A Scenic Trip as a Spaceship at 0.999...c, and questions that arise

[IroAppe]

Okay, that's very fascinating.

Please take some time to learn the basics. You have very fundamental misunderstandings.

I will do exactly that. You have given me many impulses today, education on the types of reference frames, coordinate systems, especially comoving coordinates, Lorentz Transformations, four-vectors and invariant quantities.

I've learned, that most videos on the topic seem to skip many of these important concepts that are incredibly useful to describe those scenarios. I have learned, that before, I thought I knew at least most of the concepts conceptually, now I understand that there are concepts that I haven't even touched that could answer a lot of my questions.

But still, thank you a lot for those last few answers! They answered a lot of what I have to learn to understand cosmology better. And I'm more intrigued than ever before.

 

[PAllen]

One thing I like to do is, stop the acceleration of the expansion rate of the universe. Because these statements still have to hold true, if the expansion rate remains constant.

There is no cosmological horizon without accelerated expansion.

 

[Nugatory]

I will do exactly that. You have given me many impulses today, education on the types of reference frames, coordinate systems, especially comoving coordinates, Lorentz Transformations, four-vectors and invariant quantities.

You could do much worse than starting with Taylor and Wheeler’s book “Spacetime Physics” - the first edition is available free online.

 

[IroAppe]

There is no cosmological horizon without accelerated expansion.

Aren't objects far away still moving away faster even with a constant expansion of space-time? Isn't the rate of expansion the speed at which two points at a defined distance move away from each other? If you take points at twice that distance, and per one unit of that distance the expansion rate is constant, then those two units-distanced points will still move away from each other twice as fast.

Or another analogy, if you stretch the ends of a rubber band at 10 times the speed of light, then the point exactly in the middle will still stay stationary, and the more you move to one of the ends of the band, the faster your motion will be. The point in the middle of the band is us, and all the other points are the objects that we see moving away from us. (Of course, this applies for every point in this universe).

Am I confusing acceleration and speed here?

 

[jbriggs444]

Or another analogy, if you stretch the ends of a rubber band at 10 times the speed of light,

There is a well known problem that envisions a rubber band that is one light year long. One end is held stationary. The other end moves away at an extremely high speed. Such as 10 times the speed of light.

An ant begins crawling on the rubber band at a speed of one centimeter per year. Can the ant ever reach the far end of the rubber band? The surprising answer is "Yes".

The ant is carried along with the expansion. If he gets a fraction of the way there, he stays at least that fraction of the way there.

If you analyze the ant's motion for this it is the sum of a harmonic series. He gets some fraction of the way in the first year (goal 1 light year away). He gets another 1/11 of that fraction on the second year (goal 11 light years away). He gets another 1/21 of that fraction on the third year (goal 21 light years away). He gets another 1/31 of that fraction in the next year. And so on. That is roughly a harmonic series. The sum of a harmonic series is infinite. So no matter how small a fraction he gained on that first year, he eventually makes it to 100%. The partial sums of a harmonic series increase roughly as the logarithm of the number of terms. So it takes exponentially many years to get there. Something like $e^{\frac{1}{\text{first year fraction}}}$. Don't wait up.

For an exponential expansion (fixed expansion rate in velocity per unit time per unit distance) the ant can never get there. After one year, he is looking at the same problem, but the goal line is 10 light years minus one centimeter farther away.

The infinite series thing does not save us this time. The infinite sum of a decaying geometric series is finite.

 

[Halc]

Aren't objects far away still moving away faster even with a constant expansion of space-time?

Yes. The rubber band thing illustrates that pretty well. The post by jbriggs444 just above is spot on. Constant expansion is like the end of the rubber band moving away at constant speed.

The current rate of expansion is something near 70 km/sec/mpc. Constant expansion does not mean that the rate expressed that way is constant. It means that something a megaparsec away moving away at 70 km/sec will continue to move away at 70 km/sec forever, even though its 2 mpc away in 14 billion more years. Accelerating expansion means that some distant galaxy will gain recession speed from us.

So the expansion rate might be 70 km/sec/mpc now, and will eventually go down to something like 57 km/sec/mpc and become a constant of sorts, which will indicate exponential expansion, not linear expansion.

If you take points at twice that distance, and per one unit of that distance the expansion rate is constant, then those two units-distanced points will still move away from each other twice as fast.

As measured in cosomological coordinates (comoving frame, proper distance/speed), yes. In such coordinates, speed is more of a rapidity and adds with normal addition (*), not relativistic velocity addition like you'd use with inertial coordinates. So there's nothing funny about recession rates at arbitrarily high multiples of c.

(*) Not as simple as that, but it works for objects with negligible peculiar velocity like pretty much any galaxy.

 

[A.T.]

Am I confusing acceleration and speed here?

I think you are confusing points of space with objects moving locally relative to them.

Here is the rubber band scenario for non-accelerated expansion, where you can always reach any point, given enough time, no matter how much faster than your local speed it initially recedes from your start point (as mentioned by @jbriggs444):
https://en.wikipedia.org/wiki/Ant_on_a_rubber_rope