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A general scalar cosmological perturbation has four scalar metric functions: a lapse perturbation, a scalar shift, a spatial curvature perturbation and a scalar spatial shear. The two scalar functions in a cosmological gauge transformation can set the shift and shear to zero. For nondegenerate spatial modes this gives conformal Newtonian gauge, leaving the two potentials $\psi$ and $\varphi$. The scalar traceless spatial Einstein equation makes their difference proportional to scalar anisotropic stress. In general relativity, with no scalar anisotropic stress and the usual boundary conditions, $\psi =\varphi$. Pressure alone does not spoil equality; free-streaming relativistic species can do so through their anisotropic stress.

Write $\Psi =(\varphi +\psi )/2$, the Weyl lensing potential. A conformal rescaling leaves the unparameterized null geodesics unchanged, although the affine parameter changes. To first order, the conformal optical metric for scalar perturbations hasThe only nonzero connection coefficients at this order areHere spatial indices are raised with ${\delta }^{ij}$. An affine parameter $\lambda$ for this rescaled metric therefore obeysThe first line is also ${\eta }_{,\lambda \lambda }+2{\eta }_{,\lambda }d\Psi /d\lambda +2x_{,\lambda }^i{\eta }_{,\lambda }{\partial }_i\Psi =0$, since the derivative of the potential is along the ray.

Put $v^i=d{x}^i/d\eta$. The chain rule gives $x_{,\lambda \lambda }^i={\eta }_{,\lambda }^{2}d{v}^i/d\eta +v^i{\eta }_{,\lambda \lambda }$. Eliminating the affine acceleration givesThe extra spatial-gradient term inside the parentheses does not replace the spatial term already contained in the total derivative.

At reception, the null condition is $|v{|}^2=1+4{\Psi }_0$. With the specified unit propagation vector $e$, this impliesThe zero-order trajectory is $x^i=e^i(\eta -{\eta }_0)$. In every term already containing a first-order potential, we may substitute $v=e$. Defining the transverse projector $P_{\perp }^{ij}={\delta }^{ij}-e^i{e}^j$, the first-order geodesic equation becomesIntegrating with the reception position and velocity gives the first-order photon trajectory in scalar perturbations:Its first derivative at reception is $e^i+2e^i{\Psi }_0$, and differentiating twice gives the previous equation. For a first-order calculation, evaluating the integrals on the unperturbed ray is sufficient; shifting their evaluation points by the first-order displacement changes them only at second order. This is the Born approximation in cosmological lensing.

To fix the angular-deflection sign, define ${\chi }_s={\eta }_0-{\eta }_s>0$, $n_{\mathrm{obs}}=-e$, and $n_{\mathrm{src}}=x_s/|x_s|$. The latter is the unlensed coordinate direction to the source, while the former is its apparent direction measured by the observer. The longitudinal integral changes the radial distance, not the direction at this order. The transverse displacement isDividing by ${\chi }_s$ yieldsThis is the angular deflection kernel for cosmological lensing. It depends on the sum of the two metric potentials; setting them equal reduces it to the familiar twice-the-Newtonian-potential expression. With angular derivatives, the lensing potential is $2{\int }_0^{{\chi }_s}[({\chi }_s-\chi )/({\chi }_s\chi )]\Psi d\chi$, and its angular gradient is $\alpha$.

Gravitational lensing displaces apparent source positions and produces magnification and shear through angular derivatives of the deflection. Weak gravitational lensing of galaxy shapes probes the foreground mass distribution, including dark matter. Lensing also remaps the Cosmic microwave background, smoothing its acoustic features and converting part of its E-mode polarization into B modes. Strong deflections can produce multiple images, though a single first-order ray calculation does not describe that regime completely.

Let $u$ be the tangent vector to a geodesic with affine parameter $\lambda$. Its geodesic equation is ${\nabla }_{u}u=0$. For a Killing vector field $\xi$, metric compatibility givesThusThis geodesic conserved quantity from a Killing vector is independent of whether the geodesic is timelike, spacelike or null. For a timelike geodesic one can use proper time; for a null geodesic choose an affine parameter. An arbitrary nonaffine tangent rescales the displayed quantity and need not make it constant.

A Killing horizon is a null hypersurface with a Killing vector field $\xi$ as its normal and generator. Write $\xi =d/d\alpha$ along its orbits. These are null geodesics, but their Killing parameter need not be affine:The coefficient is the surface gravity for the chosen normalization; nondegeneracy means $\kappa \ne 0$. If $\lambda$ is affine and $q=d\lambda /d\alpha$, then $\xi =qd/d\lambda$ and $d(\log q)/d\alpha =\kappa$. For constant $\kappa$, $\lambda ={\lambda }_0+C{e}^{\kappa \alpha }$.

The Killing equation yields ${\partial }_{\mu }({\xi }^2)=2{\xi }^{\nu }{\nabla }_{\mu }{\xi }_{\nu }=-2{\xi }^{\nu }{\nabla }_{\nu }{\xi }_{\mu }$. On the horizon, $\boxed{{\partial }_{\mu }{\xi }^2=-2\kappa {\xi }_{\mu }}$. Multiplying $\xi$ by a positive constant multiplies $\kappa$ by that constant; physical temperature therefore needs a stated time normalization.

Locally write the null hypersurface as $u=0$, $du\ne 0$, with normal ${\ell }_{\mu }=f{\partial }_{\mu }u$, $f\ne 0$. A vector $V$ is tangent exactly when $V(u)=0$. Nullness gives $\ell (u)={\ell }^2/f=0$ on the hypersurface. Hence $\boxed{\ell \in T\mathcal{N}}$: its normal direction lies inside its tangent space.

Put $n=du$ and $q=n_{\mu }{n}^{\mu }$. Torsion freedom gives $n^{\mu }{\nabla }_{\mu }{n}_{\nu }=\frac{1}{2}{\partial }_{\nu }q$. Since $q$ vanishes on $\mathcal{N}$, all tangential derivatives vanish there and $dq=Cdu$ there for a scalar $C$. ConsequentlyThis is the unparametrized null geodesic equation. The integral curves are therefore generators of $\mathcal{N}$. An affine rescaling of a null normal removes the proportionality coefficient locally: if ${\nabla }_{\ell }\ell =\kappa \ell$, set $K=h\ell$ with $\ell (\log h)=-\kappa$.

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.

Use geometrized units and metric signature $(-+++)$, with $M>0$. In the exterior of Schwarzschild spacetime, put $f=1-2M/r$. The Schwarzschild tortoise coordinate satisfiesThe retarded and advanced null coordinates $u=t-r_{\ast }$ and $v=t+r_{\ast }$ then give $d{s}^2=-fdudv+r^{2}d{\Omega }^2$. The logarithmic divergence of $r_{\ast }$ suggests exponentiating these null coordinates. In the right exterior define the Kruskal–Szekeres coordinatesTheir product eliminates $t$:Since $du=-4MdU/U$ and $dv=4MdV/V$, the Schwarzschild metric becomesHere $r$ is an implicitly defined function of $UV$. The derivative of the right side with respect to $r$ is $-r{e}^{r/(2M)}/(4M^2)$, which is nonzero at $r=2M$. The inverse function theorem therefore makes $r(UV)$ smooth across that surface, and the coefficient of $dUdV$ tends to $-16M^2/e$. Thus the Schwarzschild event horizon is a coordinate singularity of the original chart, while this Lorentzian metric remains regular there.

Extend the Kruskal–Szekeres coordinates to all real $U,V$ with $UV<1$. The signs give two exterior regions, $U<0<V$ and $V<0<U$, a future black hole region $U,V>0$, and a past white hole region $U,V<0$. The event horizons are $U=0$ or $V=0$, intersecting at the bifurcation surface. The boundary $UV=1$ has $r=0$ and is a genuine Schwarzschild singularity, as the Kretschmann scalar $48M^2/r^6$ diverges there.

Finally, $T=(V+U)/2$ and $X=(V-U)/2$ give $UV=T^2-X^2$ and a radial metric proportional to $-d{T}^2+d{X}^2$. Hence radial null geodesics have slopes $\pm 1$, the event horizons are $T=\pm X$, and the singular boundaries are $T^2-X^2=1$. This constructs the maximal Kruskal extension; a black hole produced by collapse need not contain the second exterior or the white hole of that eternal extension.

Start with the four-dimensional Minkowski metric $d{s}^2=-d{t}^2+d{r}^2+r^{2}d{\Omega }^2$, where $r\ge 0$. Choose an arbitrary length $L>0$, and use the retarded and advanced null coordinates $u=t-r$, $v=t+r$. For the Minkowski conformal compactification, setBoth $p,q$ lie between $-\pi /2$ and $\pi /2$. Since $v\ge u$, we have $R\ge 0$; the remaining inequalities are $|T|+R<\pi$. Moreover,Multiply by the square of the conformal factor ${\Omega }_c=2\cos p\cos q/L$. The resulting metric is regular on the appropriate boundary pieces and preserves the null directions. Suppressing the angular two-spheres gives a triangular Penrose diagram with radial null geodesics at $45$ degrees.

The line $R=0$ is the ordinary timelike centre $r=0$. The upper sloping edge $T+R=\pi$ is future null infinity, reached with $v\to +\infty$ and finite $u$; the lower sloping edge $T-R=-\pi$ is past null infinity, reached with $u\to -\infty$ and finite $v$. The vertices $(T,R)=(\pi ,0)$ and $(-\pi ,0)$ are future and past timelike infinity, denoted $i^{+}$ and $i^{-}$. The vertex $(0,\pi )$ is spacelike infinity, $i^0$. These are limiting endpoints in the conformal completion, rather than ordinary physical events. In particular, finite diagram coordinates at null infinity do not imply finite physical affine parameter.

The four-dimensional radial diagram is the triangle $R\ge 0$, $|T|+R\le \pi$. If one instead draws two-dimensional Minkowski spacetime with a signed Cartesian spatial coordinate, the diagram is the full diamond. The centre is a boundary of the radial quotient, not a boundary of the physical four-dimensional Minkowski spacetime.

A null geodesic congruence is a smooth local family of null geodesics, with one generator through each point of the region being described. Choose an affine parameter $\lambda$ on each generator and write $U^a=d{x}^a/d\lambda$. Before a caustic, $U$ is a smooth, nonzero null vector field satisfying $U^a{U}_a=0$ and $U^b{\nabla }_b{U}^a=0$.

With $B^a{}_b={\nabla }_b{U}^a$, compatibility of the Levi-Civita connection with the metric givesThe geodesic equation with affine parameter gives the other contraction:Thus both contractions vanish. Nullness supplies the first identity, and affine parametrization supplies the second; a general nonaffine tangent would instead have $B^a{}_b{U}^b=\kappa U^a$.

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 null geodesics, the metric gives $dt/dr=\pm f^{-1}$ in the exterior. Define the tortoise coordinate byAn outgoing future radial ray has $dt/dr=f^{-1}$, so $u=t-r_{\star }$ is constant; an ingoing ray has $dt/dr=-f^{-1}$, so $v=t+r_{\star }$ is constant. These describe the null trajectories independently of the choice of affine parameter.

For the tortoise coordinate of a five-dimensional charged black hole, with $a=r_{+}$, $b=r_{-}$ and $\Delta =a^2-b^2$, partial fractions giveThus a useful explicit tortoise coordinate isThe logarithms signal failure of the static coordinates at the horizons, rather than a curvature singularity there.

The radial null geodesics satisfyThe ingoing family has $v=\text{constant}$, with $r$ decreasing toward the future; its tangent $-{\partial }_r$ is affinely parametrized because ${\Gamma }^a{}_{rr}=0$. The outgoing family hasAway from the Schwarzschild event horizon, integration gives $v=2r_{\ast }+\text{constant}$. These are null geodesics up to reparametrization: a null direction in the two-dimensional radial geometry is automatically pregeodesic, and the angular geodesic equations are satisfied by constant angles. On the Schwarzschild event horizon the outgoing family instead has $r=2M$, with tangent proportional to ${\partial }_v$.

In a Finkelstein diagram, plot $T=v-r$ vertically and $r$ horizontally. Ingoing rays obey $T+r=\text{constant}$; outgoing rays obeyOutside the Schwarzschild event horizon, outgoing rays increase $r$; on it they remain at $r=2M$; inside it they decrease $r$ even though $v$ increases. Thus both radial future null directions point toward smaller $r$ inside the black hole. Every future timelike direction lies between these null directions, so it also moves toward smaller $r$ there.

Arrows show future propagation. The vertical red line is the Schwarzschild event horizon; the black boundary at $r=0$ is the Schwarzschild singularity.

A spacetime is geodesically complete if every maximal geodesic has an affine parameter ranging over all of $\mathbb{R}$. For timelike geodesics this is equivalent to unbounded proper time in both directions. An extendible geodesic segment can be prolonged in the same spacetime; an inextendible geodesic cannot. A finite coordinate endpoint need not imply finite affine parameter.

For the Kruskal spacetime, use $U,V$ withIn the right exterior $U<0<V$. A truncated ray $U=U_0<0$, $V_1<V<V_2$ is an extendible geodesic of radial null type: neither artificial endpoint is a spacetime boundary. A future ray $U=U_0>0$ in the black hole reaches $UV=1$, hence $r=0$, and is inextendible geodesic and future null-geodesically incomplete. Its Killing energy $E\ne 0$ gives $dr/d\lambda =-|E|$, so the Schwarzschild singularity occurs at finite affine parameter. The maximal continuation toward the past supplies the other half of this same null geodesic.

There is no inextendible, complete radial timelike geodesic in positive-mass Kruskal spacetime. This requested example is impossible as printed. For a radial timelike geodesic, the conserved Killing energy and normalization giveIf $E^2<1$, $r$ has at most one turning point, a maximum $r_{\max }=2M/(1-E^2)$; a maximal trajectory runs from the white hole Schwarzschild singularity to the Schwarzschild singularity. If $E^2\ge 1$, there is no finite turning point; one end can lie at infinity but the other reaches $r=0$. The exceptional $E=0$ trajectory through the bifurcation surface also reaches $r=0$ in both time directions. Near $r=0$,whose integral is finite. Constant-$r$ radial timelike curves are accelerated, not geodesics.

Two plausible repairs have different meanings. Removing “radial” permits a complete circular timelike geodesic at $r=R>6M$, with nonzero angular momentum and proper time ranging over $\mathbb{R}$. Replacing “timelike” by “null” permits a complete horizon null geodesic: $U=0$, $V\in \mathbb{R}$, with $V$ an affine parameter. The Penrose diagram shows both repairs explicitly, together with the two valid requested examples; the circular trajectory is only a radial projection and is labelled as nonradial.

The horizontal boundaries are the Schwarzschild singularities, diagonal dashed lines are the Killing horizons, and outer diagonal edges are null infinity.

For a regular causal curve, orient its tangent $Z$ toward the future; reversing its parametrization does not change $d\varphi /dt$. Since $dt(Z)>0$, use $t$ as parameter and set $\Omega =d\varphi /dt$. The causal inequality givesBoth transverse terms are nonnegative in the exterior, so necessarilyBecause $g_{\varphi \varphi }>0$ and the discriminant is $4D>0$, this is equivalent toEquality requires $dr/dt=d\theta /dt=0$ and a null tangent. The limiting vectors are the two local azimuthal null directions; a curve following one at constant radius need not be a null geodesic. For a curve with radial or polar motion these angular bounds are necessary, while the full causal inequality can impose stricter bounds.

The second law of black-hole mechanics says that the total area of future event horizon cross-sections cannot decrease toward the future, assuming the classical Einstein field equations, the null energy condition, suitable global predictability and regularity, and here the stipulated future completeness of its generators.

A future event horizon is an achronal boundary generated by null geodesics. If a smooth generator had $\theta <0$, the null focusing theorem would produce a conjugate point to a spacelike surface within finite affine parameter. Future completeness lets the generator reach it, but beyond such a point it could no longer remain on the achronal boundary, a contradiction. Hence $\theta \ge 0$ wherever the null expansion is defined. A transported area element obeysEvery generator present on an early cut continues to a later cut without decreasing its area contribution. New generators can join at past endpoints, adding area; future endpoints in a regular spacetime are excluded. ThereforeThis argument also covers the sum of initially disconnected event horizons of the black holes in a merger. Nonsmooth crease sets require the standard regularity treatment; they do not provide a classical area-decrease loophole under these hypotheses. Quantum negative-energy flux can violate the classical null energy condition, so this is a classical black-hole area theorem, not an unconditional law for evaporating black holes.

Use an affine parameter $\lambda$ along a generator of a null hypersurface. Its null twist vanishes by hypersurface orthogonality. The screen metric is positive definite, so ${\hat{\sigma }}^{ab}{\hat{\sigma }}_{ab}\ge 0$. The Einstein field equations and null energy condition imply $R_{ab}{U}^a{U}^b\ge 0$, giving from the Null Raychaudhuri equationStarting with ${\theta }_0<0$, the null expansion stays negative as long as the congruence remains regular. Set $z=-1/\theta >0$. Then $z^{\prime }={\theta }^{\prime }/{\theta }^2\le -1/2$, soA finite negative null expansion cannot persist to $\lambda -{\lambda }_0=2/|{\theta }_0|$. Thus the null focusing theorem givesThis is a focal breakdown of the transverse congruence, provided the generator and regular geometry extend that far; it is not by itself a curvature singularity or a statement that the individual null geodesic terminates there.