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Transport two connecting vectors between neighbouring rays so that they commute with $U$, and project them onto the screen. Their infinitesimal transverse lengths and angles evolve according to the symmetric part of the optical tensor. The trace part gives uniform fractional dilation, while the trace-free null shear stretches one transverse direction and contracts the other, changing a circular beam to an ellipse without changing its area to first order. With zero null twist, there is no antisymmetric rotation of the beam.

If $q_{AB}$ is the metric on an infinitesimal transverse element, then $d{q}_{AB}/d\lambda =2{\hat{B}}_{(AB)}$. Its area $A$ therefore obeysThus positive null expansion means a beam's cross-sectional area increases, and negative expansion means it decreases; null shear records the directional distortion rather than this overall area change.

For the radial null pair above, the screen-space projector selects the angular tangent plane, with metric $q_{AB}=r^2{\gamma }_{AB}$. The ingoing covector is $U_a=-E(dv{)}_a$, and ${\Gamma }^v{}_{AB}=-r{\gamma }_{AB}$. Hence the optical tensor isIt is purely symmetric and proportional to the screen metric, givingFor $E=1$, the null expansion is $-2/r$. The area check $A\propto r^2$ and $dr/d\lambda =-E$ gives $A^{\prime }/A=-2E/r$, in agreement with the spherical null expansion in ingoing coordinates. In vacuum, $d\theta /d\lambda =-2E^2/r^2=-{\theta }^2/2$ also satisfies the Null Raychaudhuri equation; the ingoing beam focuses at the singularity rather than remaining future complete.

Choose the affine tangent $k^a$ to a null geodesic congruence, and an auxiliary null vector ${\ell }^a$ with $k\cdot \ell =-1$. The screen-space projectorannihilates $k$ and $\ell$ and restricts to a positive-definite two-dimensional metric in four spacetime dimensions. Neighboring transverse rays have relative displacement governed by the optical tensorIts irreducible decomposition defines the three requested optical quantities:The null expansion $\theta$ is the fractional rate of change of transverse area, $d\log A/d\lambda$. The null shear $\sigma$ changes a beam's shape at fixed area to first order; it is symmetric and trace-free. The null twist $\omega$ is antisymmetric and measures local rotation of the transverse rays; it vanishes for a hypersurface-orthogonal null geodesic congruence. In $D$ dimensions replace $1/2$ by $1/(D-2)$. These definitions use the trace convention for null expansion, rather than the alternative screen-averaged convention.

Take an affinely parametrized null geodesic congruence, ${\nabla }_{k}k=0$. Choose the auxiliary null vector and a transverse orthonormal screen basis to undergo parallel transport along each ray. For the unprojected tensor $C_{ab}={\nabla }_b{k}_a$, the product rule and the Ricci identity yieldHere the Riemann curvature tensor convention is the one in Question 1; its last-index antisymmetry converts the covector commutator to the displayed curvature term. Because $C_{ab}{k}^a=C_{ab}{k}^b=0$, inserting the screen-space projector in the product term adds only terms killed by these contractions. Parallel propagation makes differentiation commute with the screen projection. Therefore, in screen indices $A,B$,This is the optical matrix evolution equation. Its trace has curvature contraction $q^{ab}{R}_{acbd}{k}^c{k}^d=R_{cd}{k}^c{k}^d$: the additional terms in $q^{ab}-g^{ab}$ vanish by curvature antisymmetry. Write $B=(\theta /2)q+\sigma +\omega$ using the null expansion, null shear, and null twist. Symmetric--antisymmetric cross terms have zero trace, $\sigma$ is trace-free, andThe minus sign in the last trace comes from ${\omega }_{ab}=-{\omega }_{ba}$ and the positive screen metric. Taking the trace therefore derives the Null Raychaudhuri equation:For a nonaffine tangent obeying ${\nabla }_{k}k=\kappa k$, the same product-rule derivation adds $+\kappa \theta$ on the right; the projected derivative of $\kappa$ multiplies $k_a$ and vanishes. In $D$ dimensions the quadratic null expansion term is $-{\theta }^2/(D-2)$.

A null geodesic congruence is a smooth family of null geodesics filling a region without intersections there. For affine tangent $k$, the null expansion is the trace of the optical tensor: $\theta =d(\log A)/d\lambda$ for an infinitesimal pencil's area. Positive expansion means spreading; negative expansion means focusing.

For hypersurface-orthogonal generators in four dimensions, the Null Raychaudhuri equation readsThe null twist vanishes. The null energy condition implies $R_{ab}{k}^a{k}^b\ge 0$ through the Einstein field equations. An initial ${\theta }_0<0$ would force a focal point within affine distance at most $2/|{\theta }_0|$. A future-complete generator cannot develop such a point while remaining on an achronal event horizon. Thus $\boxed{\theta \ge 0}$ under the energy and global predictability/future-completeness hypotheses of Hawking's area theorem. Quantum energy-condition violations or failure of those global assumptions remove the conclusion.

The event horizon is the global boundary $\partial J^{-}({\mathcal{I}}^{+})$ and depends on the whole future. An apparent horizon on a chosen spacelike slice is the outermost marginally outer trapped surface, ordinarily ${\theta }_{\mathrm{out}}=0$, ${\theta }_{\mathrm{in}}<0$. It is quasi-local and slice-dependent; its tube need not be null.

For a shell arriving at $v=0$, the final horizon is $r=2M$. Before arrival, outgoing flat-space rays obey $dr/dv=1/2$. Tracing this horizon backward gives $r=2M+v/2$ until its beginning at $r=0$, $v=-4M$. No apparent horizon exists in that earlier flat region, explicitly illustrating the event horizon's dependence on future collapse.

The screen-space projector annihilates both null vectors: $P^a{}_b{U}^b=P^a{}_b{N}^b=0$, and satisfies $P^2=P$. Its image is the two-dimensional spacelike screen orthogonal to $U,N$. The screen metric isIt is positive definite on that screen. Projecting both indices of $B_{ab}={\nabla }_b{U}_a$ gives the optical tensor ${\hat{B}}_{ab}=P_a{}^c{P}_b{}^d{B}_{cd}$.

Define its three parts byThus ${\hat{B}}_{ab}=\frac{1}{2}\theta q_{ab}+{\sigma }_{ab}+{\omega }_{ab}$. The null expansion $\theta$ is its trace and equals $d\log \mathcal{A}/d\lambda$ for the infinitesimal beam area $\mathcal{A}$. The null shear $\sigma$ is symmetric and trace-free, measuring shape change at fixed first-order area. The null twist $\omega$, also called rotation, is antisymmetric and measures failure of screen directions to remain hypersurface-orthogonal. In $D$ dimensions replace $1/2$ by $1/(D-2)$. Here and below expansion means the trace, rather than its average over the screen dimensions.

A future trapped surface is a closed spacelike two-surface for which both future orthogonal null expansions are strictly negative. Multiplying a future null vector normal by a positive function multiplies its null expansion by that function, so the signs are invariant under allowed normalization changes.

Here ${\theta }_X<0$ for every $r_0>0$. Thus the Reissner-Nordstrom trapped spheres are exactly those with $f(r_0)<0$. Since $r_{+}>r_{-}>0$,At either horizon ${\theta }_Y=0$, so the spheres are marginal, rather than strictly trapped surfaces. For $r_0>r_{+}$ or $0<r_0<r_{-}$, ${\theta }_Y>0$ and this future-trapping condition fails. In particular, being inside the outer event horizon alone is insufficient to make every sphere trapped in the charged solution.

For each of the two normal directions, launch a null geodesic congruence orthogonally from the sphere with initial tangent $U=X$ or $U=Y$. For $U=X$ take initial auxiliary $N=Y$; for $U=Y$ take initial $N=X$, and then parallel transport $N$ as in part (a). On the sphere the resulting screen-space projector projects onto its angular tangent space, whose metric is $q_{AB}=r^2{\gamma }_{AB}$.

For any radial normal $K$, the spherical null expansion in ingoing coordinates follows directly from area variation:Equivalently $\mathcal{A}\propto r^2$ gives $K(\log \mathcal{A})=2K(r)/r$. ConsequentlySpherical symmetry also makes the null shear and null twist zero for these radial congruences.

There is a small parametrization distinction. The smooth field $X=-{\partial }_r$ is affine, since ${\Gamma }^a{}_{rr}=0$. The natural smooth field $Y={\partial }_v+(f/2){\partial }_r$ obeys ${\nabla }_{Y}Y=(f^{\prime }/2)Y$. To meet part (a)'s affine convention, use an affine rescaling of a null normal along the outgoing generators, with scaling equal to one on $S$. Its angular derivatives on $S$ then leave the projected derivative unchanged. The displayed null expansions are therefore precisely those for the affinely launched congruences, even though that convenient global expression for $Y$ is nonaffine away from its initial sphere.

For an affinely parametrized null geodesic congruence, put $B^a{}_b={\nabla }_b{U}^a$. Differentiating along a ray and commuting covariant derivatives givesusing $[{\nabla }_c,{\nabla }_b]V^a=R^a{}_{dcb}{V}^d$. The geodesic acceleration term vanishes. Since $U$ is null and affine, $B^a{}_b{U}^b=0$ and $U_a{B}^a{}_b=0$. Its trace is therefore the screen trace $\theta$, and the trace of its square equals that of the optical tensor squared. Taking the trace, with $R_{db}=R^a{}_{dab}$, yieldsNow decompose the optical tensor into null expansion, null shear and null twist. The symmetric trace-free part is orthogonal to the trace and to the antisymmetric part, while ${\hat{\omega }}_{ab}{\hat{\omega }}^{ba}=-{\hat{\omega }}_{ab}{\hat{\omega }}^{ab}$. HenceSubstitution proves the Null Raychaudhuri equation:The squares use the positive-definite screen metric. Thus null shear contributes to focusing, whereas nonzero null twist has the opposite sign.

A null geodesic congruence is a smooth family of null geodesics filling a region without intersections there, with one tangent direction assigned smoothly at every point. Choose an affine parameter $\lambda$ and tangent $U^a$, so $U^a{U}_a=0$ and $U^b{\nabla }_b{U}^a=0$. Choose a second null vector $N^a$ with $U\cdot N=-1$. The screen-space projector isIt annihilates $U,N$ and restricts to a positive-definite metric on the two-dimensional transverse screen. Project the covariant derivative of the tangent to obtain the optical tensorIts decomposition defines the three optical quantities:The null expansion is the fractional rate of change of a transverse area, $\theta =A^{-1}dA/d\lambda$. The null shear changes a beam's shape without changing its area to first order. The null twist is its antisymmetric transverse deformation. The factor $1/2$ uses four spacetime dimensions.

For the spherical null expansions in double-null coordinates, the cross-sectional area of a symmetry sphere is $A=4\pi r^2$. For the two affine null geodesic congruences, the definition of null expansion therefore givesThere is no extra derivative of $f$ in these formulas: the null expansion differentiates the transverse area, not the normalization of the tangent. Equivalently, the projected metric is $q_{AB}=r^2{\gamma }_{AB}$ and its optical tensor is $(k(r)/r)q_{AB}$, or $(\ell (r)/r)q_{AB}$ for the other congruence. It is purely trace, so both null shears vanish.

The null twist vanishes because the covector tangents are gradients of $-U$ and $-V$, and hence their exterior derivatives vanish. Each congruence is orthogonal to a family of null hypersurfaces. This also identifies the Frobenius theorem reason for zero null twist.

A spherical trapped surface has both future null expansions negative. Thus the initial sphere has $r_{,U}<0$ and $r_{,V}<0$. The result of the previous part already gives $r_{,U}<0$ at every regular future sphere in $D^{+}(\Sigma )$.

Apply the same Null Raychaudhuri equation to $k=f^{-1}{\partial }_V$. The exchanged-coordinate double-null focusing identity isAlong $U=U_0$, increasing $V$ is future-directed. ThereforeBoth null expansions remain negative, proving persistence of spherical trapping along this future null direction. As usual, the conclusion concerns every $V_1>V_0$ for which the sphere exists in the regular chart in $D^{+}(\Sigma )$; the inequality does not assert existence of such spheres beyond a focusing singularity.

The Physical-process first law for a rotating black hole, in four-dimensional general relativity with $G=1$, isIt applies to a small neutral matter influx into an initially stationary, nonextremal black hole that settles to another stationary state. The background surface gravity and angular velocity are $\kappa$ and ${\Omega }_H$, and $\delta E,\delta J$ are the Killing energy and angular momentum delivered through the event horizon. Other conserved charges are held fixed. The perturbation must be small enough that its horizon generators do not develop caustics, and all equalities below are to first order.

Let $t^a$ be the stationary Killing vector field normalized at infinity, and ${\psi }^a$ the axial Killing vector field with $2\pi$-periodic orbits. The horizon generator is ${\xi }^a=t^a+{\Omega }_H{\psi }^a$. Choose an affine tangent $k^a$ and parameter $\lambda$ with zero at the past stationary limiting section, so affine horizon-generator scaling gives, on the background horizonThe background null expansion and null shear vanish, and null twist vanishes for the hypersurface-orthogonal horizon generators. Linearizing the Null Raychaudhuri equation and using the Einstein field equations givesThe quadratic null expansion and null shear terms are second order. With background cross-sectional measure $d{A}_0$, the first-order area change is $\delta A={\int }_{\mathcal{H}}\delta \theta d\lambda d{A}_0$. Multiplying the evolution equation by $\lambda$ and integrating by parts gives the first-order horizon-area response to matter fluxThe boundary term $[\lambda \delta \theta ]$ vanishes: $\lambda =0$ at the past limiting section, and the final stationary boundary condition gives zero expansion in the settled future. Equivalently, assume a sufficiently localized influx with the required late-time decay.

The energy and angular-momentum fluxes have signsThese correspond to the particle conventions $E=-p\cdot t$ and $J=p\cdot \psi$. Their combination isThis proves the requested Physical-process first law of black-hole mechanics. For charged infall the corresponding law contains an additional horizon-potential term ${\Phi }_H\delta Q$; the energy-minus-angular-momentum version assumes that work term is absent.

A future trapped surface is a smooth compact spacelike two-surface without boundary whose two future-directed orthogonal null expansions are strictly negative. In four-dimensional general relativity, the Penrose singularity theorem states: a time-oriented globally hyperbolic spacetime with a noncompact Cauchy hypersurface, a trapped surface, and the null convergence condition $R_{ab}{k}^a{k}^b\ge 0$ for every null vector is future null-geodesically incomplete. With the Einstein field equations, the null energy condition implies this null convergence condition; a cosmological constant drops out of the null contraction.

The Kruskal spacetime is an example. In its black hole interior, use future null normals in Ingoing Eddington-Finkelstein coordinates:For a round sphere of areal radius $r$, its area is $4\pi r^2$ and its null expansions areBoth are negative for $0<r<2M$. The vacuum Einstein field equations give $R_{ab}=0$, and a two-ended Kruskal Cauchy hypersurface is noncompact. The future radial null geodesics reaching $r=0$ in finite affine parameter provide precisely the incompleteness predicted by the Penrose singularity theorem. A trapped surface at $r<2M$ is strictly trapped; the horizon sphere has one zero null expansion instead.