Lorenz (Harmonic) Gauge

The harmonic gauge is not a historical curiosity — its generalized form is the working spine of numerical relativity. Naively, the exact harmonic condition $\Box_g x^\mu=0$ turns the full nonlinear Einstein equations into a manifestly hyperbolic system $g^{\alpha\beta}\partial_\alpha\partial_\beta g_{\mu\nu}=(\text{lower order})$, but for decades that system resisted stable numerical evolution. The breakthrough came in 2005, when Frans Pretorius used a generalized harmonic formulation — adding gauge source functions $H^\mu$ and constraint damping — to compute the first stable simulation of two black holes spiralling together and merging. The waveform models that LIGO, Virgo, and KAGRA match against their data descend from this framework without being literal solutions of it: the linearized equation $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ and its far-zone quadrupole formula fix the leading inspiral, post-Newtonian and effective-one-body expansions of the full Einstein equations carry the inspiral further, and the strong-field merger comes from numerical relativity solving the nonlinear equations outright — delivered to searches through calibrated or surrogate models. The residual gauge freedom is operational, not academic — fixing it to the transverse-traceless frame is simply the most convenient way to describe the underlying physical observable, the gauge-invariant tidal strain that stretches one detector arm while squeezing the perpendicular one. The same equation grounds the orbital decay of binary pulsars, where the Hulse–Taylor system matches the quadrupole prediction to better than a percent, and the identification of the graviton as the massless spin-2 quantum whose two helicity states are the quantized echo of the classical $10\to2$ count.

Two pictures converge on the same condition. The field-theoretic picture is electromagnetism told with an extra index: the potential $A_\mu$ has a redundancy $A_\mu\to A_\mu-\partial_\mu\lambda$, Maxwell's equations carry a mixing term $\partial_\mu(\partial^\nu A_\nu)$, and imposing $\partial^\mu A_\mu=0$ deletes it to leave $\Box A_\mu=-\mu_0 J_\mu$. In linearized gravity $h_{\mu\nu}$ plays $A_\mu$, the gauge vector $\xi^\mu$ plays $\lambda$, and $\partial^\mu\bar h_{\mu\nu}=0$ deletes the mixing to leave $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$. The geometric picture is deeper and names the gauge harmonic: it asks that each coordinate function satisfy the covariant wave equation $\Box_g x^\mu=0$, so the mesh is as smooth as the curvature permits. Linearizing that demand hands back the same Lorenz condition.

The condition is engineered, not lucky. Written in the trace-reversed variable $\bar h_{\mu\nu}=h_{\mu\nu}-\tfrac12\eta_{\mu\nu}h$, the linearized Einstein tensor is a single clean d'Alembertian plus three terms, each carrying a factor $\partial^\alpha\bar h_{\alpha\,\cdot}$ — the divergence of $\bar h$ on one index. Set that one bundle to zero and all three mixing terms vanish together; imposing $\partial^\mu h_{\mu\nu}=0$ on the raw perturbation would leave a stray $\tfrac12\partial_\nu h$ behind and fail. The trace reversal is the bookkeeping that lines the gauge variable up with the geometry. The gauge is always attainable because reaching it means solving a flat-space wave equation $\Box\xi_\nu=(\text{known})$, which always has a retarded solution. The coordinate freedom is four functions — the components of $\xi^\mu$ — exactly the number of conditions in $\partial^\mu\bar h_{\mu\nu}=0$. That matched count is also why a residual freedom survives: any harmonic $\xi^\mu$ with $\Box\xi^\mu=0$ changes nothing in the condition, and in vacuum that leftover is precisely what strips the ten components down to the two transverse-traceless polarizations a gravitational wave carries.

Linearized gravity rests on one modeling choice — that spacetime is nearly flat — and everything below unfolds from it. Write the metric as a fixed flat stage plus a weak field, and keep only terms of first order in the perturbation.

where: $\eta_{\mu\nu}$ — Minkowski metric, signature $(-,+,+,+)$; $h_{\mu\nu}$ — small symmetric perturbation, kept to first order. Reading: because the field is weak, indices are raised and lowered with $\eta$, and $\eta$-derivatives replace covariant ones. The variable that tidies every equation is not $h_{\mu\nu}$ but its trace-reversed cousin — named for the fact that taking its trace flips the sign in four dimensions.

where: $\bar h_{\mu\nu}$ — trace-reversed perturbation; $h$ — the trace $h^\mu{}_\mu$; note $\bar h\equiv\eta^{\mu\nu}\bar h_{\mu\nu}=h-\tfrac12\cdot4\cdot h=-h$. Reading: $\bar h_{\mu\nu}$ and $h_{\mu\nu}$ carry identical information, but $\bar h_{\mu\nu}$ is the combination the Einstein tensor naturally speaks in. The defining condition can be stated geometrically first: demand that each coordinate be a harmonic function of the metric. This is the covariant wave equation applied to the coordinate labels themselves.

where: $\Gamma^\mu_{\alpha\beta}$ — the Christoffel connection; the middle expression uses $\partial_\beta x^\mu=\delta^\mu_\beta$. Reading: harmonic coordinates are the ones for which the contracted connection $g^{\alpha\beta}\Gamma^\mu_{\alpha\beta}$ vanishes — the geometric content of "de Donder gauge". Linearizing equation (3) turns the contracted connection into the divergence of the trace-reversed field, so the geometric demand becomes the field-theoretic Lorenz condition — four equations, one per free index.

where: $\partial^\mu=\eta^{\mu\alpha}\partial_\alpha$ — the flat-space divergence. Reading: this is the two-index copy of the electromagnetic $\partial^\mu A_\mu=0$, which is why it wears both names — harmonic (geometric) and Lorenz (field-theoretic). The payoff is visible in the linearized Einstein tensor: written in $\bar h_{\mu\nu}$ it is one clean d'Alembertian plus three terms, and every one of the three carries a divergence of $\bar h$.

where: $G^{(1)}_{\mu\nu}$ — Einstein tensor to first order; $\Box=\eta^{\alpha\beta}\partial_\alpha\partial_\beta=-\partial_t^2+\nabla^2$ — flat d'Alembertian. Reading: the last three terms are the coupling that keeps the ten equations entangled — and all three vanish under equation (4). Imposing the gauge and inserting the field equations $G^{(1)}_{\mu\nu}=8\pi G\,T_{\mu\nu}$ leaves the centerpiece of the subject: ten decoupled, sourced wave equations.

where: $T_{\mu\nu}$ — stress-energy tensor (source); $G$ — Newton's constant, $c=1$. Reading: each component of $\bar h_{\mu\nu}$ is driven only by the matching component of $T_{\mu\nu}$, with no cross-coupling — ten scalar wave equations sharing one operator. Under a gauge change the trace-reversed field shifts by an extra trace term relative to $h_{\mu\nu}$ — the term that later makes the divergence collapse to a single d'Alembertian.

where: $\xi^\mu$ — infinitesimal generator of $x^\mu\to x^\mu+\xi^\mu$. Reading: to reach the gauge solve $\Box\xi_\nu=\partial^\mu\bar h^{\rm old}_{\mu\nu}$ (always solvable); the residual freedom that preserves the gauge is any harmonic $\xi^\mu$ with $\Box\xi^\mu=0$. The decoupled equation is solved by the retarded Green function — the metric as an integral of the source over its past light cone, the gravitational retarded potential.

where: $t-|\mathbf x-\mathbf x'|$ — retarded time (with $c=1$). Reading: the field at a point is built from the source configuration on its past light cone, so gravity propagates causally at $c$, exactly as the electromagnetic retarded potential does. Finally, consistency of the gauge with the field equations demands a conserved source — the gravitational analogue of charge conservation in electromagnetism.

where: $\partial^\mu T_{\mu\nu}$ — divergence of stress-energy. Reading: this is the linearized shadow of the contracted Bianchi identity $\nabla^\mu G_{\mu\nu}=0$; without it the wave equation and the gauge would be mutually inconsistent.

The single equation $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ contains several familiar theories as special cases. Reading across the table confirms that the harmonic gauge is the common root of Newtonian gravity, gravitational radiation, and the graviton.

1 · How weak is weak? The metric perturbation from Earth to a neutron star (fully worked)

Use the static limit of the harmonic-gauge equation to (a) recover Newtonian gravity and (b) tabulate the size of the metric perturbation $|h_{00}|$ at the surface of four objects, marking where linearization finally fails. Use $GM_\odot=1.327\times10^{20}\ \mathrm{m^3\,s^{-2}}$, $c=2.998\times10^{8}\ \mathrm{m\,s^{-1}}$.

Step 1. Specialize the field equation. For a static, slowly moving source the dominant component is $T_{00}=\rho c^2$ and all time derivatives vanish, so $\Box\to\nabla^2$. The $00$ component of $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$ becomes $\nabla^2\bar h_{00}=-16\pi G\,\rho$ in units $c=1$.

Step 2. Introduce the Newtonian potential. Write $\bar h_{00}=-4\Phi$. Substituting, $\nabla^2(-4\Phi)=-16\pi G\rho$, which is $\nabla^2\Phi=4\pi G\rho$ — exactly Poisson's equation for Newtonian gravity. The trace-reversed variable is what makes the factor of $4$ come out clean.

Step 3. Read off the metric. Undo the trace reversal: with only $\bar h_{00}$ significant, $\bar h=\eta^{00}\bar h_{00}=-\bar h_{00}=4\Phi$, so $h_{00}=\bar h_{00}-\tfrac12\eta_{00}\bar h=-4\Phi+2\Phi=-2\Phi$. Hence $g_{00}=-(1+2\Phi/c^2)$ and the surface perturbation is $|h_{00}|=2|\Phi|/c^2=2GM/(Rc^2)$.

Step 4. Put in the numbers. Evaluating $|h_{00}|=2GM/(Rc^2)$ object by object:

Step 5. Read the trend. From Earth to the Sun the perturbation grows by three orders of magnitude yet stays at parts in $10^6$, and even a white dwarf keeps $|h|\sim10^{-4}$ — the harmonic gauge and linear theory are superbly accurate throughout the Solar System, with neglected terms of order $(10^{-6})^2\sim10^{-12}$. Only at the neutron-star surface, $|h_{00}|\approx0.34$, does the expansion parameter approach unity and linearization break down. Linear harmonic-gauge gravity is exact enough for everything short of a compact object, where the full nonlinear equations take over.

2 · A plane gravitational wave: speed, wavelength, and two polarizations

A vacuum wave is $\bar h_{\mu\nu}=A_{\mu\nu}\,e^{ik_\alpha x^\alpha}$. Find its propagation speed, its wavelength at $f=100$ Hz (the LIGO band), and the number of independent amplitudes after the gauge is fully exploited.

Step 1. The wave equation fixes the speed. Substituting into $\Box\bar h_{\mu\nu}=0$ gives $k_\alpha k^\alpha A_{\mu\nu}=0$, so $k^\alpha$ is null: $\omega=|\mathbf k|c$ and the wave travels at exactly $c$. A symmetric $A_{\mu\nu}$ begins with $\mathbf{10}$ independent components.

Step 2. The wavelength. With $\omega=2\pi f$ and $|\mathbf k|=\omega/c$, the wavelength is $\lambda=c/f=(2.998\times10^8)/100=\mathbf{3.0\times10^6\ m}\approx3000\ \mathrm{km}$ — a wave longer than the Earth is wide, which is why detection needs kilometre-scale interferometers reading phase, not aperture.

Step 3. The Lorenz condition removes four. $\partial^\mu\bar h_{\mu\nu}=0$ becomes $k^\mu A_{\mu\nu}=0$: the amplitude is transverse to the propagation direction. These are four constraints, leaving $10-4=\mathbf{6}$.

Step 4. The residual gauge removes four more, and names the survivors. Harmonic vectors $\xi^\mu\propto e^{ik\cdot x}$ satisfy $\Box\xi^\mu=0$ and shift $A_{\mu\nu}$ without leaving the gauge; spending them on the transverse-traceless conditions $A^\mu{}_\mu=0$ and $A_{0\mu}=0$ removes four more, leaving $6-4=\mathbf{2}$. For a wave along $z$ these are $A_{xx}=-A_{yy}$ (the plus mode) and $A_{xy}=A_{yx}$ (the cross mode). A gravitational wave travels at $c$ with $\lambda=3000$ km at 100 Hz and carries exactly two transverse-traceless polarizations.

The Lorenz condition wears a second name — harmonic, or de Donder — and behind that name is a Belgian mathematical physicist who gave general relativity one of its most durable coordinate choices, then spent the rest of his career founding a different branch of physics entirely.

References. Misner, Thorne & Wheeler, Gravitation (1973), §18.1–18.2 · Wald, General Relativity (1984), §4.4 · Carroll, Spacetime and Geometry (2004), §7.1–7.2 · Schutz, A First Course in General Relativity, 2nd ed., §8.3 · Hartle, Gravity (2003), Ch. 21 · Maggiore, Gravitational Waves, Vol. 1 (2008), §1.1–1.2. All equations and conventions use signature $(-,+,+,+)$ with $G^{(1)}_{\mu\nu}=8\pi G\,T_{\mu\nu}$ and $c=1$. Key identities to keep in view: the trace identity $\bar h=-h$ in four dimensions (eq. 2); the contracted-connection identity $g^{\alpha\beta}\Gamma^\mu_{\alpha\beta}=\partial_\beta\bar h^{\mu\beta}+O(h^2)$, which gives $\Box_g x^\mu=0\Rightarrow\partial^\mu\bar h_{\mu\nu}=0$ (Derivation 1); the vanishing of all three mixing terms under $\partial^\mu\bar h_{\mu\nu}=0$, yielding the coefficient $-16\pi G$ (Derivation 2); the residual count $10-4-4=2$ (Derivation 3); the Newtonian normalization $\bar h_{00}=-4\Phi\Rightarrow\nabla^2\Phi=4\pi G\rho$ with $h_{00}=-2\Phi$; and the surface perturbations $|h_{00}|=2GM/Rc^2$ (Earth $1.4\times10^{-9}$, Sun $4.2\times10^{-6}$, white dwarf $2.5\times10^{-4}$, neutron star $0.34$). The alternative name "de Donder gauge" follows de Donder's 1921 formulation of harmonic coordinates; the electromagnetic name is due to Ludvig Lorenz (1867), not H. A. Lorentz. No numerical constants beyond $G$ and $c$ are asserted.

Prerequisites. The Linearized Einstein Equations · Metric Perturbation and the Trace-Reversed Field · Gauge Freedom in Linearized Gravity. Leads to. The Gravitational Wave Equation · The Transverse-Traceless Gauge · The Quadrupole Formula · Helicity and the Spin-2 Graviton.

Ask the tutor: Ask me to show how the harmonic-coordinate demand $\Box_g x^\mu=0$ linearizes into $\partial^\mu\bar h_{\mu\nu}=0$, or to count a gravitational wave's polarizations from ten components down to two.