I Confusion regarding acceleration in SR

[Janus]

I feel that I understand the twin paradox perfectly when looking at the signaling diagram and the plane-of-simultanity diagram seperate. But I can't figure out how they can be compatible with each other.My question is: If they see the same thing in the andromeda galaxy. Then how can the person approaching the Andromeda galaxy even justify that his present is simultaneous to the fleet already moving? I mean, when calculating your simultaneous universe I assume that you can use the formula:

What the time is now in Andromeda = The time I observe right now in Andromeda + The time that the andromeda people experience that it took for light to reach me.

No, the formula is, T now in Andromeda = T I observe + time it took the light to reach me as measured by me from the distance Andromeda was when the light left it x the time dilation factor for Andromeda's speed.

If you and and Andromeda are approaching each other, then Andromeda was further from you when the light left then it is when you see the light. If you are receding from each other, then it was closer to you when the light left it than it is when you see the light. Thus while two people passing each other at the same speed relative to Andromeda, but in different directions will agree that Andromeda is the same distance away from them at the moment they pass, they will not agree as to how far Andromeda was from them when the light left Andromeda, and thus how long it took the light to make the trip.
The time dilation factor for Andromeda will be the same for both, So if they take the time they see, How long the light has traveled according to them, and factor in time dilation, they get what time it is now at Andromeda according to them, and come up with different answers as to what time it is at Andromeda when they pass each other.

To illustrate, here are the space-time diagrams for our approaching and receding observers.

Approaching:

Red line is Andromeda, Green is approaching observer, Yellow line is the light traveling from Andromeda and reaching observer when he passes receding observer (blue line). The black arrow is the distance between observer and Andromeda when the light left. The white line marks off what time it is "now" at Andromeda according to our observer when he sees the light.

Receding:

Same meaning for the Lines. We see that Andromeda was closer to our Observer when the light left, and thus took less time to reach the observer. Thus the "Now" line for the observer crosses Andromeda's line at a different point then it did for the approaching observer ( between ticks 1 and 2 instead of between ticks 2 and 3.)

 

[sophiecentaur]

In the cylindrical universe there would be a preferred frame globally.

Could you translate that please? Which would be the 'preferred frame"?

 

[pervect]

My question is: If they see the same thing in the andromeda galaxy. Then how can the person approaching the Andromeda galaxy even justify that his present is simultaneous to the fleet already moving? I mean, when calculating your simultaneous universe I assume that you can use the formula:

What the time is now in Andromeda = The time I observe right now in Andromeda + The time that the andromeda people experience that it took for light to reach me.

This doesn't look right to me. Possibly I'm misunderstanding your question.

Let's suppose that "I" use coordinates in an unprimed frame, (t,x). We'll forget about y and z for now, they aren't relevant to the problem.

Let's suppose that the Andromedean's use the coordinates in a primed frame, (t', x').

Then if I'm understanding your words correctly, the time now in Andromeda = t'
And the time I observer right now in Andromada = t

A big concern I have is what you mean by the notion of "now". You haven't clarified which frame the notion of "now" you are using applies to. This is usually indicative of not understanding the relativity of simultaneity. If it is an accidental omission on your part, it's still not at all clear what you think you mean when you say "now", and that issue seems to be close to the heart of the problem you are struggling with.

If we assume that I have translated your words correctly into mathematics, then we can apply the Lorentz transform and write:

$$t' = \gamma(t - v\,x/c^2)$$

This gives us the "time now in the Andromeda frame", t', as a function of the time now in my frame, t, and the distance now in my frame, x. We could do some math to try to get rid of the x (distance in my frame) and replace it with x' (distance in the Andromedean frame). I actually did so, but I don't want to get into the details unless it appears to be useful, which at this point it doesn't.

Your words are suggesting

$$t' = t + x'/c$$

And this just isn't right. And more importantly I don't know why you think it would be.

 

[sophiecentaur]

Well, the point of introducing the cylindrical universe was just to show that acceleration isn't really necessary for the paradox; differential aging can happen even when nobody accelerates.

Except that the whacky geometry of space seems to change everything - not the least because, as you say, there will be multiple images of bob out there. But the possibility of both twins going in opposite directions and meeting on the other side has to introduce the possibility of a real paradox which cannot happen in flat space. Is my reasoning about that flawed? In flat space, the only 'odd 'thing' is that each twin will observe the other as aging slower. They will never stand together unless one of them does some accelerating to bring them together and there will definitely be an older and a younger twin. In your cylindrical universe they both start their engines and accelerate apart. The visual ageing rates will be different. When they pass each other on the other side they will actually see each other as younger. Now what happens if they both apply retro rockets and land back on Earth? Will their body clocks be back in sync with both of them being younger than the staff on Earth?

 

[Orodruin]

But the possibility of both twins going in opposite directions and meeting on the other side has to introduce the possibility of a real paradox which cannot happen in flat space

It doesn’t. Does the fact that you can draw different lines between two points on a torus mean that there is an ambiguity in the length of those lines? Also, the cylindrical universe is flat.

 

[alejandromeira]

To illustrate, here are the space-time diagrams for our approaching and receding observers.

Approaching:

Red line is Andromeda, Green is approaching observer, Yellow line is the light traveling from Andromeda and reaching observer when he passes receding observer (blue line). The black arrow is the distance between observer and Andromeda when the light left. The white line marks off what time it is "now" at Andromeda according to our observer when he sees the light.

Receding:

Same meaning for the Lines. We see that Andromeda was closer to our Observer when the light left, and thus took less time to reach the observer. Thus the "Now" line for the observer crosses Andromeda's line at a different point then it did for the approaching observer ( between ticks 1 and two instead of between ticks 2 and 3.)

Thanks a lot.
I will to work whit these diagrams for understand what they mean.

 

[jbriggs444]

If they see the same thing in the andromeda galaxy. Then how can the person approaching the Andromeda galaxy even justify that his present is simultaneous to the fleet already moving?

It's just a coordinate choice. It has no measurable consequences. There is nothing to justify. The launching of the Andromedan battle fleet is neither in our past nor our future light cone.

The boundaries of our past and future light cones are coordinate independent. Those are physical and observable. The event of the Andromedan battle fleet's launch is in neither cone. The hyper-plane of simultaneity is an artifact of coordinate choice, not physical at all. It is only a matter of accounting.

When does the bill for the launch cost for the Andromedan battle fleet come due? It is insured at launch? That's is a question for accountants and the insurance men, not a question about physical reality.

 

[Kontilera]

Thank you!


I just read your posts and I think its clear now.
I will contemplate on this and seee if I have any futher questions, but I think I understand my missunderstanding now.

So, again: Thank you!

 

[A.T.]

Could you translate that please? Which would be the 'preferred frame"?

See links here:
https://www.physicsforums.com/threads/the-cosmological-twin-paradox.51197/#post-361621

 

[sophiecentaur]

See links here:
https://www.physicsforums.com/threads/the-cosmological-twin-paradox.51197/#post-361621

Perhaps you could translate those links instead and help with identifying and explaining the 'preferred frame'. Those links introduce so much other stuff that my paths is getting more divergent than the convergent I was hoping for.

 

[A.T.]

Perhaps you could translate those links instead and help with identifying and explaining the 'preferred frame'. Those links introduce so much other stuff that my paths is getting more divergent than the convergent I was hoping for.

The links are specifically about it, as are many threads here already. Here the PDF of the Weeks article:
https://www.math.uic.edu/undergraduate/mathclub/talks/Weeks_AMM2001.pdf

 

[sophiecentaur]

The links are specifically about it, as are many threads here already. Here the PDF of the Weeks article:
https://www.math.uic.edu/undergraduate/mathclub/talks/Weeks_AMM2001.pdf

Thank you.

 

[rrogers]

It is really hard to textile questions like this with words. Draw diagrams.

Yes, Minkowsky diagrams with light cones. Since the light paths are invariant pick any two coordinate systems "moving" and "stationary" and compare "events"; where the cones intercept world lines. You get wave crests (that start at different times) or clock ticks or whatever (birthdays..).