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The Null Raychaudhuri equation is the light-ray version of the Raychaudhuri equation associated with Amal Kumar Raychaudhuri. His original work concerned relativistic cosmology.^[ref] For the normal null geodesic congruence before its first focal point, null twist vanishes. With an affine parameter $\lambda$, the four-dimensional equation reduces to the displayed ordinary differential equation. The null shear contraction is nonnegative on the transverse screen space, and the null convergence condition makes the other part of $Q$ nonnegative. Consequently ${\theta }^{\prime }\le -{\theta }^2/2$.

For $\theta (0)={\theta }_0<0$, put $w=-1/\theta >0$. ThenA regular negative null expansion therefore cannot persist beyond $\lambda =2/|{\theta }_0|$ if the null geodesic extends that far. The null focusing theorem turns this estimate into a focal-point bound. The number refers to an affine parameter, not the proper time of a light ray, which is zero.

For an original soluble model, prescribe constant $Q$ along one twist-free beam; this does not specify an entire solution of the Einstein field equations. Introduce a positive affine scale $L$ and the dimensionless variablesFor an infinitesimal area $A$ define $a=\sqrt{A/A_0}$ before the first zero. Since $y=2a^{\prime }/a$, where primes now mean $d/du$, the same model becomesTwo independently checkable solutions areThe nonnegative extra focusing term moves the first zero earlier. Direct substitution into both ordinary differential equations verifies the solutions; numerical integration of the nonlinear equation provides a separate comparison with the linear beam equation.

A focal point makes this beam description degenerate, but individual null geodesics may continue smoothly through it. Even converging light in Minkowski spacetime can form a caustic. Converting local focusing into null geodesic incompleteness requires the global step below.

If $\tilde{g}={\Lambda }^{2}g$, a null geodesic with affine parameter $\tau$ for $g$ has affine parameter $\tilde{\tau }$ for $\tilde{g}$ when $d\tilde{\tau }/d\tau =C{\Lambda }^2$ along the curve. Indeed the conformal change of the Levi-Civita connection gives ${\tilde{\nabla }}_{V}V=2V(\log |\Lambda |)V$; rescaling $V$ by $d\tau /d\tilde{\tau }$ cancels this tangential acceleration. Constant $C>0$ preserves the direction of parametrization. Agreement of unparametrized null geodesics does not imply agreement of their affine parameters or completeness.

A spherical surface with areal radius $R>3m$, where $m=GM/c^2$, has apparent angular radius $\psi$ for a static exterior observer at $r_0>R$. Its grazing null geodesic has impact parameter $b=R/\sqrt{1-2m/R}$. Projecting the photon tangent into the observer's local rest frame gives $\sin \psi =b\sqrt{1-2m/r_0}/r_0$, yielding the displayed result. The condition $R>3m$ ensures the grazing ray escapes, since the surface is outside the photon sphere. For $R=4m$, the distant image radius is $\sqrt{2}$ times the flat-space prediction.

In a first-order light-deflection calculation, evaluate the Weyl lensing potential and its gradients on the unperturbed null geodesic. The position correction is first order, so inserting it into an already first-order potential produces a second-order correction. This approximation omits repeated deflection and lens-lens coupling.

The conformal time $T=2\arctan (e^t)-\pi /2$ satisfies $\sin T=\tanh t$ and $\cos T=\mathrm{sech}t$. Multiplying the physical metric by ${\cos }^{2}T$ gives a finite conformal compactification with periodic spatial coordinate. Its past and future conformal boundaries are spacelike circles. Null geodesics have slopes $d\chi /dT=\pm 1$.

Let $X$ be a Conformal Killing vector field with conformal Killing factor $\lambda$, and let $U$ be the tangent vector of a geodesic with affine parameter $s$. The product rule, metric compatibility and geodesic equation giveEvery null geodesic therefore has the conserved quantity $g(X,U)$. On a timelike geodesic with proper time as parameter, the derivative is $-\lambda$ in metric signature $(-,+,+,+)$. With a nonaffine parameter satisfying ${\nabla }_{U}U=fU$, an extra term $fg(X,U)$ appears; the conserved expression for a null geodesic is tied to its affine parameter normalization.

Rescale the perturbed cosmological metric by $a^{-2}(1+2\varphi )$. To first order, its spatial part is Euclidean and only its lapse is perturbed. The unparameterized null geodesics are unchanged. The nonzero linear connection coefficients are ${\hat{\Gamma }}_{00}^0=2{\partial }_{\eta }\Psi$, ${\hat{\Gamma }}_{0i}^0=2{\partial }_i\Psi$ and ${\hat{\Gamma }}_{00}^i=2{\partial }^i\Psi$. The affine parameter is that of the rescaled metric, not necessarily that of the original spacetime.

The difference of the two Levi-Civita connections is ${\tilde{\Gamma }}^a{}_{bc}-{\Gamma }^a{}_{bc}={\delta }_b^a{\partial }_c\log \Omega +{\delta }_c^a{\partial }_b\log \Omega -g_{bc}{g}^{ad}{\partial }_d\log \Omega$. Along a null geodesic with tangent $k$, the last term vanishes and the remaining acceleration is $2k(\log \Omega )k$. It is parallel to the tangent, so changing the affine parameter removes it. Thus a positive conformal factor preserves unparametrized null geodesics, although it generally does not preserve timelike or spacelike geodesics.

On the universal cover, $-\pi /2<\psi <\pi /2$ and $t\in \mathbb{R}$. The two conformal boundaries are timelike. Null geodesics are $t\pm \psi =\text{constant}$ and can travel from either boundary to the center in finite coordinate time, despite taking infinite physical affine parameter to reach the boundary. There is no event horizon for the complete central static observer. The spacetime is not globally hyperbolic, unlike the global conformal cylinder of two-dimensional de Sitter spacetime; well-posed field evolution requires boundary conditions as well as initial data.

A conjugate point to a spacelike surface occurs where normal geodesics develop a focal degeneracy: the transverse Jacobi field map from initial points on the surface ceases to be invertible. Collapse of the transverse area makes the null expansion diverge. Beyond a first such point, the normal null geodesic cannot continue as a generator of the achronal boundary of the surface's future.

For one conserved constant-$w$ perfect fluid in a flat expanding FLRW metric, $H(z)=H_0(1+z{)}^{3(1+w)/2}$. Integrating the radial null geodesic giveswhen $w\ne -1/3$. Its continuous coasting limit is $D_L=c(1+z)\log (1+z)/H_0$. Expansion gives $D_L=(c/H_0)[z+(1-3w)z^2/4+O(z^3)]$, agreeing with the deceleration parameter $q_0=(1+3w)/2$. Curvature, additional independently evolving fluids or a varying $w$ change this single-fluid expression.

A stationary time-coordinate change $t^{\prime }=t+F(r)$ leaves both the areal radius and the time-translation Killing vector field unchanged. The conserved Killing energy and angular momentum of a null geodesic therefore yield the same radial equation in either chart, ${\dot{r}}^2+h^2(1-2\mu /r)/r^2=c^2{k}^2$. The effective potential and radial turning points agree, even when the coordinate time velocities and horizon regularity differ.

In an Einstein-de Sitter universe normalized by $a_0=1$, the Friedmann equation gives $H(z)=H_0(1+z{)}^{3/2}$. Integrating the radial null geodesic gives $r(z)=c{\int }_0^{z}d{z}^{\prime }/H(z^{\prime })=2(c/H_0)[1-(1+z{)}^{-1/2}]$. The luminosity distance is $(1+z)r(z)$, yielding the displayed expression. Near the observer it is $(c/H_0)[z+z^2/4+O(z^3)]$; the limiting radial comoving distance is finite even though the luminosity distance diverges.

For two comoving observers in an Einstein-de Sitter universe, light observed at redshift $z$ today fixes their separation through the past null geodesic. A newly sent signal needs the same comoving distance for every one-way leg. Since $a=(t/t_0{)}^{2/3}$, each leg adds a fixed conformal time interval, producing the displayed reception time after $n$ legs when replies are immediate. At $z=1$ the first round-trip wait is $[(3-\sqrt{2}{)}^3-1]t_0$. There is no finite count limit because future conformal time diverges, though infinitely many legs require infinite cosmic time.

For an equatorial null geodesic of the Kerr metric, $E=-p_t$ and $L=p_{\varphi }$ are geodesic conserved quantities from Killing vectors. In units $G=c=1$, $\Delta =r^2-2Mr+a^2$, and an affine parameter gives the displayed equation. It follows by solving the stationary and axial momentum equations for $\dot{t},\dot{\varphi }$ and substituting them into the null condition. A zero of its radial polynomial is only a turning point; an exterior circular null geodesic also requires the derivative of that polynomial to vanish with $E,L$ held fixed.

Radial outgoing null geodesics in the interior of Oppenheimer-Snyder collapse obey $d\chi /d\eta =1$. Trace the ray crossing the boundary at $({\eta }_H,{\chi }_0)$ backward: it starts at the centre at $\eta ={\eta }_H-{\chi }_0$. This ray is the event horizon; it can exist inside the cloud before the surface crosses $R=2M$. An event horizon is determined by global escape to future null infinity, rather than by a local stopwatch signal.

For $f=1-2M/r$ and squared areal radius $C=r(r-b)$, an equatorial null geodesic obeys ${\dot{r}}^2=E^2[1-B^{2}f/C]$, with impact parameter $B=L/E$. At $b=2M$, $f/C=1/r^2$ in the exterior. Its supremum is $1/(4M^2)$ at the null singular boundary, rather than at a regular exterior photon sphere. Rays with $B>2M$ turn at $r=B$; rays with $B<2M$ reach the boundary. Assuming those rays are absorbed gives the displayed geometric-optics cross-section despite zero limiting horizon area.

For a captured nonradial null geodesic with $u=1/r$, the large-$u$ orbit equation is $(du/d\varphi {)}^2\sim 2\mu u^3$. Integration on the inward branch givesThe curvature singularity is reached at finite azimuth. This does not prevent a near-critical ray from winding repeatedly near the photon sphere before its final plunge. A radial ray has constant azimuth, so this asymptotic coordinate parametrization is inapplicable to it.

For reception at the origin with unit propagation direction $e$, the null geodesic condition fixes $v({\eta }_0)=(1+2{\Psi }_0)e$. The perturbative spatial acceleration is $x^{i\prime \prime }=2e^{i}d\Psi /d\eta -2({\delta }^{ij}-e^i{e}^j){\partial }_j\Psi$. Its integrated trajectory is $x^i=e^i(\eta -{\eta }_0)+2e^i{\int }_{{\eta }_0}^{\eta }\Psi d{\eta }^{\prime }-2{\int }_{{\eta }_0}^{\eta }(\eta -{\eta }^{\prime })({\delta }^{ij}-e^i{e}^j){\partial }_j\Psi d{\eta }^{\prime }$. Potentials in these already first-order integrals can be evaluated on the straight background ray.

The future horismos is the part of the causal future outside the chronological future. For a compact set in a globally hyperbolic spacetime, it equals the boundary of the chronological future. Its normal null geodesic generators cannot have passed a conjugate point to a spacelike surface when the initial set is that surface.