I In what formation does this simple block universe exist?

[student34]

How so? It is the universe that we live in. The Euclidean space that you imagine is “more real” is an approximation valid only when relativistic effects are small enough to ignore (which, I’ll grant, is just about all of our lived experience).

Your argument is analogous to an argument that the curvature of the Earth is a mathematical fiction because we don’t directly experience it, we only use it for long-distance navigational calculations and the like.

Ok, I promise that I will stop giving my opinions. That seems to frustrate. I will stay with logical issues and things that I still do not understand.

 

[student34]

Why?

Because it is true that there is only one set of parallel lines. But it is also true that there are 2 sets of parallel lines, or at least one set of parallel lines, on the same 2d plane as in my examples.

 

[PeterDonis]

Because it is true that there is only one set of parallel lines.

And in each of your diagrams in the OP, there is also only one set of parallel lines (I assume you are referring to the pair of blue lines). The two diagrams are two different diagrams of the same thing, and each diagram has the same number of parallel lines as the thing it's a diagram of. What's the problem?

it is also true that there are 2 sets of parallel lines, or at least one set of parallel lines, on the same 2d plane as in my examples.

This doesn't make sense. I think you need to take more time to think through what you are saying before you say it.

 

[Ibix]

Because it is true that there is only one set of parallel lines. But it is also true that there are 2 sets of parallel lines, or at least one set of parallel lines, on the same 2d plane as in my examples.

Not really. Remember that your diagrams are Minkowski analogs for Euclidean rotations. Are you going to tell me that two parallel streets are different streets if I rotate my Google Map viewpoint?

 

[student34]

Yes. Flat spacelike slices through Minkowski spacetime are Euclidean. That's why Euclidean geometry is a thing. But timelike slices can't be Euclidean because if they were, turning around and going backwards in time would be as easy as turning round and going backwards in space. In fact, there'd be nothing to call "time" at all - it'd just be a fourth spatial dimension. The block universe cannot be Euclidean.

That just seems to be a limitation of the consciousness. The block, as I am told, is a 4d static structure. Nothing is moving (except our experiences/consciousness of the block universe). Is there anything else that you know of that makes time a different dimension than the spatial dimensions? This might help me.

 

[PeterDonis]

Is there anything else that you know of that makes time a different dimension than the spatial dimensions?

You measure spatial distance with rulers. You measure "distance in time" with clocks.

In the spacetime metric, this is reflected by the fact that timelike squared intervals and spacelike squared intervals have opposite signs.

 

[PeterDonis]

The block, as I am told, is a 4d static structure.

This is one particular interpretation of SR, but not the only possible one. The fact that timelike intervals are physically different from spacelike intervals does not depend on it.

 

[Ibix]

That just seems to be a limitation of the consciousness.

Huh? You think that it's a limitation of the conciousness that I can't just go back and catch the train I missed this morning? Seriously?

 

[Dale]

I am trying really hard to understand this in its entirety. For heavens sake, I would like to move on too and explore other things.

I wish you would explain why what I was nonsense.

It is nonsense because it is simply completely false that "at some point the 2 sets of parallel lines in my example have to converge to become 1 set of parallel lines" and there is no reason for you to claim that it is so.

You were also throwing in other earlier nonsense claims about it being impossible to embed a 1D line in a 2D graph.

Where are you getting this nonsense?

If you want us to say anything more useful than "nonsense" then you need to start engaging with what we are actually saying and not just saying any random thing that pops into your head. If you have something objectionable then you need to explain your own thought process in detail so that we can understand why you are claiming the nonsense that you are claiming.

I still have no idea why you think "at some point the 2 sets of parallel lines in my example have to converge to become 1 set of parallel lines" because you just made the nonsense statement without any explanation about your thought process and how it related to the thing that you quoted.

You also have not responded to my post 29. I put considerable effort into that post to explain something complicated as clearly as possible. Please go back and work through that post until you understand it or ask here for specific clarification.

 

[Dale]

Try answering the question I asked you in post #43.

Yes, this is a huge problem with @student34

They say something bizarre with no explanation about the thought process that led to the bizarre statement. Then they expect us to somehow be able to understand and address their concern

 

[student34]

And in each of your diagrams in the OP, there is also only one set of parallel lines (I assume you are referring to the pair of blue lines). The two diagrams are two different diagrams of the same thing, and each diagram has the same number of parallel lines as the thing it's a diagram of. What's the problem?This doesn't make sense. I think you need to take more time to think through what you are saying before you say it.

Ok I did not explain myself well.

Is it true that the parallel lines are different distances apart in each diagram on the same 2d plane?

 

[student34]

Not really. Remember that your diagrams are Minkowski analogs for Euclidean rotations. Are you going to tell me that two parallel streets are different streets if I rotate my Google Map viewpoint?

But these 2 structures are roughly of the same scale and orientation. Isn't everything equal except, of course , for the 2 structures?

 

[PeterDonis]

Is it true that the parallel lines are different distances apart in each diagram on the same 2d plane?

They are different coordinate distances apart. But coordinate distances are not necessarily the same as actually measured distances.

In the particular diagrams you drew in the OP, someone who was at rest in the left frame (the frame in which the red line is at rest, i.e., vertical) would indeed measure the two blue lines to be closer together than someone who was at rest in the right frame (the frame in which the blue lines are at rest, i.e., vertical). But for someone who was at rest in the left frame, the objects following the blue worldlines would be moving, so it would take some ingenuity to set up a way of measuring the distance between them. Whereas, for someone who was at rest in the right frame, the objects following the blue worldlines would be at rest, so the distance between them can be measured by just putting a ruler between them. So it should not be surprising a priori that these two very different measurement processes might give different results.

 

[student34]

You don’t need more dimensions, you just need a different metric. The metric is used to determine the actual distance or interval along a path, in other words, the geometry. For simplicity, let’s just talk about two dimensional spaces.

A normal Euclidean space has the usual metric from the Pythagorean theorem: $$ds^2=dx^2+dy^2$$ Since this is the metric of a piece of paper, you can draw Euclidean figures on a piece of paper without distortion.

You can also change coordinates in the Euclidean plane to polar coordinates. In that case the metric becomes $$ds^2 = dr^2 + r^2 d\theta^2$$ Although it is not so obvious, you can also draw Euclidean figures without distortion on a piece of paper with polar coordinates. The metric has a different algebraic formula, but the geometry is the same. All that has changed are the coordinates we are using.

In contrast, a unit sphere is a curved 2 dimensional space that has the following metric $$ds^2= d\phi^2 + \sin^2(\phi) d\theta^2$$ As you can see, this metric is approximately the Euclidean metric near $\phi=\pi/2$ and approximately the polar metric near $\phi=0$. However, because it is approximate this will have some distortion. You can map each point on a sphere to a point on a paper, but there will be some geometric distortion. Distances on the paper will not match distances on the sphere.

Finally, when we go to (2D) spacetime in natural units the metric becomes Minkowski’s $$ds^2=-dt^2+dx^2$$ This also cannot be represented exactly on a piece of paper. For instance, if $dt=dx$ then Minkowski’s $ds^2=0$, which doesn’t happen for distinct points on a sheet of paper.

Nevertheless, just as a map can be a valid representation of the Earth despite the distorted geometry, so a spacetime diagram can be a valid representation of Minkowski spacetime despite the distorted geometry.

A spacetime diagram is not the same as spacetime, just as a map is not the same as the earth. But a spacetime diagram can be a valid and useful representation of spacetime, just as a map is a valid and useful representation of the earth.

I have read this, and I understand it. This is definitely helpful.

 

[student34]

Huh? You think that it's a limitation of the conciousness that I can't just go back and catch the train I missed this morning? Seriously?

Yes, but if your consciousness goes back in time, everything about you goes back, brain, thoughts, body, memories etc. So you would have missed the morning train again, and you would think it was for the first time.

 

[student34]

They are different coordinate distances apart. But coordinate distances are not necessarily the same as actually measured distances.

In the particular diagrams you drew in the OP, someone who was at rest in the left frame (the frame in which the red line is at rest, i.e., vertical) would indeed measure the two blue lines to be closer together than someone who was at rest in the right frame (the frame in which the blue lines are at rest, i.e., vertical). But for someone who was at rest in the left frame, the objects following the blue worldlines would be moving, so it would take some ingenuity to set up a way of measuring the distance between them. Whereas, for someone who was at rest in the right frame, the objects following the blue worldlines would be at rest, so the distance between them can be measured by just putting a ruler between them. So it should not be surprising a priori that these two very different measurement processes might give different results.

I was told that the distances actually are different, even my physics textbook says this. This seems to be an actual implication of GR. What I wanted to know is whether or not these 2 sets of parallel lines exist on the same 2d plane.

Instead of talking about what its measurement would be, can we just talk about what it is, just so we don't have to explain so much?

 

[Dale]

I have read this, and I understand it. This is definitely helpful.

Excellent. Since you understand this then let me show you how it answers this question:

Is it true that the parallel lines are different distances apart in each diagram on the same 2d plane?

They are different distances apart according to the Euclidean metric $ds^2=dx^2+dy^2$. They are the same interval apart according to the Minkowski metric according to the Minkowski metric $ds^2=-dt^2+dx^2$. So the change in distance in the diagram is just a distortion of the map, like the fact that Greenland looks bigger than Australia on a Mercator map.

 

[student34]

Excellent. Since you understand this then let me show you how it answers this question:

They are different distances apart according to the Euclidean metric $ds^2=dx^2+dy^2$. They are the same interval apart according to the Minkowski metric according to the Minkowski metric $ds^2=-dt^2+dx^2$. So the change in distance in the diagram is just a distortion of the map, like the fact that Greenland looks bigger than Australia on a Mercator map.

It is nice to see how the Minkowski metric works on the graph. But I have always been told and I have read many times that the distances between the two worldlines would actually be different and not just be a distortion. In other words, the worldlines on my example would exist just like they appear.

But something tells me that I might be misunderstanding you.

 

[Dale]

It is nice to see how the Minkowski metric works on the graph. But I have always been told and I have read many times that the distances between the two worldlines would actually be different and not just be a distortion. In other words, the worldlines on my example would exist just like they appear.

But something tells me that I might be misunderstanding you.

The $dx$ are actually different in both diagrams. The $ds^2= -dt^2+dx^2$ are the same.

Length contraction is about the $dx$. Remember the sausage example. There is only one sausage, but you can cut slices of different widths by cutting diagonally.

 

[student34]

The $dx$ are actually different in both diagrams. The $ds^2= -dt^2+dx^2$ are the same.

Length contraction is about the $dx$. Remember the sausage example. There is only one sausage, but you can cut slices of different widths by cutting diagonally.

Ok I just want to be clear. The distances between the parallel lines in the diagrams are not just distortions, they would exist like they appear, right?

 

[PeterDonis]

if your consciousness goes back in time, everything about you goes back, brain, thoughts, body, memories etc.

What are you basing this on? Personal theories and speculations are off topic here.

 

[PeterDonis]

I was told that the distances actually are different, even my physics textbook says this.

What textbook?

As for "actually are" different, that depends on what "actually" means. The word "real" and its various cognates are not scientific terms. Physics can predict the results of measurements, but it cannot tell you what is "real".

What I wanted to know is whether or not these 2 sets of parallel lines exist on the same 2d plane.

I can't answer this question because I don't know what it means. I suspect you don't either. This is why I have advised you to think more carefully about what you are saying before you say it.

I can tell you what relativity says: relativity says that there is one spacetime, it has a particular geometry, and in the particular spacetime geometry we are talking about in this thread, there is one red worldline and there are two blue worldlines, and the two diagrams you drew in the OP are two different viewpoints of this same spacetime geometry, in the same general sense as two different drawings of the same object from different orientations.

But I don't know if that answers your question or not.

 

[PeterDonis]

Instead of talking about what its measurement would be, can we just talk about what it is, just so we don't have to explain so much?

How would you measure the distance between two objects that are both moving relative to you, in the same direction, at the same speed?

 

[Dale]

Ok I just want to be clear. The distances between the parallel lines in the diagrams are not just distortions, they would exist like they appear, right?

The $dx$ is not distorted. The $ds^2=-dt^2+dx^2$ is distorted.

I am not certain what you are referring to as “the distances between the parallel lines”.

 

[student34]

What textbook?

I took a first year physics course in university and still have the textbook. We had a couple chapters on relativity, but we did not delve into it very much.

 

[student34]

How would you measure the distance between two objects that are both moving relative to you, in the same direction, at the same speed?

I am not sure.

 

[student34]

I am not certain what you are referring to as “the distances between the parallel lines”.

The distance between the two parallel lines in the left diagram is less than the distance between the distance between the two parallel lines in the right diagram.

 

[Dale]

The distance between the two parallel lines in the left diagram is less than the distance between the distance between the two parallel lines in the right diagram.

Sure. But whether it is distorted or not depends on whether you are referring to the horizontal distance or the perpendicular distance. The horizontal distance $dx$ is not distorted, it is different both on the diagram and in spacetime (this is length contraction). The perpendicular distance $ds^2=-dt^2+ dx^2$ is distorted, it is different in the diagram but the same in reality (this is the one configuration of spacetime geometry, independent of frames).

Telling me that the one on the left is less than the one on the right still doesn’t clarify which distance you mean when you ask if the distance is distorted.

 

[PeterDonis]

I took a first year physics course in university and still have the textbook.

So again, what textbook? I'm not asking you to describe where you got it. I'm asking you what book it is, i.e., what is its title, and who wrote it?

 

[PeterDonis]

I am not sure.

Then take some time to think about it.

 

[student34]

Sure. But whether it is distorted or not depends on whether you are referring to the horizontal distance or the perpendicular distance. The horizontal distance $dx$ is not distorted, it is different both on the diagram and in spacetime (this is length contraction). The perpendicular distance $ds^2=-dt^2+ dx^2$ is distorted, it is different in the diagram but the same in reality (this is the one configuration of spacetime geometry, independent of frames).

Telling me that the one on the left is less than the one on the right still doesn’t clarify which distance you mean when you ask if the distance is distorted.

I am having trouble understanding this because I thought this simple block universe that I illustrated exists exactly as we see it from our computers minus the Minkowsky geography. So if this is true, then we can just see which perpendicular distance between the parallel lines is wider. But you say that it is the same in reality.

I can't believe how hard I am struggling with this.

 

[student34]

So again, what textbook? I'm not asking you to describe where you got it. I'm asking you what book it is, i.e., what is its title, and who wrote it?

Sears and Zemansky's University Physics: Young and Freedman 13th Edition

 

[PeterDonis]

I thought this simple block universe that I illustrated exists exactly as we see it from our computers minus the Minkowsky geography.

The Minkowski "geography" is part of the "simple block universe" that your diagrams illustrate. They are two diagrams of the same "geography" from different viewpoints, and that "geography" has a Minkowski geometry.

 

[student34]

The Minkowski "geography" is part of the "simple block universe" that your diagrams illustrate. They are two diagrams of the same "geography" from different viewpoints, and that "geography" has a Minkowski geometry.

Ok, that is what I thought, thanks.

Is one set of parallel lines the same parallel lines as the other set, or are they different slices of a 2d Minkosky object?

 

[Ibix]

Is one set of parallel lines the same parallel lines as the other set, or are they different slices of a 2d Minkosky object?

They are the same set of parallel lines. The Minkowski "distance" (the interval) between them along a line that is perpendicular (in a Minkowski sense) to them is the same in both your diagrams. The spatial distance between them differs for the same reason that the horizontal distance between a pair of lines in a Euclidean space varies as you rotate them.

 

[Dale]

I thought this simple block universe that I illustrated exists exactly as we see it from our computers minus the Minkowsky geography.

What do you think "minus the Minkowski geometry" means? It means exactly this, some distances and some angles are not going to be the same in real spacetime vs the diagram. I don't know what else you think that would mean.

 

[student34]

They are the same set of parallel lines. The Minkowski "distance" (the interval) between them along a line that is perpendicular (in a Minkowski sense) to them is the same in both your diagrams. The spatial distance between them differs for the same reason that the horizontal distance between a pair of lines in a Euclidean space varies as you rotate them.

So assuming a static structure (which I am told may or may not be the case for a block universe), there exists a 2d structure (because I have been told that there are only 4 dimensions in our universe, therefore I am assuming 2 dimensions in the OP example) that has the same line in different parts of its structure.

I don't understand how this simple universe can exist in only 2 dimensions. *And really, I do not know how this structure can logically exist mathematically or realistically.

 

[student34]

What do you think "minus the Minkowski geometry" means? It means exactly this, some distances and some angles are not going to be the same in real spacetime vs the diagram. I don't know what else you think that would mean.

But isn't the diagram a face/side/slice of the block?

 

[Ibix]

Do you understand that if I draw two maps of my town with north in different directions then the streets will not point in the same directions? Do you understand that this does not mean that there are two sets of streets pointing in different directions in reality? Do you understand that I've chosen only to represent the ground on the maps, even though there are tunnels underground all over the place here? That I know that those tunnels are there, and I've chosen not to draw them because right at the moment I don't care about anything not in the 2d plane of the surface of the Earth?

If so, what's so hard about the concept of two different maps of a 2d slice through spacetime? They just look different because you've used different projections to draw them, the same as I would have used different rotations to draw my maps.

 

[Dale]

I can't believe how hard I am struggling with this.

I think a substantial part of that struggle is due to your tendency to go off on strange and unhelpful tangents instead of focusing on the core issues. In this thread alone we started with the question about whether or not both diagrams represent spacetime equally well. After some initial explanation about maps and distorted geometry you went off on the following irrelevant tangents:

Post 13-26: we need more dimensions (this is a completely random thought disconnected from anything previous which should have just been dismissed with a single "no", but which you pursued for quite some time)

Post 30 - 35: embedding lower dimensional spaces in higher dimensional spaces. You expressed this as a restriction that was never mentioned as a requirement by anyone else.

Post 35- 46: Minkowski space is imaginary. Complete waste of time, unnecessarily provocative and unhelpful tangent.

Post 40-52: parallel lines converge. This could have been ok if you had been clear what you meant from the beginning.

Post 55-65: Limitation of consciousness. Completely irrelevant to the rest of the thread.

With post 61 you got back on track and have been more or less on track with the exception of 65. But overall posts 13-61 are wasted effort for both you and the other participants. That is the majority of this thread. If you would focus instead of jumping off at tangents then you would make much more progress with less effort.

I would recommend to
1) stop making editorial comments of any type (limitations of consciousness, Minkowski space is imaginary)
2) when you make a new idea that we tell you is wrong, don't continue arguing it, just move on (need more dimensions, embedding)
3) be as clear as you can when you describe an issue (parallel lines converge)

If you do those then you will struggle less. Currently you are like a hiker trying to go up a mountain but leaving the path to chase squirrels. Of course you are getting tired! Spend your energy on making progress. It requires focus and discipline, but you will find it easier overall.

 

[Dale]

But isn't the diagram a face/side/slice of the block?

A slice, yes. The block universe doesn't have faces/sides. That would imply a definite boundary.

But again, the geometry in the spacetime slice is governed by the Minkowski metric, not the Euclidean metric of the diagram.

 

[student34]

Do you understand that if I draw two maps of my town with north in different directions then the streets will not point in the same directions? Do you understand that this does not mean that there are two sets of streets pointing in different directions in reality? Do you understand that I've chosen only to represent the ground on the maps, even though there are tunnels underground all over the place here? That I know that those tunnels are there, and I've chosen not to draw them because right at the moment I don't care about anything not in the 2d plane of the surface of the Earth?

If so, what's so hard about the concept of two different maps of a 2d slice through spacetime? They just look different because you've used different projections to draw them, the same as I would have used different rotations to draw my maps.

In your example, you only changed the orientation of the map; you did not change anything on the map. Here we are changing where the roads are on the map. I do not understand what the tunnels are analogous of.

 

[student34]

I think a substantial part of that struggle is due to your tendency to go off on strange and unhelpful tangents instead of focusing on the core issues. In this thread alone we started with the question about whether or not both diagrams represent spacetime equally well. After some initial explanation about maps and distorted geometry you went off on the following irrelevant tangents:

Post 13-26: we need more dimensions (this is a completely random thought disconnected from anything previous which should have just been dismissed with a single "no", but which you pursued for quite some time)

Post 30 - 35: embedding lower dimensional spaces in higher dimensional spaces. You expressed this as a restriction that was never mentioned as a requirement by anyone else.

Post 35- 46: Minkowski space is imaginary. Complete waste of time, unnecessarily provocative and unhelpful tangent.

Post 40-52: parallel lines converge. This could have been ok if you had been clear what you meant from the beginning.

Post 55-65: Limitation of consciousness. Completely irrelevant to the rest of the thread.

With post 61 you got back on track and have been more or less on track with the exception of 65. But overall posts 13-61 are wasted effort for both you and the other participants. That is the majority of this thread. If you would focus instead of jumping off at tangents then you would make much more progress with less effort.

I would recommend to
1) stop making editorial comments of any type (limitations of consciousness, Minkowski space is imaginary)
2) when you make a new idea that we tell you is wrong, don't continue arguing it, just move on (need more dimensions, embedding)
3) be as clear as you can when you describe an issue (parallel lines converge)

If you do those then you will struggle less. Currently you are like a hiker trying to go up a mountain but leaving the path to chase squirrels. Of course you are getting tired! Spend your energy on making progress. It requires focus and discipline, but you will find it easier overall.

I am definitely going to try harder not to go off on tangents and other derailments.

 

[student34]

A slice, yes. The block universe doesn't have faces/sides. That would imply a definite boundary.

But again, the geometry in the spacetime slice is governed by the Minkowski metric, not the Euclidean metric of the diagram.

Ok, so does this mean that the 2 slices/diagrams do not exist as they appear to us, being flat 2d Euclidean slices?

 

[Ibix]

Here we are changing where the roads are on the map.

But roads are in different places on a map when I rotate it! A road in one corner of one map might be at the bottom of the other. Why is this suddenly a problem for you when I draw two Minkowski diagrams?

The only difference is that the Minkowski diagram distorts the reality because the reality cannot be drawn on a Euclidean plane.

Ok, so does this mean that the 2 slices/diagrams do not exist as they appear to us, being flat 2d Euclidean slices?

The slices in reality are Minkowski planes. To the extent they "look like" anything they "look like" a universe with one spatial dimension and one timelike dimension. The Minkowski diagrams are Euclidean planes. They look like flat planes.

 

[PeterDonis]

In your example, you only changed the orientation of the map; you did not change anything on the map.

Wrong. If you draw a map of some area of the Earth's surface that contains roads, with north at the top, and a second map of the same area with west at the top, the roads will be in different places on the two maps even though they are at the same places in the actual area the maps are describing.

Once again, I think you need to really, really, really think a lot harder about what you are saying before you say it. You seem to be creating a lot of problems for yourself that are totally unnecessary because you are just saying things that pop into your head without even taking any time to think them through and consider whether they are really true or whether they even make sense.

 

[Dale]

Ok, so does this mean that the 2 slices/diagrams do not exist as they appear to us, being flat 2d Euclidean slices?

Correct. For example, on the diagram we measure those angles with a protractor, but in spacetime we measure those angles with a speedometer or a radar gun. On the diagram, all distances $ds^2=dx^2+dy^2$ are measured with a ruler, but in spacetime intervals $ds^2=-dt^2+dx^2$ are only measured with rulers if $ds^2>0$ while if $ds^2<0$ then the interval is measured with a clock.

The diagram is an accurate map of spacetime, but like maps of the earth, the geometry is not identical. There is a 1-to-1 map between points in the diagram and events in spacetime, but the metric is different so distances and angles can be different between the map and the physical world.

 

[student34]

But roads are in different places on a map when I rotate it! A road in one corner of one map might be at the bottom of the other. Why is this suddenly a problem for you when I draw two Minkowski diagrams?

I think there is ambiguity when you say "But roads are in different places on a map when I rotate it!". This does not seem true by how I am interpreting it.

 

[student34]

Wrong. If you draw a map of some area of the Earth's surface that contains roads, with north at the top, and a second map of the same area with west at the top, the roads will be in different places on the two maps even though they are at the same places in the actual area the maps are describing.

Once again, I think you need to really, really, really think a lot harder about what you are saying before you say it. You seem to be creating a lot of problems for yourself that are totally unnecessary because you are just saying things that pop into your head without even taking any time to think them through and consider whether they are really true or whether they even make sense.

I took what was said to mean something else.

 

[PeterDonis]

I think there is ambiguity when you say "But roads are in different places on a map when I rotate it!". This does not seem true by how I am interpreting it.

Then you need to think very, very carefully how you are interpreting it.

For example, suppose I have a 1 square mile area of the Earth with one road through it that runs north-south in a straight line. If I draw a map of this area with north at the top, the road on the map runs top to bottom on the map. If I draw a map of the same area with west at the top, the road on the map runs right to left on the map. The road is "in different places" on these two maps under the simple, obvious interpretation of those words--top to bottom is different from right to left. And this is the same sense in which you say the lines are "in different places" in the two diagrams of spacetime that you drew in the OP of this thread. So under this simple, obvious interpretation of "in different places", the two cases work out the same and there is no problem.

Now you want to interpret "in different places" in some other way, so that the road on the two maps I described above is not "in different places" on the two maps, even though it runs top to bottom on one and right to left on the other. You can of course interpret words however you want, but you should stop and think very, very carefully about what you are doing and why you want to do it. If you do adopt this new interpretation of "in different places", then the simple, obvious implication of this new interpretation, whatever it is, is that the lines in the two diagrams you drew in your OP are not "in different places" either. So the two cases still work out the same and there is no problem.

You, however, seem to think that there is a problem: but I submit that the only reason you think that is that you have not fully thought through the implications of what you are saying, but are just, as I said before, saying whatever things pop into your head without taking the time to consider whether they are actually true or even make sense. And so this thread has gone on for a hundred posts and you are still confused, because you are insisting on confusing yourself desipte all our efforts to help you.