summarized in Fig. \[figgeo\]. $d^\mu$ is the direction of observation as defined in Eq. (\[dec-k\]). The spatial sections of the FL region can be described either by comoving spherical coordinates $(\chi,\theta,\varphi)$ or, when the spatial sections are Euclidean, by comoving Cartesian coordinates $(X,Y,Z)$.

The hole is characterized by its comoving spatial position $X_{\rm h}^i$, in terms of the FL coordinates, and its mass $M$, or equivalently its comoving radius $\rho_{\rm h}$. Note that, contrary to Subsec. \[subsec:spacetime\_patches\], it is no longer denoted $\chi{_{\text{h}}}$, in order to avoid confusion with the radial comoving coordinate of the center of the hole.

A photon enters into the hole with wave vector $k^\mu_{\rm in}$, exits from it with wave vector $k^\mu_{\rm out}$, and reaches the observer with wave vector $k^\mu_0$. We respectively denote ${\cal E}_{\rm in}$, ${\cal E}_{\rm out}$ and ${\cal E}_0$ the associated events. The coordinates of the first two can be expressed either with respect to FL, e.g. as $(T_{\rm in},X^i_{\rm in})$ in Cartesian coordinates, or with respect to the hole, e.g. as $(t_{\rm in},r_{\rm in},\theta_{\rm in},\varphi_{\rm in})$ in the Kottler spherical coordinate system.

Our calculations go backward in time. Starting from $\mathcal{E}_0$, we first determine $\mathcal{E}{_{\text{out}}}$, $\boldsymbol{\mathcal{W}}_{\rm FL}(v_{\rm out};v{_{\text{obs}}})$, and second 