Classical shear memory and the leading displacement approximation

Apply [F5] to each smooth tensor component $C_{AB}(u,x)$ at the fixed angular point. Since $N_{AB}={\mathrm{\partial }}_u{C}_{AB}$, $C_{AB}(u_2,x)-C_{AB}(u_1,x)={\int }_{u_1}^{u_2}{N}_{AB}(u,x)du$. This is the first identity.

In the adapted frame of [A2], the plane-wave curvature supplier gives $R^{(1)A}{}_{00B}=\frac{1}{2}{\mathrm{\partial }}_u^2{H}^A{}_B$ ([F3]); the Jacobi equation of [F4] with initial screen value $s_0^A{e}_A$ and zero initial derivative therefore has this forcing at first order, since the flat background curvature is zero and the basis is constant. Thus its formal first-order response obeys exactly the defining equation for $\delta s^A$ in the Statement. No finite-amplitude error estimate is used.

For the quantitative clause set $T:=u_2-u_1>0$. Choose a compact state neighborhood of $({\gamma }_0(u),{\mathrm{\partial }}_0)$, $u\in [u_1,u_2]$, whose position projection lies inside the specified open tube, with positive margins about this compact graph. By [F6] and [F8], the Christoffels are smooth jointly and vanish at $\lambda =0$, so their norms on this neighborhood are at most $C\mathrm{\mid }\lambda \mathrm{\mid }$ for small $\mathrm{\mid }\lambda \mathrm{\mid }$. The coordinate geodesic system is $x^{\mathrm{\prime }}=v$, $v^{\mathrm{\prime }a}=-{\mathrm{\Gamma }}^a{}_{bc}(\lambda ,x)v^b{v}^c$. By [F7] it has a local smooth solution from the stated unit initial velocity. As long as it stays in this neighborhood, the bounded velocity range gives $\mathrm{\mid }v-{\mathrm{\partial }}_0\mathrm{\mid }\le \mathrm{\mid }v(u_1)-{\mathrm{\partial }}_0\mathrm{\mid }+C_1\mathrm{\mid }\lambda \mathrm{\mid }T$ and $\mathrm{\mid }x-{\gamma }_0\mathrm{\mid }\le T\sup \mathrm{\mid }v-{\mathrm{\partial }}_0\mathrm{\mid }$. Choose $\mathrm{\mid }\lambda \mathrm{\mid }$ small enough to keep both bounds strictly inside the chosen margins. The solution cannot leave the neighborhood. A putative finite maximal endpoint would have a state limit there, since its derivative is bounded; [F7] at that interior limit extends the solution, a contradiction. Thus it exists on the full interval. A finite subdivision and [F7] give smooth dependence on $\lambda$ throughout it. The unit timelike norm is preserved by [F8], and ${\mathrm{\partial }}_0$ remains timelike on a smaller fixed tube for small $\mathrm{\mid }\lambda \mathrm{\mid }$, fixing the future cone.

Integrate the linearized equation twice, using $\delta s(u_1)={\mathrm{\partial }}_u\delta s(u_1)=0$, to obtain $\delta s^A(u_2)=\frac{1}{2}{\int }_{u_1}^{u_2}(u_2-v){\mathrm{\partial }}_v^2{H}^A{}_B(v)s_0^{B}dv\mathrm{.}$ Integration by parts gives $\delta s^A(u_2)=\frac{1}{2}[\mathrm{\Delta }{H}^A{}_B-(u_2-u_1){\mathrm{\partial }}_u{H}^A{}_B(u_1)]s_0^B$. Since $H=C\mathrm{/}r+\rho$, the endpoint condition $N_{AB}(u_1)=0$ and the $K_1{r}^{-2}$ bound give ${\mathrm{\partial }}_{u}H(u_1)=O(r^{-2})$; the $K_0{r}^{-2}$ bound gives $\mathrm{\Delta }H=\mathrm{\Delta }C\mathrm{/}r+O(r^{-2})$. This proves the stated displacement estimate, with constants depending on $u_2-u_1$ and $K_0,K_1$.

Integrating the linearized equation once gives ${\mathrm{\partial }}_u\delta s^A(u_2)=\frac{1}{2}({\mathrm{\partial }}_u{H}^A{}_B(u_2)-{\mathrm{\partial }}_u{H}^A{}_B(u_1))s_0^B$. At both endpoints the leading derivative is $N^A{}_B\mathrm{/}r=0$, and the $K_1{r}^{-2}$ bound controls the remainder derivatives. This proves the endpoint velocity estimate.

Smooth Gram-Schmidt at the initial point gives the chosen spatial orthonormal frame, since its projected Gram matrix is positive at $\lambda =0$ and remains positive nearby. Parallel transport solves $E^{\mathrm{\prime }}{}^a=-{\mathrm{\Gamma }}^a{}_{bc}{v}^b{E}^c$, a linear equation with bounded smooth coefficients along the compact geodesic. Local solutions from [F7] extend over the interval: the equation gives $(\mathrm{\mid }E{\mathrm{\mid }}^2{)}^{\mathrm{\prime }}\le 2C\mathrm{\mid }E{\mathrm{\mid }}^2$, so differentiating $e^{-2C(u-u_1)}\mathrm{\mid }E{\mathrm{\mid }}^2$ bounds $\mathrm{\mid }E\mathrm{\mid }$ by $\mathrm{\mid }E(u_1)\mathrm{\mid }{e}^{CT}$ and prevents escape at a finite endpoint, and local uniqueness patches a finite subdivision. They depend smoothly on $\lambda$ and remain orthonormal and orthogonal to the geodesic tangent by [F8]. The Jacobi initial-data result [F4] gives a unique field $J_{\lambda }$ with the stated covariant rest data. It is an actual variation field: choose an initial chart curve with tangent $J_{\lambda }(u_1)$, parallel-transport the initial timelike velocity along it, and issue geodesics from those data. The same compact-state argument as step 1.3 supplies the resulting smooth geodesic variation on the whole interval for sufficiently small variation parameter. Torsion-freeness gives initial covariant derivative zero, so uniqueness in [F4] identifies its variation field with $J_{\lambda }$.

Write $x_{\lambda }\in {\mathbb{R}}^3$ for the spatial components of $J_{\lambda }$ in that parallel frame; its time component is zero because $g(J_{\lambda },{\gamma }_{\lambda }^{\mathrm{\prime }})$ is affine with zero initial value and derivative by [F4]. Its exact equation is $x_{\lambda }^{\mathrm{\prime }\mathrm{\prime }}=K_{\lambda }{x}_{\lambda },(K_{\lambda }{)}^i{}_j=g_{\lambda }(E_{i,\lambda },R_{\lambda }({\gamma }_{\lambda }^{\mathrm{\prime }},E_{j,\lambda }){\gamma }_{\lambda }^{\mathrm{\prime }}),x_{\lambda }(u_1)=x_0:=(s_0^1,s_0^2,0),x_{\lambda }^{\mathrm{\prime }}(u_1)=0.$ The coefficient is smooth jointly. At $\lambda =0$ the curvature vanishes identically, so curve and frame derivatives make no contribution to its first $\lambda$ derivative. The sign convention $R(T,J)T$ in [F4] and the positive $R^{(1)A}{}_{00B}=\frac{1}{2}{H}^{\mathrm{\prime }\mathrm{\prime }}{}^A{}_B$ in [F3] give $K_{\lambda }=\lambda {\mathcal{K}}_1+{\lambda }^2{L}_{\lambda },({\mathcal{K}}_1{)}^A{}_B=\frac{1}{2}{H}^{\mathrm{\prime }\mathrm{\prime }}{}^A{}_B,$ with all first-order longitudinal entries zero. On a compact parameter interval $\mathrm{\mid }\lambda \mathrm{\mid }\le {\delta }_0$ one can take $L_{\lambda }={\int }_0^1(1-t){\mathrm{\partial }}_{\lambda }^2{K}_{t\lambda }dt$, so $\mathrm{\parallel }{L}_{\lambda }\mathrm{\parallel }\le \mathrm{\ell }$ uniformly on $[u_1,u_2]$. Put $\kappa :=\sup \mathrm{\parallel }{\mathcal{K}}_1\mathrm{\parallel }$ and $b:=\kappa +{\delta }_0\mathrm{\ell }$.

Shrink $\delta \le {\delta }_0$ so $\delta T^{2}b\mathrm{/}2\le 1\mathrm{/}2$. The exact integral equation $x_{\lambda }(u)=x_0+{\int }_{u_1}^u(u-v)K_{\lambda }(v)x_{\lambda }(v)dv$ gives $\sup \mathrm{\mid }{x}_{\lambda }\mathrm{\mid }\le 2\mathrm{\mid }{s}_0\mathrm{\mid }$ and $\sup \mathrm{\mid }{x}_{\lambda }-x_0\mathrm{\mid }\le T^{2}b\mathrm{\mid }\lambda \mathrm{\mid }\mathrm{\mid }{s}_0\mathrm{\mid }$ for $\mathrm{\mid }\lambda \mathrm{\mid }\le \delta$. Define $x_1(u):={\int }_{u_1}^u(u-v){\mathcal{K}}_1(v)x_{0}dv$ and $e:=x_{\lambda }-x_0-\lambda x_1$. Subtracting the two integral equations yields $e(u)={\int }_{u_1}^u(u-v)[\lambda {\mathcal{K}}_1(v)(x_{\lambda }(v)-x_0)+{\lambda }^2{L}_{\lambda }(v)x_{\lambda }(v)]dv\mathrm{.}$ Consequently $\sup \mathrm{\mid }e\mathrm{\mid }\le \frac{T^2}{2}(\kappa T^{2}b+2\mathrm{\ell }){\lambda }^2\mathrm{\mid }{s}_0\mathrm{\mid },\sup \mathrm{\mid }{e}^{\mathrm{\prime }}\mathrm{\mid }\le T(\kappa T^{2}b+2\mathrm{\ell }){\lambda }^2\mathrm{\mid }{s}_0\mathrm{\mid }\mathrm{.}$ These are uniform bounds for this fixed family, tube and interval. The constants have respectively dimension zero and inverse length. No estimate in terms of $\sup \mathrm{\mid }H\mathrm{\mid }$ alone was used.

The screen part of $x_1$ is exactly $\delta s$ by step 1.2 and the zero initial data. Step 2.1 bounds $\mid \delta s(u_2)-\frac{\mathrm{\Delta }C}{2r}{s}_0\mid \le (K_0+\frac{T{K}_1}{2})r^{-2}\mathrm{\mid }{s}_0\mathrm{\mid },$ while step 2.2 bounds $\mathrm{\mid }\delta s^{\mathrm{\prime }}(u_2)\mathrm{\mid }\le K_1{r}^{-2}\mathrm{\mid }{s}_0\mathrm{\mid }$. Here $K_0,K_1$ are the profile-remainder constants from the Statement, distinct from the tidal matrix ${\mathcal{K}}_1$ in step 3.1. Combining with step 4.1 proves both quantitative endpoint formulas and their claimed uniformity. In particular, vanishing news at the two endpoints removes the leading $r^{-1}$ velocity response while retaining the stated finite-radius and amplitude errors. The estimate concerns the specified local normalized family only; it does not compare arbitrary finite-amplitude Bondi metrics to their plane-profile limit.