Undergrad How Does Special Relativity Affect Perceived Distances Between Planets?

[dwspacetime]

Here's a spacetime diagram showing Tom and Mary. It's drawn using Tom's frame, so Tom is at rest.
View attachment 336789
If you haven't come across spacetime diagrams before, they are "maps" of spacetime, with time (as defined in this frame) drawn up the page and space (again, as defined in this frame) across it. Tom, therefore, appears as a vertical line (red on the diagram) because he doesn't change his position in this frame. Mary is a slanted line (blue on the diagram) because she is moving. Notice that Tom has been placed at 1ly, and Mary reaches 4ly from him at t=1. Let's add a line showing Tom's idea of "all of space at the moment that Mary is 4ly away from him":
View attachment 336790
Tom's idea of space is the fine red line, and you can see that Mary is 4ly away. However, Mary does not share this definition of "space" because she does not share the same notion of "at the same time". Her version of "all of space at the same time" is shown as a fine blue line here:
View attachment 336791
Notice that the fine blue line passes through Mary at the same time as the red line, but it does not pass through Tom at the same time as the red line. This is why they measure different distances - they are measuring along different slices of spacetime, and they are doing that because they don't share a notion of "at the same time". The interval along the fine blue line between the two thick lines is 2.4 ly.

To complete the picture, we can also draw the same graph in Mary's frame:
View attachment 336792
Now Mary is represented by a vertical line and her "all of space" is a horizontal line. You can read off the horizontal axis that the spacing is 2.4 ly, but again, Tom is measuring something different.

Your problem is that you keep hoping you can pretend the different slopes of the two fine lines don't matter. They do.

Sorry I just came back on this. I disagree. All your explanations is based on that you already decided that Tom is stationary which already eliminates the paradox. Let's forget about the two planets and Tom and Mary are moving towards each other or Tom is moving towards Mary. We can do the same as above but think the red line is Mary and the blue line is Tom. What we get "NOW"? I guess the planets just confused you. they can be any other moving things as Jane and Dawn or you can image another two planets C and D are moving relative to A and B. or A and B are moving relative C and D.

 

[PeroK]

A and B are moving relative C and D.

The simplest scenario is that A and B pass each other with a relative speed of $0.8c$.

In the rest frame of A: when A's clock reads 4 seconds, B's clock reads 2.4s.

In the rest frame of B: when B's clock reads 4 seconds, A's clock reads 2.4s. Or, alternatively, when B's clock reads 2.4s, A's clock reads 1.44s.

Are you on board with that?

 

[dwspacetime]

The simplest scenario is that A and B pass each other with a relative speed of $0.8c$.

In the rest frame of A: when A's clock reads 4 seconds, B's clock reads 2.4s.

In the rest frame of B: when B's clock reads 4 seconds, A's clock reads 2.4s. Or, alternatively, when B's clock reads 2.4s, A's clock reads 1.44s.

Are you on board with that?

I do not know what you try to say. yes I get that. So what is wrong with making the red line Mary? if the red line is Mary Tom is 4 lys instead of 2.4 lys away

 

[PeroK]

I do not know what you try to say. yes I get that.

Okay, so it follows that:

In the rest frame of A: when A's clock reads 4s, B is 3.2 light seconds from A (and B's clock reads 2.4s).

In the rest frame of B: when B's clock reads 2.4s, A is 1.92 light seconds from B (and A's clock reads 1.44s).

Are you okay with that?

 

[Ibix]

All your explanations is based on that you already decided that Tom is stationary

No - it's all based on your personal choice to start the experiment when Mary is 4ly away from Tom using Tom's rest frame. That choice that you made is what creates the asymmetry in this set up. There is no paradox.

Histspec's diagram in #28 illustrates it nicely, using a frame where Tom and Mary are moving at equal and opposite speeds. There is asymmetry because you chose to key "time zero" off an event on Mary's worldline, but Tom's "time zero" is only the same as Mary's at that one event.

Again: the asymmetry is introduced by you in your experimental setup. It's got nothing to do with me "deciding Tom is at rest". If that is not clear to you then study the spacetime diagrams until it becomes clear. Probably the most important thing to grasp is the difference between Tom's "time zero" and Mary's "time zero". That's the bit that evades a lot of people, and that failing is where most paradoxes spring from.

 

[dwspacetime]

The simplest scenario is that A and B pass each other with a relative speed of $0.8c$.

In the rest frame of A: when A's clock reads 4 seconds, B's clock reads 2.4s.

In the rest frame of B: when B's clock reads 4 seconds, A's clock reads 2.4s. Or, alternatively, when B's clock reads 2.4s, A's clock reads 1.44s.

Are you on board with that?

the reason it is a paradox is because we can not decide which is stationary unless using a reference frame independent of Tom and Mary which are the planets you unconsciously take which already eliminates the paradox and all the rest of spacetime conversion is just a waste of time.

 

[PeroK]

the reason it is a paradox is because we can not decide which is stationary unless using a reference frame independent of Tom and Mary which are the planets you unconsciously take which already eliminates the paradox and all the rest of spacetime conversion is just a waste of time.

I take it from that you are not okay with my simple analysis. Can you say what you think is wrong with my analysis? There are no planets in my analysis. Only A and B moving relative to each other. No one is "stationary".

 

[Ibix]

the reason it is a paradox is

There is no paradox.

we can not decide which is stationary unless using a reference frame independent of Tom and Mary

None of them is stationary in any absolute sense; that's one of the postulates on which the whole theory is based! So you can choose to work in a frame where either Tom or Mary or neither to be stationary. All three choices have been illustrated in this thread, and PeroK is trying to walk you through it with numbers.

 

[dwspacetime]

Okay, so it follows that:

In the rest frame of A: when A's clock reads 4s, B is 3.2 light seconds from A (and B's clock reads 2.4s).

In the rest frame of B: when B's clock reads 2.4s, A is 1.92 light seconds from B (and A's clock reads 1.44s).

Are you okay with that?

no. I disagree. what is wrong with saying

In the rest frame of B: when B's clock reads 4s, A is 3.2 light seconds from B (and A's clock reads 2.4s).

or simply put, just simply replacing A with B, all you said still stands? why not. When Tom measures Mary should X lys to reach him. Mary should measure Tom should take X lys to reach her too.

 

[PeroK]

no. I disagree. what is wrong with saying

In the rest frame of B: when B's clock reads 4s, A is 3.2 light seconds from B (and A's clock reads 2.4s).

That's true also.

or simply put, just simply replacing A with B, all you said still stands?

Yes, you can swap A and B and all statements are also true. The scenario is symmetrical between A and B.

 

[dwspacetime]

There is no paradox.

None of them is stationary in any absolute sense; that's one of the postulates on which the whole theory is based! So you can choose to work in a frame where either Tom or Mary or neither to be stationary. All three choices have been illustrated in this thread, and PeroK is trying to walk you through it with

There is no paradox.

None of them is stationary in any absolute sense; that's one of the postulates on which the whole theory is based! So you can choose to work in a frame where either Tom or Mary or neither to be stationary. All three choices have been illustrated in this thread, and PeroK is trying to walk you through it with numbers.

if I choose Mary as stationary as red line as you drew. Tom should be 4 liys away?

 

[Ibix]

why not.

The relativity of simultaneity: Tom and Mary do not have a shared global notion of what "all of space, now" means. This is what all of the fine red and blue lines on the diagrams show - each one is the slice of spacetime they call "space, now". And, as the diagram in #28 illustrates, those different definitions of "space, now" mean that "distance, now" is different.

This point seems to be the hardest part of relativity for people to grasp, and failing to understand it is how almost all "paradoxes" arise.

 

[PeroK]

Okay, so it follows that:

In the rest frame of A: when A's clock reads 4s, B is 3.2 light seconds from A (and B's clock reads 2.4s).

In the rest frame of B: when B's clock reads 2.4s, A is 1.92 light seconds from B (and A's clock reads 1.44s).

Are you okay with that?

To emphasise the point. It also follows that:

In the rest frame of B: when B's clock reads 4s, A is 3.2 light seconds from B (and A's clock reads 2.4s).

In the rest frame of A: when A's clock reads 2.4s, B is 1.92 light seconds from A (and B's clock reads 1.44s).

 

[dwspacetime]

The relativity of simultaneity: Tom and Mary do not have a shared global notion of what "all of space, now" means. This is what all of the fine red and blue lines on the diagrams show - each one is the slice of spacetime they call "space, now". And, as the diagram in #28 illustrates, those different definitions of "space, now" mean that "distance, now" is different.

This point seems to be the hardest part of relativity for people to grasp, and failing to understand it is how almost all "paradoxes" arise.

OK if Mary is stationary as red. does the chart you drew above showing Tom as blue is 4 lys away? why not

 

[Ibix]

View attachment 337735
OK if Mary is stationary as red. does the chart you drew above showing Tom as blue is 4 lys away? why not

Yes it would. But Tom would not say Mary was 4ly away at the same time in that setup, because "same time" means different things to the two of them.

 

[dwspacetime]

Yes it would. But Tom would not say Mary was 4ly away at the same time in that setup, because "same time" means different things to the two of them.

when exacty Mary can say that? I think that is what puzzles me if you are correct

 

[dwspacetime]

Yes it would. But Tom would not say Mary was 4ly away at the same time in that setup, because "same time" means different things to the two of them.

I havent gone through all the replies. I just realized that the distance btw planet A and B are not the same to Tom and Mary. distance is also relative. I guess I consider d btw A and B are the same or absolute as 4 lys to both of them. A and B are stationary with Tom but is not with Mary.

 

[dwspacetime]

Yes it would. But Tom would not say Mary was 4ly away at the same time in that setup, because "same time" means different things to the two of them.

let me ask another question. if both Tom and Mary leave planet A or the same spot at the same time and do all kinds of travels through spacetime with all kinds of speeds (including 0) and accelerations and finally meet again. who is younger?

 

[PeroK]

I havent gone through all the replies. I just realized that the distance btw planet A and B are not the same to Tom and Mary. distance is also relative. I guess I consider d btw A and B are the same or absolute as 4 lys to both of them. A and B are stationary with Tom but is not with Mary.

You don't need planets in your scenario. Planets suggest a natural frame of reference in which the planets are at rest. You don't want that.

Likewise, a spacetime diagram picks out a frame of reference - determined by the specific axes you draw.

That's why I was trying to get you towards a scenario with just A and B moving relative to each other. That scenario has two equally naturals frames of reference - the rest frames of A and B. That allows you to examine the symmetry between A and B.

 

[Ibix]

when exacty Mary can say that? I think that is what puzzles me if you are correct

That depends what you mean by "when" - there are several definitions available. Your confusion comes from the fact that Tom and Mary are using different definitions when they measure different separations "at the same time" and you are implicitly assuming they're using the same one. If they were using the same one then there would be a paradox.

Tom and Mary are going to meet at some point. Both will say that five years before that (by their own clocks) the other one was four light years away. But both will also say that, at the same time they are making this claim the other's clock would show only three years to the meeting. This is how there's no paradox - Tom knows that Mary isn't measuring the same distance at the same time.

if both Tom and Mary leave planet A or the same spot at the same time and do all kinds of travels through spacetime with all kinds of speeds (including 0) and accelerations and finally meet again. who is younger?

You don't need a planet, just for them to meet twice. Pick any inertial frame and write down their velocity as a function of time. Then compute each of their ages using $\int\sqrt{1-v^2/c^2}dt$, where the limits of the integrals are the meeting times specified using your arbitrarily chosen frame. The lower value is the younger one.

 

[dwspacetime]

You don't need planets in your scenario. Planets suggest a natural frame of reference in which the planets are at rest. You don't want that.

Likewise, a spacetime diagram picks out a frame of reference - determined by the specific axes you draw.

That's why I was trying to get you towards a scenario with just A and B moving relative to each other. That scenario has two equally naturals frames of reference - the rest frames of A and B. That allows you to examine the symmetry between A and B.

i guess the reason I was puzzled is because I thought when tom at A and Mary at B the distance is between them is Absolut

That depends what you mean by "when" - there are several definitions available. Your confusion comes from the fact that Tom and Mary are using different definitions when they measure different separations "at the same time" and you are implicitly assuming they're using the same one. If they were using the same one then there would be a paradox.

Tom and Mary are going to meet at some point. Both will say that five years before that (by their own clocks) the other one was four light years away. But both will also say that, at the same time they are making this claim the other's clock would show only three years to the meeting. This is how there's no paradox - Tom knows that Mary isn't measuring the same distance at the same time.

You don't need a planet, just for them to meet twice. Pick any inertial frame and write down their velocity as a function of time. Then compute each of their ages using $\int\sqrt{1-v^2/c^2}dt$, where the limits of the integrals are the meeting times specified using your arbitrarily chosen frame. The lower value is the younger one.

I kind think it has nothing to do with "when". the distance btw A and B is 4 lys for Tom nonmatter when since they are stationary to Tom. but to Mary both A and B are moving towards her, it is always 2.4 lys btw them whenever.
According to your equation, to use either Tom or mary as reference, Tom or Mary would have a v=0, the other one is always younger.

.

 

[Ibix]

According to your equation, to use either Tom or mary as reference, Tom or Mary would have a v=0, the other one is always younger.

They don't meet twice in the scenario we are discussing, so one of the limits of the integral isn't defined. If they do meet twice, at least one of them must accelerate so at least one of them can't always have speed zero in the arbitratily chosen inertial frame.

 

[PeroK]

i guess the reason I was puzzled is because I thought when tom at A and Mary at B the distance is between them is Absolute

It's the "when" that is not absolute. That's what I was trying to emphasise.

Although I see now that you have moved on to a different question altogether.

 

[Orodruin]

TL;DR Summary: Relative distance

Quite simple question. There are two planets A and B, they are 4 light years apart measured by tom on planet A. Say both A and B remain stationary. Mary flys 0.8c to tom on planet A. When she passes planet B she should see Tom or planet A 2.4 light years away, am I correct? What about tom now? Does he see both planet B and Mary 4 light years away? Thanks

Misconceptions regarding special relativity in general and Minkowski space in particular can often be illuminated by considering the Euclidean space equivalent. What you essentially need to know are what the statements correspond to in Euclidean space. Here, in particular, the following is relevant:

The Euclidean statement corresponding to ”When A finds B to be x away” is ”for a straight line A, the line orthogonal to A crossing the line B at a distance of x”. For brevity, let’s just call x this ”the orthogonal distance from A to B”. Note that only the direction of A is important for this definition.

Let’s consider the equivalent of your scenario.

- Draw two parallel lines A and B. (These correspond to your planets)
- Draw a line T on top of A. (This corresponds to Tom)
- Draw a line M that is not parallel to A and B. (This corresonds to Mary) This line necessarily crosses both A and B.
- Draw the line orthogonal to T which crosses M at a distance of 4. Call this line C and its intersection with M you call MC.
- Determine the orthogonal distance from M to T at the point MC. (Ie, draw a line from MC that is orthogonal to M, find how long it must be to intersect T)

You should find that the result in the last step is not 4. This is completely analogous to your setup, just in a different geometry. Whether the result is smaller or larger than 4 will differ between the geometries because of a sign in the geometry definition, but the geometries are similar enough for this to be a direct analogy.

Now remove lines A and B. Does this change the result?

edit: here is a diagram

Your claim is essentially the Minkowski space equivalent of saying that C has the same length as D.

 

[dwspacetime]

They don't meet twice in the scenario we are discussing, so one of the limits of the integral isn't defined. If they do meet twice, at least one of them must accelerate so at least one of them can't always have speed zero in the arbitratily chosen inertial frame.

I am trying to simply it by spreading the acceleration to speed. to meet again some of the speeds have to be negative. but since is v square, so it doesnt affect the result.

my original question is still a paradox. I was assuming A and B stay stationary to Tom. in the meantime we can also assume planet C and D are stationary with Mary 4 lys apart. Then to Tom C and D are 2.4 years apart.... so "NOW" we have 4 planets.

 

[PeroK]

my original question is still a paradox.

It's not a paradox. There's only your failure to understand the basics of SR.

 

[Orodruin]

to meet again the some of the speeds have to be negative

Pet peeve: Speed is never negative. It is the absolute value of velocity (which may be negative depending on direction and chosen coordinates).

 

[dwspacetime]

Pet peeve: Speed is never negative. It is the absolute value of velocity (which may be negative depending on direction and chosen coordinates).

that is fine. I am trying to tell you they will meet after all kinds of speed. since somebody says they dont meet again assuming no one turning back. if that is true, Tom travels from A to B and wait for mary from A to B also. they do meet again without turning back. if we consider either one of them as a reference they other one will have a speed which will result a less number in time. so the other one is always younger

 

[dwspacetime]

It's not a paradox. There's only your failure to understand the basics of SR.

my failure of understanding your whatever. regardless "when", or whatever "when" you applies to Tom it shoudl applies to Mary. without a reference independent from them whatever assumption you made to one of them you can just switch btw them

 

[dwspacetime]

It's not a paradox. There's only your failure to understand the basics of SR.

at whatever "when" Mary is 4 lys away to Tom there is another "when" the totally opposite is also true