Wave-Gauge Local Existence

Wave gauge turns the vacuum Einstein equations into a quasilinear system for which classical Sobolev energy estimates give short-time existence. In three spatial dimensions, the transparent metric theorem takes $h\in H^s$ and $K\in H^{s-1}$ with $s>5/2$. Constraint and gauge propagation then promote the reduced solution to an Einstein development, unique geometrically rather than in arbitrary coordinates.

Required background. Analysis for Mathematical Relativity supplies Sobolev and iteration estimates, The Einstein Equation supplies the geometric system, and PDEs and Green Functions for Gravitation supplies quasilinear wave theory.

Helpful background. Causality and Global Structure explains Cauchy developments and domains of dependence.

A local existence theorem

Let $\Sigma$ be a smooth three-manifold without boundary. Consider vacuum data with $\Lambda=0$, $(h_{ij},K_{ij})$, satisfying

$$h\in H^s_{\mathrm{loc}}, \qquad K\in H^{s-1}_{\mathrm{loc}}, \qquad s>\frac52,$$

and the vacuum constraints, with any asymptotic behavior controlled separately. Use

$$K_{ij}=-h_i{}^\mu h_j{}^\nu\nabla_\mu n_\nu.$$

Then every point of $\Sigma$ has a neighborhood whose data admit a vacuum development. For compact $\Sigma$, or under suitable uniform and asymptotic hypotheses, there is a common interval $(-T,T)$ and a spacetime metric with representative regularity

$$g\in C^0_tH^s_x\cap C^1_tH^{s-1}_x$$

in wave coordinates. The solution depends continuously on the data in the topology specified by the theorem, and smoother data yield a smoother development. Any two such developments are locally isometric near the embedded initial hypersurface.

This is a local theorem: $T$ depends on controlling norms, hyperbolicity constants, gauge data, and coordinate coverage. It does not assert geodesic completeness or give the maximal lifespan. The solution space is a space of metric perturbations relative to a smooth reference geometry when a noncompact background is used. The classical reduction, local construction, uniqueness, and continuous dependence are described separately in Friedrich and Rendall 2000, §2.4 and §§5.1–5.3.

Hypotheses behind the statement

Assume first that $h$ is uniformly positive definite in the chosen local charts and that the lapse and shift initialize a Lorentzian metric whose $t$-slices are spacelike. On a noncompact slice, assume uniform local bounds and the weighted asymptotic conditions needed to choose one lifespan rather than unrelated pointwise lifespans. For matter, assume a compatible hyperbolic reduction at the same derivative level and matter constraints implying $\nabla^\mu T_{\mu\nu}=0$.

The data must satisfy the constraints as equalities in the Sobolev class. Approximate constraints do not become exact through evolution. For the displayed reduction, use smooth prescribed gauge sources $F(x,g)$ independent of metric derivatives. Dependence on $\partial g$ would produce new second derivatives when $F$ is differentiated and needs a new principal-symbol analysis.

A limitation of the coordinate theorem is visible even in flat spacetime: a poorly chosen harmonic-coordinate map may cease to be injective while the Minkowski geometry remains smooth. The theorem then continues the solution in another admissible chart; it does not interpret failure of that coordinate representation as curvature singularity. Likewise, $s>5/2$ is sufficient for this proof architecture but does not imply that solutions below it cannot exist.

Proof architecture

The proof has five linked stages.

1. Complete the spacetime Cauchy data. Choose initial lapse, shift, and gauge-source data with the required regularity. For a concrete local choice, set the initial lapse to $1$, shift to $0$, and use ordinary harmonic coordinates with zero gauge source. Then

$$\begin{gathered} g_{00}=-1,\qquad g_{0i}=0,\qquad g_{ij}=h_{ij},\\ \partial_tg_{ij}=-2K_{ij},\qquad \partial_tg_{00}=2K,\qquad \partial_tg_{0i}=h_{i\ell}h^{jk}\Gamma(h)^\ell{}_{jk}. \end{gathered}$$

The last two conditions set $g^{\alpha\beta}\Gamma^\mu{}_{\alpha\beta}=0$ initially; they are gauge initialization, not extra geometric constraints. The minus sign in $\partial_tg_{ij}$ follows from the declared sign of $K_{ij}$. This is the source construction translated to that sign; see Friedrich and Rendall 2000, §5.1, displayed initial metric data.

For the classical smooth vacuum argument, Wald 1984, §10.2, pp. 261–262 similarly uses the still-undetermined initial time derivatives of $g_{0\mu}$ to impose the harmonic coordinate condition. Its local rescaling argument explains why the theorem is local for arbitrary smooth data, rather than restricted to globally small perturbations of flat data.

2. Reduce the equation. For gauge constraint $C^\mu$, solve

$$R_{\mu\nu}-\nabla_{(\mu}C_{\nu)}=0$$

in its trace-adjusted equivalent form. Its principal part is

$$-\frac12g^{\alpha\beta}\partial_\alpha\partial_\beta g_{\mu\nu},$$

so the system is quasilinear hyperbolic while $g$ remains Lorentzian and the slices spacelike.

3. Iterate linear equations. Given $g^{(m)}$, solve a linear wave equation whose coefficients are frozen at $g^{(m)}$ to obtain $g^{(m+1)}$. Finite-order energy estimates give a common time interval and a uniform $H^s$ bound. Sobolev embedding $H^s\hookrightarrow C^1$ for $s>5/2$ controls coefficient derivatives and nonlinear products.

4. Prove convergence and uniqueness. Differences of successive iterates are estimated in one lower norm, avoiding a demand for an uncontrolled top derivative. Contraction in $C_tH^{s-1}\cap C_t^1H^{s-2}$, combined with the uniform top-order bound, produces a solution. The lower-order difference estimate gives fixed-gauge uniqueness. Strong continuity and continuous dependence at the top norm use a separate smoothing and approximation argument; they do not follow from weak compactness alone.

5. Recover Einstein’s equation. The contracted Bianchi identity gives

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

The geometric constraints and compatible gauge data imply $C|_\Sigma=0$ and $\nabla_nC|_\Sigma=0$. Wave-equation uniqueness gives $C=0$, so the reduced metric solves $R_{\mu\nu}=0$.

The last stage is logically indispensable. Existence for the reduced system alone would be existence for a gauge-modified equation.

Threshold and scope

The condition $s>5/2$ is a classical sufficient threshold, not a universal sharp one. It provides pointwise control of first metric derivatives by direct Sobolev embedding. Geometric low-regularity theorems use additional structure and different norms; they require separate statements.

Likewise, a timelike boundary changes the problem: incoming characteristic fields need boundary conditions and corner compatibility. Matter changes it when its evolution is not hyperbolic at the same regularity or when $\nabla^\mu T_{\mu\nu}=0$ does not follow from the matter equations. These extensions cannot be obtained by merely adding a source term.

Check your understanding

Why must both $C^\mu|_\Sigma$ and $\nabla_nC^\mu|_\Sigma$ vanish?

Solution

The subsidiary equation is second-order hyperbolic. Zero value without zero normal derivative does not give zero Cauchy data: a nonzero derivative launches a nontrivial gauge-constraint wave. The Einstein constraints and compatible gauge initialization supply both conditions, after which uniqueness forces $C^\mu=0$.

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.