# What is observed as rate of change of proper distance with time due to space-expansion to a binary star approaches, then exceeds the speed of light?

Aliens on planet M observe a binary star at distance D with a 10 day "clockwise" revolution period. All other massive objects are FAR away and can be ignored.

At time=0, the center of mass of the binary star has zero velocity in planet M's reference frame.

Over a LONG period of time, due only to the expansion of space, the proper distance between M and the binary star increases such that the rate of change of proper distance with time (a) approaches c, (b) equals c, and (c) then exceeds c.

MANY generations of Aliens are observing this take place with unrealistically powerful telescopes and sensors of gravitation fields, what do they observe at time points (a), (b), and (c)? In terms of:

(1) Visible light

(2) Gravitation Fields

(3) Time dilation, i.e. revolution period

Would anything change between time point (c), and time point (d) a LONG time after (c)?

Thank you, ProfRob, for your question.

i) If the acceleration of space-expansion is positive due to the cosmological constant, then if D is large enough that gravitation between M and the binary star is negligible, wouldn't the distance between M and the binary star begin to increase even without initial velocity? I am using a purely theoretical example to try to isolate the effects space expansion from any other variables, such as actual velocity. If this doesn't work, then using a very small initial velocity away is fine.

ii) These unrealistically powerful gravitation sensors would be able to detect fluctuations in gravitational fields and/or spacetime curvature due to the binary stars being positioned horizontally, vertically, rotating clockwise, etc at huge distances. Here I am trying to isolate if there is any difference between what observers would see in terms of light versus gravity.

Thank you again, ProfRob, for clarifying (1) and (3) of my questions and for the link.

Though similar to my previous question, this one has a bit more depth. Just so I understand, let me repeat ProfRob's accurate and correct answer with a little more detail.

(a) As the binary star approaches the Hubble sphere the light, effects observed from the binary star at the time it reaches planet M would be as follows:

(1) Observed light intensity will fall by more than 1/(D*D) due to Derivation of the Lorentz transform of brightness ($dP/d\Omega$)

(2) There will be time dilation, but the effect will be less (closer to 1) then the Lorentz factor, since the light emitted from the binary star will continuously be moving through space expanding away from us a slower speeds than the binary star itself

(3) Will be redshifted

(b) As the binary star crosses the Hubble sphere, effects (1), (2), and (3) above will just continue to increase, but since the Hubble sphere is not a horizon nothing special happens here, and this will continue until the binary star crosses the "Past Light Cone"

(c) After crossing the "Past Light Cone" for a given time T, this light cannot be observed by the Aliens on planet M at corresponding time T'; however, if the aliens wait long enough they will observe this light with effects (1), (2), and (3) ever increasing

(d) If the binary star were at the event horizon, only if the Aliens waited an infinite amount of time would their observation of the binary star (1) become infinitely dim (2) Binary revolution Motion would stop, and (3) become infinitely red shifted; however, since it is impossible to wait an infinite amount of time, the Aliens will still be stuck at (c), just as ProfRob's answer suggests.

(e) After the binary crosses the event horizon, the light emitted by the binary star will never be observed by the Aliens more matter how long they wait.

• Questions. How can the recession velocity be zero at some time and then non-zero at a later time? Do you mean that it is "small"? What is a gravitational field detector and what does it measure? – ProfRob Apr 26 at 20:07

In the cosmological-constant-dominated future phase of ΛCDM cosmology, the point where the distant object's recessional speed equals $$c$$ coincides with the point where it crosses the cosmological horizon. This doesn't mean that it suddenly disappears; it just means that the aliens only see its history up to that point, redshifted into the indefinite future.
In general FLRW cosmologies, the point on the worldline where the recessional speed exceeds $$c$$ has no particular significance, and nothing special appears to happen there. Recessional speeds are defined in such a way that the speed $$c$$ isn't special, as I said in my other answer.