Bell's rockets and thread: constant chart separation and the criterion for longitudinal expansion of the translated congruence

(The rocket tangents and chart separation.) For fixed $\xi$ the coordinate functions of $x_{\xi }$ are $t\mapsto ct$, $t\mapsto q(t)+\xi$, and the two constants; differentiating with [L13] gives $x_{\xi }^{\mathrm{\prime }}(t)=(c,q^{\mathrm{\prime }}(t),0,0)$, whose squared norm is $-c^2+q^{\mathrm{\prime }2}=-c^2(1-{\beta }^2)=-c^2\mathrm{/}{\gamma }^2<0$, with time component $c>\mathrm{\mid }{q}^{\mathrm{\prime }}\mathrm{\mid }$; hence $x_{\xi }^{\mathrm{\prime }}$ is future timelike by [L4] and $x_{\xi }$ is $C^2$ by [L3]. The chart separation of labels ${\xi }_1,{\xi }_2$ is $(q(t)+{\xi }_2)-(q(t)+{\xi }_1)={\xi }_2-{\xi }_1$, in particular $L$ for the two rockets; the chart velocity and acceleration are $q^{\mathrm{\prime }}$ and $q^{\mathrm{\prime }\mathrm{\prime }}$ for every label.

(The translated congruence data.) By [L10] the longitudinal unit rest vector of $n$ is $s_{\parallel }=\gamma (\beta ,1,0,0)$, horizontal ($\eta (n,s_{\parallel })=0$) with $\eta (s_{\parallel },s_{\parallel })=1$, and the first-order deformation satisfies $\theta (s_{\parallel },s_{\parallel })={\gamma }^3{q}^{\mathrm{\prime }}{q}^{\mathrm{\prime }\mathrm{\prime }}\mathrm{/}{c}^3$, nonzero exactly where $q^{\mathrm{\prime }}{q}^{\mathrm{\prime }\mathrm{\prime }}\mathrm{\ne }0$ and changing sign with that product; when $q$ is smooth these first-order expressions agree with the smooth congruence definitions by [L9], and Born rigidity is $\theta =0$.

(Common proper time and four-velocity.) By step 1.1 each worldline is $C^1$ future timelike, so its proper-time function is $\tau (t)={\int }_0^t{c}^{-1}\sqrt{-\eta (x_{\xi }^{\mathrm{\prime }},x_{\xi }^{\mathrm{\prime }})}dv={\int }_0^t\gamma (v{)}^{-1}dv$ by [L6] and step 1.1, the same function for every $\xi$. The four-velocity is $U=x_{\xi }^{\mathrm{\prime }}\mathrm{/}{\tau }^{\mathrm{\prime }}=\gamma (c,q^{\mathrm{\prime }},0,0)$ by [L5], with $\eta (U,U)={\gamma }^2(-c^2+q^{\mathrm{\prime }2})=-c^2$.

(The four-dimensional extension.) Since $\tau$ is strictly increasing and $C^1$ with ${\tau }^{\mathrm{\prime }}=1\mathrm{/}\gamma >0$, it has a $C^1$ inverse $t(\sigma )$ on $c\tau (J)$ with $dt\mathrm{/}d\sigma =\gamma \mathrm{/}c$ by [L6]; because $q$ is $C^2$, $\gamma$ is $C^1$ and $dt\mathrm{/}d\sigma =(\gamma \circ t)\mathrm{/}c$ is $C^1$, so $t$ is $C^2$. The image of $\mathrm{\Phi }$ is the open set of events with $t\in J$, $x^1-q(t)\in I$, $x^2\in Y$ and $x^3\in Z$, with inverse $(\sigma ,\xi ,y,z)=(c\tau (t),x^1-q(t),x^2,x^3)$. Thus $\mathrm{\Phi }$ is injective and $C^2$ in all variables, with ${\mathrm{\partial }}_{\xi }\mathrm{\Phi }=(0,1,0,0)$, ${\mathrm{\partial }}_y\mathrm{\Phi }=(0,0,1,0)$, ${\mathrm{\partial }}_z\mathrm{\Phi }=(0,0,0,1)$ and ${\mathrm{\partial }}_{\sigma }\mathrm{\Phi }=(cdt\mathrm{/}d\sigma ,q^{\mathrm{\prime }}dt\mathrm{/}d\sigma ,0,0)=(\gamma ,\gamma \beta ,0,0)=n$, using the chain rule [L13]; the four partial derivatives are linearly independent, so $\mathrm{\Phi }$ is an immersion. The field $n=\gamma (1,\beta ,0,0)$ has $\eta (n,n)={\gamma }^2(-1+{\beta }^2)=-1$ and time component $\gamma >0$ with $\mathrm{\mid }(n{)}_s\mathrm{\mid }=\gamma \mathrm{\mid }\beta \mathrm{\mid }<\gamma$, so it is a future unit timelike field, $C^1$ because $q$ is $C^2$.

(Clause (a).) Steps 1.1 and 1.2 give: every member $x_{\xi }$ is $C^2$ future timelike with squared tangent norm $-c^2\mathrm{/}{\gamma }^2$; the common proper-time function is $\tau (t)={\int }_0^t{\gamma }^{-1}$; the four-velocity is $U=\gamma (c,q^{\mathrm{\prime }},0,0)$ with $\eta (U,U)=-c^2$; all labels share the chart velocity $q^{\mathrm{\prime }}$ and chart acceleration $q^{\mathrm{\prime }\mathrm{\prime }}$; and the chart separation of any two labels is constant in $t$.

(Clause (b): the material-family analysis applies.) By step 3.1 the map $\mathrm{\Phi }$ is an injective $C^2$ immersion with ${\mathrm{\partial }}_{\sigma }\mathrm{\Phi }=n(\mathrm{\Phi })$, so the material metric $G$ of [L8] is defined on $B$; the material direction $e:={\mathrm{\partial }}_{\xi }\mathrm{\Phi }=(0,1,0,0)$ is constant, and restricting $\mathrm{\Phi }$ to $y=y_0$, $z=z_0$ gives the prescribed thread worldlines. The longitudinal rest vector of the family is $s_{\parallel }$ of step 1.2 and $\theta (s_{\parallel },s_{\parallel })={\gamma }^3{q}^{\mathrm{\prime }}{q}^{\mathrm{\prime }\mathrm{\prime }}\mathrm{/}{c}^3$; when $q$ is smooth the smooth-congruence branch of [L9] is recovered.

(Clause (e): contrast with the Born-rigid Rindler congruence.) By [L10] the Rindler congruence $n={\rho }^{-1}{\mathrm{\partial }}_{\theta }$ is Born rigid, so its rest slices $\theta =\text{const}$ are isometric with constant internal rest separations; the proper accelerations of its flow lines are $c^2\sqrt{\eta (a_{\mathrm{c}\mathrm{g}},a_{\mathrm{c}\mathrm{g}})}=c^2\mathrm{/}\rho$ by [L11] and [L10], which differ for distinct radial labels. In the translated family, by contrast, all members have equal chart velocity and acceleration profiles (step 3.2) and constant chart separation $L$, yet $\theta (s_{\parallel },s_{\parallel })={\gamma }^3{q}^{\mathrm{\prime }}{q}^{\mathrm{\prime }\mathrm{\prime }}\mathrm{/}{c}^3$ is nonzero wherever $q^{\mathrm{\prime }}{q}^{\mathrm{\prime }\mathrm{\prime }}\mathrm{\ne }0$ (step 1.2), so the congruence is not Born rigid there. Hence equal chart-frame accelerations and constant chart separation are distinct from Born rigidity and from constant internal rest metric.

(Clause (c): the material length and its rate.) For $e=(0,1,0,0)$ and $n$ of step 3.1, $\eta (n,e)=\gamma \beta$, so by the identities of [L8] and [L9] $h(e,e)=\eta (e,e)+\eta (n,e{)}^2=1+{\gamma }^2{\beta }^2={\gamma }^2$, and $h(e)\mathrm{/}\gamma =\gamma (\beta ,1,0,0)=s_{\parallel }$; also $e\mathrm{/}\gamma \mathrm{\ne }{s}_{\parallel }$ in general, but horizontality of $\theta$ gives $\theta (e\mathrm{/}\gamma ,e\mathrm{/}\gamma )=\theta (s_{\parallel },s_{\parallel })$. Hence $G_{11}=h(e,e)=\gamma (t{)}^2$ and $\mathcal{L}(t)={\int }_0^L\sqrt{G_{11}}d\xi =\gamma (t)L$. By the evolution formula of [L8] and the logarithm derivative of [L14], ${\mathrm{\partial }}_{\tau }{G}_{11}=2c\theta (e,e)$ and ${\mathrm{\partial }}_{\tau }\log (\mathcal{L}\mathrm{/}L)=\frac{1}{2}{\mathrm{\partial }}_{\tau }{G}_{11}\mathrm{/}{G}_{11}=c\theta (e,e)\mathrm{/}{\gamma }^2=c\theta (s_{\parallel },s_{\parallel })={\gamma }^3{q}^{\mathrm{\prime }}{q}^{\mathrm{\prime }\mathrm{\prime }}\mathrm{/}{c}^2$; multiplying by $\mathcal{L}=\gamma L$ gives $d\mathcal{L}\mathrm{/}d\tau ={\gamma }^{4}L{q}^{\mathrm{\prime }}{q}^{\mathrm{\prime }\mathrm{\prime }}\mathrm{/}{c}^2$. With $q^{\mathrm{\prime }}=v$, $\gamma =(1-v^2\mathrm{/}{c}^2{)}^{-1\mathrm{/}2}\ge 1$ by [L7], so $\mathcal{L}=\gamma L\ge L$ with equality exactly for $v=0$; the chart separation remains $L$ by step 1.1, so the material length is constant exactly when $q^{\mathrm{\prime }}{}^2$ is constant and otherwise has the stated nonzero rate wherever $q^{\mathrm{\prime }}{q}^{\mathrm{\prime }\mathrm{\prime }}\mathrm{\ne }0$.

(Clause (d): the material reading.) Under the stipulated reading that the thread elements follow the flow lines, the internal length at common flow parameter is $\mathcal{L}(t)=\gamma (t)L$ by step 5.1, so the initial value is $\gamma (0)L$ and the relative factor is $\gamma (t)\mathrm{/}\gamma (0)$; it stretches where $\gamma (t)>\gamma (0)$, shrinks where $\gamma (t)<\gamma (0)$, and is equal where the factors agree. In the special case $q^{\mathrm{\prime }}(0)=0$ one has $\gamma (0)=1$ and (by $\mathrm{\mid }{q}^{\mathrm{\prime }}\mathrm{\mid }<c$) $\gamma (t)>1$ whenever $q^{\mathrm{\prime }}(t)\mathrm{\ne }0$ by [L7]. No elasticity or failure criterion is available in the given data, so neither breaking nor a definite tension follows; the statement asserts only the kinematic material reading.

(Clause (f): scope.) Every claim was derived for the prescribed $C^2$ family and the stated material reading, using only the cited definitional and kinematic suppliers: no elasticity, material failure criterion, force law, measurement or empirical input was used; the thread is an ideal material reading of the translated congruence, and no claim about real threads or rockets is made. All computations are finite-dimensional and no choice principle was used.