Bootstrap and Geometric Continuation

A global stability proof begins with a local solution and assumes provisional bounds on a maximal bootstrap interval. It succeeds only if the equations improve every assumed constant strictly, while a continuation theorem shows that the controlled quantities exclude finite-time breakdown. The final step is geometric: extending a coordinate solution for all coordinate time is not yet the same as proving causal geodesic completeness.

Required background. Analysis for Mathematical Relativity supplies continuity arguments and energy hierarchies; The Einstein Equation supplies constraints and gauge propagation; PDEs and Green Functions for Gravitation supplies local continuation.

Helpful background. Causality and Global Structure clarifies maximal developments and completeness.

A bootstrap interval

Let a local solution exist on $[0,T_0]$. Choose norms that control the quantities needed by the local theorem—for example a top-order weighted energy $E_N$, a lower-order energy $E_{N-2}$, pointwise decay $P$, gauge variables, and a noncollapse or foliation quantity. Define

$$T_*=\sup\left\{T: E_N(t)\le4C_0\varepsilon^2(1+t)^\delta, \quad E_{N-2}(t)+P(t)\le4C_0\varepsilon^2 \text{ on }[0,T] \right\}.$$

Local theory and small initial data make $T_*>0$. The constants are intentionally larger than the data bounds. On $[0,T_*)$, derive improved estimates such as

$$E_N(t)\le2C_0\varepsilon^2(1+t)^\delta, \qquad E_{N-2}(t)+P(t)\le2C_0\varepsilon^2.$$

Continuity then shows the bootstrap inequalities remain valid beyond any finite endpoint. Without the factor-two margin, equality at $T_*$ would not contradict maximality.

Closing a hierarchy without derivative loss

The top-order energy often grows mildly because commutators place one derivative on the evolving metric. Lower-order energies decay more strongly and supply the $L^\infty$ factors in nonlinear products. A typical error is arranged as

$$\int_0^t \lVert\partial g(s)\rVert_{L^\infty} E_N(s)\,ds,$$

not as a product requiring an uncontrolled $(N+1)$st derivative. Null structure and weighted vector-field estimates make $\lVert\partial g\rVert_{L^\infty}$ integrable or nearly integrable, with the permitted $(1+t)^\delta$ loss absorbing a borderline logarithm.

Gauge and constraints need their own bounds. In wave coordinates the gauge vector satisfies a homogeneous subsidiary wave equation, but its coefficients are the evolving metric. In a geometric foliation, the lapse solves an elliptic equation and must remain positive. A bootstrap that estimates curvature but never controls the chosen coordinates or lapse cannot invoke its local continuation theorem.

The full Minkowski argument implements this interlocking hierarchy through optical, curvature, connection, and foliation estimates; the relevant global closure occupies Christodoulou–Klainerman 1993, Chs. 8–17. Its organization shows why no single top-order energy can replace the lower-order decay and elliptic estimates.

Continuation criteria

Every local theorem has a conditional extension statement: if specified norms remain finite and the metric remains Lorentzian/nondegenerate, the solution extends. A reduced-coordinate criterion may involve Sobolev norms of metric components. A geometric criterion instead uses curvature, injectivity or volume radius, the second fundamental form of a foliation, and its lapse.

Klainerman and Rodnianski prove, in their vacuum constant-mean-curvature setting with compact slices and the regularity hypotheses of the theorem, that the foliation can be continued while the deformation tensor of its unit normal remains uniformly controlled. In lapse notation the consequential pieces are the second fundamental form and the spatial gradient of the logarithmic lapse; see Klainerman–Rodnianski 2010, Theorem 1.1 and eqs. (4)–(6), pp. 346–347. The criterion assumes no perturbative smallness, but it is tied to that foliation and theorem class.

The logical use is by contradiction. If $T_*<\infty$, the improved bootstrap bounds control the breakdown quantities up to $T_*$. The continuation theorem produces a solution beyond $T_*$, while continuity preserves the weaker bootstrap inequalities there. This contradicts the definition of $T_*$.

Coordinate singularities are not geometric breakdown. A harmonic coordinate chart can fail while curvature remains regular, and a foliation can develop a bad lapse before the spacetime becomes inextendible. A proof either changes gauge or demonstrates that its gauge-control estimates exclude this possibility.

From global existence to completeness

Once the solution exists for all $t\ge0$, causal completeness requires control of affine parameters. Let $\gamma$ be a future causal geodesic with affine tangent $V=d\gamma/d\lambda$. Write $N$ for the lapse and $e=-g(V,n)\gt0$ for its normal-observer energy. Since $n_\mu=-N\partial_\mu t$, one has

$$e=N\frac{dt}{d\lambda},\qquad \frac{d\lambda}{dt}=\frac{N}{e}.$$

For example, if the estimates show $N\ge N_0\gt0$ and $e(t)\le C(1+t/t_0)^\delta$ with $t_0\gt0$ and $\delta\le1$, then

$$\lambda(t)-\lambda(0) \ge\frac{N_0}{C}\int_0^t(1+s/t_0)^{-\delta}\,ds \longrightarrow\infty.$$

That proves future affine completeness. Merely showing $t\to\infty$ does not: a lapse that collapses too rapidly could correspond to finite proper or affine length.

An exact counterexample to the coordinate-time inference is $g=-t_0^2e^{-2t}dt^2+dx^2+dy^2+dz^2$, with dimensionless $t\ge0$. The change $\tau=t_0(1-e^{-t})$ identifies it with a flat slab $0\le\tau\lt t_0$. A vertical timelike geodesic reaches $t=\infty$ in proper time $t_0$. The slab is extendible Minkowski geometry, so this example shows incomplete coordinate coverage, not a curvature singularity.

Asymptotic stability requires another step. The estimates must identify limiting charges, radiation fields, or a modulated background and show convergence in a named norm. Global boundedness alone is not asymptotic convergence.

Solved diagnostic: strict improvement closes the interval

Starting from $A(t)\le4\varepsilon$ and the estimate $A(t)\le\varepsilon+C A(t)^2$, find a smallness condition giving $A(t)\le2\varepsilon$. Why is the strict margin needed?

Solution

Suppose the bootstrap assumption is $A(t)\le4\varepsilon$ and the estimates give

$$A(t)\le\varepsilon+C A(t)^2.$$

Using the assumed bound on the right,

$$A(t)\le\varepsilon+16C\varepsilon^2.$$

If $\varepsilon\le(16C)^{-1}$, then $A(t)\le2\varepsilon$, a strict improvement. If the estimate had only returned $A(t)\le4\varepsilon$, continuity would permit the norm to touch the boundary of the bootstrap set, and no extension past $T_*$ would follow.

This argument is abstract, but its inputs are the substantive work: weighted energy, null cancellations, gauge control, and a theorem whose breakdown quantities those estimates actually bound.

References

• Christodoulou, Demetrios, and Sergiu Klainerman. The Global Nonlinear Stability of the Minkowski Space. Princeton Mathematical Series 41. Princeton University Press, 1993. DOI.
• Klainerman, Sergiu, and Igor Rodnianski. “On the Breakdown Criterion in General Relativity.” Journal of the American Mathematical Society 23 (2010): 345–382. DOI.