How can the distance duality relation be derived in a model-independent way?

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The discussion centers on the difficulty of deriving the distance duality relation, which connects angular diameter distance and luminosity distance. The original paper by Etherington from 1933 is noted as challenging for those unfamiliar with the concepts. The user seeks references or simpler explanations that derive this relation in a model-independent manner. There is a lack of responses, indicating a gap in accessible resources on this topic. The need for clearer, more approachable derivations of the distance duality relation is emphasized.
aymer
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Hi ,

I need help from someone who has worked out the distance duality relation (between angular diameter distance and the luminosity distance).I am having trouble deriving it and the Etherington paper is a bit too tough for me where it was first discussed (1933). In one of the reviews its written that it comes when you use the transformation of solid angle under velocity transformation..
 
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Its strange that nobody replied to this...I just want a reference as to wherever an attempt was made to derive the relation in a model/background independent way...i tried reading the Etherington paper (he was the first to introduce it I guess) but its a bit too technical for me...
 

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