Can a twin paradox occur if one twin never changes direction or stops?

[phellen]

Hi Everyone,

I've managed to confuse myself with a mutilated thought experiment. A half twin paradox. Imagine a twin, Paul travels away from Earth at 8/10c while peter stays at home. Because of time dilation paul reaches a planet by his clock, in one year. Peter registers Pauls arrival at around 1 and a half years. Now if Paul does not change reference frames and continues in steady motion, peter moves at the same speed relative to Paul as Paul does relative to peter. So when Paul arrives at the planet he reads 1 year on his clock, and Peter reads 1.5 years for Pauls arrival, and yet when Paul arrives he reads Peters clock as just short of one year.
Basically I am just not sure what happens in the case of the twin paradox where the twin that leaves Earth neither stops nor changes direction. Is there some kind of simultaneity involved or are they each (relative to the other) in a range of times for each instant.

Many thanks
P

 

[The_Duck]

Let me try to get the numbers right just to be precise. It sounds like Paul is traveling a distance of 1.33 light-years. In Peter's reference frame the journey takes 1.33/0.8 = 1.67 years. But Paul's Lorentz factor is 1/sqrt(1-0.8^2) = 1.67, so for Paul the subjective time of the journey is 1.67/1.67 = 1 year.

The core of the issue you are talking about is the "relativity of simultaneity." When two events occur far apart, different reference frames disagree on whether or not they are simultaneous. Here the two events are:

1. Peter's clock on Earth reads 1.67 years.
2. Paul arrives at the distant planet.

In Peter's reference frame these events are simultaneous. In Paul's reference frame these events are not simultaneous. Instead, in Paul's reference frame the following two events are simultaneous:

1. Peter's clock on Earth reads 0.6 years.
2. Paul arrives at the distance planet.

So which events are simultaneous depends on the reference frame. For this reason, it doesn't make sense to ask "what time does Peter's clock read when Paul arrives at the distant planet?" until you have specified a reference frame. You are really asking "what event on Earth is simultaneous with Peter's arrival at the distant planet?" but there is simply no fact of the matter about which events are simultaneous unless and until you specify a reference frame to work in. But all inertial reference frames are equally valid, so by choosing to work in a specific reference frame you are just choosing a *convention* about which events you are going to call simultaneous.

On the other hand, if two events occur at the same place there is no ambiguity about whether or not they are simultaneous. That is why in the traditional discussion of the twin paradox, the traveling twin returns to Earth. Then both twins are in the same place at the end, and we can ask whether the event "twin 1's clock on Earth reads 1 year" is simultaneous with the event "twin 2's clock on Earth reads 1 year" and get a definite answer.

 

[phellen]

Thanks Duck :) this is a great reply. Just as a follow up, if we synchronise three clocks, one at Pauls location, one at the planet and one in between (and they are all in the same rf as Peter), what will they read when Paul reaches the planet?

 

[dauto]

They will all read 1.67 years because that's how long the trip takes on Peter's rf, but in Paul's rf they will read different things because of the relativity of simultaneity.

 

[ghwellsjr]

Thanks Duck :) this is a great reply. Just as a follow up, if we synchronise three clocks, one at Pauls location, one at the planet and one in between (and they are all in the same rf as Peter), what will they read when Paul reaches the planet?

Paul's moving in Peter's rf, did you mean Peter? If you meant Paul, then which of Paul's locations did you mean?

 

[phellen]

I'm sorry, I did mean Peter..

 

[Ibix]

It depends. Whose frame are you talking about?

One of the things that always trips people (me included) when they start learning relativity is that "what time is it?" has no unique answer. In Newtonian physics, if everyone syncs watches then their watches are synced forever (give or take the accuracy of the clocks). This means that the question "what time does Peter's watch show?" and the question "what time does Paul's watch show?" have the same answer - in fact, the whole "Peter and Paul" bit is redundant. I can just ask "what time is it?" and not worry about who answers me.

This is not the case in relativity. Due to time dilation effects, Peter's watch may not be synchronised with Paul's, so you always need to specify who you are asking - "what time is it according to Paul?" for example. You aren't doing this... You asked "what time do the three clocks show when Paul reaches the planet?". That won't do. You need to ask "what time do the three clocks show when Paul reaches the planet according to Paul?" or "...according to Peter?".

Dauto has already answered that in part. According to Peter, sitting at home, the clocks all show the same time (he'll have to adjust for the lightspeed delay in what he actually sees, but that's a separate issue).

According to Paul, however, the clocks will show different times. This is because the clocks were never synchronised according to Paul. They will all show the same elapsed time for his trip (though not what Peter calculated), but since they all read different times when he left, they all read different times when he arrives.

"Simultaneous" is not a global concept in relativity. Peter set the the three clocks so that they showed zero simultaneously according to him. Paul does not agree that they show zero simultaneously.

 

[ghwellsjr]

It depends. Whose frame are you talking about?

Didn't he say Peter's frame?

 

[Ibix]

Didn't he say Peter's frame?

I don't think so. He said (in #3) that the clocks were at rest in Peter's frame, and that they were synchronised (I inferred that they are to be synchronised in Peter's frame). But he just asked what the clocks read "when Paul reaches the planet" - implying that he's still thinking of a universally agreed definition of "when Paul reaches the planet". I don't think #6 fixes that.

Everyone ought to agree on what the clock at the planet says when Paul reaches it, but the other clock times are frame (and, if we feel like complicating this even more, simultaneity convention) dependent.

 

[phellen]

This is great, thanks guys. One more thing, if we are working out times and distances according to Pauls rf, do we apply length contraction to the distance between the two (as if Peter had moved). Thanks to the FAQs I am getting the twin paradox, but I am not sure about this case whereby one twin moves and neither change rfs. So in the case above, we started with the distance between Peter and the planet according to Peter, worked out the lorentz contraction and time dilation because of pauls motion. Then it seems that when paul arrives he sees Peters clock at 0.6 years because of the simultaneity of rel. But could we not have begun by assuming that Paul was still and peter was moving toward a planet, in which case Paul would read Peters clock as 1 year and Peter read Pauls as 0.6. I suppose I am aksing if symmetry is destroyed somehow, perhaps by the planet, or if it can be explained by the doppler effect as can the twin paradox, where the return Journey clears things up.

 

[ghwellsjr]

Hi Everyone,

I've managed to confuse myself with a mutilated thought experiment. A half twin paradox. Imagine a twin, Paul travels away from Earth at 8/10c while peter stays at home. Because of time dilation paul reaches a planet by his clock, in one year. Peter registers Pauls arrival at around 1 and a half years.

Here is a spacetime diagram to illustrate your scenario. It conforms to the numbers you provided in your OP and the numbers that The Duck provided for you in his post #2 except that I'm using months instead of years. Peter on the Earth is depicted in blue and the planet is green. The diagram is for the common earth-planet rest frame. Paul is depicted in red. The dots represent the passage of months for each observer/object:

Now here is the same scenario transformed to Paul's rest frame:

Note that the time on Peter's clock when Paul arrives at the planet is 7.2 months (0.6 years times 12 months/year) just as The Duck pointed out.

In a later post, you asked about Length Contraction. Yes, the distance between the Earth and the planet in Paul's rest frame is contracted from 16 light-months (1.33 light-years times 12) by the factor of 0.6 which comes out to be 9.6 light months. Can you see this at the Coordinate Time of 0?

Now if Paul does not change reference frames and continues in steady motion, peter moves at the same speed relative to Paul as Paul does relative to peter. So when Paul arrives at the planet he reads 1 year on his clock, and Peter reads 1.5 years for Pauls arrival, and yet when Paul arrives he reads Peters clock as just short of one year.
Basically I am just not sure what happens in the case of the twin paradox where the twin that leaves Earth neither stops nor changes direction. Is there some kind of simultaneity involved or are they each (relative to the other) in a range of times for each instant.

Many thanks
P

Here is a spacetime diagram showing Paul continuing past the planet:

Now you can see that everything (except the planet which we can ignore) is symmetrical between Peter and Paul. Just like in Peter's frame, when Paul's clock read 12 months (counting the red dots), Peter's clock read 20 months and we see that in Paul's frame, when Peter's clock read 12 months (counting the blue dots), Paul's clock read 20 months.

Does that answer your concern in this thread?

 

[phellen]

Thats great, I think I may have been mistakenly contracting the length between planets in Peters reference frame. These graphs are great I'm going to look for some graph drawing software,
Thanks again
P