The Lorenz Gauge in Linearized Gravity

The Lorenz-gauge wave equation is not a historical curiosity — it is the working front end of gravitational-wave astronomy. Every waveform template that LIGO, Virgo, and KAGRA match against their data begins life as a solution of $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$, expanded in the far zone into the quadrupole formula and pushed to high post-Newtonian order for the slow inspiral of two compact objects. The 2015 detection of GW150914 — black holes of about $36$ and $29$ solar masses merging $1.3$ billion light-years away, peak strain $\sim10^{-21}$ — was recognized because theorists had computed, decades in advance, exactly what such a signal should look like with this machinery. The residual gauge freedom is not academic either: fixing it to the transverse-traceless frame is what tells a detector that a passing wave stretches one arm while squeezing the perpendicular one. The same equation grounds the post-Newtonian expansion behind the orbital decay of binary pulsars — the Hulse–Taylor system matches the quadrupole prediction to better than a percent — and the relativistic corrections that keep GPS accurate, whose clocks live in exactly the $|h_{00}|\sim10^{-9}$ regime of the worked example. Harmonic coordinates, the exact-theory cousin of the Lorenz gauge, are also the standard choice in numerical relativity for keeping the evolution equations well-posed as two black holes spiral together.

The cleanest way to see the Lorenz gauge is to hold electromagnetism in the other hand. There, the four-potential $A_\mu$ carries a gauge redundancy $A_\mu\to A_\mu-\partial_\mu\lambda$; Maxwell's equations written for $A_\mu$ contain a troublesome $\partial_\mu(\partial^\nu A_\nu)$ term that mixes the components, and imposing $\partial^\mu A_\mu=0$ deletes it, leaving $\Box A_\mu=-\mu_0 J_\mu$. Linearized gravity is the same story told with one more index. The metric perturbation $h_{\mu\nu}$ plays the role of $A_\mu$, the gauge vector $\xi^\mu$ plays the role of $\lambda$, and the Lorenz condition $\partial^\mu\bar h_{\mu\nu}=0$ is the tensor copy of $\partial^\mu A_\mu=0$. The payoff is identical in form and even in the constant out front: a decoupled wave equation with a conserved source.

Why trace-reverse first? Because the linearized Einstein tensor contains both $h_{\mu\nu}$ and its trace $h$, and the combination whose divergence appears in every unwanted cross term is not $h_{\mu\nu}$ but $\bar h_{\mu\nu}=h_{\mu\nu}-\tfrac12\eta_{\mu\nu}h$. Setting the divergence of that bundle to zero deletes all four mixing terms at a stroke; imposing $\partial^\mu h_{\mu\nu}=0$ on the raw perturbation would not. The trace reversal is bookkeeping that lines the variables up with the geometry. The gauge is always attainable because reaching it means solving a wave equation $\Box\xi_\nu=(\text{known source})$, which on flat spacetime always has a retarded solution. The coordinate freedom is four functions' worth — the four 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: gauge vectors with $\Box\xi^\mu=0$ change nothing in the Lorenz condition, and in vacuum that leftover strips the ten components down to the two physical polarizations of a gravitational wave.

Linearized gravity begins with one modeling choice — that spacetime is nearly flat — and every equation below unfolds from it. We write the metric as a flat background plus a small perturbation and drop everything beyond first order.

where: $\eta_{\mu\nu}$ — flat Minkowski metric, signature $(-,+,+,+)$; $h_{\mu\nu}$ — small symmetric perturbation, kept to first order. Reading: gravity lives as a weak field on a fixed flat stage, so indices are raised and lowered with $\eta$, not $g$. The field variable that tidies the equations is not $h_{\mu\nu}$ itself but its trace-reversed cousin — literally so, since taking the trace flips the sign, as shown just below.

where: $\bar h_{\mu\nu}$ — trace-reversed perturbation; $h$ — the trace $h^\mu{}_\mu$; note $\bar h=-h$. Reading: $\bar h_{\mu\nu}$ and $h_{\mu\nu}$ carry the same information, but $\bar h_{\mu\nu}$ is the combination the Einstein tensor is naturally written in. Linearized gravity is a field defined only up to a gauge transformation, exactly as in electromagnetism. An infinitesimal coordinate change acts on the perturbation as follows and leaves all curvature invariants untouched.

where: $\xi^\mu$ — infinitesimal gauge vector; $\xi_\nu=\eta_{\nu\alpha}\xi^\alpha$ — its lowered form. Reading: two perturbations differing by $\partial_\mu\xi_\nu+\partial_\nu\xi_\mu$ describe the same spacetime in different coordinates. Translating that rule into the trace-reversed variable adds one extra trace term — the term that makes $\bar h_{\mu\nu}$ transform so conveniently under a divergence.

where: $\partial^\alpha\xi_\alpha$ — four-divergence of the gauge vector. Reading: the extra $\eta_{\mu\nu}\partial^\alpha\xi_\alpha$ is what, under a divergence, combines the derivatives of $\xi$ into a single d'Alembertian. The Lorenz gauge sets the divergence of $\bar h_{\mu\nu}$ to zero — four conditions, one per free index $\nu$.

where: $\partial^\mu=\eta^{\mu\alpha}\partial_\alpha$ — the flat-space divergence. Reading: the tensor analogue of the electromagnetic $\partial^\mu A_\mu=0$, also called the harmonic or De Donder gauge. The payoff is visible in the linearized Einstein tensor: written in $\bar h_{\mu\nu}$ it is one clean d'Alembertian plus three terms, each carrying a divergence of $\bar h_{\mu\nu}$.

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 mixing that keeps the ten equations coupled — and all three vanish in Lorenz gauge. Imposing the gauge and inserting $G^{(1)}_{\mu\nu}=8\pi G\,T_{\mu\nu}$ 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 by the matching component of $T_{\mu\nu}$, with no coupling between components. Reaching the gauge from an arbitrary perturbation means solving one wave equation for the gauge vector — always possible on flat spacetime.

where: $\bar h^{\rm old}_{\mu\nu}$ — the perturbation before the gauge change. Reading: whatever divergence you start with, this manufactures the coordinate shift that cancels it; a residual freedom $\Box\xi^\mu=0$ then remains. Finally, the decoupled equation is solved by the retarded Green's 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 $c=1$); the field at a point is built from the source configuration on its past light cone. Reading: gravity propagates causally at $c$, exactly as the electromagnetic retarded potential does.

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 is a good check that the Lorenz gauge really is the common root of Newtonian gravity, gravitational waves, and everything between.

1 · The Newtonian limit at the surface of the Sun (fully worked)

Take the Sun as a static source and use the Lorenz-gauge equation to (a) recover Newton's law of gravity and (b) estimate the size of the metric perturbation at the solar surface, checking that linearization is justified. Use $GM_\odot=1.327\times10^{20}\ \mathrm{m^3\,s^{-2}}$, $R_\odot=6.96\times10^{8}\ \mathrm{m}$, $c=2.998\times10^{8}\ \mathrm{m\,s^{-1}}$.

Step 1. Specialize the field equation. For a static, slowly moving source the dominant stress-energy component is $T_{00}=\rho c^2$ (mass-energy density) 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 $\nabla^2\bar h_{00}=-\dfrac{16\pi G}{c^2}\,T_{00}=-16\pi G\,\rho$ in units where $c=1$.

Step 2. Introduce the Newtonian potential. Write $\bar h_{00}=-4\Phi$ (with $\Phi/c^2$ the dimensionless potential). Substituting, $\nabla^2(-4\Phi)=-16\pi G\rho$, which rearranges to $\boxed{\nabla^2\Phi=4\pi G\rho}$ — exactly Poisson's equation for Newtonian gravity. The trace-reversed variable is essential here: had we used $h_{00}$ directly, the trace terms would have spoiled the clean factor of $4$.

Step 3. Read off the metric. Undoing the trace reversal gives $h_{00}=\bar h_{00}-\tfrac12\eta_{00}\bar h$. With only $\bar h_{00}$ non-negligible one finds $h_{00}=-2\Phi$, so $g_{00}=-(1+2\Phi/c^2)$ — the standard weak-field time-time metric, the piece that produces gravitational time dilation.

Step 4. Put in the numbers. At the solar surface the potential is $\Phi/c^2=GM_\odot/(R_\odot c^2)$:

The perturbation is therefore $|h_{00}|=2|\Phi|/c^2=\mathbf{4.2\times10^{-6}}$.

Step 5. Check linearization. Since $|h_{00}|\approx4\times10^{-6}\ll1$, the weak-field expansion is superbly justified at the Sun's surface — the linear theory is accurate to parts in $10^{6}$, and the neglected second-order terms are of order $(10^{-6})^2\sim10^{-12}$. The Lorenz gauge reproduces Newton exactly in the static limit, and $|h|\sim10^{-6}$ confirms that the Solar System is deep in the weak-field regime. (For comparison, the same estimate at Earth's surface gives $|h_{00}|\approx1.4\times10^{-9}$, and only at a neutron-star surface, $|h|\sim0.4$, does linearization finally break down.)

2 · Counting the polarizations of a plane gravitational wave

A gravitational wave in vacuum is $\bar h_{\mu\nu}=A_{\mu\nu}\,e^{ik_\alpha x^\alpha}$. Show how the wave equation, the Lorenz condition, and the residual gauge cut the ten amplitudes $A_{\mu\nu}$ down to two.

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

Step 2. 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 linear constraints (one per $\nu$), leaving $10-4=\mathbf{6}$.

Step 3. The residual gauge removes four more. Harmonic vectors $\xi^\mu=i\varepsilon^\mu e^{ik_\alpha x^\alpha}$ satisfy $\Box\xi^\mu=0$ automatically and shift $A_{\mu\nu}$ without leaving Lorenz gauge. The four functions $\varepsilon^\mu$ let us impose the transverse-traceless conditions $A^\mu{}_\mu=0$ (traceless) and $A_{0\mu}=0$ (purely spatial), removing four more amplitudes. That leaves $6-4=\mathbf{2}$.

Step 4. Name the survivors. For a wave along $z$, the two remaining amplitudes are $A_{xx}=-A_{yy}$ (the plus mode $+$) and $A_{xy}=A_{yx}$ (the cross mode $\times$). A gravitational wave has exactly two physical polarizations, both transverse and traceless, in one-to-one correspondence with the $10\to6\to2$ counting of the second figure.

The gauge that tames linearized gravity is named for a nineteenth-century Danish physicist most people have never heard of, and whose name is routinely misspelled into that of a more famous contemporary. The story is worth telling because it is a small monument to how physics remembers — and misremembers — the people who build its tools.

References. Misner, Thorne & Wheeler, Gravitation, §18.1–18.2 · Wald, General Relativity, §4.4 · Carroll, Spacetime and Geometry, §7.1 · Schutz, A First Course in General Relativity, §8.3 · Hartle, Gravity, §21.4 · Maggiore, Gravitational Waves, Vol. 1, §1.2 (linearized theory and gauge fixing). Key checkpoints in the derivations: the trace identity $\bar h=-h$ holds in four dimensions (equation 2); all three mixing terms vanish under $\partial^\mu\bar h_{\mu\nu}=0$ (Derivation 1); the $\partial_\nu\partial^\mu\xi_\mu$ pieces cancel exactly in the divergence of $\delta\bar h_{\mu\nu}$ (Derivation 2, Step 2); the coefficient $-16\pi G$ follows from $G_{\mu\nu}=8\pi G\,T_{\mu\nu}$ with $c=1$; the Newtonian normalization $\bar h_{00}=-4\Phi\Rightarrow\nabla^2\Phi=4\pi G\rho$ recovers the correct Poisson equation; and the solar-surface numbers are $\Phi/c^2=2.12\times10^{-6}$, $|h_{00}|=4.2\times10^{-6}$ (Worked Example 1). The historical attribution of the gauge to Ludvig Lorenz (not H. A. Lorentz) is due to J. D. Jackson.

Prerequisites. The Weak-Field Approximation · The Einstein Field Equations · The Einstein Tensor · Local Conservation of Energy-Momentum · The Stress-Energy Tensor. Leads to. The Newtonian Limit · Gravitational Waves · Post-Newtonian Expansion.

Ask the tutor: Ask me to walk through why the four "mixing" terms in the linearized Einstein tensor all vanish under $\partial^\mu\bar h_{\mu\nu}=0$, or to count a gravitational wave's polarizations from ten components down to two.