea, and the gravitational wave all trace to the same short papers of 1916, written within months of the full field equations.

Origins · 1 minAlbert Einstein and how this was discovered

Albert Einstein

1879–1955 · the linearized approximation and gravitational waves

Having presented the full nonlinear field equations in November 1915, Einstein did not wait for exact solutions before mining them for physics. In June 1916, in "Näherungsweise Integration der Feldgleichungen der Gravitation," he introduced the move that defines this article: write the metric as $\eta_{\mu\nu}$ plus a small $h_{\mu\nu}$ , keep first order, impose a gauge condition, and reduce the equations to wave equations. From that reduction he read off gravitational waves travelling at the speed of light — a sign error initially gave three polarizations, corrected to the modern two by 1918, when he also derived the quadrupole formula. His weak-field metric was the tool for the classic tests: the $1.75''$ deflection confirmed by Eddington in 1919, Mercury's perihelion precession, and the gravitational redshift. He remained ambivalent about his own waves for decades, in 1936 even submitting (then withdrawing) a paper arguing they did not exist. The theory he built as a mere approximation became, a century later, the language of gravitational-wave astronomy: the templates that matched GW150914 in 2015 are, at leading order, solutions of the very equation he wrote in 1916.

History

1915

— Einstein presents the full nonlinear field equations of general relativity

1916

— Einstein linearizes the equations and predicts gravitational waves

1918

— Lense and Thirring compute frame dragging by a rotating mass

1919

— Eddington's eclipse expedition measures the 1.75″ deflection of starlight

2015

— LIGO records GW150914, gravitational waves observed directly

Going further

Texts. Misner, Thorne & 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. 21 · Weinberg, Gravitation and Cosmology (1972), Ch. 10. Key results to keep in view: the linearized Ricci tensor follows from first-order Christoffel symbols; in Lorenz gauge the Einstein tensor reduces to $G_{\mu\nu}=-\tfrac12\Box\bar h_{\mu\nu}$ ; the Newtonian limit gives $h_{00}=-2\Phi$ and the geodesic equation reduces to $-\nabla\Phi$ ; and the linearized Riemann tensor is gauge invariant. Numerical anchors used throughout: Earth-surface $\varepsilon=6.96\times10^{-10}$ , $g=9.82\ \mathrm{m\,s^{-2}}$ , solar-limb deflection $1.75''$ , $\gamma-1\sim2\times10^{-5}$ (Cassini), Gravity Probe B $37\pm7$ mas/yr, GW150914 (2015).

Prerequisites. Einstein Field Equations · Christoffel Symbols and the Riemann Tensor · Special Relativity and Minkowski Spacetime · Stress-Energy Tensor. Leads to. Newtonian Limit of General Relativity · Deflection of Light by the Sun · Shapiro Time Delay · Gravitomagnetism · Frame Dragging and the Lense–Thirring Effect · Parametrized Post-Newtonian (PPN) Formalism.

Ask the tutor: Ask me to expand any branch of the master equation — the Newtonian limit, frame dragging, or the transverse-traceless gravitational wave — into a full derivation with worked numbers.

What does it actually mean for h_μν to be "small"?

It means the dimensionless field strength $\varepsilon=GM/(rc^2)$ is far below one, since $|h_{\mu\nu}|\sim\varepsilon$ . At the Earth's surface $\varepsilon\approx7\times10^{-10}$ ; at the Sun's $\approx2\times10^{-6}$ ; even at a neutron-star surface only $\approx0.2$ . Linearization keeps first order in $\varepsilon$ and drops $\varepsilon^2$ , so its fractional error is roughly $\varepsilon$ — which is why the theory is essentially exact throughout the Solar System and strains only near compact objects. The smallness is not an aesthetic assumption but a controlled expansion parameter: you always know how large the neglected terms are.

Why introduce the trace-reversed perturbation h̄_μν at all?

Purely to make the algebra collapse. Written in the raw $h_{\mu\nu}$ , the linearized Einstein tensor is four unwieldy derivative clusters, because the trace terms in the Ricci scalar tangle with those in $G_{\mu\nu}=R_{\mu\nu}-\tfrac12\eta_{\mu\nu}R$ . Defining $\bar h_{\mu\nu}=h_{\mu\nu}-\tfrac12\eta_{\mu\nu}h$ absorbs exactly those trace pieces, so after the Lorenz gauge only a single clean $\Box\bar h_{\mu\nu}$ survives. It is an invertible change of variable — no physics lost — and $h_{\mu\nu}$ is recovered by reversing the trace.

How is the gauge freedom here the same as in electromagnetism?

Structurally it is identical. The potential is defined only up to $A_\mu\to A_\mu+\partial_\mu\lambda$ , since that leaves $F_{\mu\nu}$ unchanged; the perturbation is defined only up to $h_{\mu\nu}\to h_{\mu\nu}-\partial_\mu\xi_\nu-\partial_\nu\xi_\mu$ for any vector $\xi^\mu$ — an infinitesimal coordinate change — since that leaves the linearized Riemann tensor unchanged. One scalar of freedom becomes one vector of freedom, and just as $\partial^\mu A_\mu=0$ gives $\Box A_\mu=-\mu_0 j_\mu$ , the condition $\partial^\mu\bar h_{\mu\nu}=0$ gives $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ .

If the field is so weak, why does gravity feel so strong to us?

Because what we feel is not $h_{\mu\nu}$ but its gradient. The perturbation at the Earth's surface is only $\sim10^{-9}$ , yet it produces $g=GM/R^2\approx9.8\ \mathrm{m\,s^{-2}}$ , because $\Phi=-GM/r$ varies over the Earth's radius — a tiny warp spread over thousands of kilometres still has a steep local slope. It is the derivative, not the height, that drives the motion, so gravity is weak as a distortion and strong as a force at once.

Does linearized gravity predict gravitational waves, or do you need the full theory?

It predicts them outright. In vacuum and the transverse-traceless gauge the master equation reduces to $\Box h^{TT}_{ij}=0$ , whose solutions are ripples propagating at $c$ with two polarizations; the retarded solution (6) applied to a time-varying source gives the quadrupole formula for the radiated power. The full nonlinear theory is needed only for the strong-field sources — two black holes in their final orbits — and their back-reaction; the wave reaching your detector is, to superb accuracy, a solution of the linear equation.

Can I always reach the Lorenz gauge, and is it unique once reached?

You can always reach it, and it is not quite unique. From any $h_{\mu\nu}$ , the divergence $\partial^\mu\bar h_{\mu\nu}$ shifts by $-\Box\xi_\nu$ , so solving $\Box\xi_\nu=\partial^\mu\bar h_{\mu\nu}$ — always possible — sets it to zero. But the condition does not exhaust the freedom: any further $\xi^\mu$ with $\Box\xi^\mu=0$ preserves it. That residual freedom is exactly what you spend to reach the transverse-traceless gauge for gravitational waves, removing the unphysical polarizations — just as the residual gauge in electromagnetism lets you set $A_0=0$ for radiation.