& \Phi_{00}
\end{pmatrix}
+
\begin{pmatrix}
- {\rm Re}\,\Psi_0 & {\rm Im}\,\Psi_0 \\
  {\rm Im}\,\Psi_0 & {\rm Re}\,\Psi_0
\end{pmatrix}$$ with $$\label{ricweyl}
 \Phi_{00} \equiv -\frac12 R_{\mu\nu} k^\mu k^\nu,\quad
 \Psi_0 \equiv -\frac{1}{2} C_{\mu\nu\alpha\beta} \, \sigma^\mu
 k^\nu k^\alpha \sigma^\beta,$$ and where $\sigma^\mu \equiv s_1^{\mu} - {\rm i} \, s_2^{\mu}$.

### Notions of distance

Since the light beam converges at the observation event, we have $\xi^A(v=0)=0$. The linearity of the Sachs equation then implies the existence of a $2\times 2$ matrix ${\cal D}\indices{^A_B}$, called Jacobi matrix, such that $$\label{jacobimat}
\xi^A(v) = {\cal D}\indices{^A_B}(v) \left(\frac{{{\rm d}}\xi^B}{{{\rm d}}v}\right)_{v=0}.$$ From Eq. , we immediately deduce that this matrix satisfies the Jacobi matrix equation $$\label{gde2}
\frac{{{\rm d}}^2}{{{\rm d}}v^2}\,{\cal D}\indices{^A_B}={\cal R}\indices{^A_C} \, {\cal D}\indices{^C_B},$$ with initial conditions $$\label{gde2b}
{\cal D}\indices{^A_B}(0) =0,
\qquad
\frac{{{\rm d}}{\cal D}\indices{^A_B}}{{{\rm d}}v}(0)=\delta^A_B.$$ We shall also use the short-hand notation $\boldsymbol{\xi}=(\xi^A)$ and $\boldsymbol{\mathcal D}=({\cal D}\indices{^A_B})$ so that Eq. (\[gde2\]) reads ${{\rm d}}^2\boldsymbol{\mathcal D}/{{\rm d}}v^2=\boldsymbol{\mathcal R}\cdot\boldsymbol{\mathcal D}$, with $\boldsymbol{\mathcal D}(0)=0$ and $\boldsymbol{\dot{\mathcal D}}(0)=\boldsymbol{1}$.

Since the Jacobi matrix relates the shape of a light beam to its “initial” aperture, it is naturally related to the various notions of distance used in astronomy and cosmology. The *angular distance* $D{_{\text{A}}}$ is defined by comparing the emission cross-sectional area ${{\rm d}}^2 S_{\rm 