I have posted a few times about Etherington’s Reciprocity Theorem in cosmology, largely in connection with the Hubble constant tension – see, e.g., here.

The point is that if the Universe is described by a space-time with the Robertson-Walker Metric (which is the case if the Cosmological Principle applies in the framework of General Relativity) then angular diameter distances and luminosity distances can differ only by a factor of *(1+z) ^{2}* where

*z*is the redshift:

*D*.

_{L}=D_{A}(1+z)^{2}I’ve included here some slides from undergraduate course notes to add more detail to this if you’re interested:

The result *D _{L}=D_{A}(1+z)^{2}* is an example of Etherington’s Reciprocity Theorem and it does not depend on a particular energy-momentum tensor; the redshift of a source just depends on the scale factor when light is emitted and the scale factor when it is received, not how it evolves in between.

Etherington’s theorem requires light rays to be described by *null* geodesics which would not be the case if photons had mass, so introducing massive photons would violate the theorem. It also requires photon numbers to be conserved, so some mysterious way of making photons disappear might do the trick, so adding some exotic field that interacts with light in a peculiar way is another possibility, as is the possibility of having a space-time with torsion, i.e. a non-Riemannian space-time.

Another possibility you might think of is to abandon the Robertson-Walker metric. We know that the Universe is not exactly homogeneous and isotropic, so one could appeal to the gravitational lensing effect of lumpiness to provide a departure from the simple relationship given above. In fact a inhomogeneous cosmological model based on GR does not in itself violate Etherington’s theorem, but it means that the relation *D _{L}=D_{A}(1+z)^{2}* is no longer global. In such models there is no way of defining a global scale factor a(t) so the reciprocity relation applies only locally, in a different form for each source and observer. In order to test this idea one would have to have luminosity distances and angular diameter distances for each source. The most distant objects for which we have luminosity distance measures are supernovae, and we don’t usually have angular-diameter distances for them.

Anyway, these thoughts popped back into my head when I saw a new paper on the arXiv by Holanda et al, the abstract of which is here:

Here we have an example of a set of sources (galaxy clusters) for which we can estimate both luminosity and angular-diameter distances (the latter using gravitational lensing) and thus test the reciprocity relation (called the cosmic distance duality relation in the paper). The statistics aren’t great but the result is consistent with the standard theory, as are previous studies mentioned in the paper. So there’s no need yet to turn the Hubble tension into torsion!

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