Sharp radial focusing in flat Minkowski space: equality in the Raychaudhuri bound without a singularity

(The field is a smooth unit future timelike geodesic congruence.) On $O$ the function $\rho$ is smooth and strictly positive, and the components $W^a=(e^a-q^a)\mathrm{/}\rho$ are smooth; by [L2], $\eta (W,W)=-1$ and $W$ is future-directed relative to the time-orientation field $N={\mathrm{\partial }}_0$. It remains to verify the geodesic condition. Put $y:=e-q$, so $y^0=T-t$ and $y^i=-z^i$. From ${\rho }^2=(T-t{)}^2-\mathrm{\mid }z{\mathrm{\mid }}_{\mathrm{e}}^2$ one computes ${\mathrm{\partial }}_0\rho =-(T-t)\mathrm{/}\rho =y_0\mathrm{/}\rho$ and ${\mathrm{\partial }}_i\rho =-z^i\mathrm{/}\rho =y_i\mathrm{/}\rho$, hence ${\mathrm{\partial }}_b\rho =y_b\mathrm{/}\rho$ with $y_b={\eta }_{bc}{y}^c$; also ${\mathrm{\partial }}_b{y}^a=-{\delta }_b^a$. Therefore ${\mathrm{\partial }}_b{W}^a={\mathrm{\partial }}_b\left(\frac{y^a}{\rho }\right)=-\frac{{\delta }_b^a}{\rho }-\frac{y^a{y}_b}{{\rho }^3}=-\frac{{\delta }_b^a}{\rho }-\frac{W^a{W}_b}{\rho }\mathrm{.}$ Using $y^b{y}_b=\eta (y,y)=-{\rho }^2$, $W^b{\mathrm{\partial }}_b{W}^a=\frac{y^b}{\rho }(-\frac{{\delta }_b^a}{\rho }-\frac{y^a{y}_b}{{\rho }^3})=-\frac{y^a}{{\rho }^2}-\frac{y^a(y^b{y}_b)}{{\rho }^4}=-\frac{y^a}{{\rho }^2}+\frac{y^a}{{\rho }^2}=0.$ Since the metric-selected connection of the flat model has vanishing Christoffel symbols in this chart, ${\mathrm{\nabla }}_{W}W=0$; with [L3] the field $W$ is a smooth future unit timelike geodesic congruence. This proves (i).

(The flow lines.) For fixed $p\in O$ define $\gamma (s):=e-(1-s\mathrm{/}\rho (p))(e-p)$ for $s\in [0,\rho (p))$. Then $\gamma (0)=p$, $\gamma$ is affine in $s$ with derivative ${\gamma }^{\mathrm{\prime }}(s)=(e-p)\mathrm{/}\rho (p)=W(p)$, a constant vector; by [L1] straight coordinate lines are geodesics, so $s\mapsto \gamma (s)$ is a geodesic of the model with ${\gamma }^{\mathrm{\prime }}(s)=W(p)$. Its $\rho$-value is $\rho (\gamma (s))=\sqrt{-\eta (e-\gamma (s),e-\gamma (s))}$; since $e-\gamma (s)=(1-s\mathrm{/}\rho (p))(e-p)$, one gets $\rho (\gamma (s))=(1-s\mathrm{/}\rho (p))\rho (p)=\rho (p)-s$, which is positive for $s<\rho (p)$ and vanishes at $s=\rho (p)$; in particular $\gamma (s)\in O$ for $0\le s<\rho (p)$ and $\gamma (\rho (p))=e$ in the ambient affine space $A$. Also $W(\gamma (s))=(e-\gamma (s))\mathrm{/}\rho (\gamma (s))=(e-p)\mathrm{/}\rho (p)=\gamma \mathrm{\prime }(s)$, so it is an integral curve. Therefore $\gamma$ is the maximal forward flow line of $W$ through $p$, with $e$ as its limiting focal event at finite affine parameter $\rho (p)$. The same straight line continues for all real $s$ and is a geodesic of $A$ defined on all of $\mathbb{R}$ by [L1]. This proves (ii).

(The kinematic quantities.) By [L1] the connection is flat and $B_{ab}={\mathrm{\nabla }}_b{W}_a={\mathrm{\partial }}_b{W}_a$; from the computation in step 1.1, with $W_a=y_a\mathrm{/}\rho$, $B_{ab}={\mathrm{\partial }}_b\left(\frac{y_a}{\rho }\right)=-\frac{{\eta }_{ab}}{\rho }-\frac{y_a{y}_b}{{\rho }^3}=-\frac{1}{\rho }({\eta }_{ab}+W_a{W}_b)=-\frac{h_{ab}}{\rho }\mathrm{.}$ In particular $B$ is symmetric, so ${\omega }_{ab}=B_{[ab]}=0$, and $B_{ab}{W}^b=-h_{ab}{W}^b\mathrm{/}\rho =0$. For the expansion, [L4] and $g^{ab}{h}_{ab}=3$ give $\theta =g^{ab}{B}_{ab}=-\frac{1}{\rho }{g}^{ab}{h}_{ab}=-\frac{3}{\rho }$; then the shear is ${\sigma }_{ab}=B_{(ab)}-\frac{1}{3}\theta h_{ab}=-\frac{h_{ab}}{\rho }+\frac{1}{\rho }{h}_{ab}=0$. This proves (iii).

(The focusing bound is attained and the congruence does not extend.) Let $\gamma ={\gamma }_p$ be the flow line of step 2.1 and write $\theta (s):=\theta (\gamma (s))$. By step 2.2 and $\rho (\gamma (s))=\rho (p)-s$, $\theta (s)=-3\mathrm{/}(\rho (p)-s)$ for $0\le s<\rho (p)$. Hence ${\theta }_0:=\theta (p)=-3\mathrm{/}\rho (p)<0$ and $\frac{1}{-\theta (s)}=\frac{\rho (p)-s}{3}=\frac{1}{-{\theta }_0}-\frac{s}{3},$ so the focusing bound of [L5] holds with equality for $s_0=0$. To see directly that this congruence field cannot extend continuously across its focal event $e$, use the translation identification of tangent spaces of $A={\mathbb{R}}^4$ and set $v_1:={\mathrm{\partial }}_0{\mathrm{\mid }}_e$ and $v_2:=\sqrt{2}{\mathrm{\partial }}_0{\mathrm{\mid }}_e+{\mathrm{\partial }}_1{\mathrm{\mid }}_e$. The flat metric gives $\eta (v_i,v_i)=-1$ and $\eta (v_i,N_e)<0$ for $i=1,2$, while $v_1\mathrm{\ne }{v}_2$. Put $p_i:=e-v_i\in O$. Their flow lines have ${\gamma }_{p_i}(s)=e-(1-s)v_i$ for $0\le s<1$ and $W({\gamma }_{p_i}(s))=v_i$; both base curves tend to $e$ as $s\uparrow 1$, but the two limits of $W$ are distinct. A continuous extension of $W$ to $e$ would have to take both values $v_1$ and $v_2$, which is impossible. Thus this smooth congruence has no continuous, hence no smooth, extension through $e$; the individual affine geodesics still continue in $A$. This proves the local nonextension directly, independent of continuation beyond the region to which the hypotheses of [L5] apply.

(No singularity.) By [L1] the curvature and the Ricci tensor of the model vanish identically, every maximal geodesic of the model is a straight coordinate line with domain all of $\mathbb{R}$, and at $s=\rho (p)$ all flow lines pass through the single event $e$; the field $W$ is not defined at $e$ because $\rho (e)=0$. Hence the finite-time breakdown of the congruence in step 3.1 is exactly the focal crossing of the flow lines at $e$, while the spacetime and each individual geodesic continue; the example therefore shows that the local focusing theorem's conclusion is sharp and must be read as a statement about the congruence, not as a singularity. This proves (v) and completes the example.