Round spheres in Minkowski spacetime

On $S$, $q=R^{2}d{\mathrm{\Omega }}^2$. For tangent $X,Y$, flat-space differentiation gives ${\mathrm{\nabla }}_X{\mathrm{\partial }}_r=X\mathrm{/}R$ and ${\mathrm{\nabla }}_X{\mathrm{\partial }}_t=0$. Therefore ${\chi }_{\mathrm{\ell }}=q\mathrm{/}R$ and ${\chi }_k=-q\mathrm{/}R$, and since the screen dimension is two, ${\theta }_{\mathrm{\ell }}=2\mathrm{/}R$ and ${\theta }_k=-2\mathrm{/}R$. Direct inner products give $g(\mathrm{\ell },\mathrm{\ell })=g(k,k)=0$ and $g(\mathrm{\ell },k)=-1-1=-2$; both fields are future because their $t$ components are $1$.

In Cartesian coordinates the inward geodesic is ${\gamma }_{\omega }(\lambda )=(\lambda ,(R-\lambda )\omega )$. Varying $\omega$ in a tangent direction $X\in T_{R\omega }(R{\mathbb{S}}^2)$ gives the Jacobi field $J_X(\lambda )=(1-\lambda \mathrm{/}R)X$. Its screen map is therefore $A_{\lambda }=(1-\lambda \mathrm{/}R)\mathrm{id}$ for $0\le \lambda <R$, with first rank loss at $R$. Hence $B=-\mathrm{id}\mathrm{/}(R-\lambda )$, $\theta =-2\mathrm{/}(R-\lambda )$, $\sigma =0$, and $\mathrm{Ric}=0$. Differentiating $\theta$ verifies the displayed equation directly, and $2\mathrm{/}\mathrm{\mid }\theta (0)\mathrm{\mid }=2\mathrm{/}(2\mathrm{/}R)=R$. Thus the explicit focal parameter equals the comparison value; under its Countable Choice convention, [F2] gives the same sharp bound.

Fix $R<\lambda <3R$. The point $p_{-}=(0,-R\omega )\in S$ and $q_{\lambda }={\gamma }_{\omega }(\lambda )$ have time separation $\lambda$ and spatial separation $\mathrm{\mid }(R-\lambda )\omega -(-R\omega )\mathrm{\mid }=\mathrm{\mid }2R-\lambda \mathrm{\mid }<\lambda \mathrm{.}$ The straight segment from $p_{-}$ to $q_{\lambda }$ is future timelike because its time component is positive and its squared interval is $-{\lambda }^2+(2R-\lambda {)}^2<0$. Thus $q_{\lambda }\in I^{+}(S)$. Under Countable Choice and Dependent Choice this also agrees with [F3] for parameters sufficiently close to $R$; the displayed segment proves the stronger stated range $R<\lambda <3R$ directly, without either choice assumption.