It is customary in gravitational lensing problems, to project both the background source and the deflecting mass (e.g. a background quasar, and a foreground galaxy acting as a lens) in a plane.
Then, the lensing problem can be regarded as a mapping between the unlensed source plane, and the lensed image plane. In such transformations, the Jacobian evaluated at a point of the source plane, expresses how an infinitesimal area located around that point increases.
Lens mass and mass distribution, relative positions and distances involved give rise to different scenarios. The special case in which the distortions are too small to be resolved by telescopes, is called microlensing regime. Typically, a dark, unseen object like a floating planet, happens to cross transversally in front of a background star.
The image of the background star suffers amplification and distortions that are unresolved, but a change in brightness is detected, with a very typical light curve shape. The measured light curve of a microlensing event can be related to physical parameters of the problem, because the change in brightness of a lensed image can be modelled simply by dividing the area of the lensed image by that of the unlensed source image. If that can be done, it is because the mean surface flux of the image equals that of the source.
That is, gravitational lensing can make a tiny source appear bigger in the sky but in plain terms, every square inch of the image has the same brightness of every square inch of the source. Here comes my question, because that seems to me rather counter-intuitive and, when I try to find a rigorous justification to it, I find the same arcane sentence in each book, in each review, in each paper I have seen:
'Because of Liouville's theorem, gravitational lensing conserves surface brightness'
(... and therefore the magnification is found by dividing the subtended area of the image by that of the source). Every single author I have read, drops that sentence as if it were something very obvious, and quickly goes into other questions.
I have tried to trace-back the origin of the idea, by consulting the bibliography of every book or document in which that thing is stated. Interestingly, I have recognized sort of a fingerprint of obscure points like this one, a patter that is repeated in many of the documents, as if some authors didn't understand and merely copied from each other, developing and personalizing only the parts they understand in between.
I have rigorously developed and resolved every one of the dark points in that pattern, but this one remains unresolved. Is it perhaps something obvious? How is Liouville's theorem applied to photons along null geodesics? I will accept an appropiate link or paper reference as a good answer.