we assume that a static axially symmetric thin test disc lies in the equatorial plane $\theta=\pi/2$ of the Schwarzschild black hole outside some radius greater than the horizon radius, and that the photons emanate from it in the perpendicular directions and follow geodesics. Due to the axial symmetry, each spatial point above the disc is then crossed by two rays,[^2] except for the symmetry axis $\theta=\pi/2$ where at each point all the rays starting from a certain circular loop meet in a “caustic", as illustrated in Fig. \[vertical-photons\].

![The flux of free test photons emitted perpendicularly from the Schwarzschild equatorial plane into the “upper" half-space, as plotted in coordinates $(\pm r\sin\theta,r\cos\theta)$ for a typical “meridional" plane through the symmetry axis. Through each spatial location pass two rays, except for the axis where each point is crossed by photons coming from an entire equatorial circle. Photons starting from below $r=3M$ fall into the black hole (the grey circle) and are not shown. In all the plots the axis values are given in the units of $M$.[]{data-label="vertical-photons"}](vertical-flux.eps){width="\columnwidth"}

Writing the metric in the standard Schwarzschild form $${\rm d}s^2=-N^2{\rm d}t^2+\frac{{\rm d}r^2}{N^2}+r^2({\rm d}\theta^2+\sin^2\theta\,{\rm d}\phi^2)$$ with $N^2=1-\frac{2M}{r}$, our photons with zero axial angular momentum have non-zero 