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This excellent answer to Could dark matter exist in the Universe in the form of sufficiently dense objects? includes the following image and description:

Black Hole light paths Light from the background galaxy circles a black hole an increasing number of times, the closer it passes the hole, and we therefore see the same galaxy in several directions (credit: Peter Laursen).

and illustrates three strong gravitational lensing light paths from a source to an observer. In this not-drawn-to-scale illustration, each path is shown reaching the observer from a close, similar direction, but they represent very different trajectories; one is a simple bend, the second makes a complete ~360° loop around the black hole, and the third a double ~720° loop-de-loop.

Conservation of etendue is how optics folks view conservation of phase space (Liouville's theorem).

Familliar examples might be understood from the impossibility of concentrating blue sky with a magnifying glass, or that a wall does not appear darker when you walk away from it despite $1/r^2$.

Question: How well conserved is etendue in extreme gravitational lensing scenarios?

In "normal lensing scenarios" like the use of telescopes, they can never increase the apparent luminance of an extended object no matter how large the aperture. But in some gravitational lensing scenarios and real-world searches, dramatic brightening of objects occurs.

Are those, or the scenario in the illustration examples of breakdown in conservation of etendue?

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    $\begingroup$ Its not phasespace that is conserved but distributions in phasespace. Around every point in phasespace the density of points closeby is constant. Which doesnt mean the points cant smear out. $\endgroup$
    – Deschele Schilder
    Jul 31, 2021 at 11:07
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    $\begingroup$ @pela my questions are a lot more successful here than they are there. Because the question rate is so high there, Physics SE folks will often scan for the low-hanging fruit and avoid questions that don't have textbook answers or answers that can be plucked from recent papers. I'm also not sure that "How well conserved is..." will fly there, they are pretty quick with the "unclear what you're asking" close votes; if you are not pretty sure just what kind of answer you'll expect, it only takes three question closers to shut everything down assuming the insta-close experiment is still running. $\endgroup$
    – uhoh
    Jul 31, 2021 at 14:16
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    $\begingroup$ @uhoh Fair enough, I also rarely go there anymore, actually. $\endgroup$
    – pela
    Jul 31, 2021 at 16:54
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    $\begingroup$ @uhoh Not a bad idea. Another way would be to subdivide into fields, simply to make each one smaller. If we can have astrophysics/astronomy as a separate site, I guess we could also have mechanics, solid state, fluid dynamics, etc. $\endgroup$
    – pela
    Jul 31, 2021 at 21:21
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    $\begingroup$ I would like to add to @DescheleSchilder's excellent point that, in terms of wave optics, this distribution-conservation (etendue) only holds under the paraxial condition, in that paraxial-Helmholtz equation = Fresnel diffraction = constant in etendue = the phase-space representation, and this "amount" of etendue depends on the wavelength. I cannot comment more on the question itself due to my lack of expertise in gravitational lensing. $\endgroup$
    – WDC
    Aug 1, 2021 at 8:36

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