Generalized Harmonic Evolution

Generalized harmonic evolution separates coordinate choice from the principal gravitational dynamics. Four gauge-source functions prescribe the contracted connection, and the reduced Einstein equations then have the principal part of ten coupled quasilinear wave equations for the metric components. A solution of the reduced system is a solution of Einstein’s equations only when the harmonic constraints vanish. Their homogeneous subsidiary wave equation supplies the bridge, provided the Einstein constraints and harmonic constraints are compatible initially and any timelike boundary admits no incoming constraint violation. The result is local well-posed evolution for the stated wave reduction, not a theorem of global existence or geometric uniqueness for arbitrary boundaries.

Required background. Partial differential equations and Green functions supplies quasilinear wave equations, energy estimates, and boundary data. The Einstein equation supplies the covariant equations recovered on the constraint surface.

Helpful background. Hypersurface normals and tangential projectors supplies normal derivatives and Cauchy data. Formulations of General Relativity distinguishes reduced equations from Einstein solutions. Differential equations reviews uniqueness for homogeneous evolution equations.

Gauge sources and the harmonic constraint

Define the contracted Christoffel symbol with a lowered index by

$$\Gamma_\mu=g^{\rho\sigma}\Gamma_{\mu\rho\sigma},$$

and prescribe smooth gauge-source functions $H_\mu$. This page uses the convention

$$\mathcal C_\mu=H_\mu-\Gamma_\mu.$$

Thus the generalized harmonic condition is $\mathcal C_\mu=0$, or $\Gamma_\mu=H_\mu$. Since $\Box_g x^\mu=-\Gamma^\mu$, the corresponding coordinate functions satisfy $\Box_gx^\mu=-H^\mu$. Ordinary harmonic coordinates are the special case $H_\mu=0$. The coordinate condition, its densitized inverse-metric form, and its residual freedom are derived in Carroll 1997, §4, equations (4.84)–(4.91), pp. 123–124, PDF.

For the simplest quasilinear-wave theorem, assume that $H_\mu=H_\mu(x,g)$ depends smoothly on position and the metric but not on derivatives of the metric. More general derivative-dependent or independently evolved gauge drivers are admissible only after showing that their coupled principal symbol retains the required hyperbolicity. Set $\Lambda=0$ here and require a nondegenerate Lorentzian metric with spacelike initial hypersurface. For matter, $\nabla^\mu T_{\mu\nu}=0$ is necessary for the subsidiary argument, but it does not by itself make the coupled matter evolution well posed; that must be checked for the selected matter equations.

Although $\Gamma_\mu$ is not a covector under an arbitrary coordinate change, the gauge sources can be assigned the same inhomogeneous transformation law, so that their difference $\mathcal C_\mu$ is a covector. Alternatively one can state the reduction in a fixed chart or relative to a reference connection. Prescribing $H_\mu$ is thus a coordinate-gauge prescription, not the addition of a physical covector field to Einstein gravity.

The reduced wave equations

Use the trace-reversed Einstein equation and add a derivative of the harmonic constraint:

$$R_{\mu\nu} +\nabla_{(\mu}\mathcal C_{\nu)} =8\pi G\left(T_{\mu\nu}-\frac12g_{\mu\nu}T\right).$$

The principal part of the Ricci tensor in coordinates is

$$R_{\mu\nu} \simeq -\frac12g^{\rho\sigma}\partial_\rho\partial_\sigma g_{\mu\nu} +\partial_{(\mu}\Gamma_{\nu)}.$$

Because $\mathcal C_\mu=H_\mu-\Gamma_\mu$, the connection-derivative terms cancel in the reduced equations:

$$-\frac12g^{\rho\sigma}\partial_\rho\partial_\sigma g_{\mu\nu} +\partial_{(\mu}H_{\nu)} +Q_{\mu\nu}(g,\partial g,H) =8\pi G\left(T_{\mu\nu}-\frac12g_{\mu\nu}T\right),$$

where $Q_{\mu\nu}$ contains no second derivatives of the metric. The characteristic covectors of the metric block therefore obey

$$g^{\rho\sigma}\xi_\rho\xi_\sigma=0.$$

These are the null covectors of the evolving metric for this second-order wave block. Gauge and constraint fields still have to be identified when a first-order reduction and a boundary normal are chosen. The linearized harmonic calculation reduces the vacuum equations to ordinary wave equations, as shown in Carroll 1997, §6, equations (6.15)–(6.23), pp. 146–147, PDF.

The wave form gives a standard local Cauchy theory for sufficiently regular data in Sobolev spaces above the dimension-dependent threshold required by the chosen quasilinear theorem. The data must include a complete Lorentzian metric and its normal derivative, satisfy the Einstein Hamiltonian and momentum constraints, and be chosen so that the gauge condition is compatible with the lapse–shift components. Merely specifying six spatial metric components is not complete generalized harmonic data.

The subsidiary system

It is useful to use the trace-adjusted reduced equation

$$G_{\mu\nu}-8\pi G T_{\mu\nu} +\nabla_{(\mu}\mathcal C_{\nu)} -\frac12g_{\mu\nu}\nabla_\rho\mathcal C^\rho=0,$$

which is equivalent to the trace-reversed form above. Take a divergence. The contracted Bianchi identity, stress-energy conservation, and the commutator of covariant derivatives give

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

This is a homogeneous linear wave equation in $\mathcal C_\mu$ once the reduced metric solution is known. If

$$\left.\mathcal C_\mu\right|_\Sigma=0, \qquad \left.n^\rho\nabla_\rho\mathcal C_\mu\right|_\Sigma=0,$$

then uniqueness gives $\mathcal C_\mu=0$ in the domain of dependence. The reduced equation consequently becomes Einstein’s equation there.

The normal derivative is not an independent extra restriction on compatible Einstein data. To see this at a point of $\Sigma$, choose an orthonormal frame with $e_0=n$. If $\mathcal C_\mu=0$ throughout $\Sigma$, its tangential derivatives vanish there. Write $E_{\mu\nu}=G_{\mu\nu}-8\pi GT_{\mu\nu}$. The normal components of the trace-adjusted reduced equation give

$$E_{00}=-\frac12\nabla_0\mathcal C_0, \qquad E_{0i}=-\frac12\nabla_0\mathcal C_i.$$

The Hamiltonian and momentum constraints set $E_{00}=E_{0i}=0$, supplying the vanishing normal derivative required by the subsidiary wave problem.

On a domain with timelike boundary, zero initial subsidiary data are insufficient. One must impose constraint-preserving boundary conditions that eliminate incoming subsidiary fields and satisfy compatibility conditions where the initial surface meets the boundary. For a frozen half-space harmonic system, Kreiss, Reula, Sarbach, and Winicour 2007, §3, equations (11)–(14) and Theorem 3, pp. 7–10, PDF proves a frequency-uniform interior and boundary estimate under explicit spectral conditions on the boundary matrices. The paper also explains the local quasilinear extension and notes that generalized harmonic forcing does not alter the principal part.

Checks and limitations

Three checks distinguish a genuine Einstein evolution from a solution of only the reduced equations. First, evaluate $\mathcal C_\mu$ from the metric and the declared $H_\mu$ on the initial slice. Second, verify its normal derivative directly or through the Einstein constraints. Third, at a boundary decompose the subsidiary wave system into incoming and outgoing fields and show that the boundary data set the incoming constraint fields consistently; imposing all components of $\mathcal C_\mu$ without a flux analysis can overdetermine the metric problem.

The source functions control coordinates, not physical curvature. A poor choice can create steep coordinate gradients or cease to define a useful chart even while the local wave system remains hyperbolic. Conversely, a convenient gauge does not supply asymptotic conditions, radiative boundary data, corner compatibility, or a global continuation theorem. Constraint-damping terms are lower-order modifications of the subsidiary system and require their own background-dependent analysis.

Exercise

Linearize vacuum generalized harmonic evolution about Minkowski spacetime with $H_\mu=0$. Show that the trace-reversed perturbation obeys a wave equation and that the linearized harmonic constraint obeys a homogeneous wave equation. Explain why vanishing constraint data on one slice are not enough unless the normal derivative also vanishes.

Solution

Write $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ and define

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

To first order,

$$\Gamma^{(1)}_\mu =\partial^\nu h_{\mu\nu}-\frac12\partial_\mu h =\partial^\nu\overline h_{\mu\nu}, \qquad \mathcal C^{(1)}_\mu=-\partial^\nu\overline h_{\mu\nu}.$$

The reduced vacuum equation is

$$\Box_\eta\overline h_{\mu\nu}=0.$$

Taking a divergence and commuting constant partial derivatives gives

$$\Box_\eta\mathcal C^{(1)}_\mu =-\partial^\nu\Box_\eta\overline h_{\mu\nu}=0.$$

A wave equation requires both position and velocity data. A solution can have $\mathcal C^{(1)}_\mu=0$ initially but nonzero time derivative and then become nonzero immediately. Vanishing of both the constraint and its normal derivative, together with no incoming boundary data when a boundary is present, selects the unique zero solution.

References

• Carroll, Sean M. Lecture Notes on General Relativity. 1997. arXiv:gr-qc/9712019.
• 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. arXiv:0707.4188.