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

[IroAppe]

Hello there!

I have a few questions that lead up to one significant problem that's been in my head for a long time. Many physics students will probably have had the same questions during their studies

The scenario is a spaceship, that travels away from Earth at 0.999...c . Due to time dilation, from your perspective as a traveler, it will take a shorter time to get to those targets, depending on how close to c you get, while from the Earth's perspective, it will still take all these years.

So let's say goodbye to our friends and family on Earth , and begin the trip! Let's begin with the questions:

1. We travel to Andromeda: Will you see a different observable universe there, since you are still seeing the same distance into all directions?
2. When you travel to the distant galaxies that we observe from Earth, that are also moving away from us at that 0.999...c speed: Will it seem like you are slowing down, since you are traveling through the voids where the expansion of the universe takes effect, and start catching up with these objects that are moving away from us incredibly fast?
3. When you eventually arrive at those distant galaxies, will you be stationary there, relative to the frame of reference of those galaxies?
4. Dependent on question 1: At those galaxies, if you see another observable universe, doesn't that contradict that the light from beyond the observable universe will never be able to reach us? Or does that not apply to objects that are traveling fast, essentially moving to another part in spacetime?
5. If you decide to accelerate to 0.999...c again in reference to those galaxies, due to the relativistic effects, you won't actually travel faster than the speed of light away from Earth, correct? It will be the same answer, as two objects that are approaching each other at 0.9c.
6. The final question: If you continue at 0.999c, what will you see? What eventually stops you from leaving the observable universe?

The thing is: I know, that it's not possible to leave the observable universe. If you look at it from the Earth's inertial reference frame, it all makes sense. But from the perspective of the traveler, I just can't explain to myself, what exactly it is, that would stop me from doing so. And: What I would see as a traveler, while attempting to doing it.

I'm very curious. I hope I'll learn a lot. Perhaps you can give me a few keywords and impulses, perhaps I have to read up a lot of stuff first, before I am able to understand this. But it has tickled me since forever, how this works.

Thank you for every answer in advance :)
IroAppe

 

[PeroK]

For the current model of an expanding universe, there is a distance beyond which light will never reach Earth. And vice versa: light from Earth will never reach a galaxy beyond that current distance. And, therefore, these galaxies can never be reached by a spacecraft from Earth.

This is called the cosmological  event horizon. See:

https://en.m.wikipedia.org/wiki/Cosmological_horizon

 

[IroAppe]

The thing is, I know this. From the perspective of the inertial reference frame of the Earth, it is clear. I know about the cosmological event horizon, using light that has to travel from either the galaxies beyond the horizon, or from us to that point, it's clear to see how that is impossible with the expansion of space-time faster than the speed of light beyond the horizon.

That's why I am so confused, because I can't find the one or multiple reason/s, why it won't work from the perspective of the travelling spaceship in incremental steps.

If:

1. You are able to reach the farthest galaxies inside our cosmological horizon.
2. And the inertial frame of reference is the same there as for us, including all laws of physics and another cosmological event horizon, whose center is at those distant galaxies.

Then:

1. You can get there, and it will be the same as here on Earth.
2. Meaning, that from there, you have galaxies inside your cosmological event horizon, that are not inside the cosmological event horizon of the Earth.
3. That means, that you can get to those new galaxies, which from the perspective of the Earth, should never be possible.

It feels like one of these "trick proofs" in Mathematics, where you derive that 1=2, but only because there was some mistake that you did along the way. Somewhere along this chain of statements that follow each other, there has to be an error of reasoning, or misunderstanding of the laws of Physics, that are violated while doing so.

But what is the purpose of doing this, if the derivation is so much easier from the inertial reference frame of the Earth? The purpose is, to get a picture, how relativistic trips look like, and what we would see, if we tried to escape the horizon. It is the same curiosity as for depictions of falling into a black hole, only for the cosmological event horizon. I admit, it's a recent obsession of mine

 

[PeroK]

The thing is, I know this. From the perspective of the inertial reference frame of the Earth, it is clear.

There are no global inertial reference frames in GR. Only local inertial reference frames.

I know about the cosmological horizon, and using light that has to travel from either the galaxies beyond the horizon, or from us to that point, it's clear to see how that is impossible with the expansion of space-time faster than the speed of light beyond the horizon.

The expansion of space is not a speed. It's an expansion rate per unit distance.

That's why I am so confused, because I can't find the one or multiple reason/s, why it won't work from the perspective of the travelling spaceship in incremental steps.

If you travel one year from Earth at near light speed, then the distance to a distant galaxy may have increased by more than one light year in that time. So, you are further from the galaxy than when you started. So is a light pulse from Earth.

That's a simplified analysis of what's going on.

The rest of your analysis suffers from too many misconceptions, as explained above.

 

[PeroK]

PS what we are using here is cosmological comoving coordinates. That does not constitute a global inertial reference frame.

 

[IroAppe]

There are no global inertial reference frames in GR. Only local inertial reference frames.

I know. The Earth's inertial reference frame is not in any way more special than the traveler's inertial reference frame. That is, why it has to be impossible from the perspective of both of those inertial reference frames. I just find it much easier to see that, if I look at it from the perspective of Earth's inertial reference frame.

The expansion of space is not a speed. It's an expansion rate per unit distance.

I know. Perhaps that was worded unclearly. What I mean: The further you get away from Earth, the faster objects travel away from us. Increasing the radius from Earth, eventually you reach a point where objects are moving away faster from the Earth than the speed of light, where they are unreachable to us.

If you travel one year from Earth at near light speed, then the distance to a distant galaxy may have increased by more than one light year in that time. So, you are further from the galaxy than when you started. So, is a light pulse from Earth.

I understand, but that's from the perspective of the Earth's inertial reference frame, which I understand.

The paradox in my mind is:

• From Earth, you have galaxies that are near the horizon, but still in reach.
• If you get there, won't you have a new horizon? And won't you be stationary in reference to those galaxies around you, even though you travelled away from Earth close to the speed of light?
• What then stops you from getting to those new galaxies?

 

[Nugatory]

I know. The Earth's inertial reference frame is not in any way more special than the traveler's inertial reference frame. That is, why it has to be impossible from the perspective of both of those inertial reference frames.

You are misunderstanding @perok’s point. The issue is not that one IRF might be somehow more special, it is that there are no global inertial reference frames - there is no inertial reference frame that includes the earth and the distant galaxies.

 

[IroAppe]

You are misunderstanding @perok’s point. The issue is not that one IRF might be somehow more special, it is that there are no global inertial reference frames - there is no inertial reference frame that includes the earth and the distant galaxies.

I think, I understand this point now, thanks. But I don't yet understand the implication on my apparent paradox. With "distant galaxies", do you mean those galaxies that are still inside the Earth's horizon, or those that are outside the Earth's horizon?

 

[IroAppe]

PS what we are using here is cosmological comoving coordinates. That does not constitute a global inertial reference frame.

Okay, I think I have to read more about coordinates, and I guess, the metric tensor, that I've heard a couple of times. If I understand how the coordinates are defined and manipulated by all these actions in the inertial reference frame of the traveler, I might understand the logical fallacy in the deduction of these statements.

 

[PeroK]

I think, I understand this point now, thanks. But I don't yet understand the implication on my apparent paradox. With "distant galaxies", do you mean those galaxies that are still inside the Earth's horizon, or those that are outside the Earth's horizon?

The point about the horizon is that it would take a light signal an "infinitely" long time to reach it. From that point of view, the horizon is similar to a light signal trying to catch a constantly accelerating partricle. A signal sent $n$ years minus one second would eventually catch the particle, but a light signal sent $n$ years plus one second never would. And a light signla sent after precisely $n$ years is on the horizon/limit/boundary between the two.

The other critical point is that the calculation of the horizon relies on galaxies moving with the Hubble flow. In practice this means that gravitationally bound galactic clusters are either all beyond the horizon or all below the horizon. You wouldn't get the scenario that a distant light signal could reach the Sun but not Alpha Proxima, for example. In practice, on a more local level, the mass in the universe is clumped. There won't be a horizon within a clump, as it were. The local structure needs to be factored in on top of the generally global accelerating expansion.

 

[PeterDonis]

• From Earth, you have galaxies that are near the horizon, but still in reach.
• If you get there, won't you have a new horizon?

Yes, but it won't include any galaxies that weren't in your original horizon. You are forgetting that it takes time for you to go from Earth to that distant galaxy--call it galaxy D--that was near Earth's horizon but still within reach. You are thinking of galaxy D's horizon, as compared to Earth's, as it is at the time you set out from Earth; at that time, yes, there will be galaxies within galaxy D's horizon that are not within Earth's horizon. But by the time you reach galaxy D, many galaxies will have moved out of galaxy D's horizon and won't be reachable from galaxy D any more; and any galaxy that was within galaxy D's horizon, but not Earth's, when you set out, will be among those.

 

[PeroK]

Okay, I think I have to read more about coordinates, and I guess, the metric tensor, that I've heard a couple of times. If I understand how the coordinates are defined and manipulated by all these actions in the inertial reference frame of the traveler, I might understand the logical fallacy in the deduction of these statements.

You may not be ready for this mathematically, but try this wonderful Insight:

https://www.physicsforums.com/insights/inflationary-misconceptions-basics-cosmological-horizons/

 

[IroAppe]

Yes, but it won't include any galaxies that weren't in your original horizon. You are forgetting that it takes time for you to go from Earth to that distant galaxy--call it galaxy D--that was near Earth's horizon but still within reach. You are thinking of galaxy D's horizon, as compared to Earth's, as it is at the time you set out from Earth; at that time, yes, there will be galaxies within galaxy D's horizon that are not within Earth's horizon. But by the time you reach galaxy D, many galaxies will have moved out of galaxy D's horizon and won't be reachable from galaxy D any more; and any galaxy that was within galaxy D's horizon, but not Earth's, when you set out, will be among those.

Thank you! That's it. I think, that was exactly the fallacy.

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.

Even if the expansion rate is constant, those distant galaxies from the perspective of galaxy D, of course still move away. Now if I visualize it, it would probably just work out.

But wait, wouldn't that violate causality, since you can get to galaxy D and hear from the recordings of an Alien civilization there, what those distant galaxies were? No: Because, were is exactly the keyword. All that galaxy D would have of their distant galaxies, would be old, outdated information. Even information that you bring yourself from Earth, which would by then be beyond the horizon, would be outdated.

And: You can never take that information back to Earth. So the statement stays true, that no light or information from those distant galaxies of galaxy D, will ever be able to reach Earth.

I think that cleared up a lot. Thank you very much!

 

[IroAppe]

So that's very interesting: You yourself can get information about these distant galaxies of galaxy D. But it's a one-way trip. But still: Is that valid? Is it valid, that someone starting on Earth, by travelling, will be able to get information about galaxies beyond Earth's event horizon - even though they will never be able to return to Earth?

It's like: You lose some access to information, you gain some access to other information. That's an interesting aspect of moving inertial reference frames in reference to others.

 

[PeroK]

So that's very interesting: You yourself can get information about these distant galaxies of galaxy D. But it's a one-way trip. But still: Is that valid? Is it valid, that someone starting on Earth, by travelling, will be able to get information about galaxies beyond Earth's event horizon - even though they will never be able to return to Earth?

Why not?

 

[PeterDonis]

Is it valid, that someone starting on Earth, by travelling, will be able to get information about galaxies beyond Earth's event horizon - even though they will never be able to return to Earth?

Yes, but as you noted, this information will be "old" information--it will be information from the distant past of those other galaxies.

 

[IroAppe]

Now, there's still the question of that, in relation to length contraction and time dilation.

Traveling at a speed of arbitrarily close to c, from the perspective of the travelers, they can pass billions of lightyears in a single second. Even though from the perspective of the Earth, that journey will take billions of years.

How does this work approaching the event horizon (that I now learned, is essentially the same for the Earth, you as a traveler, and galaxy D)?

Alright, I think I can make an attempt to answering my own question: The expansion of the universe will speed up in the same way, correct? So essentially, those unreachable galaxies will still move away from you, no matter how close to c you get, and how much the time dilation plays into it.

What about length contraction? I guess, the speedup of the expansion of space-time will be a greater effect than that of the length contraction?

Since from the inertial reference frame of the Earth it is so, I can deduct, that it also has to be so, while you travel close to c, since all IRFs have to be valid.

What is the purpose of asking this question? Now I have a visual picture of how it would look like, if we attempted such a trip.

And basically, by going arbitrarily close to c, we could skip to the end of the universe (time-wise), right? So if we want to travel incredibly fast (from the perspective of a traveler close to c) through the universe using that effect of time dilation and length contraction to our advantage, it will come with a disadvantage of "using up time" way faster.

 

[PeterDonis]

length contraction and time dilation

These concepts assume that inertial frames exist that cover the entire scenario, which, as you have already been told, is not the case here.

 

[IroAppe]

These concepts assume that inertial frames exist that cover the entire scenario, which, as you have already been told, is not the case here.

Yes, I remember that, but I do not understand that concept, nor its implications. That was the one topic here, that I had the least understanding about. Do you have some keywords for me to read up about it?

I will now start with this, perhaps it's inside there: https://www.physicsforums.com/insights/inflationary-misconceptions-basics-cosmological-horizons/

The only thing I know so far is, that global reference frames do not exist. Then the question is, what even are global reference frames, and what are their implications? I think that's a good start.

 

[PeterDonis]

I do not understand that concept

You don't understand what concept? The fact that there are no global inertial frames in a curved spacetime?

The best way to build an understanding of that is to build an understanding of basic GR. Sean Carroll's online lecture notes on GR are one good source for that.

 

[PeterDonis]

The only thing I know so far is, that global reference frames do not exist.

No, global inertial frames do not exist in a curved spacetime. But global non-inertial frames do. The standard FRW coordinates used in cosmology are an example: they are global coordinates on the entire universe, but they are not an inertial frame.

 

[IroAppe]

No, global inertial frames do not exist in a curved spacetime. But global non-inertial frames do. The standard FRW coordinates used in cosmology are an example: they are global coordinates on the entire universe, but they are not an inertial frame.

Ah, that clears up a lot of things. I suppose, that accelerated frames are non-inertial frames. But that shows me, that I lack some understanding of those concepts, that seems essential to understanding the effects of time dilation and length contraction. Thank you, I will definitely read Sean Carroll's online lecture notes on GR!

These concepts assume that inertial frames exist that cover the entire scenario

Especially that seems to be an essential connection, that I didn't know about before. And right now, I even don't understand the sentence, nor its implications. For example: What if such IRFs don't exist in that case? Does that show that it's a scenario that is not possible? But then: Why exactly, where is the point, where it becomes impossible?

 

[PeterDonis]

I suppose, that accelerated frames are non-inertial frames.

Yes, but not the only kind.

 

[PeterDonis]

What if such IRFs don't exist in that case? Does that show that it's a scenario that is not possible?

No. It just means there is no inertial frame that covers the entire scenario, i.e., that includes all of the events and worldlines in the scenario. That will happen in any curved spacetime where the scenario includes events that are separated far enough so that the effects of spacetime curvature cannot be neglected.

 

[IroAppe]

But aren't they still embedded in the IRF? For example, the IRF of Earth includes all events that are inside its event horizon. You can say that objects that cross the event horizon are receding faster than c, due to the curvature of space. You can also say, that objects that cross the event horizon of a black hole, are receding faster than c, due to the curvature of space. Isn't everything, that accessible from the Earth, describable inside the Earth's IRF?

Or do you say, that it breaks exactly down AT the event horizon. That is interesting. For example, one of my initial questions were, if at the time you arrive at galaxy D near the event horizon, you would be stationary in reference to galaxy D, since you've caught up to it. Then you'd have to accelerate again, which is not an IRF anymore. And by the time, you reach >c in reference to the Earth, you would have left the Earth's event horizon. Which should not be possible. Or is it possible, if you can never return again? Well no, it should be impossible in principle.

 

[PeroK]

But aren't they still embedded in the IRF? For example, the IRF of Earth includes all events that are inside its event horizon.

There is not even an IRF covering the surface of the Earth! If you analyse a falling object at a single location on the Earth's surface, then you can use a local IRF and Newton's second law. But, to explain gravity across the globe, you need to patch together all the local IRF's into a global non-IRF.

 

[IroAppe]

Alright, and time dilation and length contraction are not described for non-inertial references frames?

So in order to describe such a trip, you'd need to constantly construct global non-IRFs, and split them into other local IRFs, to get an answer?

 

[PeroK]

Alright, and time dilation and length contraction are not described for non-inertial references frames?

Time dilation and length contraction are potentially important when you first learn about SR, as they give you something to get a grip on. But, as you progress, they are less and less useful. The concepts that do become useful are the Lorentz Transformation, four-vectors and invariant quantities. You are unlikely to use time dilation or length contraction in a GR or cosmology course.

So in order to describe such a trip, you'd need to constantly construct global non-IRFs, and split them into other local IRFs, to get an answer?

You need an appropriate coordinate system. On the cosmological scale, comoving coordinates are often the most useful. A local IRF is useful only really to describe local events.

 

[PeterDonis]

aren't they still embedded in the IRF?

What IRF?

the IRF of Earth includes all events that are inside its event horizon

There is no such "IRF of Earth".

Please take some time to learn the basics. You have very fundamental misunderstandings. Correcting them would amount to teaching you a basic course in GR, and that is beyond the scope of a PF discussion.

 

[PeterDonis]

in order to describe such a trip, you'd need to constantly construct global non-IRFs, and split them into other local IRFs, to get an answer?

No. You can use one, single, global coordinate chart to describe the entire trip. It just won't be an inertial frame.

 

[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.