O. Wucknitz [1,2]
We calculate magnifications and amplifications (in photon number and energy flux density) for general lensing scenarios not limited to regions close to the optical axis. In this way the formalism is naturally extended from tangential planes for the source and lensed images to complete spheres. We derive the lensing potential theory on the sphere and find that the Poisson equation is modified by an additional source term that is related to the mean density and to the Newtonian potential at the positions of observer and source. This new term generally reduces the magnification and leads to violations of the theorem far from the optical axis, ensuring conservation of the total photon number received on a sphere around the source.
This discussion does not affect the validity of the focusing theorem, in which the unlensed situation is defined to have an unchanged affine distance between source and observer. The focusing theorem does not contradict flux conservation, because the mean total magnification (or amplification) directly corresponds to different areas of the source (or observer) sphere in the lensed and unlensed situation. We argue that a constant affine distance does not define an \emph{astronomically} meaningful reference.
By exchanging source and observer, we confirm that magnification and
amplification differ according to Etherington's reciprocity law, so that
surface brightness is no longer strictly conserved. At this level we also
have to distinguish between different surface brightness definitions that
are based on photon number, photon flux, and energy flux.
Key words:
Gravitational lensing -- Cosmology: miscellaneous
This is the abstract for my
paper about the subject. For more details, click
here.