Hypersurface and null geometry

Assume Countable Choice for the smooth tangent-bundle convention in the Lorentzian suppliers below. Let $(M,g,N)$ be a smooth time-oriented Lorentzian four-manifold of signature $(-,+,+,+)$, and let $\iota :S\hookrightarrow M$ be a smooth embedded two-manifold without boundary whose induced form $q:={\iota }^{\ast }g$ is positive definite. Write ${\mathcal{N}}_{p}S:=\{v\in T_{p}M:g_p(v,X)=0\text{ for every }X\in T_{p}S\}\mathrm{.}$ Then $\mathcal{N}S$ is a smooth rank-two Lorentzian vector bundle. Around each $p\in S$ there are smooth future null normal fields $\mathrm{\ell },k$ with $g(\mathrm{\ell },\mathrm{\ell })=g(k,k)=0,g(\mathrm{\ell },k)=-2.$ On an overlap where two such ordered pairs select the same future null ray, they differ by a smooth boost ${\mathrm{\ell }}^{\mathrm{\prime }}=e^f\mathrm{\ell }$, $k^{\mathrm{\prime }}=e^{-f}k$; if their ray labels are exchanged, the same rule holds after interchanging the labels. No global null normal pair is asserted.

Here a smooth null hypersurface $H\subset M$ means a smooth embedded three-dimensional submanifold without boundary such that, at every $p\in H$, the restriction $g_p{\mathrm{\mid }}_{T_{p}H}$ is positive semidefinite of rank two. Its radical is the one-dimensional subspace $\mathrm{Rad}(g_p{\mathrm{\mid }}_{T_{p}H}):=\{v\in T_{p}H:g_p(v,w)=0\text{ for every }w\in T_{p}H\}\mathrm{.}$ A smooth future null normal field on $H$ is a smooth nowhere-zero section $\mathrm{\ell }$ of this radical line, with $g_p({\mathrm{\ell }}_p,N_p)<0$. Thus $T_{p}H={\mathrm{\ell }}_p^{\perp },(T_{p}H{)}^{\perp }=\mathrm{Rad}(g_p{\mathrm{\mid }}_{T_{p}H})=\mathbb{R}{\mathrm{\ell }}_p;$ in particular, the normal space of a null hypersurface is a null line, not a Lorentzian two-plane. A spacelike section $S$ of $H$ is a smooth embedded two-manifold $S\subset H$ such that $q:=g{\mathrm{\mid }}_{TS}$ is positive definite and $T_{p}H=T_{p}S\oplus \mathbb{R}{\mathrm{\ell }}_p(p\in S)\mathrm{.}$ The condition says that $S$ is transverse to the null generators. The null generator curves are integral curves $\gamma$ of $\mathrm{\ell }$, meaning $\dot{\gamma }=\mathrm{\ell }\circ \gamma$; when $D_{\mathrm{\ell }}\mathrm{\ell }=0$, their parameter is affine. For fields defined only along $H$ or a surface $S$, covariant derivatives ${\mathrm{\nabla }}_X\mathrm{\ell }$ use the pullback of the metric-selected connection.

For a smooth future null normal field $\mathrm{\ell }$ on $H$, the quotient ${\mathcal{S}}_p:={\mathrm{\ell }}_p^{\perp }\mathrm{/}\mathbb{R}{\mathrm{\ell }}_p,p\in H,$ is a smooth rank-two bundle with positive metric ${\bar{q}}_p([X],[Y]):=g_p(X,Y)\mathrm{.}$ The normal line is $\mathbb{R}{\mathrm{\ell }}_p$; the auxiliary null direction used in the proof below is not a second normal to $H$. If $\mathrm{\ell }$ is affinely geodesic, meaning $D_{\mathrm{\ell }}\mathrm{\ell }=0$ along its integral curves, the map $B_p:{\mathcal{S}}_p\to {\mathcal{S}}_p,[X]\mapsto [{\mathrm{\nabla }}_X\mathrm{\ell }]$ is well-defined and self-adjoint with respect to $\bar{q}$, that is, $\bar{q}(Bu,v)=\bar{q}(u,Bv)$. On a spacelike section $S$ of $H$, the quotient map $T_{p}S\to {\mathcal{S}}_p$, $X\mapsto [X]$, is an isometry; it is an isomorphism by transversality, and $\bar{q}([X],[Y])=q(X,Y)$. For a smooth future null normal $\mathrm{\ell }$ to a spacelike surface $S$, define ${\chi }_{\mathrm{\ell }}(X,Y):=g({\mathrm{\nabla }}_X\mathrm{\ell },Y)(X,Y\in T_{p}S);$ where ${\mathrm{\nabla }}_X\mathrm{\ell }$ is computed with the pullback of the metric-selected connection to $S$. This is a well-defined symmetric bilinear form. Along an affine generator, for a screen section $[X]$, the screen derivative ${\mathcal{D}}_{\mathrm{\ell }}[X]:=[D_{\mathrm{\ell }}X]$ is independent of the representative and preserves the screen metric in the sense that $\mathrm{\ell }(\bar{q}([X],[Y]))=\bar{q}({\mathcal{D}}_{\mathrm{\ell }}[X],[Y])+\bar{q}([X],{\mathcal{D}}_{\mathrm{\ell }}[Y])\mathrm{.}$ For a smooth function $f$ on the section, the rescaling ${\mathrm{\ell }}^{\mathrm{\prime }}=e^f\mathrm{\ell }$, $k^{\mathrm{\prime }}=e^{-f}k$ of a normalized null normal pair is a boost; it obeys ${\chi }_{e^f\mathrm{\ell }}=e^f{\chi }_{\mathrm{\ell }}\mathrm{.}$

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

Embedded submanifolds and slice charts

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

Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields

Covariant derivative along a curve

Sylvester's law of inertia: every real symmetric form is congruent to $\mathrm{diag}(I_p,-I_q,0_r)$, and $(p,q,r)$ is unique

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

The screen inverse and every trace on a null hypersurface are taken on the positive quotient ${\mathrm{\ell }}^{\perp }\mathrm{/}\mathbb{R}\mathrm{\ell }$ (or its identified tangent plane on a spacelike section). The induced form on all of $TH$ is degenerate and has no inverse. A global ordered pair of future null normals requires an additional trivialization; the local boost law above does not assert one.

Assume Countable Choice for the smooth tangent-bundle and time-orientation conventions of the Lorentzian suppliers. Let $(M,g,N)$ be a smooth time-oriented Lorentzian four-manifold of signature $(-,+,+,+)$. A spacelike two-surface is a nonempty smooth embedded two-manifold $S\subset M$ without boundary whose induced metric $q:={\iota }^{\ast }g$ is positive definite, where $\iota :S\hookrightarrow M$ is the inclusion. It is closed when it is additionally compact. The constructions below apply to any such spacelike two-surface; compactness is not required. Its normal bundle is $\mathcal{N}S=(TS{)}^{\perp }$. Work on a nonempty open set $U\subseteq S$ carrying a smooth ordered pair of future null normal fields $\mathrm{\ell },k$ with $g(\mathrm{\ell },\mathrm{\ell })=g(k,k)=0$ and $g(\mathrm{\ell },k)=-2$; such pairs exist locally by the preceding lemma. No global pair is part of the definition unless one is explicitly supplied.

For tangent fields $X,Y$ on $U$, define the null second fundamental form, expansion and shear by ${\chi }_{\mathrm{\ell }}(X,Y):=g({\mathrm{\nabla }}_X\mathrm{\ell },Y),{\theta }_{\mathrm{\ell }}:={\mathrm{tr}}_q{\chi }_{\mathrm{\ell }},{\sigma }_{\mathrm{\ell }}:={\chi }_{\mathrm{\ell }}-\frac{{\theta }_{\mathrm{\ell }}}{2}q\mathrm{.}$ Here $\mathrm{\nabla }$ is the metric-selected connection; its derivative of a section along $S$ means the pullback connection, equivalently the derivative of any smooth ambient extension along tangent directions. The preceding lemma establishes extension independence, symmetry of ${\chi }_{\mathrm{\ell }}$, and positive definiteness of the screen metric. A normal-frame boost is the change $\mathrm{\ell }\mapsto e^f\mathrm{\ell }$, $k\mapsto e^{-f}k$ for a smooth real function $f$ on $U$. It leaves $q$ fixed; ${\chi }_{\mathrm{\ell }},{\theta }_{\mathrm{\ell }},{\sigma }_{\mathrm{\ell }}$ scale by $e^f$.

A null geodesic hypersurface congruence from $U$ along the chosen initial normal field $\mathrm{\ell }$ is a smooth map $F:U\times J\to M$, where $J\subseteq \mathbb{R}$ is a nonempty open interval containing $0$, such that $F(p,0)=p$ and ${\mathrm{\partial }}_{\lambda }F(p,0)={\mathrm{\ell }}_p$ for every $p\in U$, each curve ${\gamma }_p(\lambda ):=F(p,\lambda )$ is future-directed and null, and its tangent $\mathrm{\ell }:={\mathrm{\partial }}_{\lambda }F$ obeys the affine geodesic equation $D_{\lambda }\mathrm{\ell }=0$ along every curve ${\gamma }_p$; here $D_{\lambda }$ is the covariant derivative along a curve from Covariant derivative along a curve. For each $p\in U$, let $J_{\mathrm{r}\mathrm{e}\mathrm{g}}(p)$ be the connected component containing $0$ of the parameter values for which $d(F_{\lambda }{)}_p:T_{p}U\to T_{F(p,\lambda )}M$ is injective and the pullback form $(X,Y)\mapsto g(d(F_{\lambda }{)}_{p}X,d(F_{\lambda }{)}_{p}Y)$ is positive definite on $T_{p}U$. Require, for every $\lambda \in J$, that its variation directions satisfy $g_{F(p,\lambda )}(\mathrm{\ell }(p,\lambda ),d{F}_{(p,\lambda )}(X,0))=0(X\in T_{p}U)\mathrm{.}$ Thus its variation directions are orthogonal to $\mathrm{\ell }$. For $\lambda \in J_{\mathrm{r}\mathrm{e}\mathrm{g}}(p)$, the map $F$ has rank three locally near $(p,\lambda )$: its two section directions are spacelike and orthogonal to the nonzero null vector $\mathrm{\ell }$. Its local image is a null hypersurface, with tangent space ${\mathrm{\ell }}^{\perp }$ and induced radical $\mathbb{R}\mathrm{\ell }$, as in Local null normal pairs and the positive screen quotient. The parameter $\lambda$ is affine. On each section in $J_{\mathrm{r}\mathrm{e}\mathrm{g}}$, the screen is $\mathcal{S}={\mathrm{\ell }}^{\perp }\mathrm{/}\mathbb{R}\mathrm{\ell }$ with metric $\bar{q}([X],[Y])=g(X,Y)$; it is the positive two-dimensional quotient, identified isometrically with the tangent plane of that section by projection. The induced form on the full null hypersurface has radical $\mathbb{R}\mathrm{\ell }$ and has no inverse. All traces and inverses in this definition are taken on the screen.

All screen bundles associated with this congruence are understood as bundles over the parameter domain $U\times J$: the notation ${\mathcal{S}}_{F(p,\lambda )}$ denotes the branch determined by $(p,\lambda )$, so self-intersections do not identify distinct screen fibers. For each $\lambda \in J$, the screen Jacobi map $A_{\lambda }:T_{p}S\to {\mathcal{S}}_{F(p,\lambda )}$ sends $X$ to the screen class of the variation field $J_X(\lambda ):=d{F}_{(p,\lambda )}(X,0)$; it is invertible for $\lambda \in J_{\mathrm{r}\mathrm{e}\mathrm{g}}(p)$. A focal instant is a positive parameter in $J$ where $A_{\lambda }$ loses rank; the first focal instant, if it exists, is the least such parameter after $0$. Along an affine generator the screen derivative is ${\mathcal{D}}_{\mathrm{\ell }}[X]:=[D_{\mathrm{\ell }}X]$; it is independent of the representative, by the preceding lemma. On $J_{\mathrm{r}\mathrm{e}\mathrm{g}}$, the derivative $A_{\lambda }^{\mathrm{\prime }}$ is taken using ${\mathcal{D}}_{\mathrm{\ell }}$, and the optical endomorphism is $B_{\lambda }:=A_{\lambda }^{\mathrm{\prime }}{A}_{\lambda }^{-1}$. Its optical expansion is $\theta :=\mathrm{tr}B$, and its optical shear is $\sigma :=B-\frac{1}{2}\theta {\mathrm{i}\mathrm{d}}_{\mathcal{S}}$. Since the congruence is normal to a null hypersurface, $B$ is self-adjoint. Its antisymmetric part $\omega :=(B-B^{\ast })\mathrm{/}2$, called the twist, therefore vanishes. At $\lambda =0$, $B$ is the endomorphism $q^{-1}{\chi }_{\mathrm{\ell }}$ and these optical quantities agree with ${\theta }_{\mathrm{\ell }},{\sigma }_{\mathrm{\ell }}$.

No orientation of $M$, $S$, or the screen is required for these constructions. Indeed, if two orthonormal screen frames are related by a real $2\times 2$ matrix $O$ with $O^{\mathsf{T}}O=I_2$, then the endomorphism matrices obey $\widehat{B}=O^{-1}BO$ and $\widehat{\sigma }=O^{-1}\sigma O$. Cyclicity of the matrix trace gives $\mathrm{tr}\widehat{B}=\mathrm{tr}B$ and $\mathrm{tr}({\widehat{\sigma }}^2)=\mathrm{tr}({\sigma }^2)=\mathrm{\mid }\sigma {\mathrm{\mid }}_{\bar{q}}^2$; the last equality follows by summing the squared entries of the symmetric shear matrix in an orthonormal frame. This includes $\det O=-1$. The screen inner products and the rank of $A_{\lambda }$ are likewise invariant under changes of frame, so neither the index-form contractions nor the focal condition uses a signed screen area form. The preceding lemma constructs normal pairs using the supplied time orientation and only a local spacelike complement; changing the complement's sign interchanges the two future null rays. No global ordered pair follows. If an orientation is additionally supplied, the same formulas apply on that oriented subclass.

Local null normal pairs and the positive screen quotient

Covariant derivative along a curve

The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors

The Jacobi equation, geodesic deviation and the initial-value theory of Jacobi fields

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Use $K(X,Y)=g({\mathrm{\nabla }}_{X}n,Y)$ for a spacelike hypersurface with future unit normal $n$, and use $R(X,Y)Z={\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z\mathrm{.}$ Thus the Jacobi equation in the supplied convention is $D_{\mathrm{\ell }}^{2}J=R(\mathrm{\ell },J)\mathrm{\ell }$ and the screen curvature endomorphism is $[X]\mapsto [R(X,\mathrm{\ell })\mathrm{\ell }]$. The stated data are geometric definitions and make no claim about an Einstein equation, stress-energy tensor, energy condition or the physical validity of a model.

For a physical dimensional reading in SI, use length-valued coordinates $x^0=ct$ and a dimensionless metric; take the affine parameter $\lambda$ in metres. Then $\mathrm{\ell }$ and $k$ are dimensionless tangent components, ${\chi }_{\mathrm{\ell }},{\theta }_{\mathrm{\ell }},{\sigma }_{\mathrm{\ell }}$ have units ${\mathrm{m}}^{-1}$, and curvature and $\mathrm{Ric}(\mathrm{\ell },\mathrm{\ell })$ have units ${\mathrm{m}}^{-2}$. A constant boost rescales $\lambda$ inversely, as recorded in the companion calculations.

The schematic shows one tangent-space section at $p\in S$. The horizontal line represents a two-dimensional spacelike tangent plane in a two-dimensional drawing; the upward arrows are the two future null rays in its Lorentzian normal plane. The labels ${\lambda }_{\mathrm{\ell }},{\lambda }_k$ increase to the future along their respective affine generators. The figure illustrates the convention $g(\mathrm{\ell },k)=-2$ and asserts neither a global pair over $S$ nor any statement used by a proof.

Assume Countable Choice for the Lorentzian supplier conventions. Let $(M,g,N)$ be a smooth time-oriented Lorentzian four-manifold, let $S$ be a smooth spacelike two-surface, and let $F:U\times J\to M$ be a smooth null geodesic hypersurface congruence from an open section $U\subseteq S$, as defined in Null normal pairs and screen geometry. Write $\mathrm{\ell }={\mathrm{\partial }}_{\lambda }F$ and fix $p\in U$. Before the first focal instant let $A_{\lambda }:T_{p}S\to {\mathcal{S}}_{F(p,\lambda )}$ be the screen Jacobi map, $B=A^{\mathrm{\prime }}{A}^{-1}$ the optical endomorphism, $\theta =\mathrm{tr}B$ its expansion, and $\sigma =B-\frac{1}{2}\theta \mathrm{i}\mathrm{d}$ its shear. Let $\mathrm{Ric}$ be the Ricci tensor in The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors. Then, along the generator $\gamma (\lambda )=F(p,\lambda )$ and before the first focal instant, $\frac{d\theta }{d\lambda }=-\frac{{\theta }^2}{2}-\mathrm{\mid }\sigma {\mathrm{\mid }}_{\bar{q}}^2-\mathrm{Ric}(\mathrm{\ell },\mathrm{\ell })\mathrm{.}$ If $\mathrm{Ric}(\mathrm{\ell },\mathrm{\ell })\ge 0$ along the generator and $\theta (0)<0$, then a first focal instant occurs by affine parameter ${\lambda }_{\ast }=\frac{2}{\mathrm{\mid }\theta (0)\mathrm{\mid }},$ provided the smooth congruence and generator are defined through ${\lambda }_{\ast }$. Here a focal instant means that the screen Jacobi map loses rank; the conclusion is local along this generator and asserts no curvature singularity.

Null normal pairs and screen geometry

The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

The conclusion is a conditional geometric deduction from the Ricci inequality and the specified congruence. It does not derive $\mathrm{Ric}(\mathrm{\ell },\mathrm{\ell })\ge 0$ from matter assumptions, invoke an Einstein equation, or infer a spacetime singularity from a focal point.

Assume Countable Choice for the smooth Lorentzian supplier conventions. Let $(M,g,N)$ be a smooth time-oriented Lorentzian four-manifold, and let $S\subset M$ be a smooth embedded two-manifold without boundary with positive induced metric $q:=g{\mathrm{\mid }}_{TS}$. Choose $p\in S$, a smooth future null normal field $\mathrm{\ell }$ near $p$ in $S$, and $L>0$. Let $\gamma :[0,L]\to M$ be smooth up to its endpoints, future-directed, affine and null, with $\gamma (0)=p$, $\dot{\gamma }(0)=\mathrm{\ell }(p)$, and $\dot{\gamma }$ normal to $S$ at $p$. For each $\lambda \in [0,L]$, put $u_{\lambda }:=N_{\gamma (\lambda )}\mathrm{/}\sqrt{-g_{\gamma (\lambda )}(N_{\gamma (\lambda )},N_{\gamma (\lambda )})}$, ${\alpha }_{\lambda }:=-g_{\gamma (\lambda )}(\dot{\gamma }(\lambda ),u_{\lambda })>0$, and ${\mathrm{\ell }}_{\lambda }:=\dot{\gamma }(\lambda )$. Define $n_{\lambda }:={\alpha }_{\lambda }^{-1}{u}_{\lambda }-\frac{1}{2{\alpha }_{\lambda }^2}\dot{\gamma }(\lambda ),E_{\lambda }:=\mathrm{span}\{\dot{\gamma }(\lambda ),n_{\lambda }{\}}^{\perp }\mathrm{.}$ Then $n_{\lambda }$ is null with $g(\dot{\gamma },n_{\lambda })=-1$, and the quotient map $E_{\lambda }\to {\mathcal{S}}_{\lambda }$, $X\mapsto [X]$, is an isometry onto the positive screen ${\mathcal{S}}_{\lambda }:={\mathrm{\ell }}_{\lambda }^{\perp }\mathrm{/}\mathbb{R}{\mathrm{\ell }}_{\lambda }$; the spaces $E_{\lambda }$ form a smooth rank-two screen subbundle along $\gamma$. The quotient metric is $\bar{q}([X],[Y]):=g(X,Y)$. Let $\mathrm{\nabla }$ be the metric-selected connection and $R$ its curvature with convention $R(X,Y)Z={\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z$. At $\lambda =0$, define ${\chi }_{\mathrm{\ell }}(X,Y):=g({\mathrm{\nabla }}_X\mathrm{\ell },Y)$ for $X,Y\in T_{p}S$ and identify $T_{p}S$ with ${\mathcal{S}}_0$ by the isometry $X\mapsto [X]$. A screen field is a smooth section of $\mathcal{S}$ along $\gamma$; write $D_{\mathrm{\ell }}V$ for its screen covariant derivative along ${\mathrm{\ell }}_{\lambda }$, and set $\mathcal{R}V:=[R(V,{\mathrm{\ell }}_{\lambda }){\mathrm{\ell }}_{\lambda }]$. If $V,W$ are screen fields with $V(0),W(0)\in T_{p}S$ under the stated isometry and $V(L)=W(L)=0$, define $I(V,W):={\int }_0^L(\bar{q}(D_{\mathrm{\ell }}V,D_{\mathrm{\ell }}W)-\bar{q}(\mathcal{R}V,W))d\lambda +{\chi }_{\mathrm{\ell }}(V(0),W(0))\mathrm{.}$ Then $I$ is a well-defined symmetric bilinear form. For every such $V$ there are $\epsilon >0$ and a smooth map $F:(-\epsilon ,\epsilon )\times [0,L]\to M$ whose sections $F_s$ have screen variational field $V$, with $F_s(0)\in S$, $F_s(L)=\gamma (L)$ and $F_0=\gamma$. For this family, with $E(s):=\frac{1}{2}{\int }_0^{L}g({\mathrm{\partial }}_{\lambda }{F}_s,{\mathrm{\partial }}_{\lambda }{F}_s)d\lambda ,$ the energy second variation is $E^{\mathrm{\prime }\mathrm{\prime }}(0)=I(V,V)\mathrm{.}$

Null normal pairs and screen geometry

The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors

Embedded submanifolds and slice charts

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors

The geodesic initial-value problem and the exponential map of the metric-selected connection

For SI interpretation use $x^0=ct$, dimensionless metric components and $\lambda$ in metres. The index form has units of length when a variation field is measured in metres. Its boundary sign is tied to ${\chi }_{\mathrm{\ell }}(X,Y)=g({\mathrm{\nabla }}_X\mathrm{\ell },Y)$.

Assume Countable Choice and Dependent Choice, as required by the supplied causal-curve and push-up results. Let $(M,g,N)$ be a smooth time-oriented Lorentzian four-manifold. Let $S$ be a smooth embedded two-manifold without boundary whose induced metric $q:=g{\mathrm{\mid }}_{TS}$ is positive definite, and let $p\in S$. Choose an open neighborhood $U\subseteq S$ of $p$ and a smooth future null normal field $\mathrm{\ell }$ along $U$, so $g({\mathrm{\ell }}_q,Y)=0$ for every $q\in U$ and $Y\in T_{q}S$. Let $\gamma :[0,L_1]\to M$ be smooth up to its endpoints, with $\gamma (0)=p,\dot{\gamma }(0)={\mathrm{\ell }}_p,g(\dot{\gamma },\dot{\gamma })=0,g(\dot{\gamma },N)<0,D_{\lambda }\dot{\gamma }=0.$ Here $D_{\lambda }$ is covariant differentiation along the curve. Put ${\chi }_{\mathrm{\ell }}(X,Y):=g({\mathrm{\nabla }}_X\mathrm{\ell },Y)$ as in Null normal pairs and screen geometry, and set $B_0:T_{p}S\to T_{p}S$ by $q(B_{0}X,Y)={\chi }_{\mathrm{\ell }}(X,Y)$ for all $X,Y\in T_{p}S$. The screen along $\gamma$ is ${\mathcal{S}}_{\lambda }:=\dot{\gamma }(\lambda {)}^{\perp }\mathrm{/}\mathbb{R}\dot{\gamma }(\lambda ),$ with its positive metric and screen derivative ${\mathcal{D}}_{\lambda }$ from Null normal pairs and screen geometry. Define ${\mathcal{R}}_{\lambda }[X]:=[R(X,\dot{\gamma })\dot{\gamma }]\mathrm{.}$ A normal screen Jacobi field is a smooth section $J$ of $\mathcal{S}$ that satisfies ${\mathcal{D}}_{\lambda }^{2}J+{\mathcal{R}}_{\lambda }J=0$ and the initial conditions $J(0)=[X]$, ${\mathcal{D}}_{\lambda }J(0)=B_{0}X$ for some $X\in T_{p}S$. A focal instant ${\lambda }_0\in (0,L_1]$ is a parameter at which such a field with $X\mathrm{\ne }0$ has $J({\lambda }_0)=0$.

Put $I^{+}(S):={\bigcup }_{p\in S}{I}^{+}(p)$, $J^{+}(S):={\bigcup }_{p\in S}{J}^{+}(p)$, and $E^{+}(S):=J^{+}(S)\setminus I^{+}(S)$, using the causal-reachability conventions of Causal curves, chronological and causal reachability, and inextendibility.