The spatially flat FRW metric in Cartesian co-ordinates is given by: $$ds^2 = -dt^2 + a^2(t)(dx^2 + dy^2 + dz^2)$$
As I understand it, since the metric does not depend on the spatial co-moving co-ordinates $x,y,z$ then there are Killing vectors in the $x,y,z$ directions.
Does this imply that the 3-momentum of a free particle is conserved when measured with respect to the $x,y,z$ co-ordinates? (In terms of expanding proper distances I presume that the particle would seem to lose velocity)
Does this also imply that the 3-momentum of a photon is conserved when measured with respect to the $x,y,z$ co-ordinates?
If the 3-momentum of the photon is conserved then, as $E=pc$ for photons, does this imply that its energy is conserved as well?