Question about length contraction

[Nathan123]

I have seen in SR that regular time and regular distance is equal to slower time and less distance because of length contraction.
Say a person on Earth sees a ship moving towards it. This reference frame has regular time and regular length and it views the ship's reference frame as having slower time and less distance because the ship sees everything moving so the distance is contacted.

My issue with this is that for the ship's reference frame, it is the opposite. The ship's reference frame has regular time and less distance because it sees everything moving so distance is contracted and it views the Earth's reference frame as having slower time and regular distance.

So what is the point of converging time and distance for the Earth reference frame when for the ship it is the exact opposite?

 

[Ibix]

My issue with this is that for the ship's reference frame, it is the opposite. The ship's reference frame has regular time and less distance because it sees everything moving so distance is contracted and it views the Earth's reference frame as having slower time and regular distance.

This is not correct. The point of the principle of relativity is that you may always regard yourself at rest - so the spaceship also measures the Earth's clocks and rulers as ticking slowly and contracted because it regards the Earth as moving. The situation is symmetric. If that seems paradoxical look up the relativity of simultaneity.

So what is the point of converging time and distance for the Earth reference frame when for the ship it is the exact opposite?

"Converging" is not a term I recognise in this context. But generally there isn't a point to this kind of thing. It's just the way things are.

 

[Nathan123]

This is not correct. The point of the principle of relativity is that you may always regard yourself at rest - so the spaceship also measures the Earth's clocks and rulers as ticking slowly and contracted because it regards the Earth as moving.

That is not true for length contraction. Length contraction is for moving things. The space ship's reference frame always has distance between ship and Earth contracted because it sees everything moving towards it. The Earth only sees the ship moving so it never has distance contraction between ship and earth

 

[Ibix]

That is not true for length contraction. Length contraction is for moving things. The space ship's reference frame always has distance between ship and Earth contracted because it sees everything moving towards it. The Earth only sees the ship moving so it never has distance contraction between ship and earth

All rulers are length contracted when viewed from a frame in which they are moving. The distance from Earth to some point fixed with respect to the Earth is defined by rulers at rest with respect to the Earth, so it is length contracted when viewed from the ship. The same can be said (in reverse) of a distance measured from the ship to a point fixed with respect to the ship.

I think you are adopting the Earth's frame when measuring the distance from the Earth to the ship, and not realising that you can adopt the ship's frame to do it. The difference of opinion boils down to the relativity of simultaneity, as I said. I suggest writing down the coordinates of the Earth and ship at some chosen time in the Earth frame and transforming them into the ship frame. What do you notice about the time coordinates?

 

[PeroK]

That is not true for length contraction. Length contraction is for moving things.

There is no such thing as a "moving thing". All motion is relative. The ship is moving in the Earth's reference frame; and, the Earth is moving in the ship's reference frame.

 

[Nugatory]

That is not true for length contraction. Length contraction is for moving things. The space ship's reference frame always has distance between ship and Earth contracted because it sees everything moving towards it. The Earth only sees the ship moving so it never has distance contraction between ship and earth

You are misunderstanding how length contraction works, almost certainly because you are overlooking the relativity of simultaneity.

To get rid of the confusion, we have to be more precise about exactly which distances we're talking about. Suppose that we could attach a kilometer long ruler to the ship and another one to the earth, pointing towards one another. As the ship flies towards the Earth the ship is lined up with the end of the Earth ruler; using the frame in which the Earth and its ruler are at rest we will say that the distance between the ship and the Earth is one kilometer Because of length contraction, the ship's ruler will be less than one kilometer long so its tip will not yet have reached the earth; if the relative speed between ship and Earth is .6c the tip of the ship ruler will line up with the 200 meter mark on the Earth ruler and we will say that at the same time that the ship is one kilometer from Earth the tip of the ship ruler is 200 meters from the earth. Clearly this result is consistent with the ship ruler being length-contracted by a factor of .8 while the Earth ruler is not length-contracted.

Now let's consider the situation using the frame in which the ship is at rest and the Earth is moving towards the ship. At the same time that the ship is lined up with the tip of the Earth ruler the end of the ship ruler is not lined with the 200 meter mark on the Earth ruler. That is, the two events "ship lines up with tip of Earth ruler" and "tip of ship ruler lines up with 200 meter mark on Earth ruler" are happen at the same time using the frame in which the Earth is at rest, but do not happen at the same time using the frame in which the ship is at rest. This is the relativity of simultaneity - "at the same time" is different in different frames - which is essential for making sense of relativity. In fact, using the frame in which the ship is at rest, at the same time that the ship lines up with the tip of the Earth ruler the tip of the ship ruler is sticking out 200 meters past the far end of the Earth ruler, and the end of the Earth ruler lines up with the 800 meter mark on the ship ruler. This result is consistent with the Earth ruler being length-contracted by a factor of .8 while the ship ruler is not length-contracted.

The situation is completely symmetrical and both descriptions are equally correct. Which one you prefer depends on whether you prefer to think of the ship at rest while the Earth moves towards it, or the Earth at rest while the ship moves toward it.

 

[Nathan123]

There is no such thing as a "moving thing". All motion is relative. The ship is moving in the Earth's reference frame; and, the Earth is moving in the ship's reference frame.

That is not true. Each reference frame is at rest for itself. It then looks out and sees things moving around. Sometimes it sees not just things moving around but it sees everything moving. Try looking out your car window, you will see everything moving toward you. In any case, whatever you view moving will be contracted.

 

[Nathan123]

You are misunderstanding how length contraction works, almost certainly because you are overlooking the relativity of simultaneity.

To get rid of the confusion, we have to be more precise about exactly which distances we're talking about. Suppose that we could attach a kilometer long ruler to the ship and another one to the earth, pointing towards one another. As the ship flies towards the Earth the ship is lined up with the end of the Earth ruler; using the frame in which the Earth and its ruler are at rest we will say that the distance between the ship and the Earth is one kilometer Because of length contraction, the ship's ruler will be less than one kilometer long so its tip will not yet have reached the earth; if the relative speed between ship and Earth is .6c the tip of the ship ruler will line up with the 200 meter mark on the Earth ruler and we will say that at the same time that the ship is one kilometer from Earth the tip of the ship ruler is 200 meters from the earth. Clearly this result is consistent with the ship ruler being length-contracted by a factor of .8 while the Earth ruler is not length-contracted.

This much I was able to follow. Let me try to present the situation clearly and explain what is bothering me. I will ignore the length contraction of the ship itself because it is not significant to the issue I am having.

Note that every situation has 4 calculations:
1. My perspective (regular time for me).
2. My perspective of a different reference frame (whereby its time is slower for me).
3. The other person's reference frame (regular time for him).
4.The other person's perspective of my reference frame (whereby my time is slower for him).

1. I sit on Earth and see a ship coming towards Earth really fast. It is say 1 million miles away (regular distance). Yes, the ship is contracted, but I am interested in how long it will take to get here. So the ship's contraction is fairly meaningless. The main point is the distance between me and the ship (that is very far away) and the speed. So let's say it will get to Earth in 1 year. In my reference frame, there is no time dilation and no distance contraction of the space between me and the ship.
2. Now I compare the ship's reference frame to mine. I calculate that it will reach Earth from that perspective at the same time as the previous calculation. Even though it takes 1 year to get here and that reference frame's time is slower than mine, that reference frame's distance is also less because it sees the whole space as moving. So regular time/regular distance = slower time/shorter distance.

The problem comes with the third and fourth calculation that everyone seems to ignore.

3. The person on the ship does not have slower time for himself. He only has slower time for me. For himself, he has regular time. But he still sees everything coming towards him, so there is distance contraction for him regardless. This perspective has the best of everything to get to Earth fast. It has regular time and also has distance contraction. So in this perspective, the ship will get here faster than 1 year since it not only has regular time but also distance contraction.
4. When the guy on the ship compares my perspective to his, I have slower time but no distance contraction. So this calculation has the ship taking longer than 1 year to get here.

So it's nice to align things from my perspective and say that regular time/regular length = slower time/shorter length. But this does not work for the guy on the ship. He will have regular time and shorter distance and he will view that I have slower time and regular distance.

I think the whole issue revolves around the point I am making that each situation has 4 calculations not 2. Everyone seems to ignore 3 and 4, and that is where the problem lies.

 

[Mentz114]

There is no contradiction. Your points 3 and 4 are superfluous. 1 and 2 are connected by a Lorentz transformation. I've attached two space time diagrams. The 'regular' frames of the two receeding onjects is plotted. The ticks on the worldlines are the clock ticks. Note that the ticks are spaced wider on the tilted worldline.

 

[Nathan123]

There is no contradiction. Your points 3 and 4 are superfluous. .

My whole issue arises from calculation 3 and 4. My whole point is that it is not superfluous. So saying it's superfluous does not answer my question, when I am trying very hard to show that it is not superfluous.

 

[Mentz114]

My whole issue arises from calculation 3 and 4. My whole point is that it is not superfluous. So saying it's superfluous does not answer my question, when I am trying very hard to show that it is not superfluous.

If you swap 'My' with 'the other persons' they are the same - which is what the LT does - as the diagram shows.

 

[Grinkle]

That is not true.

You didn't make any case that I can see for @Nugatory 's post being incorrect. Its not at all clear to me which of your comments you believe contradict his post.

 

[Nathan123]

You didn't make any case that I can see for @Nugatory 's post being incorrect. Its not at all clear to me which of your comments you believe contradict his post.

He said there is no such thing as a moving thing. There sure is. You are not moving in your own perspective, but other things are, and those other things are length contracted.

 

[Grinkle]

My whole issue

Is your issue:

You don't understand how clocks synch up when two frames finally come to rest with each other
If so, read this - https://www.physicsforums.com/insights/geometrical-view-time-dilation-twin-paradox/

You don't believe its possible for two observers to both see each other's clocks running more slowly than their own clock
If so, not sure where to point you - there are numerous summaries of SR to look at.

 

[Grinkle]

He said there is no such thing as a moving thing. There sure is.

When you say there is, you are implicitly assuming the existence of a preferred "at rest" frame against which something can be judged to be moving.

A statement as to whether an object is moving or at rest is only valid in the context of a specific reference frame, it is not valid in any absolute frame-free context.

 

[Nathan123]

If you swap 'My' with 'the other persons' they are the same - which is what the LT does - as the diagram shows.

They are not the same.
I pointed out that length contraction is always for the ship's perspective, since it sees everything moving.
Time dilation is only for the other perspective compared to mine, but not for his for himself.
So:
1. Earth guy for himself. No time dilation. No length contraction.
2. Earth guy's view of ship. Yes time dilation. Yes length contraction.
3. Ship guy for himself. No time dilation. Yes length contraction.
4. Ship guy's view of earth. Yes time dilation. No length contraction.

You can see that 1 and 2 are not equivalent to 3 and 4.

 

[Nathan123]

When you say there is, you are implicitly assuming the existence of a preferred "at rest" frame against which something can be judged to be moving.

A statement as to whether an object is moving or at rest is only valid in the context of a specific reference frame, it is not valid in any absolute frame-free context.

No I am not assuming anything. I just said that whoever sees something moving will see it contracted. Nothing to argue about on this.

 

[Grinkle]

I am not assuming anything

If you don't comprehend that a claim "there is such a thing as a moving object" implicitly assumes a preferred frame, it is not possible to progress further in understanding SR.

 

[Nathan123]

Is your issue:

You don't understand how clocks synch up when two frames finally come to rest with each other
If so, read this - https://www.physicsforums.com/insights/geometrical-view-time-dilation-twin-paradox/

You don't believe its possible for two observers to both see each other's clocks running more slowly than their own clock
If so, not sure where to point you - there are numerous summaries of SR to look at.

I don't think what you are saying is my issue at all.
I am showing 4 calculations.
1. Mine
2, His compared to mine.
3. His for himself.
4. Mine compared to his (for him).
I show how 1 and 2 are opposite of 3 and 4.
Since I saw an explanation of length contraction to explain how 1 and 2 coincide, I am bothered by that 3 and 4 do not coincide.

 

[Grinkle]

I don't think what you are saying is my issue at all.

Fair enough - maybe I just don't understand what you are saying. Seems like a twin paradox question to me, but I could be wrong.

In any case, reading the twin paradox article wouldn't be a waste of time!

 

[Nathan123]

If you don't comprehend that a claim "there is such a thing as a moving object" implicitly assumes a preferred frame, it is not possible to progress further in understanding SR.

In my original post, I specifically said that the ship sees everything moving so the distance is contacted. I was clearly speaking from it's reference frame seeing everything move.

 

[Nathan123]

Fair enough - maybe I just don't understand what you are saying. Seems like a twin paradox question to me, but I could be wrong.

In any case, reading the twin paradox article wouldn't be a waste of time!

This isn't the twin paradox. Since I saw that distance contraction counteracts time dilation to coincide nicely, I am bothered that it does not coincide from the perspective of the guy on the ship.

 

[Grinkle]

the ship sees everything moving

I agree with the above (except for unimportant exceptions of things that are co-moving with the ship), and I contend it is different and much less strong a claim than saying "there is such a thing as a moving object".

I think you would agree that for any object that is moving in the frame of the ship, I can pick a frame for which the object is at rest, and there is no absolute frame to resolve the discrepancy - both are valid claims in the context of the right reference frame.

 

[Nathan123]

I agree with the above (except for unimportant exceptions of things that are co-moving with the ship), and I contend it is different and much less strong a claim than saying "there is such a thing as a moving object".

I think you would agree that for any object that is moving in the frame of the ship, I can pick a frame for which the object is at rest, and there is no absolute frame to resolve the discrepancy - both are valid claims in the context of the right reference frame.

I don't think we disagree about relative movement. I was trying to point out that the ship will always see "everything" moving, so it will always have distance contraction in that perspective.

Time dilation works differently. It only applies to my comparison of your reference to mine (2 and 4). It does not apply to mine for myself or yours for yourself (1 and 3). Length contraction applies to your reference whether it's you for yourself or mine to yours (2 and 3). That is why 1 and 2 coincide, while 3 and 4 diverge. And that is what I am bothered about.

 

[Mentz114]

They are not the same.
I pointed out that length contraction is always for the ship's perspective, since it sees everything moving.
Time dilation is only for the other perspective compared to mine, but not for his for himself.
So:
1. Earth guy for himself. No time dilation. No length contraction.
2. Earth guy's view of ship. Yes time dilation. Yes length contraction.
3. Ship guy for himself. No time dilation. Yes length contraction.
4. Ship guy's view of earth. Yes time dilation. No length contraction.

You can see that 1 and 2 are not equivalent to 3 and 4.

Length contraction and time dilation are frame dependent illusions caused by loss of simultaneity.
Every clock runs at 1 sec per sec and everyone is at rest in their own coordinates. Lengths do not change.

The LT and proper time are the only useful things.

 

[Nathan123]

Length contraction and time dilation are frame dependent illusions caused by loss of simultaneity.
Every clock runs at 1 sec per sec and everyone is at rest in their own coordinates. Lengths do not change.
.

Time dilation and length contraction are frame dependent in different ways, which is the basis of my issue.

 

[Mentz114]

Time dilation and length contraction are frame dependent in different ways, which is the basis of my issue.

Why should they be the same ?

On the ST diagram each observer sees the other clock apparently running slower in a symmetric way but the number of ticks remains the same.
Length contraction is a comparison between measuring both ends simultaneously and non-simultaneously. It has no physical significance.

 

[Janus]

This much I was able to follow. Let me try to present the situation clearly and explain what is bothering me. I will ignore the length contraction of the ship itself because it is not significant to the issue I am having.

Note that every situation has 4 calculations:
1. My perspective (regular time for me).
2. My perspective of a different reference frame (whereby its time is slower for me).
3. The other person's reference frame (regular time for him).
4.The other person's perspective of my reference frame (whereby my time is slower for him).

1. I sit on Earth and see a ship coming towards Earth really fast. It is say 1 million miles away (regular distance). Yes, the ship is contracted, but I am interested in how long it will take to get here. So the ship's contraction is fairly meaningless. The main point is the distance between me and the ship (that is very far away) and the speed. So let's say it will get to Earth in 1 year.

1,000,000 miles in one year only works out to be ~114 mph, fast for a car maybe, but hardly a relativistic speed

In my reference frame, there is no time dilation and no distance contraction of the space between me and the ship.

In other words, you are using your own clock and the 1,000,000 miles is as measured by your measuring stick.

2. Now I compare the ship's reference frame to mine. I calculate that it will reach Earth from that perspective at the same time as the previous calculation. Even though it takes 1 year to get here and that reference frame's time is slower than mine, that reference frame's distance is also less because it sees the whole space as moving. So regular time/regular distance = slower time/shorter distance.

[I'm not even sure why you are even bringing distance into this. The ship took 1 yer to reach you by your clock, and the ship clock, having run slow, during that time ticked off less than a year. The slower time/shorter distance bit is invalid because you are trying to divide something measured in one frame by something measured in another. This is called "frame mixing" and is a no-no.

The problem comes with the third and fourth calculation that everyone seems to ignore.

3. The person on the ship does not have slower time for himself. He only has slower time for me. For himself, he has regular time. But he still sees everything coming towards him, so there is distance contraction for him regardless. This perspective has the best of everything to get to Earth fast. It has regular time and also has distance contraction. So in this perspective, the ship will get here faster than 1 year since it not only has regular time but also distance contraction.

the ship frame measures the distance and time by his own measuring stick/clock. He measures the distance as shorter than 1,000,000 miles, so he naturally measures the trip as taking less than 1 year by his own clock.

4. When the guy on the ship compares my perspective to his, I have slower time but no distance contraction. So this calculation has the ship taking longer than 1 year to get here.

Your clock will run slow compared to the ship clock as measured from the ship, thus the ship will measure your clock as having advanced even less than his during the trip. I don't no where you got the longer than a year from.[/quote]

So it's nice to align things from my perspective and say that regular time/regular length = slower time/shorter length. But this does not work for the guy on the ship. He will have regular time and shorter distance and he will view that I have slower time and regular distance.

I think the whole issue revolves around the point I am making that each situation has 4 calculations not 2. Everyone seems to ignore 3 and 4, and that is where the problem lies.[/QUOTE]

No one is ignoring anything.
What you are missing is the relativity of simultaneity. If we assume that your clock starts at 0 when the ship is exactly 1,000,000 miles away. And the ship clock reads zero as it passes that point (assume we have a buoy sitting out 1,000,000 miles from Earth as measured from the Earth and the ship clock reads 0 when it passes it), then:
According to you, the ship clock starts reading zero when your clock reads zero, and as the ship travels between the buoy and yourself, your clock advances by 1 year and the ship clock advances by less than 1 year. Thus your clock will read 1 year and the ship clock will read less than 1 year when you meet.
According to the ship, its clock reads 0 when it passes the buoy, which is less than 1,000,000 miles from you. However, your clock will not read 0 at that moment, but some time after 0, due to the relativity of simultaneity. The ship clock advances less than one year and reads less than 1 year when it meets up with you. Your clock advance even less than that, but because it started out some time past 0 when you passed the buoy, that time plus the time is advanced will equal 1 year and your clock will read 1 yr while the ship clock reads less than 1 year when the ship and you meet up. The same conclusion you came to.

 

[Nathan123]

[I'm not even sure why you are even bringing distance into this. The ship took 1 yer to reach you by your clock, and the ship clock, having run slow, during that time ticked off less than a year. The slower time/shorter distance bit is invalid because you are trying to divide something measured in one frame by something measured in another. This is called "frame mixing" and is a no-no. the ship frame measures the distance and time by his own measuring stick/clock. He measures the distance as shorter than 1,000,000 miles, so he naturally measures the trip as taking less than 1 year by his own clock. Your clock will run slow compared to the ship clock as measured from the ship, thus the ship will measure your clock as having advanced even less than his during the trip. I don't no where you got the longer than a year from.

What you are missing is the relativity of simultaneity. If we assume that your clock starts at 0 when the ship is exactly 1,000,000 miles away. And the ship clock reads zero as it passes that point (assume we have a buoy sitting out 1,000,000 miles from Earth as measured from the Earth and the ship clock reads 0 when it passes it), then:
According to you, the ship clock starts reading zero when your clock reads zero, and as the ship travels between the buoy and yourself, your clock advances by 1 year and the ship clock advances by less than 1 year. Thus your clock will read 1 year and the ship clock will read less than 1 year when you meet.
According to the ship, its clock reads 0 when it passes the buoy, which is less than 1,000,000 miles from you. However, your clock will not read 0 at that moment, but some time after 0, due to the relativity of simultaneity. The ship clock advances less than one year and reads less than 1 year when it meets up with you. Your clock advance even less than that, but because it started out some time past 0 when you passed the buoy, that time plus the time is advanced will equal 1 year and your clock will read 1 yr while the ship clock reads less than 1 year when the ship and you meet up. The same conclusion you came to.

When I wrote slower time/shorter distance I wasn't dividing. let me rephrase.
[No time dilation and regular distance] is equivalent to [yes time dilation and distance contraction]. The point here is that from my perspective, whether I calculate for myself or whether I calculate for the ship's frame of reference in relation to mine, the end result is the same. The ship will reach Earth simultaneously for both calculations because the slower time (not a whole year passed) is offset by the shorter distance.

Note, I am not trying to be smug and argue. I am trying to understand. I will be more than happy when I see what I am missing. So far, I have not found it. But perhaps with your last paragraph, I will.

I have been making the point that calculation 3 and 4 do not coincide as do 1 and 2, but they actually diverge.

Let's try to clarify this.

When the ship reaches Earth from its own perspective, less than 1 year have passed, because of distance contraction.
When the ship calculates my perspective, I am saying that less time has passed (so it didn't go as far) and it has more distance to go because it doesn't have length contraction.
So whereas 1 and 2 (mine for me, his for me) reach Earth simultaneously, 3 & 4 (his for him, mine for him) do not.

This is not the same as relatively of simultaneity which states that my frame for me and your frame for me don't always align.

 

[bahamagreen]

I don't think we disagree about relative movement. I was trying to point out that the ship will always see "everything" moving, so it will always have distance contraction in that perspective.

Time dilation works differently. It only applies to my comparison of your reference to mine (2 and 4). It does not apply to mine for myself or yours for yourself (1 and 3). Length contraction applies to your reference whether it's you for yourself or mine to yours (2 and 3). That is why 1 and 2 coincide, while 3 and 4 diverge. And that is what I am bothered about.

Nathan123, I think the question you are posing is identical to the famous one about the dilemma of the relativistic muons. The link goes to an examination of this question comparing the different frames of reference.

http://hyperphysics.phy-astr.gsu.edu/hbase/Relativ/muon.html

 

[PeterDonis]

I am not trying to be smug and argue. I am trying to understand.

Then do the math. You are confusing yourself because you are trying to reason using vague ordinary language instead of precise math. You need to stop doing that.

This is not the same as relatively of simultaneity which states that my frame for me and your frame for me don't always align.

That's not what relativity of simultaneity says. Do the math. Explicitly write down coordinates in one frame for all events of interest. Then use the Lorentz transformation to obtain coordinates for the same events in the other frame. Then look at the coordinates in the two frames to see how time dilation, length contraction, and relativity of simultaneity actually work.

 

[Nathan123]

Then do the math. You are confusing yourself because you are trying to reason using vague ordinary language instead of precise math. You need to stop doing that.
That's not what relativity of simultaneity says. Do the math. Explicitly write down coordinates in one frame for all events of interest. Then use the Lorentz transformation to obtain coordinates for the same events in the other frame. Then look at the coordinates in the two frames to see how time dilation, length contraction, and relativity of simultaneity actually work.

I don't have the background to do the math. I am a novice trying to understand to the best of my ability. I am using common terms to try and keep it simple.

 

[PeterDonis]

I don't have the background to do the math.

Then you don't have the background to analyze the scenario you're posing correctly. The math is really not that difficult; taking the time to learn it will be a more productive use of your time than trying to reason about this using vague ordinary language.

The short answer to the questions you are posing is that you cannot just look at length contraction or time dilation in isolation. A correct analysis requires taking into account length contraction, time dilation, and relativity of simultaneity. Leaving out anyone of those three will mislead you. And trying to take all three of those things into account correctly without using math is going to be a lot harder than just learning the math so you can use it.

 

[Nugatory]

So it's nice to align things from my perspective and say that regular time/regular length = slower time/shorter length. But this does not work for the guy on the ship. He will have regular time and shorter distance and he will view that I have slower time and regular distance.

You are still failing to consider relativity of simultaneity in your steps 3 and 4. When you correct this mistake you will find that both frames work properly and symmetrically.

Using the Earth frame: When the ship is lined up with the end of the Earth ruler, it is one kilometer away from the Earth and the ship moves through that distance to reach the Earth in time ##(1 km)/.6c## seconds according to "regular time" on earth.

Using the ship frame: When the ship is lined up with the end of the Earth ruler, the Earth is .8 kilometers away from the ship and the Earth moves through that distance to reach the ship in ##(.8 km)/.6c## seconds according to a "regular time" on the ship.

This is not the same as relatively of simultaneity which states that my frame for me and your frame for me don't always align.

It is not the same as relativity of simultaneiity, but relativity of simultaneity is essential to understanding this problem. The best way to do this is to do as @PeterDonis suggests above and do the math (when you do, you will see why I chose ##v=.6c## - that value makes the arithmetic particularly easy, which is how I was able to do the 200 meter and 800 meter calculations above in my head).

But before you do that, I can point you to the reason why the relativity of simultaneity matters: You are starting from the correct statement that the point that is one kilometer from Earth in the Earth frame is 800 meters from Earth in the ship frame and assuming (whether you realize it or not), that at the moment in the Earth frame that the ship and the Earth are one kilometer apart they will be 800 meters apart in the ship frame. They aren't.

 

[Nathan123]

You are still failing to consider relativity of simultaneity in your steps 3 and 4. When you correct this mistake you will find that both frames work properly and symmetrically.

Using the Earth frame: When the ship is lined up with the end of the Earth ruler, it is one kilometer away from the Earth and the ship moves through that distance to reach the Earth in time ##(1 km)/.6c## seconds according to "regular time" on earth.

Using the ship frame: When the ship is lined up with the end of the Earth ruler, the Earth is .8 kilometers away from the ship and the Earth moves through that distance to reach the ship in ##(.8 km)/.6c## seconds according to a "regular time" on the ship.

Something seems wrong here. If it is 1 km at .6c, then .8 km takes less time, not the same amount of time. It will be equivalent to .6c of the Earth's regular time, but it will take less of its own time, which happens to be slower.

In any case, we keep coming back to 1 and 2 when I am trying to understand 3 and 4. I am hearing that relativity of simultaneity is the answer (and I'm afraid it might get overused because it seems to answer everything). But I am not sure exactly where it comes into play here.

So let's try to stick to 3 and 4 and show me where relativity of simultaneity kicks in.

3. Man on ship sees regular time for himself, no time dilation. He also sees a shorter distance to earth.
4. He compares that to the other frame that has slower time and no distance contraction.

So if he gets there before his year is up because of distance contraction, the other person in his estimation will first take at least a year, because there is no distance contraction, and it will take even longer than that in comparison to him because the other person's clock runs slower.

please tell me where relativity of simultaneity kicks in here.

If the answer is just that it's ok that the simultaneity diverges for 3 and 4 even though it does not for 1 and 2, then just say so.

The reason why saying so bothers me is because it not the standard relativity of simultaneity that applies to 1 and 2 as well. Secondly, I often see 1 and 2 so neatly explained, ignoring that 3 and 4 diverge.