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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.

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.

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 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.

If $k_a={\nabla }_{a}S$ is a null gradient, symmetry of the Hessian gives $k^a{\nabla }_a{k}_b=k^a{\nabla }_b{k}_a=\frac{1}{2}{\nabla }_b(k^2)=0$. Its integral curves are therefore affinely parametrized null geodesics, rather than merely null curves with a nonaffinity term.

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.

A null pregeodesic has a null tangent and can be parametrized as a null geodesic. In a two-dimensional Lorentzian metric, the acceleration of a regular null curve is orthogonal to its tangent and therefore proportional to it.

The retarded coordinate $u=t-r_{\ast }$ puts the radial Schwarzschild metric into $d{s}^2=-fd{u}^2-2dudr+r^{2}d{\Omega }^2$, with $f=1-2M/r$. Outgoing radial null geodesics have constant $u$. These coordinates are regular at the past horizon; an infaller crossing the future horizon has $u\to +\infty$.