Etherington's reciprocity theorem
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The Etherington distance-duality equation is the relationship between the luminosity distance of standard candles and the angular diameter distance.[1] The equation is as follows: , where is the redshift, is the luminosity distance and the angular-diameter distance.
When Ivor Etherington introduced this equation in 1933, he mentioned that this equation was proposed by Tolman as a way to test a cosmological model. Ellis proposed a proof of this equation in the context of Riemannian geometry.[2][1][3] A quote from Ellis: "The core of the reciprocity theorem is the fact that many geometric properties are invariant when the roles of the source and observer in astronomical observations are transposed". This statement is fundamental in the derivation of the reciprocity theorem.