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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 the round sphere, every normal has only $v,r$ components because the Reissner-Nordstrom metric has no mixed angular terms. The vector $X=-{\partial }_r$ is normal and null: $g_{rr}=0$. Write the second null vector normal as $Y=A{\partial }_v+B{\partial }_r$. The normalization gives $X\cdot Y=-A=-1$, so $A=1$. Its null condition then reads $-f+2B=0$. ThusThe specified future orientation of $X$, together with $X\cdot Y<0$, makes $Y$ future-directed as well. This choice fixes the reciprocal scaling freedom of the two null vectors on the sphere.

One standard form of the Penrose singularity theorem assumes a time-oriented globally hyperbolic spacetime with a noncompact Cauchy hypersurface, the null convergence condition $R_{ab}{k}^a{k}^b\ge 0$ for every null vector $k$, and a closed future trapped surface. It concludes future null geodesic incompleteness: some future-inextendible null geodesic has a finite upper endpoint of its affine parameter.

The focusing mechanism is the Null Raychaudhuri equation. The normal generators have zero null twist, soAn initial ${\theta }_0<0$ therefore gives a conjugate point to a spacelike surface within affine distance at most $2/|{\theta }_0|$, assuming the generator can be continued that far. Such a generator ceases to lie on the achronal boundary after its first focal point. Compactness of the trapped surface supplies a uniform bound for all normalized initial null normals. Future completeness would consequently make its future boundary compact. Projection along timelike curves to a connected Cauchy hypersurface is injective on the achronal boundary and has open image. Compactness makes the image closed as well; the nonempty image must therefore be the entire hypersurface, contradicting its noncompactness. This explains why the global assumptions supplement local focusing.

For Reissner-Nordstrom spacetime, the Maxwell stress-energy tensor satisfies the null energy condition; the Einstein field equations imply the required null convergence condition. The spheres in the band just found are closed and trapped. Apply the theorem to a maximal Cauchy development with a noncompact Cauchy hypersurface and containing one such sphere. That globally hyperbolic spacetime must be future null-geodesically incomplete.

The Penrose theorem at a Cauchy horizon needs care: the full maximal analytic Reissner-Nordstrom spacetime has inner Cauchy horizons and is not globally hyperbolic. It does not satisfy every hypothesis of the displayed theorem. In the exact solution some incomplete geodesics of the globally hyperbolic development reach a smoothly extendible Cauchy horizon in finite affine parameter. The theorem asserts incompleteness of the development, not that every such endpoint is a curvature singularity. The separate curvature singularity at $r=0$ does not justify silently dropping the theorem's global hypothesis.

For radial motion, take an affine parameter $\lambda$ and use dots for $d/d\lambda$. The radial null geodesics of the Vaidya metric satisfy the two radial geodesic equationsThe radial squared norm is $-f{\dot{v}}^2+2\dot{v}\dot{r}$. If $v$ is constant it vanishes, and the geodesic equation reduces to $\ddot{r}=0$. Thus constant-$v$ radial curves are null geodesics, with $r=a\lambda +b$ an affine parametrization. Future ingoing motion has $a<0$.

For the other family let $dr/dv=f/2$. Its tangent $k={\partial }_v+(f/2){\partial }_r$ is a null vector. Using the Christoffel symbols of the Vaidya metric, direct differentiation givesHence these are also null geodesics, but $v$ is generally not affine. For completeness, put $p=dv/d\lambda$ and chooseThis makes $\ddot{v}=-(M/r^2){\dot{v}}^2$. Differentiating $\dot{r}=(f/2)\dot{v}$ then gives $\ddot{r}=(f_v/2){\dot{v}}^2=-M^{\prime }{\dot{v}}^2/r$; substitution verifies the radial $r$ equation as well. Thus the null condition and both radial geodesic equations hold, not just the null condition alone.

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.

In an orthonormal frame, $F_{0i}=-E_i$ and $F_{ij}={ϵ}_{ijk}{B}_k$. The electromagnetic stress-energy tensor givesThe sign of $T_{0i}$ follows directly from $F_{0j}{F}_i{}^j=-E_j{ϵ}_{ijk}{B}_k$. Thusand $T_{00}\ge 0$. These formulas hold in every local Lorentz frame, so the criterion just proved establishes the dominant energy condition. Equality occurs for a nonzero null Maxwell field with $E^2=B^2$ and $\mathbf{E}\cdot \mathbf{B}=0$, whose energy current is null vector.

The dominant energy condition says that, for every future timelike vector $u^a$, the energy current $J^a=-T^a{}_b{u}^b$ is future causal or zero. Choose normal coordinates whose future unit time vector at $p$ is $u^a=(1,0,0,0)$. With metric signature $(-,+,+,+)$ and symmetric stress-energy tensor,A vector with these components is future causal or zero exactly whenThis proves necessity. Conversely, every future unit timelike vector can be made the time axis of an orthonormal frame by a proper orthochronous Lorentz transformation, and that frame can be extended to normal coordinates at $p$. Requiring the displayed inequality in every such chart therefore gives the dominant energy condition for every future timelike vector; positive rescaling handles nonunit vectors. Future null vectors follow by continuity if they are included in the definition. The condition must hold in every local Lorentz frame; one chart alone is insufficient.

A null geodesic congruence is a smooth family of null geodesics filling a region without crossings there. Choose a smooth future-directed tangent $U^a$ and an affine parameter $\lambda$, soThe space orthogonal to $U$ is degenerate because it contains $U$ itself. To measure genuine transverse separation, introduce an auxiliary null vector $N^a$ with $U\cdot N=-1$ and the screen-space projectorIn four dimensions $q$ has a positive-definite two-dimensional image and annihilates $U$ and $N$. The optical tensor is the transverse part of the velocity derivative,Its irreducible decomposition defines the three optical quantities:The null expansion is the fractional rate of change of a small transverse area, $\theta =d\log \delta A/d\lambda$. The symmetric trace-free null shear distorts the shape of the transverse beam without changing its area at first order. The antisymmetric null twist, also called rotation of the congruence, describes the rotational part of its transverse deformation. These are properties of the screen quotient $U^{\perp }/⟨U⟩$; the auxiliary vector chooses a convenient representative of that quotient. For an affinely parametrized null geodesic congruence, the contraction constraints also give $\theta ={\nabla }_a{U}^a$.

Parallel transport the auxiliary null vector $N$ and a screen basis along each generator of the null geodesic congruence. This keeps the screen-space projector fixed under the corresponding transported derivative. Use the curvature convention $[{\nabla }_c,{\nabla }_b]V_a=-R^d{}_{acb}{V}_d$.

With $B_{ab}={\nabla }_b{U}_a$, differentiation of the affine geodesic equation givesHere the first term comes from commuting the derivative past $U^c$, and the second from the curvature commutator. The constraints $B_{ab}{U}^a=B_{ab}{U}^b=0$ follow respectively from the null normalization and affine geodesic equation. Projecting onto the screen therefore yields the optical matrix equationThe trace of the tidal matrix is $R_{ab}{U}^a{U}^b$; extra terms from replacing the screen trace by the spacetime trace vanish by curvature antisymmetry. Taking the trace givesDecompose the optical tensor as $\hat{B}=\frac{1}{2}\theta I+\hat{\sigma }+\hat{\omega }$. The null shear is symmetric and trace-free and the null twist is antisymmetric. Their cross traces vanish, andThe negative sign in the last term is the identity $\mathrm{tr}({\hat{\omega }}^2)=-{\sum }_{I,J}{\hat{\omega }}_{IJ}^2$ on the positive-definite screen. Substitution proves the four-dimensional Null Raychaudhuri equation:The coefficient is $1/(D-2)$ in $D$ spacetime dimensions; the factor $1/2$ here belongs to the four-dimensional congruence in this question, not the five-dimensional geometry of the next question.

Evaluate the metric inner products of the proposed orthonormal tetrad to first order in the vector cosmological perturbation:The orthonormal tetrad therefore has metric signature $(-,+,+,+)$ to the required order. Put $q=ϵ/a^2$ and $b=\mathbf{e}\cdot \mathbf{B}$. The photon momentum components areThey satisfy the null vector condition to first order, with $|\mathbf{e}|=1$.

Use $\eta$ as parameter in the time component of the geodesic equation:The derivative of the direction is first order, since the background photon moves on a straight comoving line; $B_{i}d{e}^i/d\eta$ is consequently second order. With $d\mathbf{x}/d\eta =\mathbf{e}+O(B)$,The supplied Levi-Civita connection coefficients giveThe spatial gradients and all background expansion terms cancel. Since ${ϵ}^{\prime }/ϵ$ vanishes in the background, its product with $b$ is also second order. The photon energy redshift from a vector metric perturbation is thereforeAn independent check uses the covariant component $p_0=g_{0\mu }{p}^{\mu }=-ϵ+O(B^2)$. The covariant geodesic equation gives $d{p}_0/d\eta =(2p^0{)}^{-1}{\partial }_{\eta }{g}_{\alpha \beta }{p}^{\alpha }{p}^{\beta }=ϵe^i{\dot{B}}_i$; the conformal expansion term vanishes by the null vector condition and reproduces the same result.

Let $U$ be the smooth nonzero tangent along a regular null curve, and $a={\nabla }_{U}U$. Metric compatibility gives $g(U,a)=\frac{1}{2}U[g(U,U)]=0$. In a two-dimensional Lorentzian manifold, the orthogonal complement of a nonzero null vector is its own span; this follows directly in the preceding null basis. Therefore $a=\kappa (t)U$.

Consequently every regular null curve is a geodesic up to reparametrization. To obtain an affine parameter $\lambda$, write $q=dt/d\lambda >0$ and solve $dq/dt+\kappa (t)q=0$. Then ${\nabla }_{qU}(qU)=q^2(\kappa +q^{-1}dq/dt)U=0$. A solution $q=C\exp (-\int \kappa dt)$ exists locally and along any nonsingular parameter interval. The original parameter need not be affine; the literal assertion of zero covariant acceleration for every parameter would be false.

At $p$, choose local normal coordinates with $g_{\alpha \beta }(p)=\mathrm{diag}(-1,1)$. Let $e_0,e_1$ be their coordinate basis. The vectors $V=e_0+e_1$ and $W=e_0-e_1$ have zero metric tensor square, and their component matrix has determinant $-2\ne 0$. Thus two independent null vectors form a basis of the tangent space. Normal coordinates are only needed at the chosen point, not on a whole neighbourhood with flat metric.

Set $u=t-r$ and $R=r$, soWriting now $\Sigma =R^2+a^2{\cos }^2\theta$ and $\Delta =R^2+a^2$, the Minkowski metric becomesThe squared norm of the coordinate vector is its $RR$ metric component:For $a>0$ this is nonzero away from the symmetry axis, so ${\partial }_R$ is not a null vector.

For a null vector, the perfect fluid in general relativity tensor givesbecause $g_{\alpha \beta }{V}^{\alpha }{V}^{\beta }=0$. For phantom energy, $\rho +P=(1+w)\rho <0$. A nonzero $u_{\alpha }{V}^{\alpha }$ therefore makes the contraction negative, so phantom energy violates the null energy condition.

Because $U$ is a null tangent, metric compatibility givesBecause $U$ is tangent to an affinely parametrized null geodesic congruence,by the geodesic equation. Thus $B$ has no component along $U$ in either index.

For radial motion, use the geodesic LagrangianThe Euler-Lagrange equation for $t$ gives $c^2\ddot{t}+a(da/dt){\dot{\chi }}^2=0$. Since $\chi$ is a cyclic coordinate, its conjugate momentum is constant. HenceFor a massive particle, take $\tau$ as proper time; the four-velocity normalization is $c^2{\dot{t}}^2-a^2{\dot{\chi }}^2=c^2$. For a photon, use an affine parameter rather than proper time, and the corresponding tangent is a null vector. ThereforeThe normalization of the null affine parameter remains arbitrary.

Use the Minkowski metric $\eta =\mathrm{diag}(1,-1,-1,-1)$ and coordinates $x^0=ct$. At a point, a smooth path is timelike if its tangent $\dot{x}$ is a timelike vector, here meaning ${\eta }_{ab}{\dot{x}}^a{\dot{x}}^b>0$. It is lightlike if its tangent is a null vector: the same expression is zero and the tangent is nonzero. In these coordinates, a future-directed timelike tangent has spatial speed less than $c$; a lightlike tangent has speed $c$.

For a massive particle on a timelike world line, the four-velocity is $U^{\mu }=d{X}^{\mu }/d\tau =\gamma (c,\mathbf{v})$, with $U^2=c^2$. Its four-momentum is $P^{\mu }=m{U}^{\mu }=(E/c,\mathbf{p})$ and its four-force is $F^{\mu }=d{P}^{\mu }/d\tau$. Here $\tau$ is proper time. A massless trajectory does not admit this massive-particle proper-time parametrization.

The asserted classification of arbitrary orthogonal vectors is false. With the chosen Minkowski metric, $D=(0,1,0,0)$ and $G=(0,0,1,0)$ are nonzero, orthogonal and both spacelike vectors. Also $D=(1,1,0,0)$ and $G=2D$ are orthogonal null vectors. The valid statement is that a nonzero vector orthogonal to a timelike vector is spacelike. To prove it, choose the rest frame of the timelike vector, $D=(d,0,0,0)$; then $D\cdot G=d{G}^0=0$ forces $G^0=0$, so $G^2=-|\mathbf{G}{|}^2<0$.

The relativistic energy-momentum relation givesTherefore $\boxed{P^2=0⟹m=0}$. A photon is an example: $E=c|\mathbf{p}|$. A collection of photons need not itself have zero invariant mass.

For nonzero $\chi$, the outer product $X=\chi {\chi }^{†}$ is rank-one Hermitian positive semidefinite, so $\det X=0$ and $ct={\chi }^{†}\chi /2>0$. It therefore represents a future-directed null vector. Conversely any nonzero future-null Hermitian $X$ has eigenvalues $2ct$ and zero, so its positive eigenvector multiplied by $\sqrt{2ct}$ supplies $\chi$. The phase $\chi \mapsto e^{i\zeta }\chi$ leaves $X$ unchanged. A past-directed null vector needs $X=-\chi {\chi }^{†}$, rather than the positive outer product alone.

For the printed normalized spinor, direct multiplication yieldsComparing with the printed $x+iy$ convention givesIts usual spatial azimuth is $-\varphi$, not $\varphi$. The source's asserted usual-angle identification has a sign inconsistency. Either conjugating the two spinor phases or replacing the upper-right matrix entry by $x-iy$ restores the usual azimuth $\varphi$. Neither alteration is made silently here. A further positive scale factor sets the magnitude of an arbitrary future-null vector; the displayed normalized spinor fixes $ct=1/2$ in the chosen coordinate units.

The Penrose singularity theorem states that a globally hyperbolic spacetime with a noncompact Cauchy hypersurface, a closed trapped surface, and the null convergence condition is future null-geodesically incomplete. Time reversal replaces the trapped surface by an anti-trapped surface and gives past incompleteness.

In his 1965 paper Gravitational Collapse and Space-Time Singularities, Roger Penrose used trapped surfaces and global causal structure to address whether nonspherical perturbations could avert a breakdown during gravitational collapse. Quasars supplied the opening astrophysical motivation. The paper used metric signature $(+,-,-,-)$ and a stronger timelike local-energy assumption; the statement above uses the modern null convergence condition.^[ref]

For this version, take a smooth connected four-dimensional spacetime with a time orientation. A Cauchy hypersurface meets every inextendible causal curve exactly once; its existence expresses global hyperbolicity. The hypersurface must be noncompact, meaning not compact; this is a global spatial condition, not an assumption about the size of the collapsing body. The null convergence condition is $R_{ab}{k}^a{k}^b\ge 0$ for every null vector $k$. Through the Einstein field equations it follows from the null energy condition; a cosmological constant drops out of the null contraction.^[ref]

The conclusion is that at least one future-inextendible null geodesic has a finite upper endpoint of its affine parameter. It neither asserts infinite mass density or divergent curvature at every endpoint, nor says that such an endpoint is visible. Locating an event horizon requires additional global information; the theorem is not a horizon-finding rule for arbitrary spacetimes. Weak cosmic censorship conjecture is an additional issue.^[ref]

The supermassive black hole account describes the motivating observational evidence from stellar orbits at the Galactic centre. Such measurements infer a compact gravitating mass. They do not observe a trapped surface, measure a geodesic's complete future, or experimentally prove the Penrose singularity theorem.