# What would be the form of this world line?

• I
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?

Mentor
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.

• David Lewis and PeterDonis
2022 Award
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?

Mentor
Assume that the ball does not accelerate.
Then what I said in post 2 holds.

2022 Award
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?

2022 Award
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.

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• Dale
Mentor
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

• PeterDonis
2022 Award
@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.

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• Dale
Mentor
@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?

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Mentor
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)?

Mentor
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?

Homework Helper
Gold Member
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?

Mentor
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?

Mentor
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

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Mentor
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.

• cianfa72
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.

Mentor
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?

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2022 Award
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.

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• Dale and cianfa72
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.

Homework Helper
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.

• PeroK, Dale and Ibix
2022 Award
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.

• jbriggs444
Mentor
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?

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• Ibix
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

• weirdoguy
2022 Award
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.

• weirdoguy