source}$ of a source to the solid angle ${{\rm d}}\Omega_{\rm obs}^2$ under which it is observed, $${{\rm d}}^2 S_{\rm source}=D{_{\text{A}}}^2 \, {{\rm d}}\Omega_{\rm obs}^2.$$ It is related to the Jacobi matrix by $$\label{defDA}
D{_{\text{A}}} = \sqrt{\vert{\rm det}\,{\boldsymbol{\mathcal D}}(v_{\rm source})\vert},$$ where $v{_{\text{source}}}$ is the affine parameter at emission.

The *luminosity distance* $D{_{\text{L}}}$ is defined from the ratio between the observed flux $F{_{\text{obs}}}$ and the intrinsic luminosity $L{_{\text{source}}}$ of the source, so that $$L{_{\text{source}}} = 4\pi D{_{\text{L}}}^2 \, F{_{\text{obs}}} .$$ It is related to the angular distance by the following distance duality law $$D{_{\text{L}}}=(1+z)^2 D{_{\text{A}}}.$$ Hence, the theoretical determination of the luminosity distance relies on the computation of the Jacobi matrix.

Solving the Sachs equation piecewise {#sec-piece}
------------------------------------

Since we work in a Swiss-cheese universe, we have to compute the Jacobi matrix for a patchwork of spacetimes. It is tempting, in this context, to calculate the Jacobi matrix for each patch independently, and then try to reconnect them. In fact, such an operation is unnatural, because the very definition of $\boldsymbol{\mathcal{D}}$ imposes that the initial condition is a vertex point of the light beam. Thus, juxtaposing two Jacobi matrices is only possible at a vertex point, which is of course too restrictive 