Null normal pairs and screen geometry

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.