What would be the form of this world line?

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In summary, the straightness of a worldline is an invariant geometrical fact that is independent of any coordinate system, reference frame, or observer. 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. Using curved coordinates does not change the underlying geometrical facts. The presence or absence or shape of the grid lines does not change the geometry of the lines. The straightness of the worldline is
  • #36
Dale said:
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.
 
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  • #37
student34 said:
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.
student34 said:
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?
 
  • #38
student34 said:
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?

student34 said:
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.

student34 said:
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.
 
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  • #39
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.
 
  • #40
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.
 
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  • #41
student34 said:
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.

student34 said:
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.

student34 said:
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".
 
  • #42
PeterDonis said:
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?
 
  • #43
Ibix said:
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?
 
  • #44
Dale said:
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.
 
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  • #45
student34 said:
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?
 
  • #46
Ibix said:
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.
 
  • #47
student34 said:
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".

student34 said:
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?

student34 said:
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.
 
  • #48
student34 said:
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.
 
  • #49
student34 said:
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:

CartesianTable.jpg

Here is the graph using spherical coordinates:
SphericalTable.jpg


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.
 
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  • #50
PeterDonis said:
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.
 
  • #51
student34 said:
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)?
 
  • #52
student34 said:
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.
 
  • #53
PeterDonis said:
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)?
 
  • #54
Dale said:
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.
 
  • #55
Dale said:
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.
 
  • #56
student34 said:
This is what length contraction seems to be saying
No, it isn't. See below.

student34 said:
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.

student34 said:
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.
 
  • #57
student34 said:
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.
 
  • #58
student34 said:
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.
 
  • #59
Dale said:
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.
 
  • #60
student34 said:
I am really not sure.
Then how were you able to answer "no" to @Dale's question?
 
  • #61
PeterDonis said:
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.
If a ball 5 meters from you suddenly starts moving to 3 meters from you, isn't it logical, by deduction, to say that the ball has moved closer to you?
 
  • #62
student34 said:
If a ball 5 meters from you suddenly starts moving to 3 meters from you, isn't it logical, by deduction, to say that the ball has moved closer to you?
Can you describe the experiment more completely? What did you originally measure? What did you measure later?

It is cheating to say "the ball starts moving" and then conclude "the ball has moved". You've included the conclusion in the problem statement.
 
  • #63
jbriggs444 said:
Can you describe the experiment more completely? What did you originally measure? What did you measure later?
I don't know why it is so blurry, but in the diagram below, from Wikipedia, an object moves from rest, at the origin 0,0, to a high speed along the x axis. We will assume that A is a ball at rest with the object O at the origin. Let's say that the acceleration is instantaneous for simplification. Roughly speaking, the ball seems to shift or "move" closer to O, let's say from 5 meters to 4 meters.
 

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  • #64
student34 said:
I don't know why it is so blurry, but in the diagram below, from Wikipedia, an object moves from rest, at the origin 0,0, to a high speed along the x axis. We will assume that A is a ball at rest with the object O at the origin. Let's say that the acceleration is instantaneous for simplification. Roughly speaking, the ball seems to shift or "move" closer to O, let's say from 5 meters to 4 meters.
Not responsive to the question that was posed. What was measured?

No one asked "what coordinates were ascribed?"
 
  • #65
PeterDonis said:
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.
What about in the case of the muons reaching Earth due to there being less distance that the muon has to travel than if there were no length contraction?
 
  • #66
student34 said:
What about in the case of the muons reaching Earth due to there being less distance that the muon has to travel than if there were no length contraction?
The elapsed proper time for the muon is an invariant. It is the same for a high speed and time-dilated muon traversing a large distance as for a stationary muon being approached by the Earth's surface through a length-contracted atmosphere. Both descriptions yield the same result for the fraction of muons impacting upon the Earth's surface. No difference in measured results corresponds to the difference in description.
 
  • #67
jbriggs444 said:
Not responsive to the question that was posed. What was measured?

No one asked "what coordinates were ascribed?"
Object O was measured to be 5 meters from A when both are at rest with each other, and then measured to be 4 meters from A after acceleration.
 
  • #68
student34 said:
Object O was measured to be 5 meters from A when both are at rest with each other, and then measured to be 4 meters from A after acceleration.
No. It was not. You are asserting coordinates, not providing actual physical measurements.

What was actually measured? How was the measurement performed?

Example:

We took a five meter stick, lined one end up at the origin, placed a ball at the other end. The ball stayed there. We concluded that the ball was continuously 5 meters away from the origin in the lab frame.

We then took two atomic clocks, synchronized them at the origin and then slowly moved one four meters away on the lab floor.

We launched our original 5 meter stick at 60 percent of the speed of light (details available upon request) rightward from well to the left of the origin so that it would have time for any acceleration-induced stresses (*ahem*, length contraction) to relax away. We glued a ball on the right edge of the stick. We observed the left edge of the stick pass by the origin at a particular time on our atomic clock there. We recorded that time reading. We observed the ball on the stick's right edge pass by our 4 meter clock. We recorded the 4 meter clock's reading.

Our two clock readings matched.

We concluded that this ball was 4 meters away from the origin at the time when the left end of the meter stick passed the origin.

If you prefer, we can re-do the experiment for the scenario in which the lab is accelerated leftward and the meter stick is left alone...

So this time we leave the 5 meter stick on the ground and move the lab (and the origin of our coordinate system) well to the right. We set the lab in motion leftward at 60 percent of the speed of light and wait for any acceleration-induced stresses to relax away.

We set up two atomic clocks, synchronized at our lab origin and slowly move one four meters to the right.

We wait for the ball to pass by our 4 meter clock and for the left edge of the meter stick to pass by our clock at the origin and record both time stamps. We observe that they match and conclude that the ball was 4 meters away from our origin when the left end of the meter stick passed the origin.

None of this involves any coordinates. It involves measurements of distance and of time.
 
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  • #69
student34 said:
I am really not sure.
OK, maybe some other suggestions then. Let’s say that you are building a deck and need to mark a straight line on the deck. How do you physically make a reasonably straight line on a floor? (Assuming the floor is flat)

Here is just one common method How does this simple device make a straight line?
 
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  • #70
student34 said:
If a ball 5 meters from you suddenly starts moving to 3 meters from you
That is not what is happening. Again: nothing about the ball changes when you change speed. Are you going to continue to ignore this or not?

student34 said:
What about in the case of the muons reaching Earth due to there being less distance that the muon has to travel than if there were no length contraction?
That's not how we account for it in the Earth frame. In the Earth frame, we account for it by the muons being time dilated; the internal "clocks" inside the muons that determine their decay rate is slowed compared with Earth clocks.

In the muon frame, the distance Earth travels to reach them (note how I phrased that: the muons are at rest in this frame so they don't move, the Earth does) is length contracted, compared to the Earth frame. But neither the muons nor the Earth change speed in this scenario, so it is irrelevant to your question about what happens if you change speed.

student34 said:
Object O was measured to be 5 meters from A when both are at rest with each other, and then measured to be 4 meters from A after acceleration.
Wrong. You have already been told repeatedly that there are no actual measurements that correspond to these numbers.
 

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