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It is shown with several concrete examples that the Dyer-Roeder approximationis valid in spacetimes which fulfill the condition that fluctuations in theexpansion rate along a light ray locally cancels with the shear contribution tothe redshift. This is the case for standard cosmological scenarios includingperturbed FLRW spacetimes, N-body simulations and Swiss-cheese models. Withanother concrete example it is then illustrated that it is possible toconstruct statistically homogeneous and effectively statistically isotropiccosmological models which do not fulfill the condition. In this case, theDyer-Roeder approximation is invalid. Instead, the mean redshift-distancerelation can be described using a relation based on the spatial averages of thetransparent part of the spacetime.

Understanding the Dyer-Roeder approximation as a consequence of local cancellations of projected shear and expansion rate fluctuations