Gauge Degeneracy and Hyperbolic Reduction

Einstein’s equation is not manifestly hyperbolic as an equation for the metric: infinitesimal coordinate changes lie in its principal-symbol kernel. A wave gauge removes precisely this degeneracy. The reduced equations then form a quasilinear wave system, while a separate subsidiary equation proves that a reduced solution is a genuine Einstein solution when the gauge and constraint data vanish initially.

Required background. Analysis for Mathematical Relativity supplies Sobolev estimates, The Einstein Equation supplies the field equation, PDEs and Green Functions for Gravitation supplies hyperbolic equations, and Tensor Fields and Natural Operations supplies pullbacks and Lie derivatives.

Helpful background. Causality and Global Structure explains why a spacelike Cauchy hypersurface is the natural domain for initial data.

Diffeomorphism degeneracy

Let $E_{\mu\nu}=G_{\mu\nu}+\Lambda g_{\mu\nu}-8\pi G T_{\mu\nu}$. If $g$ solves $E(g)=0$ and $\Phi_s$ is a one-parameter family of diffeomorphisms, then $\Phi_s^*g$ describes the same geometry and also solves the pulled-back equation. At first order,

$$\delta_\xi g_{\mu\nu}=\mathcal L_\xi g_{\mu\nu}=2\nabla_{(\mu}\xi_{\nu)}.$$

Thus the ten coordinate components of $g_{\mu\nu}$ cannot all have independent second-order evolution equations. At a covector $k_\mu$, the principal symbol of the linearized Einstein tensor annihilates perturbations of the form $k_{(\mu}\xi_{\nu)}$. This is a structural kernel, not a physical instability. It is also why simply reading the $\Box g_{\mu\nu}$ terms in one coordinate system does not establish well-posedness.

A convenient gauge compares the Levi-Civita connection of $g$ with a fixed background connection $\widehat\nabla$. Define

$$C^\mu =g^{\alpha\beta}\bigl(\Gamma^\mu{}_{\alpha\beta} -\widehat\Gamma^\mu{}_{\alpha\beta}\bigr)-F^\mu,$$

where the prescribed gauge source is a smooth function $F^\mu(x,g)$, independent of metric derivatives. This restriction matters: differentiating a source $F(x,g,\partial g)$ can add second derivatives and change the principal symbol. More general gauge drivers require their own coupled hyperbolicity analysis. The condition $C^\mu=0$ is generalized wave gauge. With a flat coordinate background and $F^\mu=0$, it is the familiar condition $\Box_g x^\mu=0$, since $\Box_g x^\mu=-g^{\alpha\beta}\Gamma^\mu{}_{\alpha\beta}$. Parentheses on indices denote symmetrization with weight $1/2$.

The reduced Ricci tensor

The coordinate expression for the Ricci tensor separates into a wave operator and gauge terms:

$$R_{\mu\nu} =-\frac12 g^{\alpha\beta}\partial_\alpha\partial_\beta g_{\mu\nu} +\nabla_{(\mu}C_{\nu)} +Q_{\mu\nu}(g,\partial g;\widehat\nabla,F),$$

where $Q$ contains at most first derivatives of $g$. Consequently the reduced equation

$$E^{\mathrm{red}}_{\mu\nu} \equiv E_{\mu\nu}-\nabla_{(\mu}C_{\nu)} +\frac12 g_{\mu\nu}\nabla_\alpha C^\alpha=0$$

is equivalent, after taking its trace reverse in four dimensions, to

$$R_{\mu\nu}-\nabla_{(\mu}C_{\nu)} =\Lambda g_{\mu\nu} +8\pi G\left(T_{\mu\nu}-\frac12Tg_{\mu\nu}\right).$$

It is this reduced Ricci equation that has the diagonal principal part

$$-\frac12 g^{\alpha\beta}\partial_\alpha\partial_\beta g_{\mu\nu}.$$

The principal part of $E^{\mathrm{red}}$ itself also contains the trace term $\tfrac14g_{\mu\nu}g^{\rho\sigma}g^{\alpha\beta}\partial_\alpha\partial_\beta g_{\rho\sigma}$. Trace reversal is an invertible algebraic operation on symmetric tensors, so these two systems are equivalent and have the same characteristics; their component formulas must nevertheless not be confused.

Because the metric has signature $(-,+,+,+)$, $g^{\alpha\beta}\partial_\alpha\partial_\beta$ is a wave operator wherever the chosen time slices remain spacelike. The ten equations now have the same null characteristic cone as $g$. Standard quasilinear hyperbolic theory applies once initial data, gauge sources, regularity, and the nondegeneracy of the slicing are fixed. The Ricci decomposition and gauge-source reduction are given in Friedrich and Rendall 2000, §2.4, equations (2.29)–(2.39); the existence assumptions are in §5.1.

The coefficient $-1/2$ provides a useful convention check. Linearize $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ and introduce $\bar h_{\mu\nu}=h_{\mu\nu}-\tfrac12\eta_{\mu\nu}h$. In harmonic gauge $\partial^\mu\bar h_{\mu\nu}=0$, the vacuum equation becomes

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

Changing the curvature convention changes intermediate signs, but after translating the definition of $R^\rho{}_{\sigma\mu\nu}$ the physical characteristic cone is unchanged.

The calculation also counts the reduction correctly: four gauge conditions remove the four diffeomorphism directions in the unreduced symbol, while the remaining equations propagate ten coordinate components subject to four constraints. This count is a consistency check, not a count of six independent local gravitational observables; constraints and residual gauge still have to be quotiented when physical degrees of freedom are identified.

Recovering Einstein’s equation

Solving the reduced system is only the first half of the argument. Take the divergence of $E^{\mathrm{red}}_{\mu\nu}=0$, use $\nabla^\mu G_{\mu\nu}=0$, and require $\nabla^\mu T_{\mu\nu}=0$. The gauge vector obeys a homogeneous linear equation on the reduced spacetime,

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

exactly for the displayed reduction, including a source $F(x,g)$. Extra constraint-damping terms, if added to the reduced equation, would change this subsidiary equation. If the Einstein constraints and gauge choice give

$$C^\mu\big|_\Sigma=0, \qquad n^\alpha\nabla_\alpha C^\mu\big|_\Sigma=0,$$

uniqueness for this wave equation implies $C^\mu=0$ throughout the domain of dependence. Hence $E^{\mathrm{red}}_{\mu\nu}=0$ implies $E_{\mu\nu}=0$. The divergence calculation and zero-data argument follow the reduced equations in Friedrich and Rendall 2000, §2.4, immediately after equation (2.39).

The same logical distinction between reduced existence and propagation of the harmonic coordinate condition appears in Wald 1984, §10.2, pp. 260–262, equations (10.2.32)–(10.2.36). The gauge functions and curvature conventions must be translated together when comparing the displayed formulas.

Two limitations matter. First, a wave coordinate chart can fail even when the spacetime curvature is regular; gauge breakdown is not necessarily geometric breakdown. Second, a reduced solution with nonzero initial $C^\mu$, or with nonconserved stress-energy, need not solve Einstein’s equation. Hyperbolicity of the reduced equations therefore never replaces the constraint and subsidiary-system checks developed on Constraints and the Contracted Bianchi Identity.

Check your understanding

Show that a pure-gauge linear perturbation $h_{\mu\nu}=2\partial_{(\mu}\xi_{\nu)}$ does not generally satisfy harmonic gauge, and find the condition on $\xi_\mu$ that preserves it.

Solution

For a pure-gauge perturbation, the trace reverse satisfies $\partial^\mu\bar h_{\mu\nu}=\Box_\eta\xi_\nu$. Therefore a transformation preserves harmonic gauge exactly when $\Box_\eta\xi_\nu=0$. Residual wave-gauge transformations remain, but their Cauchy data are fixed when comparing two developments with the same normalized gauge.

References

• Friedrich, Helmut, and Alan D. Rendall. “The Cauchy Problem for the Einstein Equations.” In Einstein’s Field Equations and Their Physical Implications, Lecture Notes in Physics 540, 127–224. Springer, 2000. arXiv:gr-qc/0002074.
• Wald, Robert M. General Relativity. University of Chicago Press, 1984. Publisher record.

Further reading

• Ringström, Hans. The Cauchy Problem in General Relativity. ESI Lectures in Mathematics and Physics. European Mathematical Society, 2009. Publisher record.