Warp said:
I think there is a fundamental misunderstanding in all of this.
If I'm an observer and am measuring a (massive) object receding from me, according to SR I will never, ever measure said object to be receding from me faster than c, or even at c. It may approach c, and thus may red-shift to almost invisibility, but it will never reach c, and thus never become completely invisible.
However, according to GR the receding object can recede from me faster than c. It thus becomes completely unobservable from my perspective, effectively being beyond an observability horizon. And there is effectively no limit to how much faster than c it can recede. SR does not have this concept because it considers space to be linear and static.
Let me try again to get across what your category error is.
In SR, if I measure the speed of an object relative to me, it will always be less than c. However, if I measure the rate of growth of proper distance between between two objects, the result can be up to 2c in an inertial frame, and any value in a non-inertial coordinates (even though I ams still talking about growth of proper distance with respect to proper time of a fiducial observer). The flat space analog of cosmological coordinates is a non-inertial frame (known as Milne coordinates). Note that these coordinates have a cosmological horizon (and do not completely cover all of Minkowsi space). Also, note that as simple as case as a uniformly accelerating observer in SR sees a Rindler horizon form behind them, and objects beyond it become causally disconnected from them. Infinite redshift occurs as an object approaches said horizon, and there is no signal, let alone redshift possible for an object beyond the Rindler horizon.
In going to GR, we have to look more at what relative velocity means. In SR you can define it either as speed in a global inertial frame in which one of the objects is at rest. Or you can define it in terms of 4-vector comparison (dot product of two 4-velocities gives gamma of their relative speed),
relying on the fact that parallel transport in SR is path independent, thus distant vectors can be unambiguously compared. Note that it is
only relative velocity in one of these senses that is limited to c in SR, as noted above. Unfortunately, in GR, neither of these definitions work at all. Globally inertial frames do not exist; parallel transport is path dependent so there is no such thing as comparison of distant vectors. As a result, relative velocity
does not exist in GR except locally (where you can use a local inertial frame; equivalently, vectors at the same event can be compared unambiguously because parallel transport is not necessary). Of course, this local relative velocity in GR is always < c.
Globally, in GR, all you have are analogs of the coordinate dependent quantities described above, that are not limited to c in SR. These things (including recession rate)
do not correspond to SR relative velocity at all. There is a limited statement you can make globally in GR that is in the same category as SR relative velocity. That is: while the relative speed of distant objects is inherently ambiguous because of path dependence of parallel transport, no matter what path you use for parallel transport, the result of parallel transport followed by vector comparison is
always < c, with no exceptions. Thus there is no way to choose a specific value, but the range of admissable values are all < c.
