$$\label{def.z}
1+z =\frac{u{_{\text{em}}}^\mu k_\mu(v{_{\text{em}}})}{u{_{\text{obs}}}^\mu k_\mu(0)},$$ where $v$ is an affine parameter along the geodesic, so that $k^\mu={{\rm d}}x^\mu/{{\rm d}}v$, and $v=0$ at the observation event. The wave four-vector can always be decomposed into temporal and spatial components, $$\label{dec-k}
k^\mu =  (1+z)(u^\mu - d^\mu),\quad d^\mu u_\mu =0 ,\quad d^\mu d_\mu=1,$$ where $d^\mu $ denotes the spatial direction of observation. In Eq. , we have chosen an affine parameter adapted to the observer, in the sense that $2\pi \nu_0 = u{_{\text{obs}}}^\mu k_\mu(0)=1$. This convention is used in all the remainder of the article.

Light beams
-----------

### Geodesic deviation equation {#subsubsec:gde}

A light beam is a collection of light rays, that is, a bundle of null geodesics $\{x^\mu(v,\gamma)\}$, where $\gamma$ labels the curves and $v$ is the affine parameter along them. The relative behavior of two neighboring geodesics $x^\mu(\cdot,\gamma)$ and $x^\mu(\cdot,\gamma+{{\rm d}}\gamma)$ is described by their separation vector $\xi^\mu \equiv {{\rm d}}x^\mu/{{\rm d}}\gamma$. Hence, this vector encodes the whole information on the size and shape of the bundle.

Having chosen $v=0$ at the observation event—which is a vertex point of the bundle—ensures that the separation vector field is everywhere orthogonal to the geodesics, $k^\mu\xi_\mu=0$. In such conditions, the evolution of $\xi^\mu$ with 