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For smooth spatial fields $E^i,C^i{}_j$, define a torsion-free connection on $\mathbb{R}\times {\mathbb{R}}^n$ by ${\Gamma }^i{}_{00}=-E^i$, ${\Gamma }^i{}_{0j}={\Gamma }^i{}_{j0}=-C^i{}_j/2$ and all other coefficients zero. The time geodesic equation is $\ddot{t}=0$. For $\dot{t}\ne 0$, use $t$ as affine parameter; the spatial geodesic equation becomes $d^2{x}^i/d{t}^2=E^i+C^i{}_{j}d{x}^j/dt$. In dimension three, $C^i{}_j=2{ϵ}^i{}_{kj}{B}^k$ realizes $\ddot{\mathbf{x}}=\mathbf{E}+2\mathbf{B}\times \dot{\mathbf{x}}$. Such a connection need not preserve a metric; a spacetime description of these trajectories only needs affine geometry.

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.

For physical four-momentum $P^{\mu }=md{x}^{\mu }/d\tau$, the covariant-component equation is $d{P}_{\mu }/d\tau =({\partial }_{\mu }{g}_{\nu \sigma })P^{\nu }{P}^{\sigma }/(2m)$. The equation without $1/m$ instead uses a tangent parametrization, or the affine parameter $\lambda =\tau /m$ when the tangent is the physical momentum.

For $d{s}_5^2={\eta }_{ab}d{x}^{a}d{x}^b+(dy+A_{a}d{x}^a{)}^2$, with $A$ independent of $y$, an affine parameter $\lambda$ makes $\xi =dy/d\lambda +A_{a}d{x}^a/d\lambda$ and the tangent norm $\mathcal{N}=g_{AB}{\dot{x}}^A{\dot{x}}^B$ constant. The four-dimensional interval satisfiesConsequently the formula $d{\lambda }^2=d{u}^2/(1-{\xi }^2)$ assumes the additional normalization $\mathcal{N}=1$; affine parametrization alone does not set this value. With base metric signature $+---$ and $u=c\tau$, the projected curve is timelike only for $\mathcal{N}>{\xi }^2$. The normalized case gives $d\tau /d\lambda =\sqrt{1-{\xi }^2}/c$. The positive fibre term and the $+---$ base together have two positive directions; the same sign convention must be retained when interpreting the reduction.

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.

For nonzero angular constant $h$, the interior null radial equation gives ${\dot{r}}^2\sim 2\mu h^2/r^3$. Its inward branch integrates to $r^{5/2}\sim \frac{5}{2}\sqrt{2\mu }|h|({\lambda }_{\ast }-\lambda )$. Thus the singularity is at finite affine parameter and $r\propto ({\lambda }_{\ast }-\lambda {)}^{2/5}$. The zero-angular-momentum case has a different affine scaling.

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.

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.

For an unbound equatorial Schwarzschild spacetime geodesic, let $E_{\infty }$ and $L_{\infty }$ be its conserved energy and angular momentum measured relative to infinity. The ratio $J=c^2{L}_{\infty }/E_{\infty }$ has dimensions of area divided by time. For a massive test particle, $E_{\infty }=m{c}^2{\gamma }_{\infty }$ and $L_{\infty }=m\ell$, so $J=\ell /{\gamma }_{\infty }=b{v}_{\infty }$, where $\ell$ is specific angular momentum and $b$ the impact parameter. For a photon, $E_{\infty }=c{p}_{\infty }$ and $L_{\infty }=b{p}_{\infty }$, giving $J=bc$ without assigning a finite Lorentz factor or a proper time to the photon. This ratio removes the arbitrary normalization of a null affine parameter.

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.

An extendible geodesic is a geodesic segment that can be continued beyond at least one endpoint within the same manifold after retaining an affine parameter. An artificial truncation at a regular event is extendible even when its continuation eventually reaches a singularity.

A geodesic is a curve whose tangent vector is parallel along itself under the Levi-Civita connection. With an affine parameter it has zero covariant acceleration:This is the Riemannian manifold version of moving straight ahead with constant speed. Its coordinate form is the geodesic equation. In Euclidean space it becomes $\ddot{\gamma }=0$, giving straight lines. On an embedded surface with its induced metric, the ordinary acceleration of such a curve is normal to the surface: all acceleration is used to follow the surface, rather than to turn within it.

On a unit round sphere, the great circle $\gamma (t)=(\cos t,\sin t,0)$ has $\ddot{\gamma }(t)=-\gamma (t)$, which is normal to the sphere. Two points at $t=0$ and $t=\pi /3$ can be joined along that circle by the short arc of length $\pi /3$ or the long arc of length $5\pi /3$. Each admits a constant-speed geodesic parametrization, yet only the short arc realizes their great-circle distance.

Every geodesic minimizes arc length on sufficiently short subarcs. A long segment can cease to be globally minimizing, as the sphere example shows; the first loss from a starting point is described by the cut locus. Thus “locally straight” and “shortest route between these endpoints” are distinct assertions. On a circular cylinder, different winding gives the same distinction, made explicit by Riemannian distance on a circular cylinder.

This provides a useful division of work. The geodesic equation predicts a path from an initial point and tangent vector; comparing arc lengths decides which candidate solves a global shortest-route problem. Variations of energy of a curve give the equation as an Euler-Lagrange equation, offering a proof outline and a connection with classical mechanics. ^[ref].

Each Killing vector field supplies a conserved quantity for freely moving test particles: it turns a geometric symmetry into a first integral of the geodesic equation. Let $\gamma (\lambda )$ be a geodesic with affine parameter $\lambda$, tangent vector $U^{\mu }=d{x}^{\mu }/d\lambda$, and ${\nabla }_{U}U=0$. Wherever a Killing vector field $K$ is defined along the geodesic,The statement applies to Riemannian manifolds and to timelike geodesics, null geodesics and spacelike geodesics in a Lorentzian manifold. It requires an affine parameter; for a massive test particle, proper time is one choice. See Tong, section 4.3.2, equation (4.32).

For a worked example in the Euclidean plane, take $\gamma (t)=(1+t,2-t)$ with affine parameter $t$ and velocity $U=(1,-1)$. The Euclidean Killing vector fields for translation and rotation giveThe rotational field along the geodesic is $K_{\mathrm{rot}}(\gamma (t))=(t-2,1+t)$: it changes from $(-2,1)$ at $t=0$ to $(-1,2)$ at $t=1$, yet its pairing with $U$ stays $-3$. With toy mass $m=1$ and $t$ interpreted as time, the first constant is the $x$ component of linear momentum, and the second is the signed $z$ component of angular momentum about the origin. Their numerical values use toy units. The geodesic is straight; the rotation flow orbits are circles.

This conservation law follows from metric compatibility and the Killing equation:The geodesic equation kills the acceleration term, and the symmetric product $U^{\mu }{U}^{\nu }$ selects the vanishing symmetric derivative. This tensor contraction is the whole proof outline; see Carroll, equation (5.43).

In a stationary spacetime, a chosen time-translation Killing vector field $K$ gives Killing energy. With metric signature $(-+++)$, $c=1$, and a massive test particle's four-momentum $p^{\mu }=m{U}^{\mu }$, the physical normalization isHere $\Phi$ generates the axial rotational symmetry when one exists. In coordinates with $K={\partial }_t$ and $\Phi ={\partial }_{\varphi }$, these are $E=-p_t$ and $L_z=p_{\varphi }$. Both are conserved quantities along the geodesic, even if the metric has a time-angle cross term. A future timelike $K$ makes $E$ positive for a future massive particle; the scale of $E$ depends on the normalization of $K$. Not every spacetime has a global timelike Killing vector field, and a stationary field can become spacelike in an ergoregion of a stationary spacetime. Conservation on the field's domain does not imply a universal global positive energy; see Tong, sections 4.3.1-4.3.2 on the timelike-domain qualification and conserved charges.

A complete inertial timelike geodesic in Minkowski spacetime has $\mathbf{x}=\mathbf{b}+\mathbf{v}t$ with $|\mathbf{v}|<1$. Under $p=\arctan (t-r)$, $q=\arctan (t+r)$, both $p,q$ tend to the same endpoint, giving past timelike infinity and future timelike infinity. A complete null geodesic has one finite limiting null coordinate at each end and reaches past null infinity and future null infinity. Complete spacelike geodesics have $r>|t|$ asymptotically and both ends approach spacelike infinity. The endpoints have finite compactified coordinates but occur at infinite physical affine parameter. A conformal transformation preserves null paths as unparametrized geodesics; it does not generally preserve timelike or spacelike geodesic paths.

Let $\tau$ be an affine parameter for a geodesic of $\nabla$ and suppose ${\bar{\nabla }}_{X}Y-{\nabla }_{X}Y=V(X)Y+V(Y)X$. Then ${\bar{\nabla }}_{\dot{\gamma }}\dot{\gamma }=2V(\dot{\gamma })\dot{\gamma }$. For the new velocity $d\gamma /dt=\dot{\gamma }/t^{\prime }$, the acceleration is $(t^{\prime }{)}^{-2}[{\bar{\nabla }}_{\dot{\gamma }}\dot{\gamma }-(t^{\prime \prime }/t^{\prime })\dot{\gamma }]$, which vanishes under the displayed differential equation. Its nonzero solution $t^{\prime }=C\exp (2\int V(\dot{\gamma })d\tau )$ changes the parameter while preserving the curve.

For smooth stationary fields on ${\mathbb{R}}^3$, set ${\Gamma }^i{}_{tt}=-E^i$ and ${\Gamma }^i{}_{tj}={\Gamma }^i{}_{jt}={ϵ}^i{}_{jk}{B}^k$, with all other Christoffel symbols zero. This torsion-free connection makes the equation $\ddot{\mathbf{x}}=\mathbf{E}+2\mathbf{B}\times \dot{\mathbf{x}}$ a geodesic equation with time as affine parameter. Arbitrary smooth fields suffice; a metric realization imposes additional potential conditions.

For smooth spatial $\varphi$ and $g(c)=-c^2{e}^{2\varphi /c^2}d{t}^2+{\delta }_{ij}d{x}^{i}d{x}^j$, the only nonzero Christoffel symbols are ${\Gamma }^t{}_{ti}={\Gamma }^t{}_{it}=c^{-2}{\partial }_i\varphi$ and ${\Gamma }^i{}_{tt}=e^{2\varphi /c^2}{\partial }_i\varphi$. Consequently the Levi-Civita connection converges smoothly on compact sets as $c\to \infty$ although the metric itself diverges in these coordinates. The limiting torsion-free connection has only ${\Gamma }^i{}_{tt}={\partial }_i\varphi$ and gives $d^2{x}^i/d{t}^2=-{\partial }_i\varphi$ when $t$ is an affine parameter. With $R^a{}_{bcd}={\partial }_c{\Gamma }^a{}_{db}-{\partial }_d{\Gamma }^a{}_{cb}+\cdots$, its Ricci tensor is $(\Delta \varphi )dt\otimes dt$: quadratic connection terms vanish and $R^i{}_{tjt}={\partial }_j{\partial }_i\varphi$. Thus zero Ricci tensor is the Laplace equation, while full flatness would require the entire Hessian to vanish.

For an equatorial null geodesic in Schwarzschild spacetime, let $k=(1-2\mu /r)dt/d\lambda$ and $h=r^{2}d\varphi /d\lambda$ be the conserved quantities for an affine parameter $\lambda$. The null metric condition gives ${\dot{r}}^2/2+V_{\mathrm{eff}}=c^2{k}^2/2$. For $h\ne 0$, the effective potential is zero at the Schwarzschild event horizon, positive outside, and tends to zero at infinity. Its unique exterior maximum is at $r=3\mu$, with value $h^2/(54{\mu }^2)$ and second derivative $-h^2/(81{\mu }^4)$. This maximum gives the unstable photon sphere.

The impact parameter of an incoming ray is $b=|h|/(ck)$ for $k>0$. Capture requires its radial energy to exceed the maximum, namely $b<3\sqrt{3}\mu$. At equality the incoming ray approaches the circular orbit asymptotically; it does not cross the horizon. For $h=0$, the potential is identically zero and the ingoing radial ray is captured.

Null geodesic incompleteness means that some maximal geodesic with a null tangent has a finite endpoint of its affine parameter. This can indicate a removable missing region or a genuine obstruction; it does not alone assert curvature divergence.

Null infinity is the ideal boundary reached by asymptotically escaping null geodesics in a suitable conformal completion. It has future and past components. A finite location on a Penrose diagram need not represent a finite affine parameter.