Asymmetry in Length contraction?

[substitute materials]

Instead of a twin paradox, let’s just consider an inbound starship approaching Earth at relativistic speed. The traveler is on a flyby mission, he will never change speed or direction. We will disagree with the traveler on how much time will have elapsed when he passes Earth. The discrepancy can be explained by length contraction, just as the asymmetry in the twin paradox is explained. He is closer to Earth in his reference frame than he is in ours. This is the physical meaning of the spacetime diagram.

But why do we see only the traveler’s ship as length-contracted, whereas the traveler sees the entire distance between us and him as length-contracted? Doesn’t this suggest a preferred reference frame for the Earthly observer?

 

[Dale]

But why do we see only the traveler’s ship as length-contracted, whereas the traveler sees the entire distance between us and him as length-contracted? Doesn’t this suggest a preferred reference frame for the Earthly observer?

We don’t, and no.

You are missing the important factor here: the relativity of simultaneity. In the situation you describe there is no one value for the distance. It is a function of time. If you pick a different time then you get a different distance. Because of the relativity of simultaneity you will pick different times and therefore get different distances. But it is entirely symmetric.

 

[Nugatory]

But why do we see only the traveler’s ship as length-contracted, whereas the traveler sees the entire distance between us and him as length-contracted?

You are talking about "the entire distance", but as @Dale points out relativity of simultaneity makes that a somewhat slippery concept.
We can try to make it more precise: imagine that we place a buoy in space, between earth and the spaceship and at rest relative to the earth. The "total distance" is the distance between where the earth is and where the buoy is, at the same time. Likewise, the length of the spaceship is the distance between where the nose is and where the tail is, at the same time.

We consider the problem from the point of view of the ship and the earth, at the exact moment that the nose of the spaceship passes the buoy.
Using the frame in which the ship is at rest, the buoy and the earth are moving so the distance between them is length-contracted; the ship is at rest so of course it is not length-contracted.
Using the frame in which the earth and the buoy are at rest, the distance between earth and buoy is not length-contracted but the ship is moving so is length-contracted.

 

[Ibix]

Doesn’t this suggest a preferred reference frame for the Earthly observer?

The asymmetry is in your problem statement, where you've chosen to regard "the entire distance between us and him" as something naturally measured in the rest frame of the Earth. In other words, you chose to prefer one frame's measure of distance which is why your explanation prefers one frame. This was an arbitrary choice since there is nothing actually marking the other end of "the distance", and it is this choice @Nugatory makes concrete by adding a buoy. You could have made the opposite choice, and then you would need to have added a buoy at rest with respect to the ship and your explanation would have been the other way round.

A symmetric scenario with well-defined notions of distance would be a pair of planets a light year apart in their rest frame and a pair of spaceships one light year apart in their rest frame and moving at 0.6c with respect to the planets. Both frames measure each object in the other frame cross the distance in 1.67 years, but with the moving objects' clocks showing 1.33 years elapsed. Both can explain that by saying the other frame measures the "entire distance" as only 0.8 light years. As @Dale points out, these symmetric explanations are available to both because, due to the relativity of simultaneity, the two frames are actually defining different lines in spacetime to be "the distance" between a pair of objects.

 

[Demystifier]

But why do we see only the traveler’s ship as length-contracted,

Any observer sees contracted all things that move relative to this observer.

whereas the traveler sees the entire distance between us and him as length-contracted?

The distance between two things is not a thing, so it is not seen contracted. The two things in this case move relative to each other, so the distance between them changes with time. The observer comoving with the first thing will see that the distance changes with time as ##L(t)##. The observer comoving with the second thing will see that the distance changes with time as ##L'(t')##. But there is a complete symmetry, because ##L## and ##L'## are really the same functions, in the sense that ##L(u)=L'(u)## for any real number ##u##.

Doesn’t this suggest a preferred reference frame for the Earthly observer?

No.

 

[Mister T]

We will disagree with the traveler on how much time will have elapsed when he passes Earth.

You need two events to define an elapsed time between them. You have specified only one event: traveler passes Earth. You need another event, otherwise the notion of an elapsed time is meaningless.

The other responses have given you examples of the other event, and shown you the consequences.

 

[substitute materials]

So the problem is synchronizing clocks to define the beginning of the travelers journey, yes? In Ibis' 2 planet example we can use the passing of the distant planet as the starting point, but we will disagree about when that happened, correct?

If we extend the buoy from Nugatory's response to become a rope stretched between the Earth and the traveler's initial location, at rest with respect to Earth, I understand that it would appear shorter to the traveler than to the Earth observer. I can make this work in my head. However if the rope was at rest with respect to the traveler, it should appear shorter to the Earthbound observer. But if that's the case, it's tantamount to saying the Earth bound observer sees the traveler as closer than the traveler sees them, which doesn't seem right. Will they disagree, using numbers from Ibix's example, by .33 years as to when the rope actually spanned the distance between them? Is this how simultaneity solves the problem? So I've turned the twin paradox into the ladder paradox.

 

[Nugatory]

However if the rope was at rest with respect to the traveler, it should appear shorter to the Earthbound observer.

Yes, that is correct.

Will they disagree, using numbers from Ibix's example, by .33 years as to when the rope actually spanned the distance between them? Is this how simultaneity solves the problem?

I haven't checked the numbers to see if .33 is the right answer, but the principle is exactly as you say - because of relativity of simultaneity they find that the rope spanned the distance between them at different time, and of course the distance between them will be different at different time.

So I've turned the twin paradox into the ladder paradox.

Yes, that's pretty much what you've done. The ladder paradox was invented to demonstrate the relativity of simultaneity, whereas the twin paradox avoids the simultaneity issue by comparing only colocated clocks.
(There is still a simultaneity issue in the twin paradox when we ask what number appears on the earth twin's wristwatch at the same time that the traveler turns around, when we can use any of three frames - eart at rest, traveler at rest during the outbound leg, traveler at rest during the inbound leg - to define "at the same time")