Post-Newtonian Expansion

General relativity is exact but, for almost every system anyone actually wants to describe, unsolvable in closed form. Between the clean-but-wrong world of Newton and the correct-but-intractable world of full Einstein sits an enormous middle ground: planets orbiting the Sun, binary stars, the long slow spiral of two black holes before they touch. What these share is that they are bound by their own gravity and move slowly compared with light. The post-Newtonian expansion exploits exactly those two facts, writing the metric as flat spacetime plus an ordered stack of small corrections — each fainter than the last by a fixed factor — computed one rung at a time instead of solving the full nonlinear theory at once. The engine that makes a single expansion possible is the virial theorem. For a bound orbit, velocity and potential are not independent smallnesses: $v^2\sim GM/r$ locks them together, so $v^2/c^2\sim|\Phi|/c^2\equiv\epsilon^2$ and one parameter $\epsilon\sim v/c$ orders everything. The leading term reproduces Newton exactly; the first correction, one power of $\epsilon^2$ up, is "1PN" and already contains Mercury's perihelion advance, the bending and delay of light, and frame dragging. Push further and at half-integer order 2.5PN the expansion sprouts a time-asymmetric term — the signature of energy leaking irreversibly to infinity as gravitational waves. The same hierarchy that grounds every solar-system test supplies the analytic inspiral templates that let LIGO recognize two coalescing black holes in a wall of noise.

Key takeaways

The post-Newtonian (PN) expansion is a systematic weak-field, slow-motion approximation to GR: the metric and equations of motion are expanded in powers of $\epsilon\sim v/c$ , valid for systems held together by their own gravity so that $v^2/c^2\sim GM/(rc^2)$ .
The virial relation $v^2\sim GM/r$ is the crucial input — it ties velocity to potential, so one parameter controls the hierarchy and each PN order adds a relative correction of size $\epsilon^2=(v/c)^2$ .
At 0PN one recovers $g_{00}=-(1+2\Phi/c^2)$ with $\nabla^2\Phi=4\pi G\rho$ ; 1PN supplies Mercury's $43''$ /century perihelion advance, the $1.75''$ deflection of starlight, and the Shapiro delay.
The 1PN dynamics are the Einstein–Infeld–Hoffmann (EIH) equations, derived from the field equations plus the Bianchi identities — in GR the law of motion is not a separate postulate.
Radiation reaction first enters at 2.5PN, relative order $(v/c)^5$ ; the half-integer order is the fingerprint of time-asymmetry and gives the quadrupole energy-loss formula driving compact-binary inspiral.
The Parametrized Post-Newtonian (PPN) formalism repackages the 1PN metric with adjustable coefficients — the standard model-independent language for confronting GR with experiment.
Key idea: for a gravitationally bound system the virial relation $v^2/c^2\sim GM/(rc^2)\equiv\epsilon^2$ ties velocities to potentials, so a single small parameter $\epsilon\sim v/c$ orders the entire expansion — $n$ PN means corrections of relative order $\epsilon^{2n}$ beyond Newtonian gravity.

Where this lives today

The post-Newtonian expansion runs, right now, inside every gravitational-wave detection pipeline. When LIGO and Virgo record a candidate, matched filtering slides thousands of analytic inspiral templates against the data, and the low-frequency body of each is a PN waveform: the 3.5PN phase $\phi(f)$ built from the same $\dot f\propto f^{11/3}$ chirp law derived above. That phase is so sensitive that GW170817 pinned the neutron stars' masses and tidal deformabilities, while its gamma-ray burst fixed the speed of gravity to equal $c$ to one part in $10^{15}$ . In the solar system the 1PN metric, as the ten-parameter PPN formalism, is the language of every modern test: Cassini bounds $\gamma-1=(2.1\pm2.3)\times10^{-5}$ , Gravity Probe B measured the 1.5PN frame-dragging of orbiting gyroscopes, and the double pulsar PSR J0737−3039 tests the conservative and dissipative sectors at once through its orbital decay. Open frontiers: the conservative dynamics beyond 4PN, effective-one-body resummations valid closer to merger, and higher multipole fluxes for LISA and the Einstein Telescope.

The physical picture

Zoom in on the solar system with a magnifying glass whose power is $\epsilon\sim v/c$ — about $10^{-4}$ for Earth, $1.6\times10^{-4}$ for Mercury, the fastest planet. At the coarsest setting, $\epsilon^0$ , you see Newton: ellipses, closed and eternal. Turn the power up one notch — multiply by $\epsilon^2\approx2.6\times10^{-8}$ for Mercury — and a faint new layer appears: the ellipse no longer quite closes, its perihelion creeping forward. The post-Newtonian expansion resolves these layers one at a time rather than seeing the whole nonlinear picture at once (which is what numerical relativity does, at enormous cost). The reason one magnification factor works for everything is the virial theorem: in a bound orbit the kinetic smallness $v/c$ and the potential smallness $\Phi/c^2$ are not free to differ, since $v^2\sim GM/r$ welds them into one scale $\epsilon^2$ . This is what makes gravity peculiar — a charged particle can move arbitrarily fast in a given external potential, but here the mass that sources the field also sets the speed of the orbit, so the whole tower of corrections can be indexed by one number.

Figure

The post-Newtonian ladder — each rung down is a factor ε² smaller than the one above, every rung through 2PN time-symmetric (conservative) and the 2.5PN rung time-odd (radiation reaction). For Mercury ε ≈ 1.6×10⁻⁴, so one 1PN step is a relative correction of only 2.6×10⁻⁸

There is a subtle rhythm in which metric pieces appear at which order, dictated by time-reversal. Energy density and stress ( $T^{00}$ , $T^{ij}$ ) are even under $t\to-t$ , so they feed $g_{00}$ and $g_{ij}$ at even powers of $\epsilon$ ; momentum density ( $T^{0i}\propto\rho v^i$ ) flips sign and is one power of $v$ smaller, so it feeds $g_{0i}$ at odd powers from $\epsilon^3$ — the "gravitomagnetic" field, exactly as moving charge makes a magnetic field. And a genuinely half-integer order — 2.5PN, $\epsilon^5$ — is the stamp of time-asymmetry: it cannot come from any time-symmetric near-zone potential and instead represents energy flowing irreversibly outward as radiation. The second figure is where that matters most.

Figure

A compact-binary chirp — gravitational-wave emission (a 2.5PN effect) drains orbital energy as the bodies spiral in, so frequency and amplitude sweep upward with ḟ ∝ f^(11/3). The PN series describes the inspiral where v/c ≈ 0.1 to 0.3, but near merger v/c → 0.4 and numerical relativity takes over

Finally, a piece of conceptual magic: in GR you never postulate a force. Feed the field equations a slowly moving lump, impose the contracted Bianchi identities ( $\nabla_\mu G^{\mu\nu}\equiv0$ , hence $\nabla_\mu T^{\mu\nu}=0$ ), and the equations of motion — Newton plus every relativistic correction — fall out automatically. The Einstein–Infeld–Hoffmann equations are this fact made explicit.

The core equations

The whole scheme rests on one estimate: for a bound system, every smallness is the same smallness. The parameter $\epsilon$ measures it; the virial theorem is why it exists.

\epsilon\sim\frac{v}{c}\sim\sqrt{\frac{GM}{rc^2}}\sim\sqrt{\frac{|\Phi|}{c^2}} \qquad (1)

where: $\epsilon$ — dimensionless smallness parameter ordering the expansion; $v$ — characteristic internal velocity of the source; $\Phi$ — Newtonian gravitational potential, with $\Phi<0$ ; $GM/r$ — characteristic potential energy per unit mass, virial-equal to $v^2$ . Because the orbit is bound, choosing a velocity fixes the potential — there is only one knob to turn. The metric is written as Minkowski plus corrections, with a deliberate staggering of which component gets which power — the skeleton of everything that follows.

g_{00}=-1+\overset{(2)}{g}_{00}+\overset{(4)}{g}_{00}+\cdots,\quad g_{0i}=\overset{(3)}{g}_{0i}+\cdots,\quad g_{ij}=\delta_{ij}+\overset{(2)}{g}_{ij}+\cdots \qquad (2)

where: $\overset{(n)}{g}_{\mu\nu}$ — the part of $g_{\mu\nu}$ of order $\epsilon^n$ ; $\delta_{ij}$ — flat Cartesian spatial metric; $\eta_{\mu\nu}=\mathrm{diag}(-1,1,1,1)$ — the Minkowski background. Even powers in $g_{00},g_{ij}$ ; odd powers from $\epsilon^3$ in $g_{0i}$ — a pattern forced by time-reversal, derived below. At the lowest nontrivial order the machinery reproduces Newton exactly — the 0PN anchor of the whole tower.

\overset{(2)}{g}_{00}=-\frac{2\Phi}{c^2},\qquad \nabla^2\Phi=4\pi G\rho \qquad (3)

where: $\Phi$ — Newtonian potential; $\rho$ — rest-mass density of the source; $\nabla^2$ — flat-space Laplacian. The geodesic equation collapses to $\ddot{x}^i=-\partial_i\Phi$ , Newton's second law with Newton's gravity. Carrying the expansion one order up produces the full 1PN metric; in one common (standard PN) gauge it collects a new scalar potential and a vector potential alongside $\Phi$ .

ds^2=-\Big(1+\tfrac{2\Phi}{c^2}+\tfrac{2\Phi^2}{c^4}-\tfrac{2\psi}{c^4}\Big)c^2\,\mathrm{d}t^2+\tfrac{8}{c^3}\zeta_i\,c\,\mathrm{d}t\,\mathrm{d}x^i+\Big(1-\tfrac{2\Phi}{c^2}\Big)\delta_{ij}\,\mathrm{d}x^i\mathrm{d}x^j \qquad (4)

where: $\Phi$ — Newtonian potential, $\nabla^2\Phi=4\pi G\rho$ ; $\psi$ — 1PN scalar potential collecting kinetic, pressure and gravitational self-energy sources; $\zeta_i$ — vector (gravitomagnetic) potential sourced by the mass current $\rho v^i$ . The $2\Phi^2/c^4$ term is gravity gravitating on itself; the cross-term is what drags inertial frames. The vector potential obeys a Poisson equation with the mass current as source — structurally identical to magnetostatics, which is why the whole $g_{0i}$ sector is called gravitomagnetism.

\nabla^2\zeta_i=-4\pi G\,\rho\,v_i\qquad\Rightarrow\qquad \zeta_i(\mathbf{x})=G\int\frac{\rho(\mathbf{x}')v_i(\mathbf{x}')}{|\mathbf{x}-\mathbf{x}'|}\,\mathrm{d}^3x' \qquad (5)

where: $\zeta_i$ — vector potential entering $g_{0i}$ ; $\rho v_i$ — mass-current density, the source of frame dragging. Replace mass current with electric current and this is exactly the magnetostatic vector potential — hence the name. The 1PN equations of motion for a set of bodies are the Einstein–Infeld–Hoffmann equations — schematically Newton plus corrections of relative order $v^2/c^2$ , with no separately postulated force law.

\frac{\mathrm{d}^2\mathbf{x}_a}{\mathrm{d}t^2}=-\sum_{b\ne a}\frac{Gm_b(\mathbf{x}_a-\mathbf{x}_b)}{r_{ab}^3}\Big[1+\mathcal{O}\!\left(\tfrac{v^2}{c^2}\right)\Big]+\frac{1}{c^2}(\text{EIH corrections}) \qquad (6)

where: $\mathbf{x}_a$ — position of body $a$ as a point mass; $m_a$ — mass of body $a$ ; $r_{ab}=|\mathbf{x}_a-\mathbf{x}_b|$ — coordinate separation. The bracketed corrections depend on all the bodies' velocities and mutual potentials — they are what precess Mercury. The most famous single number the 1PN dynamics deliver is the secular advance of a bound orbit's pericenter.

\Delta\varpi=\frac{6\pi GM}{c^2\,a(1-e^2)}\quad\text{per orbit} \qquad (7)

where: $\Delta\varpi$ — advance of pericenter per orbit (radians); $a$ — semi-major axis; $e$ — orbital eccentricity; $M$ — total mass ( $\approx M_\odot$ for Mercury). The effect grows for tight ( $a$ small) and eccentric ( $e\to1$ ) orbits — why Mercury shows it most among the planets. At half-integer order the conservative story ends and dissipation begins: the 2.5PN energy flux is the quadrupole luminosity, the rate at which the system radiates gravitational waves.

\dot{E}_{\rm GW}=-\frac{G}{5c^5}\big\langle\dddot{Q}_{ij}\dddot{Q}_{ij}\big\rangle \qquad (8)

where: $Q_{ij}$ — trace-free mass quadrupole moment; $\dddot{Q}_{ij}$ — its third time derivative; $\langle\cdot\rangle$ — average over an orbital period. The three time derivatives and the $1/c^5$ prefactor are exactly what put this at relative order $(v/c)^5$ , i.e. 2.5PN.

Derivation APower counting and the order staggering of the metric

The goal here is to explain the most puzzling feature of the ansatz in equation (2): why $g_{00}$ and $g_{ij}$ carry only even powers of $\epsilon$ while $g_{0i}$ carries only odd powers, and to pin down the leading coefficients from first principles. The strategy is to demand that the slow-motion geodesic reproduce Newton at lowest order, then track how each metric piece is sourced and behaves under time reversal.

Step 1. Write the geodesic for a slow particle, splitting the Christoffel contractions by velocity factors: $\mathrm{d}^2x^i/\mathrm{d}t^2\simeq-\Gamma^i_{00}c^2-2\Gamma^i_{0j}c\,v^j-\Gamma^i_{jk}v^jv^k$ . This is the leading slow-motion form, not the full $t$ -parametrized geodesic equation: because $t$ is not an affine parameter there is also a term $+\big(\Gamma^0_{00}c^2+2\Gamma^0_{0j}c\,v^j+\Gamma^0_{jk}v^jv^k\big)v^i/c$ , which is harmless for the Newtonian match in Step 2 but contributes at the same 1PN order as $\Gamma^i_{jk}v^jv^k$ . Each $v^j$ is $\mathcal{O}(\epsilon c)$ , so the three terms kept carry $0,1,2$ extra powers of $\epsilon$ . Why: to recover Newton the $\Gamma^i_{00}c^2$ term must dominate, fixing the leading order of $g_{00}$ .

Step 2. Evaluate $\Gamma^i_{00}\approx\tfrac12\partial_i\,\overset{(2)}{g}_{00}$ for a static field and demand $\ddot{x}^i=-\partial_i\Phi$ . This forces $\overset{(2)}{g}_{00}=-2\Phi/c^2$ , so the first correction to $g_{00}$ sits at $\mathcal{O}(\epsilon^2)$ , the next at $\mathcal{O}(\epsilon^4)$ . Why: the Newtonian match ties the abstract expansion to measured gravity and starts the even-power counting.

Step 3. Invoke time-reversal. Under $t\to-t$ , $g_{00}$ and $g_{ij}$ are even, $g_{0i}$ is odd; their sources inherit this. $T^{00}$ (energy) and $T^{ij}$ (stress) are time-even, but $T^{0i}\propto\rho v^i$ (momentum) is time-odd and one power of $v$ smaller. Hence $g_{0i}$ cannot appear until $\mathcal{O}(\epsilon^3)$ , one order above the $\mathcal{O}(\epsilon^2)$ opening of $g_{00},g_{ij}$ . Why: this is the whole origin of the staggering — a symmetry statement about the source, not an accident.

Step 4. Close the spatial metric using the $ij$ Einstein equations in harmonic gauge, $\nabla^2\overset{(2)}{g}_{ij}=-(8\pi G/c^2)\rho\,\delta_{ij}$ , giving $\overset{(2)}{g}_{ij}=-(2\Phi/c^2)\delta_{ij}$ — the same $\Phi$ as in $g_{00}$ but isotropic. Why: consistency forces spatial and time curvature to share $\Phi$ , and it is this spatial piece that (with $g_{00}$ ) doubles the Newtonian light-bending to the observed $1.75''$ .

Go deeper · Derivation B · 1 minThe 1PN perihelion advance from the orbit equation

Derivation BThe 1PN perihelion advance from the orbit equation

The goal is to derive the perihelion advance of equation (7) directly from the relativistic orbit equation, showing exactly where the Newtonian ellipse fails to close and reducing the answer to $6\pi GM/[c^2a(1-e^2)]$ . The plan: start from the radial equation for a test particle in the 1PN (effectively Schwarzschild) field, differentiate to a driven-oscillator form, solve perturbatively, and read off the frequency shift that becomes the precession.

Step 1. Write the orbit equation in the Binet variable $u=1/r$ : $(\mathrm{d}u/\mathrm{d}\phi)^2+u^2=(2GM/L^2)u+2E/L^2+(2GM/c^2)u^3$ , with $L,E$ the specific angular momentum and energy. The cubic term is the entire relativistic correction; drop it for the exact Newtonian Kepler problem. Why: isolating it shows the effect is suppressed by $GM/(c^2r)\sim\epsilon^2$ relative to Newton.

Step 2. Differentiate with respect to $\phi$ : $\mathrm{d}^2u/\mathrm{d}\phi^2+u=GM/L^2+(3GM/c^2)u^2$ . Newton gives a pure harmonic oscillator (period $2\pi$ , closed ellipse); the extra $3GM\,u^2/c^2$ is a small anharmonic perturbation. Why: a closed orbit is a $2\pi$ -periodic solution, so any shift away from $2\pi$ is the non-closure we seek.

Step 3. Insert the near-Keplerian trial $u\approx(GM/L^2)[1+e\cos((1-\delta)\phi)]$ and match to first order; the perturbation shifts the radial frequency by $\delta=3(GM)^2/(c^2L^2)$ . Why: $\delta$ is the extra fraction of a turn the particle must sweep to return to pericenter — the precession per orbit is $2\pi\delta$ .

Step 4. Convert with the Keplerian identity $L^2=GMa(1-e^2)$ : $\Delta\varpi=2\pi\delta=6\pi(GM)^2/(c^2L^2)=6\pi GM/[c^2a(1-e^2)]$ , which is $43.0''$ /century for Mercury. Why: expressing $L$ through $a,e$ turns a dynamical quantity into catalogue orbital elements — the numeric check is in the worked examples.

Go deeper · Derivation C · 1 minWhy radiation reaction first appears at 2.5PN

Derivation CWhy radiation reaction first appears at 2.5PN

The goal is to show, by energy balance and power counting alone, that gravitational radiation reaction cannot appear before relative order $(v/c)^5$ — the half-integer 2.5PN order — and why that half-integer character is physically inevitable. The approach compares the gravitational-wave luminosity to the rate at which the orbit does mechanical work, then reads the odd power through time-reversal.

Step 1. Estimate the luminosity from the quadrupole formula: $P_{\rm GW}\sim(G/c^5)\langle\dddot{Q}_{ij}\dddot{Q}_{ij}\rangle\sim(G/c^5)(Mr^2)^2\omega^6$ , using $Q\sim Mr^2$ , three time derivatives ( $\omega^3$ each), and $\omega^2\sim GM/r^3$ . Why: the $1/c^5$ and three derivatives are why the effect is faint — no fewer than five inverse powers of $c$ , since the leading radiative multipole is the quadrupole.

Step 2. Form the ratio of radiated power to the orbital energy scale $E_{\rm bind}\,\omega$ (with $E_{\rm bind}\sim GM^2/r$ , $\omega\sim v/r$ ): it collapses to $(GM/rc^2)^{5/2}\sim(v/c)^5=\epsilon^5$ , two and a half powers of $\epsilon^2$ — hence 2.5PN. Why: comparing to the orbital energy timescale converts an absolute luminosity into a relative PN order, landing on a half-integer.

Step 3. Interpret the odd power. The near-zone potentials that dominate the lower rungs are time-even and describe conservative, reversible dynamics; the $\epsilon^5$ term is time-odd, changing sign under $t\to-t$ , and describes an irreversible process — energy leaving for good. Half-integer order is necessary but not sufficient for dissipation: $\epsilon^3$ (1.5PN) is half-integer yet purely conservative, carrying the gravitomagnetic frame dragging, and conservative half-integer terms return at 5.5PN through hereditary tail effects. Why: 2.5PN is where dissipation first bites because it is the lowest order at which a time-odd term can be sourced by energy escaping to infinity — not because every half-integer order is dissipative.

Step 4. Feed the loss back through $\dot{E}=-P_{\rm GW}$ for a circular binary; with $E(f)$ and $P_{\rm GW}(f)$ this gives the chirp law $\dot{f}\propto f^{11/3}$ — the frequency sweeping upward ever faster as the orbit shrinks. Why: this single relation is the analytic backbone of the inspiral template that let LIGO extract GW150914 from the noise.

Limiting cases

The PN hierarchy is labeled by the relative order $\epsilon^{2n}=(v/c)^{2n}$ beyond Newton; each rung has a name, a leading result, and a domain where it is the dominant new physics.

Case	Result	Meaning
0PN (Newtonian)	$g_{00}=-(1+2\Phi/c^2),\ \nabla^2\Phi=4\pi G\rho$	Recovers Newton's law; the geodesic reduces to $\ddot{x}^i=-\partial_i\Phi$ .
1PN ( $\epsilon^2$ )	$\Delta\sim(v/c)^2\sim2.6\times10^{-8}$ (Mercury)	Perihelion advance, Shapiro delay, light deflection, next-order redshift; the EIH equations live here.
1.5PN ( $\epsilon^3$ )	$g_{0i}\sim(G/c^3)\,(\text{mass current})/r$	Gravitomagnetic frame dragging (Lense–Thirring) and leading spin–orbit coupling — conservative, despite the half-integer order. (The first radiation "tail" is 1.5PN relative to the leading wave flux, but reaches the equations of motion only at 4PN.)
2.5PN ( $\epsilon^5$ )	$\dot{E}=-\frac{G}{5c^5}\langle\dddot{Q}_{ij}\dddot{Q}_{ij}\rangle$	First radiation-reaction / energy-loss term — the quadrupole formula; half-integer order signals dissipation.
Breakdown	$v\to c,\ GM/(rc^2)\to1$	Near compact-binary merger the series stops converging usefully; full numerical relativity is required.

Common trapThe PN expansion is the same thing as weak-field linearization

It is not, and conflating them is the commonest error. Linearized gravity is an expansion in the field strength $G$ (or $h_{\mu\nu}$ ) about flat space, holding nothing fixed about velocities — it describes fast-moving weak sources and freely propagating far-zone waves. The PN expansion is an expansion in $v/c$ for bound, slowly moving sources, and it deliberately keeps nonlinear terms like $\Phi^2/c^4$ that linearization throws away. The two agree only in their overlapping corner (weak field and slow motion). PN is fundamentally a near-zone method: by itself it does not describe the radiation zone, ultrarelativistic scattering, or strong-field horizons, and connecting it to the far-zone radiation needs separate matched-asymptotic machinery. When $v\to c$ near coalescence both assumptions fail together and only numerical relativity remains.

Worked examples

1 · Mercury's perihelion advance (fully worked)

Compute Mercury's 1PN perihelion advance from equation (7) and confirm $43''$ /century. Use $GM_\odot/c^2=1.477$ km, $a=5.79\times10^{10}$ m, $e=0.2056$ , period $87.969$ d.

Step 1. Sanity-check the smallness. Mercury's orbital speed is $v\approx47.9$ km/s, so $v/c=\mathbf{1.60\times10^{-4}}$ and $\epsilon^2=\mathbf{2.55\times10^{-8}}$ — why the effect is $\sim0.1''$ per orbit, not degrees.

Step 2. Geometric factor: with $e=0.2056$ , $1-e^2=0.95773$ , so $a(1-e^2)=5.79\times10^{10}\times0.95773=\mathbf{5.545\times10^{10}}$ m.

Step 3. Advance per orbit: $\Delta\varpi=6\pi(1477\ \mathrm{m})/(5.545\times10^{10}\ \mathrm{m})=\mathbf{5.02\times10^{-7}}$ rad (units cancel because $GM/c^2$ is a length).

Step 4. Per century: $36525$ d $/\,87.969$ d $=\mathbf{415.2}$ orbits, so $\Delta\varpi_{\rm century}=5.02\times10^{-7}\times415.2=2.085\times10^{-4}$ rad.

Step 5. In arcseconds ( $\times206265$ ): $\mathbf{43.0''\ \text{per century}}$ . This is the residual left after all Newtonian planetary perturbations were subtracted — unexplained from Le Verrier (1859) until Einstein (1915), the first triumph of general relativity.

2 · Shapiro delay of a radar echo grazing the Sun (fully worked)

A radar pulse bounces off a planet at superior conjunction, grazing the Sun. Estimate the extra round-trip delay from the 1PN Shapiro formula $\Delta t=(2GM/c^3)\ln(4r_Er_R/b^2)$ , with $r_E=1$ AU $=1.496\times10^{11}$ m, $r_R=1.52$ AU $=2.28\times10^{11}$ m, and $b=R_\odot=6.96\times10^8$ m.

Step 1. Prefactor: $GM_\odot/c^3=(1477\ \mathrm{m})/(2.998\times10^8\ \mathrm{m/s})=4.93\times10^{-6}$ s, so $2GM_\odot/c^3=\mathbf{9.85\times10^{-6}}$ s.

Step 2. Logarithm: $4r_Er_R/b^2=4(1.496\times10^{11})(2.28\times10^{11})/(6.96\times10^8)^2=\mathbf{2.82\times10^{5}}$ , so $\ln(2.82\times10^{5})=\mathbf{12.55}$ .

Step 3. One-way excess delay: $\Delta t=9.85\times10^{-6}\times12.55=\mathbf{124\ \mu s}$ . The pulse is delayed not because it travels farther but because the coordinate speed of light drops in the Sun's potential well.

Step 4. The observation is a round trip, so double it: $\mathbf{247\ \mu s}$ . This matches the classic Viking result of $\sim250\ \mu$ s and, via the PPN parameter $\gamma$ , is confirmed by Cassini tracking to $\sim2\times10^{-5}$ — the sharpest of the classical solar-system tests.

3 · Where the expansion breaks down: a neutron-star binary at merger

Estimate $v/c$ for two $1.4\,M_\odot$ neutron stars at the moment of contact and judge how well the PN series converges there.

Step 1. At the end of inspiral the separation approaches the innermost stable circular orbit, $r\sim6GM/c^2$ for the total mass $M\approx2.8\,M_\odot$ . The virial estimate gives $(v/c)^2\sim GM/(rc^2)\sim1/6$ , so $v/c\sim\mathbf{0.41}$ .

Step 2. The expansion parameter per order is now $\epsilon^2\sim\mathbf{0.17}$ — successive PN terms shrink by only about six, not the $10^{-8}$ of Mercury. The series is at best asymptotic here, which is why numerical relativity is required for the final orbits and merger.

Step 3. The fix is hybridization: analytic PN (4PN conservative, 3.5PN flux) or resummed effective-one-body waveforms cover the slow inspiral ( $v/c\lesssim0.3$ ), stitched to numerical-relativity merger-ringdown. This PN-plus-NR template bank is what detected GW150914 and GW170817.

Physics of the portrait

The post-Newtonian approximation is not one person's work but a century-long relay — the history of asking, ever more precisely, "how much of relativity do you need to explain what you see?"

Origins · 1 minpost-Newtonian approximation and how this was discovered

post-Newtonian approximation

the weak-field slow-motion expansion of GR in powers of v/c · relativistic celestial mechanics and gravitational waves

The post-Newtonian idea is nearly as old as general relativity. Within months of the 1915 field equations, Einstein extracted Mercury's $43''$ and the $1.75''$ deflection by keeping the leading relativistic corrections — the first post-Newtonian calculation, done before the scheme had a name. In 1916 de Sitter derived the geodetic precession of the Earth–Moon system, and in 1917–1918 Lense and Thirring worked out the gravitomagnetic frame-dragging of the $g_{0i}$ sector. The decisive step came in 1938, when Einstein, Infeld, and Hoffmann showed the trajectories need not be postulated at all — they follow from the field equations plus the Bianchi identities. From the 1960s Subrahmanyan Chandrasekhar recast the approximation as a rigorous hydrodynamic expansion and pushed it to the 2.5PN radiation-reaction order. The modern program — Thorne, Damour, Blanchet, Will and others — carried the conservative dynamics to 4PN and the flux to 3.5PN, turning Einstein's first hand-calculation into the industrial engine that produces gravitational-wave templates. What began as a check on one anomalous number in Mercury's orbit is now the backbone of an entire observational science.

History

1915

— Einstein computes Mercury's perihelion advance and light deflection from GR

1916

— de Sitter derives the relativistic geodetic precession of the Earth–Moon system

1918

— Lense and Thirring predict frame dragging from the gravitomagnetic metric

1938

— Einstein, Infeld and Hoffmann derive equations of motion from the field equations

1965

— Chandrasekhar formulates the post-Newtonian hydrodynamic expansion systematically

2014

— Blanchet's Living Review consolidates the PN program to 4PN / 3.5PN for LIGO-era templates

Going further

Texts. MTW, Gravitation, §39 · Will, Theory and Experiment in Gravitational Physics, 2nd ed., Ch. 4–5 · Poisson & Will, Gravity: Newtonian, Post-Newtonian, Relativistic, Ch. 7–9 · Wald, General Relativity, §4.4 · Carroll, Spacetime and Geometry, §5.5, §7.1 · Hartle, Gravity, Ch. 10 · Schutz, A First Course in GR, §11 · Maggiore, Gravitational Waves Vol. 1, Ch. 3–5 · Blanchet, Living Rev. Relativity 17 (2014) 2. The key numerical values used throughout: $GM_\odot/c^2=1.477$ km, $a=5.79\times10^{10}$ m, $e=0.2056$ , $415.2$ orbits/century (Mercury perihelion, $43.0''$ /century); Shapiro round-trip $\sim247\ \mu$ s and light deflection $1.75''$ ; quadrupole luminosity with $(v/c)^5$ ordering and chirp law $\dot{f}\propto f^{11/3}$ ; merger estimate $v/c\sim0.41$ from the ISCO scaling $r\sim6GM/c^2$ .

Prerequisites. Newtonian Limit of General Relativity · Weak-Field Limit and Linearized Gravity · Gravitomagnetism. Leads to. Parametrized Post-Newtonian (PPN) Formalism · Perihelion Precession of Mercury · Shapiro Time Delay · Frame Dragging and the Lense–Thirring Effect.

Ask the tutor: Ask me to walk through any rung of the PN ladder — from re-deriving Mercury's $43''$ to why the chirp goes as $f^{11/3}$ — or to contrast the post-Newtonian and linearized expansions.

Is the post-Newtonian expansion a convergent series?

Almost certainly not in the strict sense — it is best regarded as an asymptotic series. For Mercury ( $\epsilon^2\sim10^{-8}$ ) the distinction is academic: the first correction is already good to parts in $10^8$ . But near merger ( $\epsilon^2\sim0.1$ – $0.2$ ) it bites — adding more PN orders eventually stops improving the answer. This is why the field uses resummation (Padé approximants, effective-one-body) to reorganize the truncated series for the strong field, and hands analytic PN off to numerical relativity for the last few orbits.

What exactly does the "n" in nPN count?

It counts powers of $\epsilon^2=(v/c)^2$ relative to the Newtonian result. So 1PN is one factor of $(v/c)^2$ beyond Newton, 2PN is $(v/c)^4$ , and so on. Half-integer labels like 2.5PN carry an odd power, $(v/c)^5$ , and are physically special: an odd power is time-asymmetric and marks dissipative, radiative effects. Do not read "2.5PN" as "vaguely between 2PN and 3PN" — it is precise: exactly $(v/c)^5$ beyond Newton.

How can the equations of motion come out of the field equations for free?

This is one of the deepest features separating GR from Newtonian gravity. The Einstein tensor obeys a geometric identity, the contracted Bianchi identity $\nabla_\mu G^{\mu\nu}\equiv0$ , true for any metric. Since the field equations set $G^{\mu\nu}\propto T^{\mu\nu}$ , they immediately imply $\nabla_\mu T^{\mu\nu}=0$ — local energy-momentum conservation — which is the matter's equation of motion. Einstein, Infeld and Hoffmann made this explicit at 1PN: model the bodies as concentrated lumps, integrate the field equations around each, and the Bianchi identities hand back their trajectories, corrections included. In Newtonian physics you must separately postulate $\mathbf{F}=-\nabla\Phi$ ; in GR you cannot, because the geometry already contains it.

Why does gravitomagnetism appear, and is it a real magnetic-like field?

It is a genuine, measured effect, and the analogy is structural, not just poetic. The $g_{0i}$ components of the 1PN metric obey a Poisson equation sourced by the mass current $\rho v^i$ , identical in form to magnetostatics sourced by electric current. Just as a moving charge makes a magnetic field, a moving or rotating mass makes a gravitomagnetic field that drags inertial frames and precesses gyroscopes and orbits — the Lense–Thirring effect, measured by Gravity Probe B around the rotating Earth. The caveat: gravity is spin-2, not spin-1, so the analogy is exact only at this linearized 1PN level.

Does the PN expansion describe gravitational waves themselves, or only the orbit?

Both, through a division of labor. PN proper is a near-zone, slow-motion scheme — it describes the orbital dynamics close to the source, where radiation reaction first bites at 2.5PN. But the outgoing waves live in the far zone, where slow motion no longer holds; to connect them you match the near-zone PN metric to a far-zone multipole (post-Minkowskian) expansion. The waveform an interferometer records is the far-zone field, but its phase evolution is set by the PN orbital dynamics and energy balance — which is why PN accuracy in the phase determines detection sensitivity.

Why is Mercury the classic test rather than Earth or Jupiter?

Because the advance scales as $1/[a(1-e^2)]$ , it is largest for orbits that are both tight (small $a$ ) and eccentric (large $e$ ). Mercury, the innermost planet with a sizeable eccentricity ( $e=0.206$ ), wins on both counts, giving $43''$ /century — small but cleanly measurable and, crucially, unexplained by Newtonian planetary perturbations, which had been computed precisely by the mid-19th century. Earth's near-circular orbit and Jupiter's large $a$ make their advances far smaller and harder to disentangle. Today binary pulsars precess by degrees per year, but Mercury was the first and remains the textbook case.