Harmonic and Generalized Harmonic Coordinates

Harmonic coordinates turn the coordinate functions themselves into solutions of wave equations. In those coordinates, the only second derivatives of the metric in the reduced Einstein equations appear through a spacetime wave operator. Generalized harmonic coordinates replace zero on the right-hand side of the coordinate wave equations by prescribed gauge-source functions. This exposes a hyperbolic principal part, but it does not by itself prove constraint preservation, boundary well-posedness, or uniqueness of the coordinates.

Required background. The Einstein Equation supplies the trace-reversed field equation, while Killing Fields and Isometry Groups supplies infinitesimal diffeomorphisms and residual gauge transformations.

Helpful background. Hypersurface Normals and Tangential Projectors supplies initial data and normal derivatives, and PDEs and Green Functions for Gravitation supplies principal symbols, domains of dependence, and wave-equation boundary data.

Coordinate functions as scalar waves

Although $x^\mu$ labels a chart, each coordinate function can be treated as a scalar field on the chart domain. Its covariant wave operator is

$$\Box_g x^\mu =g^{\rho\sigma}\nabla_\rho\nabla_\sigma x^\mu =-g^{\rho\sigma}\Gamma^\mu{}_{\rho\sigma} \equiv-\Gamma^\mu.$$

The contracted Christoffel symbol $\Gamma^\mu$ is not a vector under an arbitrary coordinate change; that is precisely why setting it to zero is a coordinate condition rather than a covariant field equation. A chart is harmonic when

$$\Box_g x^\mu=0, \qquad\text{equivalently}\qquad \Gamma^\mu=0$$

for all four coordinate functions. The condition is local to a chart and requires initial and boundary values for $x^\mu$. It neither guarantees that the chart covers the desired spacetime region nor prevents its Jacobian from failing later.

For generalized harmonic coordinates, choose gauge-source functions $H^\mu$ and impose

$$\Box_g x^\mu=-H^\mu, \qquad C^\mu\equiv H^\mu-\Gamma^\mu=0.$$

This page fixes the sign of $C^\mu$ with these two equations. Sources that define $C^\mu=\Gamma^\mu-H^\mu$, or define $H^\mu$ with the opposite sign, describe the same gauge after every dependent formula is translated.

For the simplest hyperbolic reduction, $H^\mu$ may depend on the coordinates and metric but not on derivatives of the metric. Derivative-dependent gauge drivers can be useful, but they can change the principal symbol and must be analyzed as part of the coupled system rather than treated as harmless source terms.

The wave principal part of the metric

In a coordinate chart, the Ricci tensor can be organized as

$$R_{\mu\nu} =-\frac12 g^{\rho\sigma} \partial_\rho\partial_\sigma g_{\mu\nu} +\partial_{(\mu}\Gamma_{\nu)} +Q_{\mu\nu}(g,\partial g),$$

where $Q_{\mu\nu}$ contains at most first derivatives of the metric and the displayed form suppresses lower-order rearrangements between partial and covariant derivatives. The obstruction to a diagonal wave principal part is the derivative of $\Gamma_\mu$.

Substituting $\Gamma_\mu=H_\mu$ gives

$$R_{\mu\nu} =-\frac12 g^{\rho\sigma} \partial_\rho\partial_\sigma g_{\mu\nu} +\partial_{(\mu}H_{\nu)} +Q_{\mu\nu}(g,\partial g).$$

When $H_\mu=H_\mu(x,g)$, its derivative contributes no new second derivative of $g$. The ten metric components therefore obey a coupled quasilinear wave system whose principal covectors $k_\mu$ satisfy

$$g^{\mu\nu}k_\mu k_\nu=0.$$

This is the answer to why harmonic gauge makes Einstein’s equations wave-like: four coordinate conditions cancel the non-wave second-derivative terms in the Ricci tensor. Carroll derives the corresponding weak-field condition and wave operator in Carroll 1997, §6, pp. 142–169, PDF.

It is useful to write a reduced Einstein tensor that remains meaningful away from the gauge surface. With the convention above, define

$$\mathcal E_{\mu\nu} =G_{\mu\nu}+\Lambda g_{\mu\nu}-8\pi G T_{\mu\nu} +\nabla_{(\mu}C_{\nu)} -\frac12g_{\mu\nu}\nabla_\rho C^\rho.$$

The reduced equations are $\mathcal E_{\mu\nu}=0$. On $C^\mu=0$ they are the Einstein equations. Away from that surface, the added terms change the evolution of gauge-constraint violations while retaining the desired metric principal part. Different generalized harmonic systems add lower-order constraint-damping terms; those belong to the later evolution analysis and must not be folded silently into this definition.

The harmonic constraint and its propagation

Solving the reduced wave equations is not enough. A reduced solution represents an Einstein solution only when the harmonic constraint vanishes. Taking the divergence of $\mathcal E_{\mu\nu}=0$, using the contracted Bianchi identity and $\nabla^\mu T_{\mu\nu}=0$, gives the homogeneous subsidiary equation

$$\Box_g C_\mu+R_\mu{}^\nu C_\nu=0$$

for the undamped system. Consequently,

$$C_\mu\big|_\Sigma=0, \qquad n^\rho\nabla_\rho C_\mu\big|_\Sigma=0$$

on an initial hypersurface imply $C_\mu=0$ in its domain of dependence, provided the wave problem is unique and any incoming boundary data for the subsidiary system also vanish. The Einstein constraints, reduced equations, and gauge initial data are arranged together to enforce these conditions; merely setting $C_\mu=0$ at one instant is not a propagation proof.

For a timelike boundary, one must control incoming physical, gauge, and constraint fields consistently. A boundary condition that is stable for ten uncoupled scalar waves can still violate the Einstein subsidiary system. Kreiss, Reula, Sarbach, and Winicour give an energy-estimate treatment of the harmonic Einstein initial-boundary problem in Kreiss et al. 2007, §§2–3 and Theorem 3, pp. 3–9, PDF. The full characteristic and boundary analysis is deferred to the hyperbolicity and initial-boundary chapters.

Residual harmonic gauge

Harmonic gauge does not select one chart uniquely. Keep the geometric metric fixed and let $x^\mu$ be harmonic coordinate functions. A new set $x'{}^\mu=x^\mu+\varepsilon\xi^\mu(x)$ is harmonic exactly when

$$\Box_g^{\mathrm{sc}}\xi^\mu=0 \qquad\text{for each coordinate label }\mu,$$

provided the new coordinate Jacobian remains invertible. Here $\Box_g^{\mathrm{sc}}$ acts on each function $\xi^\mu$ as a scalar; it does not covariantly differentiate a vector index. Linearity of the scalar wave operator gives the result immediately. In the original harmonic chart this is $g^{\rho\sigma}\partial_\rho\partial_\sigma\xi^\mu=0$; see Carroll 1997, §4, pp. 123–124, PDF. The initial values and normal derivatives of these functions, together with their boundary behavior, select a representative.

A related but different condition is covariant de Donder gauge for a perturbation about a fixed background:

$$F_\mu(h)=\nabla^\nu h_{\mu\nu}-\frac12\nabla_\mu h=0.$$

Under $h_{\mu\nu}\mapsto h_{\mu\nu}+2\nabla_{(\mu}\xi_{\nu)}$, commuting covariant derivatives gives

$$\delta_\xi F_\mu =\Box_g^{\mathrm{vec}}\xi_\mu+R_\mu{}^\nu\xi_\nu.$$

That residual equation uses the covariant wave operator on a covector. The two descriptions coincide in inertial Minkowski coordinates but must not be identified on a general curved background. In particular, varying $\Gamma^\mu=g^{\rho\sigma}\Gamma^\mu{}_{\rho\sigma}$ also varies the inverse metric in the contraction. The full fixed-chart variation, including metric-dependent $H^\mu$, is derived in Residual Gauge Freedom and Boundary Data.

An exact Killing field is a special residual generator for which $\mathcal L_\xi g=0$. Generic residual generators change the metric components while preserving the harmonic condition. At an asymptotic or finite boundary, a residual generator that does not vanish may act as a physical symmetry rather than a redundancy.

A moving-coordinate check in flat spacetime

Use the flat metric and coordinate change

$$T=t, \qquad X=x+F(t)$$

from the gauge-transformation page. In the inertial chart,

$$\Box_\eta t=0, \qquad \Box_\eta x =\left(-\partial_T^2+\partial_X^2\right)\bigl(X-F(T)\bigr) =\ddot F(T).$$

Thus $(t,x)$ is harmonic exactly when $\ddot F=0$. A constant-velocity moving grid remains harmonic; an accelerating spatial relabeling does not. The metric is flat in either case, so failure of harmonic gauge is not curvature.

The contracted connection in the moving chart obeys

$$\Gamma^t=0, \qquad \Gamma^x=-\ddot F.$$

Choosing $H^x=-\ddot F$ and $H^t=0$ makes the same accelerating chart generalized harmonic. This explicitly separates geometry from gauge source: the curvature remains zero, while $H^\mu$ specifies how the coordinate labels accelerate.

Common pitfalls

Harmonic does not mean flat. Curved spacetimes admit harmonic charts, and flat spacetime admits nonharmonic charts. The condition concerns coordinate functions, not the Riemann tensor.

A wave principal part is not a complete theorem. Hyperbolicity also requires a specified reduction, admissible $H^\mu$, coefficient regularity, constraint propagation, initial data, and boundary conditions. Equivalence on $C^\mu=0$ does not control solutions away from that surface.

Four equations do not remove every residual map. The coordinate wave equations require initial and boundary data. Homogeneous solutions for $\xi^\mu$ preserve the gauge until those data or asymptotic conditions exclude them.

Exercise: residual gauge on Minkowski space

Let $h_{\mu\nu}$ be a linear perturbation of Minkowski spacetime and define its trace reverse by

$$\bar h_{\mu\nu}=h_{\mu\nu}-\frac12\eta_{\mu\nu}h.$$

Show that the linearized harmonic condition is $\partial^\nu\bar h_{\mu\nu}=0$, and determine the equation obeyed by a gauge generator $\xi_\mu$ that preserves it under $h_{\mu\nu}\mapsto h_{\mu\nu}+2\partial_{(\mu}\xi_{\nu)}$.

Solution

To first order,

$$\Gamma^{(1)}_\mu =\partial^\nu h_{\mu\nu}-\frac12\partial_\mu h =\partial^\nu\bar h_{\mu\nu}.$$

The trace-reversed perturbation changes by

$$\delta_\xi\bar h_{\mu\nu} =\partial_\mu\xi_\nu+\partial_\nu\xi_\mu -\eta_{\mu\nu}\partial_\rho\xi^\rho.$$

Taking a divergence gives

$$\partial^\nu\delta_\xi\bar h_{\mu\nu} =\Box_\eta\xi_\mu.$$

Therefore a generator preserves the harmonic condition exactly when $\Box_\eta\xi_\mu=0$. Initial or boundary data for this vector wave equation determine whether the residual transformation is allowed or removed.

References

• Carroll, Sean M. “Lecture Notes on General Relativity.” NSF-ITP/97-147, arXiv:gr-qc/9712019v1 (1997). arXiv.
• Kreiss, Heinz-Otto, Oscar Reula, Olivier Sarbach, and Jeffrey Winicour. “Well-Posed Initial-Boundary Value Problem for the Harmonic Einstein Equations Using Energy Estimates.” Classical and Quantum Gravity 24 (2007): 5973–5984. DOI.