# Explain relationship between angular diameter distance and luminosity distance, Etherington Theorem

I have a question relating to the Etherington Theorem.

The luminosity distance is defined by the equation for flux, i.e.

$F=\frac{L}{4\pi D_L^2}$

where flux is in units energy per unit time (luminosity) per unit area.

The angular diameter distance is defined by

$D_A=\theta/R$, where $\theta$ is the observed angular size measured by a telescope, and $R$ denotes the proper size of an object.

These two quantities are related by $D_L=(1+z)^2D_A$

I have never read a clear explanation for this relationship, nor have I come across a derivation. Could anyone explain to me where the redshift dependence $(1+z)^2$ comes from?

This relation is quite important, non trivial, and mathematical, and was proved by Etherington along with the other closely related theorem in this paper

I. M. H. Etherington (Philosophical Magazine ser. 7, vol. 15, 761 (1933))

This theorem only depends on photon conservation and the fact that photons only travel in null geodesics in Reimannian geometries. For a more detailed overview, read the original paper.

• Unfortunately, this article is not free for students. Are you aware of an arXiv article which discusses this result? This relationship is used very frequently in the large-scale structure and baryon acoustic oscillations BAO. May 18 '15 at 17:43
• Unfortunately, there is no free result which explains the result, though there are some which experimentally prove the result. However, Wienbergs General Relativity book apparently explains the result quite nicely. May 19 '15 at 0:25
• Can I ask you where in Weinberg's GR book? I don't find the Etherington relationship in my 1972 copy. May 19 '15 at 19:24
• I don't know, which is why I said apparently. It was mentioned in a text discussing etherington. May 19 '15 at 23:13
• Found it. Weinberg's section on Cosmography. Equations 14.4.22-14.4.23. Page 423. Also, see David Hogg's paper on "Distance measures in cosmology", arxiv.org/abs/astro-ph/9905116, Section 7 "Luminosity Distance". May 19 '15 at 23:16

See D. Hogg's Distance measures in cosmology, 2000

http://arxiv.org/abs/astro-ph/9905116

Section 7, Luminosity Distance, p. 6

$D_L=(1+z)^2 D_A$

follows because the surface brightness of a receding object is reduced by a factor $(1+z)^{−4}$, and the angular area goes down as $D^{-2}_A$.

• This doesn't explain it, it just states it. I'm still looking for the derivation. It appears it has something to do with the power output and Cosmological Time Dilation, but I still haven't found a cogent explanation of 'why'. Apr 12 '20 at 16:39