I What would be the form of this world line?

[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.

 

[Dale]

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.

Ok, this is good. So, how would you determine, physically, without coordinates, if the line on the piece of paper is straight (assuming a Euclidean space)?

 

[PeterDonis]

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.

No. No. No. No. No.

Banish the word "graph" from your mind. Get rid of it. Forget it. Abolish it. Eradicate it.

I emphasize this because, in spite of the fact that you have been told many times now to stop thinking in terms of coordinates, you keep thinking in terms of coordinates. "Graph" means "coordinates". You need to stop doing that or you will never get anywhere. And this thread will be closed because there is no point in wasting our time responding to you if you are not going to listen to the responses.

 

[student34]

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?

This, right here, is the heart of the issue. This is what length contraction seems to be saying Peter. The relative distance between the traveler and the ball changes. We know that it is in fact the ball that shifts toward the traveler on the x-axis in the diagram - how does this happen - we shouldn't be able to explain it away by distorting a graph that fits the situation. How does this increase in speed bring the ball closer to the traveler if not by accelerating it (even though an accelerometer would probably not read anything)?

 

[jbriggs444]

Ok, this is good. So, how would you determine, physically, without coordinates, if the line on the piece of paper is straight (assuming a Euclidean space)?

If it were me, I'd see if I had a pencil with which to make marks on the line and a piece of thread to stretch between the marks.

Then I'd notice something about the triangle inequality.

 

[student34]

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:

View attachment 295335
Here is the graph using spherical coordinates:
View attachment 295336

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.

This is really interesting. Thanks for the visual. I will keep this in mind.

 

[PeterDonis]

This is what length contraction seems to be saying

No, it isn't. See below.

The relative distance between the traveler and the ball changes.

The calculated distance to the ball in the traveler's rest frame changes. But there is no direct observable that corresponds with this change. It's a calculated number that depends on your choice of coordinates.

How does this increase in speed bring the ball closer to the traveler

The traveler's change in speed does nothing to the ball. The traveler's decision to choose a new frame of reference when he changes speed changes the distance he calculates to the ball, but that has nothing to do with the ball.

This is what comes of continuing to focus on coordinates despite repeated advice not to. All you're doing is confusing yourself further.

 

[PeterDonis]

We know that it is in fact the ball that shifts toward the traveler on the x-axis in the diagram - how does this happen - we shouldn't be able to explain it away by distorting a graph that fits the situation.

We're not the ones who are distorting things. You are, by continuing to harp on coordinates after being told repeatedly not to do that. There is nothing here to "explain away", because nothing happens to the ball when the traveler changes speed. All that happens is that the traveler calculates a new number for what he calls "distance to the ball" in his new reference frame.

 

[PeterDonis]

This, right here, is the heart of the issue.

No, it isn't. It's a distraction that has nothing whatever to do with how you determine whether a worldline is straight or curved without using coordinates, which is the heart of the issue.

 

[student34]

Ok, this is good. So, how would you determine, physically, without coordinates, if the line on the piece of paper is straight (assuming a Euclidean space)?

I am really not sure.

 

[PeterDonis]

I am really not sure.

Then how were you able to answer "no" to @Dale's question?