Intergalactic regatta

As a thought experiment, let's conduct a regatta involving three spaceships in intergalactic space, where they are not significantly gravitationally influenced by any nearby massive object. We conduct a "running start" where each ship completes its acceleration and turns off its engine before reaching the start line. The three ships cross the start line very near to each other and at the same instant. Ships #1, 2 and 3 cross the start line at proper velocities (<< c) of 1km/s, 2km/s, and 3km/s respectively. For the sake of simplicity, the expansion rate of the universe is coasting (neither accelerating or decelerating), with $$\Lambda (w= -1/3)$$ and $$\Omega = 1.$$

After two units of time have passed, Ship #1 decides to observe the other two ships in "comoving coordinates" which it bases on Ship #1's own proper velocity. Ship #1 views itself as stationary in its comoving coordinates; Ship #2 is then at a comoving distance of 2 km and is receding at 1km/s. Ship #3 is at a comoving distance of 4 km and is receding at 2 km/s.

So, Ship #1 is delighted to observe that the ships observe the Hubble expansion law: comoving recession velocity is exactly proportional to comoving distance.

After four units of time have passed, Ship #1 observes Ship #2 at a comoving distance of 4km, and Ship #3 at 8 km. However, the observed recession velocities remain fixed at 1km/s for Ship #2 and 2km/s for Ship #3.

The observed comoving velocities remain exactly proportional to distance. However, the comoving velocities have not increased with distance, contradicting the intuitive result.

Do Ship #1's observations reflect the generic behavior of a coasting universe? At (arbitrarily) t=10Gy, is the observed comoving recession velocity of Galaxy X the same as it is at t=20 Gy, despite the observed distance of the latter being 2X the former?

Jon

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Wallace
If the observer on Ship #1 is constructing a co-moving co-ordinate system, we can see in fact that Ship 2 and 3 are at rest with respect to these co-ordinates. In other words, as you point out, the other ships follow a Hubble law. What this means is that ships 2 and 3 are not changing there co-moving distances to ship 1. What is increasing are the proper distances, not the co-moving distances.

You ask why the co-moving velocities have not increased with distances, however for objects in a Hubble flow, the co-moving velocity is always zero (i.e. the co-moving co-ordinates are not changing).

Lets look at the maths in more detail. For the Universe you describe, the scale factor goes as $$a(t) = t$$. Now, the proper distance goes as $$r=a(t)x$$ where x in the co-moving co-ordinate. Taking the derivate of r with time we get $$\dot{r} = \dot{a}x + a\dot{x}$$ but for co-moving objects $$\dot{x}=0$$ and hence $$\dot{r} = \dot{a}x$$. We can re-write this using the original distance relationship as $$\dot{r} = \frac{\dot{a}}{a}r$$. Now, for in this case $$a(t)=t$$ and therefore our final answer is $$\dot{r} = \frac{r}{t}$$.

What this tell us is that for a given proper distance, the recession speed of Hubble flow objects at that fixed proper distance from us drops as 1/t. If you apply this to the numbers you have presented you will find that this agrees with your calculations.

So in a nutshell, the recession speed of objects at a fixed proper distances drops as 1/t in a coasting Universe, and the recession speed of objects at a fixed co-moving distance remains constant for all time. The origin of the confusion here is that you where taking proper distances to be co-moving distances in your analysis.

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Hi Wallace,

Thanks for your very clear answer. I agree that I misstated the concept of Hubble recession velocity in comoving coordinates, which should always be zero.
... therefore our final answer is $$\dot{r} = \frac{r}{t}$$.
...
So in a nutshell, the recession speed of objects at a fixed proper distances drops as 1/t in a coasting Universe....
So my thought about proper distances and proper velocities was correct. I hadn't previously realized that the proper recession velocity of "Galaxy X" will remain constant when observed after intervals of time in a coasting universe, despite the increase in proper distance over those time intervals. This conclusion is consistent with interpreting recession velocity of massive objects to be a kind of conserved kinematic momentum.

This effect means that the measured redshift for Galaxy X will remain fixed over time, which is a good thing. And for a succession of galaxies momentarily passing through a surface at any given proper distance, doppler-calculated redshifts should decrease over time compared to such measurements at the same proper distance at earlier times. Will the same result be calculated if redshift is calculated by the "classical GR" method based on the proportion by which intervening space has "expanded" during the travel time of the redshifted photon? It seems like it should, because when thought of the latter way, the rate of "expansion of space" at any given proper distance must also be declining over time. But a declining rate of spatial expansion (when using this terminology) at a given proper distance seems inconsistent with the description of a coasting universe.

As an aside I should clarify one other point about my original terminology. A universe with $$\Lambda (w= -1/3)$$ and $$\Omega = 1$$ is more accurately described as asymptotically coasting only at late times. At early times of such a universe, gravitational deceleration dominates the acceleration effect of the relatively small Lambda. My regatta needs to be conducted entirely at late times.

Jon

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Wallace