Null expansions, screen quotients and focal data

Assume the axiom of choice. Let $(M,g,N)$ be a smooth time-oriented Lorentzian four-manifold of signature $(-,+,+,+)$, with metric-selected connection $\mathrm{\nabla }$. A spacelike two-surface $S$ is a nonempty smooth embedded two-manifold without boundary whose induced metric $\gamma =g{\mathrm{\mid }}_{TS}$ is positive definite. On an open set $U\subseteq S$, choose smooth future null normal fields $L,K$ with $g(L,K)=-2$. Thus $L$ and $K$ are nonzero, orthogonal to $TS$, null, and future-directed.

For tangent fields $X,Y$ on $U$, the null second fundamental form, expansion, and shear of $L$ are respectively ${\chi }_L(X,Y)=g({\mathrm{\nabla }}_{X}L,Y)$, ${\theta }_L={\mathrm{tr}}_{\gamma }{\chi }_L$, and ${\widehat{\chi }}_L={\chi }_L-\frac{1}{2}{\theta }_L\gamma$. Here the derivative of $L$ along $S$ uses the pullback connection and the trace uses the inverse of the positive metric $\gamma$.

Suppose a smooth map $F:U\times J\to M$, with $J$ an open interval containing zero, satisfies $F(p,0)=p$, ${\mathrm{\partial }}_{\lambda }F(p,0)=L_p$, and has future null affine geodesics $\lambda \mapsto F(p,\lambda )$ as its parameter curves. Write $L={\mathrm{\partial }}_{\lambda }F$ along this map and require $g(L,dF(X,0))=0$ for every $X\in T_{p}U$. The screen at $(p,\lambda )$ is $\mathcal{S}=L^{\perp }\mathrm{/}\mathbb{R}L$, equipped with $\bar{\gamma }([V],[W])=g(V,W)$. Fibers are indexed by the parameter domain, even when distinct branches of $F$ meet.

The normal screen Jacobi map is $A_{\lambda }:T_{p}S\to {\mathcal{S}}_{(p,\lambda )}$, $A_{\lambda }X=[d{F}_{(p,\lambda )}(X,0)]$. A focal instant to $S$ on this generator is a positive $\lambda \in J$ where $A_{\lambda }$ loses rank; the corresponding image is a focal point. The first focal instant, when one exists, is the least positive focal instant.

On the connected regular interval containing $0$, where $A_{\lambda }$ is invertible and the induced section metric is positive, let ${\mathcal{D}}_L[V]=[D_{\lambda }V]$ be the screen derivative and set $B=A^{\mathrm{\prime }}{A}^{-1}$, where the prime is the screen derivative. The optical expansion is $\theta =\mathrm{tr}B$ and the optical shear is $\sigma =B-\frac{1}{2}\theta \mathrm{i}\mathrm{d}$. At $\lambda =0$ these correspond to ${\theta }_L$ and ${\widehat{\chi }}_L$ under the screen identification.

For comparison, use the following causal-reachability conventions. Future causal and future timelike curves here are locally absolutely continuous curves whose velocities are nonzero and future causal or future timelike, respectively, almost everywhere. A point $q$ belongs to $I^{+}(p)$ when a future timelike curve on a nondegenerate compact parameter interval joins $p$ to $q$. It belongs to $J^{+}(p)$ when $q=p$ or such an interval carries a future causal curve joining $p$ to $q$. Put $I^{+}(S)={\bigcup }_{p\in S}{I}^{+}(p)$, $J^{+}(S)={\bigcup }_{p\in S}{J}^{+}(p)$, and $E^{+}(S)=J^{+}(S)\setminus I^{+}(S)$. When the set of $t\ge 0$ in the parameter domain such that the entire segment $F(p,[0,t])$ lies in $E^{+}(S)$ is nonempty and has a finite supremum $c$ in that domain, call $c$ the future boundary cut parameter of this generator relative to $S$ and $F(p,c)$ its boundary cut point. An intersection of distinct generators means $F(p,\lambda )=F(q,\mu )$ for distinct parameter curves, for example $p\mathrm{\ne }q$. These are separate definitions: focal rank loss is local differential data, whereas the cut condition and intersections involve global reachability or multiple branches.

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

Metric compatibility, torsion freedom and the metric-selected connection class

Causal curves, chronological and causal reachability, and inextendibility

Null normal pairs and screen geometry

Local null normal pairs and the positive screen quotient

Null focal points exclude later horismos generators

The Axiom of Choice

The supplier for null normal planes supplies local pairs and shows that the form and quotient metric are well-defined; its screen derivative is independent of the representative. The null-screen geometry supplier identifies the initial screen with $T_{p}S$, makes $A$ invertible on the stated regular interval, and identifies the initial optical data with the surface quantities. The cut parameter is a conditional supremum definition: existence of a finite attained cut is not asserted. The focal-exclusion supplier gives the stated local causal consequence of a focal instant; it does not identify the three notions or assert that a cut or an intersection exists.