I What would be the form of this world line?

[student34]

Imagine a ball floating in space, and there is an observer at rest next to it. Then the observer goes for a trip at high speeds and returns to be at rest with the ball once again.

Would the world line of the ball be curved because of the frame of reference of the observer, or would the world line be straight as from the frame of reference of anything at rest with the ball? Or are there 2 worldlines?

 

[Dale]

You didn’t provide enough information to answer. If the accelerometer reading of the ball reads zero at all times then the ball’s worldline is straight (geodesic). If the accelerometer reading of the observer is non-zero at some times then their worldline is not straight (non-geodesic) during those times.

The straightness of a worldline is an invariant geometrical fact that is independent of any coordinate system, reference frame, or observer.

 

[Ibix]

The ball doesn't seem to undergo acceleration, so its worldline is a geodesic - a straight line if this is flat spacetime.

If the accelerating observer chooses to represent his curved coordinate system as a rectangular grid in a picture then the representation of the ball's worldline in that picture will be curved, certainly. But a distorted representation doesn't change the fact that the worldline is straight.

 

[student34]

You didn’t provide enough information to answer. If the accelerometer reading of the ball reads zero at all times then the ball’s worldline is straight (geodesic). If the accelerometer reading of the observer is non-zero at some times then their worldline is not straight (non-geodesic) during those times.

The straightness of a worldline is an invariant geometrical fact that is independent of any coordinate system, reference frame, or observer.

Assume that the ball does not accelerate.

 

[student34]

The ball doesn't seem to undergo acceleration, so its worldline is a geodesic - a straight line if this is flat spacetime.

If the accelerating observer chooses to represent his curved coordinate system as a rectangular grid in a picture then the representation of the ball's worldline in that picture will be curved, certainly. But a distorted representation doesn't change the fact that the worldline is straight.

So the "absolute form" of the worldline is straight?

 

[Dale]

Assume that the ball does not accelerate.

Then what I said in post 2 holds.

 

[Ibix]

So the "absolute form" of the worldline is straight?

I have no idea what "absolute form" is supposed to mean. A straight line is one that parallel transports its own tangent vector, and that is true of the ball's worldline, however you choose to draw it.

 

[student34]

Then what I said in post 2 holds.

But then what about the frame of reference of the accelerating observer? I thought that its observation frame of reference is just as true as any other frame of reference. Why is the straight worldline "special" in this case?

 

[Ibix]

Why is the straight worldline "special" in this case?

It isn't. The curved coordinate frame simply makes it hard to see that the line is straight, it doesn't stop it being straight. Any measurements will show that the ball is unaccelerated (edit: that is, has zero proper acceleration) whichever frame is being used.

 

[Dale]

But then what about the frame of reference of the accelerating observer? I thought that its observation frame of reference is just as true as any other frame of reference. Why is the straight worldline "special" in this case?

Frames of reference are irrelevant to the geometry. A curved line is curved (non-geodesic) and a straight line is straight (geodesic) irrespective of how you choose to draw your coordinates. Using curved coordinates does not change the underlying geometrical facts.

A straight worldline is not “special”, it is just straight. A curved worldline is not “unspecial”, it is just curved. Your choice of coordinates doesn’t change either

 

[Ibix]

@student34 - you seem to me to be confusing the representation of a line in some complicated coordinate system with the line itself. An analogy is drawing lines of latitude on the Earth. As you move away from the equator each circle has a smaller circumference, reducing towards zero at a pole. But if you draw these lines on a Mercator map each one is simply a horizontal line of the same length. Does that mean that the circles are both all of the same circumference and at the same time varying in circumference? No. It just means you are hopelessly naive if you think arbitrary measurements on a distorted representation like a Mercator projection will correspond to the real world. The circles really vary in circumference, and you can read that off the Mercator map if you know how to do it. You just can't treat a Euclidean representation of a non-Euclidean surface (or a non-Euclidean representation of a Euclidean plane) as if it were an ordinary Euclidean plane.

 

[Dale]

@student34 as an example, I can take a piece of paper, draw a straight line and a curved line and hand it to you and you can unambiguously determine which is curved and which is straight. It is a geometrical fact.

It does not matter if the paper had straight grid lines or curved grid lines or even if it had no grid lines at all. The presence or absence or shape of the grid lines does not change the geometry of the lines I drew.

 

[student34]

It isn't. The curved coordinate frame simply makes it hard to see that the line is straight, it doesn't stop it being straight. Any measurements will show that the ball is unaccelerated (edit: that is, has zero proper acceleration) whichever frame is being used.

The reason why I ask if the worldline is curved by the frame of reference of the traveler is because it seems like the position of the ball for the traveler has shifted due to length contraction.

I got this diagram from Wikipedia that shows the frame of reference of an object from rest begin moving along the x axis. When we look at the diagram below, we can see that the x' position of event A has shifted towards the traveler, origin 0,0. Now wouldn't the worldline of the ball, that we can call event A, be curved for the traveler's frame of reference. And the worldline should be straight for an observer at rest at the origin 0,0. Aren't there 2 worldlines here?

 

[Dale]

it seems like the position of the ball for the traveler has shifted

Yes, position is frame-dependent. Curvature is frame-invariant.

Aren't there 2 worldlines here?

What two worldlines? I only see a single event, A.

 

[student34]

Yes, position is frame-dependent. Curvature is frame-invariant.

What two worldlines? I only see a single event, A.

So wouldn't the worldline for the ball, located at event A, be curved for the traveler but straight for an observer at the origin at rest with the ball (event A)?

 

[Dale]

So wouldn't the worldline for the ball, located at event A, be curved for the traveler but straight for an observer at the origin at rest with A?

No, I already answered that multiple times. Asking again will not change the answer. A straight line (geodesic) is a straight line. This is a frame invariant geometric fact.

 

[sdkfz]

@student34 as an example, I can take a piece of paper, draw a straight line and a curved line and hand it to you and you can unambiguously determine which is curved and which is straight. It is a geometrical fact.

It does not matter if the paper had straight grid lines or curved grid lines or even if it had no grid lines at all. The presence or absence or shape of the grid lines does not change the geometry of the lines I drew.

If you gave me the paper and I rolled it into a cylinder, the straight line might be curved in my view, but relative to the paper it's still straight.

It's the same line.

 

[student34]

No, I already answered that multiple times. Asking again will not change the answer. A straight line (geodesic) is a straight line. This is a frame invariant geometric fact.

You seemed to reply with "yes" to my question "Aren't there 2 worldlines here?" in your post #14. I am confused. Are there 2 worldlines for the ball or not?

 

[robphy]

A Spacetime Diagram is a
position vs time graph.

In the “lab frame” (using the lab clock)
what are the positions of each particle
as a function of lab time?

The frame for an accelerated particle is more complicated and would require a more precise definition of “frame”.

 

[student34]

A Spacetime Diagram is a
position vs time graph.

In the “lab frame” (using the lab clock)
what are the positions of each particle
as a function of lab time?

Are you referring to the diagram that I posted, or the OP?

 

[Dale]

You seemed to reply with "yes" to my question "Aren't there 2 worldlines here?" in your post #14. I am confused. Are there 2 worldlines for the ball or not?

No, I very specifically and clearly replied that the position is frame dependent. There is one worldline, and it’s position is frame dependent.

Its curvature is frame independent.

 

[student34]

No, I very specifically and clearly replied that the position is frame dependent. There is one worldline, and it’s position is frame dependent.

Then I just do not understand how the worldline of the ball for the traveler is not curved. In the graph, as something begins to travel fast down x axis, doesn't the x position of the ball, located at event A, shift towards the origin? If so, how can that happen without curving the worldline of the ball for the traveler?

 

[Dale]

Then I just do not understand how the worldline of the ball for the traveler is not curved. In the graph, …

The graph is not relevant. The curvature is a frame invariant fact of the underlying spacetime geometry. It has nothing to do with any graph. It is a fact about the physical geometry.

For a moment, let’s ignore time and just think about familiar ordinary spatial geometry. I have a table, it has four legs. The table top is flat and the legs are perpendicular to the top. Each leg is straight and they all have equal lengths. These are all physical facts describing the physical geometry.

I can write down equations describing the table geometry: regions of one plane and four lines. I can use any coordinate system I like to write down those equations. I can, if I want, use spherical coordinates. I can graph the equations in those coordinates.

If I use spherical coordinates to write down the equations for my table do you think my table top is physically no longer flat? Do you think the table legs are physically no longer straight?

Of course not. The curved coordinates don’t change the geometry of the table.

Similarly in spacetime. Whether a worldline is straight or not is a question of physical geometry, just like the table leg being straight. It is measured using accelerometers. The reading of an accelerometer is a frame-invariant fact. It does not change regardless of what frame we use, just like the table top didn’t become physically curved just because you used spherical coordinates

 

[Nugatory]

Then I just do not understand how the worldline of the ball for the traveler is not curved. In the graph, as something begins to travel fast down x axis, doesn't the x position of the ball, located at event A, shift towards the origin? If so, how can that happen without curving the worldline of the ball for the traveler?

The line on the graph is not the worldline, it is a line on a graph. The line on the graph is drawn by plotting the x and t coordinates of the events on the worldline, so its appearance will depend on on how choose the x and t coordinates. But the worldline is the same set of events either way.

 

[student34]

The graph is not relevant. The curvature is a frame invariant fact of the underlying spacetime geometry. It has nothing to do with any graph. It is a fact about the physical geometry.

For a moment, let’s ignore time and just think about spatial geometry. I have a table, it has four legs. The table top is flat and the legs are perpendicular to the top. Each leg is straight and they all have equal lengths. These are all physical facts describing the physical geometry.

I can write down equations describing the table geometry: regions of one plane and four lines. I can use any coordinate system I like to write down those equations. I can, if I want, use spherical coordinates. I can graph the equations in those coordinates.

If I use spherical coordinates to write down the equations for my table do you think my table top is physically no longer flat? Do you think the table legs are physically no longer straight?

Of course not. The curved coordinates don’t change the geometry of the table.

Similarly in spacetime. Whether a worldline is straight or not is a question of physical geometry, just like the table leg being straight. It is measured using accelerometers. The reading of an accelerometer is a frame-invariant fact. It does not change regardless of what frame we use, just like the table top didn’t become physically curved just because you used spherical coordinates

Okay but the graph seems to contradict the answer given using accelerometers. I want to understand this from all angles, no pun intended.

The origin and the ball (at event A on the graph) should be parallel worldlines since they are at rest with each other. But it seems unavoidable that the change in the x' position, due to length contraction, of the ball would make the lines nonparallel in the frame of reference of the traveler moving towards the ball.

 

[Dale]

Okay but the graph seems to contradict the answer given using accelerometers.

The graph is not physical. The accelerometers are.

The origin and the ball (at event A on the graph)

Not sure what you are talking about here. There is no ball in the picture. Event A is not a ball. A ball would be a line in this graph, not a point. A point in this graph is an event, like an explosion.

Anyway, I don’t know how I could possibly be more clear in this thread. The spacetime geometry is an underlying physical fact, like the physical geometry of my table. Do you understand the concept of a physical geometry, independent of any coordinates?

If I physically draw a line on a piece of paper, do you need coordinates to determine if it is straight?

 

[Ibix]

Okay but the graph seems to contradict the answer given using accelerometers.

Just like the equal lengths of the latitude lines on a Mercator map seem to contradict the fact that each one in reality is a different length. The map is not the territory. Just because the map of spacetime drawn by an accelerating observer represents straight lines as curves it does not mean that the lines are curved. It means that the maps are distorted maps.

I mean, do you seriously believe that a person using a Mercator map can't cross the International Date Line because the left and right sides of their map aren't joined? I hope you don't. So if you are willing to accept that Mercator maps aren't straightforward representations of Earth's surface, why is it so hard to accept that maps drawn using accelerating coordinate systems aren't straightforward representations of spacetime? We can tell that the worldline of the ball is straight because accelerometers attached to it always read zero. However I choose to draw the line, those accelerometers still read zero so the line is straight.

I understand that you desperately want to believe that there is some kind of multiple reality thing going on in relativity. But this is wrong. The sooner you accept that the truth is that there can be multiple representations of one reality, the sooner you can make progress.

 

[student34]

The graph is not physical. The accelerometers are.

But the graph is real and it has real implications like when muons impact the Earth earlier than they should due to the length contraction between the Earth's upper atmosphere and the ground.

In this case, x' seems to really be the x coordinate for the traveler. And I have been told that there is no absolute frame of reference.

 

[jbriggs444]

In this case, x' seems to really be the x coordinate for the traveler.

When you use "really" and "coordinate" in the same sentence, that may indicate a difficulty with your understanding of one or the other.

What really happens is independent of the coordinates we use to describe or label it.

Coordinates are numbers we write down on pieces of paper. We can choose them after the fact without affecting the experiment that has already been run.

 

[Ibix]

But the graph is real and it has real implications like when muons impact the Earth earlier than they should due to the length contraction between the Earth's upper atmosphere and the ground.

This is completely wrong. The muons reach Earth because the "angle" (technically, the rapidity) between the worldline of the muon and that of the Earth is extreme.

It has nothing to do with any graph. How could it? Humans weren't around to draw graphs for most of the history of the universe.

 

[Dale]

But the graph is real and it has real implications like when muons impact the Earth earlier than they should due to the length contraction between the Earth's upper atmosphere and the ground.

In this case, x' seems to really be the x coordinate for the traveler. And I have been told that there is no absolute frame of reference.

The graph is not real. I am not even sure what would make you think that a graph is real.

I think that problem that you are having is that you are overly focused on coordinates. That will work reasonably for inertial topics, but as soon as you start trying to think about non-inertial topics you need to de-emphasize the coordinates and focus on the underlying geometry. If you are unwilling to do that then you may need to just stick with inertial topics only.

Also, I am curious about why you are unwilling to engage about the geometrical discussion. That is the most important part of this topic and you have seemed to ignore it completely. Why?

Do you understand the concept of a physical geometry, independent of any coordinates?

If I physically draw a line on a piece of paper, do you need coordinates to determine if it is straight?

If a builder built you a crooked wall and tried to show you a graph of the wall in some coordinates where the wall had a constant coordinate, would you then think that the wall was physically straight?

 

[student34]

This is completely wrong. The muons reach Earth because the "angle" (technically, the rapidity) between the worldline of the muon and that of the Earth is extreme.

These lecture notes suggest that I am not "completely wrong", from the University of Toronto, "... the distance that the ground travels before the muon decays is 0.6 km. But what is the thickness of the atmosphere that the muon sees? ... the atmosphere that the muon sees is 70 times thinner 0.6 km > 0.14 km and so the ground will reach the muon." from https://www.atmosp.physics.utoronto.ca/people/strong/phy140/lecture32_01.pdf

 

[Ibix]

These lecture notes suggest that I am not "completely wrong", from the University of Toronto,

And where do they say that it's because of a graph? That's the point that's wrong. The muons don't survive because they draw a spacetime diagram. They survive because of the "angle" between their worldline and that of the atmosphere - a geometric fact independent of any graph.

 

[student34]

And where do they say that it's because of a graph? That's the point that's wrong. The muons don't survive because they draw a spacetime diagram. They survive because of the "angle" between their worldline and that of the atmosphere - a geometric fact independent of any graph.

Yeah whatever. Why avoid the obvious point that I am trying to make and try to find some semantical side argument?

And I meant that the graph is real in the sense that if you were to superimpose this graph into a real situation, then it would align with what is actually happening.

 

[Ibix]

Why avoid the obvious point that I am trying to make and try to find some semantical side argument?

Because it seems to be the source of your problems. For example:

And I meant that the graph is real in the sense that if you were to superimpose this graph into a real situation, then it would align with what is actually happening.

No it wouldn't align, because a spacetime diagram is a Euclidean plane. How is it going to "align" with a Minkowski plane that doesn't even share the same rules of geometry?

Spacetime diagrams are a representation of reality. They are not the reality. That's why you can draw them using curvilinear coordinates where straight lines in reality are not represented by straight lines on the page, and still say that the lines are straight.

 

[student34]

The graph is not real. I am not even sure what would make you think that a graph is real.

I think that problem that you are having is that you are overly focused on coordinates. That will work reasonably for inertial topics, but as soon as you start trying to think about non-inertial topics you need to de-emphasize the coordinates and focus on the underlying geometry. If you are unwilling to do that then you may need to just stick with inertial topics only.

Also, I am curious about why you are unwilling to engage about the geometrical discussion. That is the most important part of this topic and you have seemed to ignore it completely. Why?

Do you understand the concept of a physical geometry, independent of any coordinates?

If I physically draw a line on a piece of paper, do you need coordinates to determine if it is straight?

If a builder built you a crooked wall and tried to show you a graph of the wall in some coordinates where the wall had a constant coordinate, would you then think that the wall was physically straight?

I do not think that I am avoiding the underlying geometry. I am trying to understand how the underlying geometry of the worldline of the ball (in the position of event A from the graph) is straight. Doesn't it seem at least counterintuitive to you that the position of the ball can change on the x' coordinate without causing a curve in its worldline? This is what I do not understand; it is all about the underlying geometry for me.

 

[Ibix]

I am trying to understand how the underlying geometry of the worldline of the ball (in the position of event A from the graph) is straight.

It parallel transports its own tangent vector, in mathematical language. Or accelerometers following that path show zero proper acceleration, in operational language.

Doesn't it seem at least counterintuitive to you that the position of the ball can change on the x' coordinate without causing a curve in its worldline?

If your lines of constant $x'$ are curved but you draw them as straight lines on your chart, why would you be at all surprised that a straight line in reality was represented by a curve on the chart? You distorted your coordinate lines - why would you expect anything else to be an undistorted representation of itself?

 

[Dale]

I do not think that I am avoiding the underlying geometry.

And yet you still haven't responded to any of the questions about underlying geometry that I have asked of you. Are you at all surprised that it seems to me like you are avoiding it?

I am trying to understand how the underlying geometry of the worldline of the ball (in the position of event A from the graph) is straight. Doesn't it seem at least counterintuitive to you that the position of the ball can change on the x' coordinate without causing a curve in its worldline?

Please go back and respond to the questions that I have asked you. Those questions are intended to cause you to think about exactly this issue. Start with spatial geometry, and once you understand the issues for spatial geometry will you be well equipped to move to spacetime geometry.

it is all about the underlying geometry for me.

Then try making a post where you do not mention reference frames or coordinates at all. Consider them completely verboten concepts for your post.

 

[PeroK]

What is a "straight line"? And why can't it be a "curve"?

PS take a look at the finite projective plane of order 2, for example.

 

[Ibix]

Look, let's try this with a simpler case. Go to Google maps and find your street. How long is your street? How long is it on the map? Is the picture on the map oriented the same as the real street? You can use a map despite all this because you understand that there's a non-trivial relationship between what's shown on the map and reality: you need to rotate, translate, and scale one to get the other. And with a bit of training you could use a Mercator map to navigate around the globe despite that it is a cut and distorted representation of the reality.

But all this seems to desert you when we get to spacetime. An inertial frame of reference is just a choice of t, x, y, and z axes. That's all. If you switch frames you simply pick a different set of axes, just like when you rotate a map on your phone you pick a different set of x and y axes. And just as a strip of land on the map might change from short and wide to long and narrow, the description of some object and duration changes - a long atmosphere and a slow-ticking clock becomes a short atmosphere and a normally ticking clock. But nothing has changed in reality! All you've done is changed the directions you are calling "time" and "space" and updated your descriptions of things to match.

And when you switch to a non-inertial frame you are picking curved lines that you intend to draw on your map as straight. Not only do you now need to rotate, translate, and scale before mapping a Euclidean plane to a Lorentz one you need to also undo the distortion you put in - much like you would have to do with a Mercator map of the Earth. Again, using a distorted map doesn't change anything in reality. It just means that you need a complicated transformation before you can interpret what is drawn on the map.

And this is why @Dale and I keep on at you about the distinction between coordinate based descriptions and reality. All of the descriptions involve some transform from reality to the description or the spacetime diagram, be it more complex or less. And you keep talking as if the many different descriptions are the important thing. They aren't. In the muon experiment the angle between worldlines is important, but which worldline you choose to draw vertically on your diagram doesn't matter, but you say "the graph is real and it has real implications". No it isn't and no it doesn't. It just affects how you describe the situation.

 

[PeterDonis]

Yeah whatever.

This attitude is going to accomplish nothing except to get your thread closed. You came here asking for help. If you keep ignoring the help you are getting, there's no point in going on.

Why avoid the obvious point that I am trying to make

Your "obvious point" is wrong. You have been repeatedly told why it is wrong. Either pay attention to what you are being told or your thread will be closed.

and try to find some semantical side argument?

The points being made in this thread, which you are failing to understand, are not a "semantical side argument". They are the crux of the issue.

I strongly suggest carefully reading, in particular, @Dale's post #38, and following his advice. If you want to understand geometry independently of coordinates and "graphs", you need to stop thinking in terms of coordinates and "graphs".

 

[PeterDonis]

If you want to understand geometry independently of coordinates and "graphs", you need to stop thinking in terms of coordinates and "graphs".

For another exercise, in addition to what @Dale asked you, consider a line and a circle on a plane. How, without using any coordinates whatever, would you confirm your intuitive guess that the line is straight and the circle is curved?

 

[student34]

It parallel transports its own tangent vector, in mathematical language. Or accelerometers following that path show zero proper acceleration, in operational language.

If your lines of constant $x'$ are curved but you draw them as straight lines on your chart, why would you be at all surprised that a straight line in reality was represented by a curve on the chart? You distorted your coordinate lines - why would you expect anything else to be an undistorted representation of itself?

The implication of the accelerometer that you and Dale brought up is very helpful; I did not know this. And thank you for that. It's another tool that I can use moving forward.

What do you mean that the "constant x' is curved"? Is it actually curved?

 

[student34]

And yet you still haven't responded to any of the questions about underlying geometry that I have asked of you. Are you at all surprised that it seems to me like you are avoiding it?

Please go back and respond to the questions that I have asked you. Those questions are intended to cause you to think about exactly this issue. Start with spatial geometry, and once you understand the issues for spatial geometry will you be well equipped to move to spacetime geometry.

Then try making a post where you do not mention reference frames or coordinates at all. Consider them completely verboten concepts for your post.

I thought those were rhetorical questions. I see you asked me, "If I physically draw a line on a piece of paper, do you need coordinates to determine if it is straight?" My answer to this is no, if we are assuming a Euclidean space.

 

[PeterDonis]

I see you asked me, "If I physically draw a line on a piece of paper, do you need coordinates to determine if it is straight?" My answer to this is no, if we are assuming a Euclidean space.

So how do you determine that it is straight, without using coordinates? Or, if you draw a curved line, such as an arc of a circle, how do you determine that it is curved without coordinates?

 

[student34]

Look, let's try this with a simpler case. Go to Google maps and find your street. How long is your street? How long is it on the map? Is the picture on the map oriented the same as the real street? You can use a map despite all this because you understand that there's a non-trivial relationship between what's shown on the map and reality: you need to rotate, translate, and scale one to get the other. And with a bit of training you could use a Mercator map to navigate around the globe despite that it is a cut and distorted representation of the reality.

But all this seems to desert you when we get to spacetime. An inertial frame of reference is just a choice of t, x, y, and z axes. That's all. If you switch frames you simply pick a different set of axes, just like when you rotate a map on your phone you pick a different set of x and y axes. And just as a strip of land on the map might change from short and wide to long and narrow, the description of some object and duration changes - a long atmosphere and a slow-ticking clock becomes a short atmosphere and a normally ticking clock. But nothing has changed in reality! All you've done is changed the directions you are calling "time" and "space" and updated your descriptions of things to match.

And when you switch to a non-inertial frame you are picking curved lines that you intend to draw on your map as straight. Not only do you now need to rotate, translate, and scale before mapping a Euclidean plane to a Lorentz one you need to also undo the distortion you put in - much like you would have to do with a Mercator map of the Earth. Again, using a distorted map doesn't change anything in reality. It just means that you need a complicated transformation before you can interpret what is drawn on the map.

And this is why @Dale and I keep on at you about the distinction between coordinate based descriptions and reality. All of the descriptions involve some transform from reality to the description or the spacetime diagram, be it more complex or less. And you keep talking as if the many different descriptions are the important thing. They aren't. In the muon experiment the angle between worldlines is important, but which worldline you choose to draw vertically on your diagram doesn't matter, but you say "the graph is real and it has real implications". No it isn't and no it doesn't. It just affects how you describe the situation.

And I am taking @Dale and your suggestions into account when I reply. I am trying to tell you both that I do not understand how the worldline for the ball (at event A on the graph I posted earlier) is not curved in the frame of reference of the traveler traveling along the x axis. That is all I need to know now. Please continue reading.

Please read my entire post.

When the traveler moves along the x' axis, the ball shifts toward the traveler. Now the worldline of the ball would seem to have to change too since contraction is not just an illusion or a distorted diagram, it is a real phenomenon.

 

[PeterDonis]

I am trying to tell you both that I do not understand how the worldline for the ball (at event A on the graph I posted earlier) is not curved in the frame of reference of the traveler traveling along the x axis. That is all I need to know now.

And the only way you are going to understand this is to understand in general how you tell whether a worldline is straight or curved without using coordinates. You have already been given the answer for spacetime: a worldline is straight if an accelerometer following that worldline reads zero, and it is curved if an accelerometer following it reads nonzero. That is obviously independent of any choice of frame.

It might help to retrain your intuition if you try to answer the analogous question about a line and arc of circle in a plane that I asked you. That is one way to get to the link with geometry, which the accelerometer answer above doesn't necessarily help a lot with. The relevant geometric concepts are "geodesic" and "path curvature".

When the traveler moves along the x' axis, the ball shifts toward the traveler. Now the worldline of the ball would seem to have to change too

No. How can the traveler change the ball's trajectory (which is what would have to happen for the ball's worldline to change) by moving himself?

contraction is not just an illusion or a distorted diagram, it is a real phenomenon.

That depends on what you mean by "real phenomenon". If you change your state of motion relative to the ball, nothing about the ball changes. The only change is in you, and the coordinates you calculate, not in the ball. If you think about physics instead of getting hung up on coordinates, this should be obvious.

 

[PeterDonis]

curved in the frame of reference

There is no such thing. The path curvature of a worldline is independent of any choice of frame. That is why we keep telling you to forget about frames and think about how you would tell whether a particular worldline is curved or straight without using frames or coordinates at all.

 

[Dale]

I meant that the graph is real in the sense that if you were to superimpose this graph into a real situation, then it would align with what is actually happening.

I wanted to address this comment. I have a table, and I have two graphs of the table. Here is the graph using Cartesian coordinates:

Here is the graph using spherical coordinates:

It is not generally true that if you superimpose a graph onto its corresponding real situation, then it would align with what is actually happening.

The fact that the table legs and table top are all curved in the second graph in no way means that they are physically curved. The physical table is flat and the physical legs are straight, regardless of the fact that the second graph makes it hard to see that fact.

 

[student34]

So how do you determine that it is straight, without using coordinates? Or, if you draw a curved line, such as an arc of a circle, how do you determine that it is curved without coordinates?

I suppose it is sufficient to define what we intend to illustrate by using a graph with the conjecture of what kind of space it's in.