The material metric of a congruence: evolution of rest separations and preservation of the internal metric under Born rigidity

(Tensoriality and smooth agreement.) Fix an inertial chart and expand the definitions: $h^a{}_b={\delta }^a{}_b+n^a{n}_b$, $h_{ab}={\eta }_{ab}+n_a{n}_b$, $a^b=n^c{\mathrm{\partial }}_c{n}^b$, ${\theta }_{ab}=\frac{1}{2}{h}_a{}^c{h}_b{}^d({\mathrm{\partial }}_c{n}_d+{\mathrm{\partial }}_d{n}_c)$, and $(L_{n}h{)}_{ab}=n^c{\mathrm{\partial }}_c{h}_{ab}+h_{cb}{\mathrm{\partial }}_a{n}^c+h_{ac}{\mathrm{\partial }}_b{n}^c$; under the inertial chart change the factors $\mathrm{\Lambda }$ and ${\mathrm{\Lambda }}^{-1}$ produced by [L8] contract pairwise with the Kronecker deltas, with the components of $n$ and of every derivative, so each displayed formula is a sum of products of components of $n$, of $\mathrm{\partial }n$ and of the constant $\eta$, multiplied by the same index factors in the new chart; this is exactly the tensorial transformation law of [L8], and no connection terms appear by [L3].

(Two first-order identities.) Differentiating the identity $\eta (n,n)=-1$ in an inertial chart and using $\mathrm{\partial }\eta =0$ of [L3] gives $2{\eta }_{ab}{n}^a{\mathrm{\partial }}_c{n}^b=0$, that is $n_b{\mathrm{\partial }}_c{n}^b=0$ for every $c$; contracting with $n^c$ and relabelling, $n_c{a}^c=n^a{\eta }_{ab}{n}^d{\mathrm{\partial }}_d{n}^b=0$, so $a$ is $\eta$-orthogonal to $n$. Similarly, $h_a{}^c{h}_b{}^d{\mathrm{\partial }}_c{n}_d={\mathrm{\partial }}_a{n}_b+n_a{a}_b$: expand the two projectors, discard the two terms containing $n^d{\mathrm{\partial }}_a{n}_d$ and $n^c{n}^d{\mathrm{\partial }}_c{n}_d$, both of which vanish by the first identity, and use $n^c{\mathrm{\partial }}_c{n}_b=a_b$.

(Independence of the family directions.) Since $\mathrm{\Phi }$ is an immersion, its differential $D\mathrm{\Phi }(s,\xi )$ is injective at every point of $J\times B$, and the four partial derivatives ${\mathrm{\partial }}_s\mathrm{\Phi },e_1,e_2,e_3$ are the columns of $D\mathrm{\Phi }$; a linear relation $\mu {\mathrm{\partial }}_s\mathrm{\Phi }+{\sum }_I{\lambda }^I{e}_I=0$ therefore has all coefficients zero. Substituting ${\mathrm{\partial }}_s\mathrm{\Phi }=n(\mathrm{\Phi }(s,\xi ))$ gives that $n(\mathrm{\Phi }(s,\xi )),e_1,e_2,e_3$ are linearly independent at every point, so in particular $e_1,e_2,e_3$ are linearly independent and $\mathrm{span}(e_1,e_2,e_3)\cap \mathrm{span}(n(\mathrm{\Phi }))=\{0\}$.

(The family is a congruence of worldlines; material fields.) For each fixed $\xi$ the curve $y_{\xi }(s):=\mathrm{\Phi }(s,\xi )$ satisfies $y_{\xi }^{\mathrm{\prime }}(s)=n(\mathrm{\Phi }(s,\xi ))$, a nonzero future timelike vector with $\eta (n,n)=-1$ by [L1]; hence $y_{\xi }$ is a $C^1$ regular future timelike curve, and it is a future timelike worldline of the model. Its proper-time function satisfies $d\tau \mathrm{/}ds=1\mathrm{/}c$ and ${\mathrm{\partial }}_{\tau }=c{\mathrm{\partial }}_s$ by [L6]; the associated four-velocity is $U=cn$ by [L7]. For each $I$, $s\mapsto e_I(s,\xi )$ is a $C^1$ $V$-valued field along $y_{\xi }$, using the canonical affine identification and the $C^1$ along-curve convention of [L4].

(The identity $L_{n}h=2\theta$.) Expanding $h_{ab}={\eta }_{ab}+n_a{n}_b$ and differentiating gives $(L_{n}h{)}_{ab}=n^c(n_a{\mathrm{\partial }}_c{n}_b+n_b{\mathrm{\partial }}_c{n}_a)+{\mathrm{\partial }}_a{n}_b+n_b{n}_c{\mathrm{\partial }}_a{n}^c+{\mathrm{\partial }}_b{n}_a+n_a{n}_c{\mathrm{\partial }}_b{n}^c$; by the vanishing identities of step 1.2 this equals ${\mathrm{\partial }}_a{n}_b+{\mathrm{\partial }}_b{n}_a+n_a{a}_b+n_b{a}_a$, which is the right-hand side of $2{\theta }_{ab}=h_a{}^c{h}_b{}^d({\mathrm{\partial }}_c{n}_d+{\mathrm{\partial }}_d{n}_c)$ computed in step 1.2; hence $(L_{n}h{)}_{ab}=2{\theta }_{ab}$ and Born rigidity $L_{n}h=0$ is equivalent to $\theta =0$.

(Metricity of $G$ and its $C^1$ regularity.) $G_{IJ}(s,\xi )=h_{ab}(\mathrm{\Phi }(s,\xi ))e_I^a(s,\xi )e_J^b(s,\xi )$ is symmetric because $h$ is symmetric by [L1]. For $w=w^I{e}_I\mathrm{\ne }0$, $G(w,w)=h(w,w)=\eta (h(w),h(w))\ge 0$ with equality if and only if $h(w)=0$, i.e. by [L5] if and only if $w\in \mathrm{span}(n(\mathrm{\Phi }))$; step 1.3 shows that this forces $w=0$, so $G$ is positive definite at every point. Regularity: $\mathrm{\Phi }$ is $C^2$, so the components of $e_I={\mathrm{\partial }}_{{\xi }^I}\mathrm{\Phi }$ and of $\mathrm{\Phi }$ are $C^1$ functions on $J\times B$; $n$ is a $C^1$ field, so the components $n_a(\mathrm{\Phi }(s,\xi ))$ are compositions of $C^1$ maps, and each $G_{IJ}$ is a sum of products of such $C^1$ functions, hence $C^1$ by [L11] applied componentwise with the partial derivatives of [L14]; when $n$ and $\mathrm{\Phi }$ are smooth all these compositions and products are smooth. In particular, for each fixed $s$, $\xi \mapsto G_{IJ}(s,\xi )$ is a $C^1$ positive-definite Riemannian metric on $B$.

(Derivative of the material directions.) On $J\times B$ the components satisfy ${\mathrm{\partial }}_s{e}_I^a={\mathrm{\partial }}_s{\mathrm{\partial }}_{{\xi }^I}{\mathrm{\Phi }}^a={\mathrm{\partial }}_{{\xi }^I}(n^a\circ \mathrm{\Phi })$, because the mixed partials of the $C^2$ map $\mathrm{\Phi }$ commute by [L10] and ${\mathrm{\partial }}_s\mathrm{\Phi }=n(\mathrm{\Phi })$; applying the chain rule of [L9] to the composite $\xi \mapsto n^a(\mathrm{\Phi }(s,\xi ))$ with the other variables fixed gives ${\mathrm{\partial }}_s{e}_I^a=({\mathrm{\partial }}_b{n}^a)(\mathrm{\Phi })e_I^b$. Equivalently, ${\mathrm{\partial }}_s{e}_I=({\mathrm{\nabla }}_{e_I}n)(\mathrm{\Phi })$ by the component formula of [L3] for $C^1$ data.

(Evolution of the material metric.) Differentiate the product $G_{IJ}=h_{ab}(\mathrm{\Phi })e_I^a{e}_J^b$ with the product rule of [L11] and the total chain rule of [L9], using ${\mathrm{\partial }}_s{\mathrm{\Phi }}^c=n^c(\mathrm{\Phi })$ and step 2.3: ${\mathrm{\partial }}_s{G}_{IJ}=(n^c{\mathrm{\partial }}_c{h}_{ab})(\mathrm{\Phi })e_I^a{e}_J^b+h_{ab}(\mathrm{\Phi })({\mathrm{\partial }}_c{n}^a)(\mathrm{\Phi })e_I^c{e}_J^b+h_{ab}(\mathrm{\Phi })e_I^a({\mathrm{\partial }}_c{n}^b)(\mathrm{\Phi })e_J^c$. Relabelling the second term to $(h_{cb}{\mathrm{\partial }}_a{n}^c)(\mathrm{\Phi })e_I^a{e}_J^b$ and the third to $(h_{ac}{\mathrm{\partial }}_b{n}^c)(\mathrm{\Phi })e_I^a{e}_J^b$ gives exactly $(L_{n}h{)}_{ab}(\mathrm{\Phi })e_I^a{e}_J^b$; by step 2.1 this is $2{\theta }_{ab}(\mathrm{\Phi })e_I^a{e}_J^b=2\theta (e_I,e_J)$, and multiplication by ${\mathrm{\partial }}_{\tau }\mathrm{/}{\mathrm{\partial }}_s=c$ from [L6] gives ${\mathrm{\partial }}_{\tau }{G}_{IJ}=2c\theta (e_I,e_J)$.

(Smooth branch.) If $n$ and $\mathrm{\Phi }$ are smooth, step 2.2 gives that $G$ is a smooth metric on $B$ for each $s$; evaluating the $C^1$ formula of the statement, $(L_{n}h{)}_{ab}=n^c{\mathrm{\partial }}_c{h}_{ab}+h_{cb}{\mathrm{\partial }}_a{n}^c+h_{ac}{\mathrm{\partial }}_b{n}^c$ agrees with that evaluation definition: substituting the bracket formula [L13] and the product rule into $n(h(X,Y))-h([n,X],Y)-h(X,[n,Y])$ cancels the derivatives of $X,Y$, leaving $(n^c{\mathrm{\partial }}_c{h}_{ab}+h_{cb}{\mathrm{\partial }}_a{n}^c+h_{ac}{\mathrm{\partial }}_b{n}^c)X^a{Y}^b$, and the identity $(L_{n}h{)}_{ab}=2{\theta }_{ab}$ of step 2.1 together with [L2] recovers the smooth congruence definitions of expansion and of Born rigidity.

(Rest length and its fractional rate.) For a nonzero label direction $e_I$ put ${\mathrm{\ell }}_I=\sqrt{G_{II}}>0$. Its $h$-normalized direction $u_I:=h(e_I)\mathrm{/}{\mathrm{\ell }}_I$ satisfies $h(u_I)=u_I$ and $h(u_I,u_I)=1$: indeed $h(h(v))=h(v)$ and $\eta (h(v),h(w))=h(v,w)$ by [L1] and [L5], with $\eta (h(e_I),h(e_I))=h(e_I,e_I)={\mathrm{\ell }}_I^2$, so $u_I$ is a horizontal unit vector. Because $\theta$ is horizontal and symmetric by [L2], $\theta (v,w)=\theta (h(v),h(w))$ for all $v,w$, so $\theta (e_I,e_I)={\mathrm{\ell }}_I^2\theta (u_I,u_I)$ and also $\theta (e_I\mathrm{/}{\mathrm{\ell }}_I,e_I\mathrm{/}{\mathrm{\ell }}_I)=\theta (u_I,u_I)$. The square root is continuous at each positive $G_{II}$, since $\mathrm{\mid }\sqrt{u}-\sqrt{v}\mathrm{\mid }=\mathrm{\mid }u-v\mathrm{\mid }\mathrm{/}(\sqrt{u}+\sqrt{v})$ and the denominator is bounded below near $v>0$. At a fixed label its difference quotient is $(G_{II}(s+\epsilon )-G_{II}(s))\mathrm{/}[\epsilon ({\mathrm{\ell }}_I(s+\epsilon )+{\mathrm{\ell }}_I(s))]$, so ${\mathrm{\partial }}_s{\mathrm{\ell }}_I={\mathrm{\partial }}_s{G}_{II}\mathrm{/}(2{\mathrm{\ell }}_I)$ by [L14]. Using step 3.1 and ${\mathrm{\partial }}_{\tau }=c{\mathrm{\partial }}_s$ gives ${\mathrm{\partial }}_{\tau }{\mathrm{\ell }}_I=(2{\mathrm{\ell }}_I{)}^{-1}{\mathrm{\partial }}_{\tau }{G}_{II}=c\theta (e_I,e_I)\mathrm{/}{\mathrm{\ell }}_I=c\theta (u_I,u_I){\mathrm{\ell }}_I$, which is the stated fractional rate.

(Born rigidity preserves the material metric exactly.) Suppose $\theta =0$ on the image of the family. Then by step 3.1 ${\mathrm{\partial }}_s{G}_{IJ}(s,\xi )=0$ for every $(s,\xi )\in J\times B$. Fix $\xi \in B$; the function $s\mapsto G_{IJ}(s,\xi )$ is $C^1$ on the interval $J$ by step 2.2, hence continuous, with zero derivative at every point of $J$, so it is constant on $J$ by [L12]. As $I,J$ and $\xi$ were arbitrary, the entire internal metric $G$ is preserved exactly along each flow line, with arbitrary label separation.

(Nonrigidity gives local nonconstancy.) If $\theta (e_I,e_J)(s_0,{\xi }_0)\mathrm{\ne }0$ at some point, then by step 3.1 ${\mathrm{\partial }}_s{G}_{IJ}(s_0,{\xi }_0)=2\theta (e_I,e_J)(s_0,{\xi }_0)\mathrm{\ne }0$; hence $G_{IJ}$ is not constant on any neighbourhood of $(s_0,{\xi }_0)$: a function that is constant near a point has vanishing derivative there. In particular the deformed metric component varies along the corresponding flow line.

(Local nature and scope.) Everything proved was computed from the given family: the metric $G$ is obtained by evaluating $h$ on the material directions $e_I$ at the common flow parameter $s$ (equivalently $\tau =s\mathrm{/}c$), so it is an internal label-space metric of the family. A finite endpoint distance on a single observer's orthogonal simultaneity hyperplane would require a separate identification of a slice with the label space; no such identification, matter model, elasticity, field equation, global rigidity theorem or boundary condition has been used, and every step is a finite-dimensional algebraic or differential computation using only the cited suppliers, so no choice principle is used.