Consider a light ray emanating from the origin of a FLRW coordinate system in a homogeneous, isotropic universe. The initial velocity of that ray will have only x0 (t) and x1 (r) components. In papers I have seen it is assumed that its velocity will continue to have zero circumferential components: x2 (θ) and x3 ([itex]\phi[/itex]), in other words that θ and [itex]\phi[/itex] are constant. A loose argument for this is that, for the geodesic to develop any circumferential components would identify a preferred direction in space, thereby contradicting the isotropy assumption. I find this unconvincing, as the 'direction' is a coordinate-dependent artifact, and hence does not necessarily have any physical significance. The isotropy assumption is a coordinate-independent statement about the nature of the spacetime, not about a particular coordinate system (well, perhaps it does contain information about the time coordinate, as it seems to state that the constant-time hypersurfaces are isotropic, but those hypersurfaces can be parameterised in an infinity of different ways, so there's nothing significant about a particular spherical choice of coordinates as in the FLRW system.). Is there a more rigorous argument as to why the null geodesic cannot have any circumferential components?