Please excuse the obscurity of the question but say there are two spheres: sphere A has a radius of 1 metre and sphere B has a radius of 2 metres. Both sphere have super- luminous velocites (they are both exactly c/H metres from us) and due to the extent of their velocities they will begin to fade. My question is that will both spheres "fade away" simultaneously or will sphere A "fade away" before sphere B or vice versa?

George Jones
Staff Emeritus
Gold Member
The concept of "superluminal" speed in this context is a red herring that, as far as I know, has no physical consequences. The only physical consequence of which I know is for the significance of a damping term for cosmological perturbations, but that is not something that we are discussing here.

Maybe some someone else, like Brian, knows some physical consequences.

Note the scare quotes above. Since speed = distance/time, the definition of "speed" depends on the definitions of "distance" and "time". The definitions used in special relativity do not generalize easily to the curved spacetimes of cosmology. The definitions used in cosmology are, however, are easily applied in special relativity. When this done, cosmological speed turns out to rapidity (also known as the velocity parameter). In special relativity (with ##c=1##), the relationship between speed ##v## and rapidity ##w## is

$$v =\tanh w.$$

Consequently, ##0<v<1## gives ##0<w<\infty##, and ##w=1## (i.e., ##w=c##) has no physical significance.

Note that even though speeds don't add algebraically in special relativity, rapidities do add algebraically in special relativity, just like the cosmological speeds in equation (20) from the paper to which Andrew linked