Continuity of finite-domain geometric observables

The finite-slab estimate [F1] with zero source difference gives $\mathrm{\parallel }{z}_n-z{\mathrm{\parallel }}_{C(J;H^{s-1})}\to 0$ from convergence of the initial data. Since $s\ge 5$, [F2] gives uniform convergence in space of the metric components and their first spatial derivatives; the first-order state includes $p_{ab}={\mathrm{\partial }}_0{g}_{ab}$ and $q_{abi}={\mathrm{\partial }}_i{g}_{ab}$, so those derivatives converge uniformly as well. Thus the metric components converge in $C^1$ on the compact spacetime slab.

The classical first-order states are $C^1$, and ${\mathrm{\partial }}_{0}g=p$, ${\mathrm{\partial }}_{i}g=q_i$, so their metric components are $C^2$. The coordinate formula in [F3] also applies at this regularity: substitution in ${\mathrm{\nabla }}_b{g}_{cd}=0$ and ${\mathrm{\Gamma }}^a{}_{bc}={\mathrm{\Gamma }}^a{}_{cb}$ verifies compatibility and torsion freedom using only first metric derivatives, and the same algebra determines the coefficients uniquely. Thus the Christoffel functions are formed from the metric and its first derivatives by finite sums, products and inversion of the metric matrix. The metric matrices lie in a compact subset of the nonsingular Lorentzian matrices, so the adjugate-over-determinant formula bounds their inverses; step 1.1 therefore gives $\mathrm{\Gamma }(g_n)\to \mathrm{\Gamma }(g)$ uniformly. The uniform $H^s$ state and time-derivative bounds in the hypotheses, together with [F2], bound all second coordinate derivatives of the metric on every compact chart neighborhood: purely spatial derivatives come from the $H^s$ bounds for $q$, and derivatives involving $x^0$ come from ${\mathrm{\partial }}_{t}p$ and ${\mathrm{\partial }}_{t}q$, which are bounded in $H^{s-1}$. Thus the Christoffel functions and geodesic vector fields have uniformly bounded first derivatives on a compact neighborhood of the lift set. A finite cover by convex coordinate balls, together with boundedness of the vector fields for pairs outside a common ball, gives one Lipschitz constant on the compact lift set. The geodesic vector fields $G_n(x,v)=(v,-\mathrm{\Gamma }(g_n)(x)[v,v])$ also converge uniformly there to $G$.

Let $Y_n=(x_n,v_n)$ and $Y=(x,v)$ be the position-velocity lifts. The coordinate covariant-acceleration formula in [F4], obtained by differentiating the velocity and adding ${\mathrm{\Gamma }}^k{}_{ij}{\dot{x}}^i{\dot{x}}^j$, uses only $C^2$ curves and continuous Christoffel functions; hence their chart components satisfy $\dot{x}=v$, $\dot{v}=-\mathrm{\Gamma }(x)[v,v]$. Their integral equations and the common Lipschitz constant $L$ give $\mathrm{\mid }{Y}_n(\lambda )-Y(\lambda )\mathrm{\mid }\le \mathrm{\mid }{Y}_n(a)-Y(a)\mathrm{\mid }+(b-a)\mathrm{\parallel }{G}_n-G{\mathrm{\parallel }}_{\mathrm{\infty }}+L{\int }_a^{\lambda }\mathrm{\mid }{Y}_n(r)-Y(r)\mathrm{\mid }dr\mathrm{.}$ The first two terms form a constant $A_n\to 0$. Apply [F5] to $u(\lambda )=\mathrm{\mid }{Y}_n(\lambda )-Y(\lambda )\mathrm{\mid }$ to get ${\sup }_I\mathrm{\mid }{Y}_n-Y\mathrm{\mid }\le A_n\exp (L(b-a))\to 0$. Since ${\dot{x}}_n=v_n$, this is exactly $C^1$ convergence of the coordinate curves.

On the common compact phase set, the velocities are uniformly bounded. Uniform metric convergence from step 1.1 and uniform curve/velocity convergence from step 3.1 imply $g_n({\dot{\gamma }}_n,{\dot{\gamma }}_n)⟶g(\dot{\gamma },\dot{\gamma })$ uniformly on $I$. The limit therefore satisfies $g(\dot{\gamma },\dot{\gamma })\le -\eta$; because all curves use the same time orientation, the limit remains future timelike. The uniform margin places all numbers $-g_n({\dot{\gamma }}_n,{\dot{\gamma }}_n)$ and their limit in a bounded interval $[\eta ,M]$ with $\eta >0$. The square-root function is Lipschitz on this interval, so the proper-time integrands converge uniformly. Their integrals on the finite interval converge, and [F6] identifies them with the stated proper times.