Curved Space-time and Relative Velocity

[PAllen]

I must strongly agree with dalespam. A somewhat silly example: suppose 'you' are a tentacled alien with eyes on the end of each tentacle. Now 'you' have at least a relevant frame for the end of each tentacle and for the brain which is in yet another location.

A frame does not exist in the real world. Any frame and any 'reasonable' coordinate system can correctly analyze any physical situation, and there is no reason to favor one over the other except for mathematical convenience.

Another example, that caused me confusion in another thread, are debates about the radiation in different frames. As certain comments from cesiumfrog made clear, what is really required (if using maxwell and classical fields only) is to model some antenna, and compute what actually happens given some (moving) charge and some (differently moving) antenna. This analysis can be done in any frame and must yield the same physical result: the antenna will or will not respond.

 

[Dale]

It is still preposterous to try to create a coordinate chart where you are simultaneously facing North, East, and South. It is just as preposterous to try to create a coordinate chart where you are simultaneously traveling at three different rapidities.

Yes, which is exactly the reason why a coordinate chart must be 1-to-1. This preposterous situation is what you get when a coordinate chart is not 1-to-1.

 

[Dale]

I must directly contradict you. It is not just a shorthand. In every sense, you are "in" that frame in which you are momentarily at rest.

If you can be "in" something then you can also be "out" of something. For SR, how can you possibly be "out" of a reference frame since it extends to infinity in space and goes forever in time. All you can be is "moving in" or "at rest in" a given coordinate system, but you cannot just be "in" or "out" of it.

If you want to calculate what light you are currently seeing, and in particular, where the events are that produced that light, you must use the coordinates of events as calculated "in" that momentarily comoving frame.

You miss the whole point of relativity. You may use any coordinate system you like, and the laws of physics are the same. That is the whole point of expressing the laws in terms of coordinate-independent geometric objects like tensors, and using frame-invariant quantities like the spacetime interval.

 

[JesseM]

I must directly contradict you. It is not just a shorthand. In every sense, you are "in" that frame in which you are momentarily at rest.

Can you be specific about what "sense" you think you are "in" a given frame? Do you disagree that I can perform calculations from the perspective of a frame other than the one where I'm at rest, and I'll get all the same predictions about coordinate-independent facts (like what I am seeing at a particular instant, see below) as I would if I used my own rest frame?

you want to calculate what light you are currently seeing, and in particular, where the events are that produced that light, you must use the coordinates of events as calculated "in" that momentarily comoving frame.

All frames make identical predictions about local facts like what light rays are reaching a particular observer at the moment their own clock reads a particular time. Do you disagree? If not, then I don't see how it matters which frame you use to "calculate what light you are currently seeing". As for "where the events are that produced that light", does "where" mean the coordinates of those events? If so then the answer simply depends on what coordinate system I choose to use, and again I see no physical reason why I am "forced" to choose my own rest frame.

 

[JDoolin]

But nobody is trying to do that, you can't be facing all directions at the same time in any particular coordinates, nor traveling at different speeds in a determinate coordinate chart, by definition a coordinate chart makes you be in a particular situation for that particular chart.
But of course depending on how you define the coordinates you can be facing north with one of them and south with the other without turning around.
This seems pretty trivial to understand, doesn't it?

Let's see if we're on the same page at all.

[URL]http://www.wiu.edu/users/jdd109/stuff/img/dolbygull.jpg​

[/URL]

Dolby and Gull have drawn lines of simultaneity in three different reference frames. These lines of simultaneity are valid

• in Region P for Barbara's outbound trip,
• in Region F for Barbara's return trip,
• in Regions I and II the Lines of simultaneity are drawn for Alex's reference frame.

Now, that's how the lines of simultaneity are drawn, but if you look at the whole graph, and the way it is laid out on the page, the whole thing is actually drawn in Alex's reference frame. No real attempt is made to draw it in any of Barbara's frames.

The difficulty is that in order to do this, you would have to create an animation. You can't represent all of those frames in a single diagram.

Here are three snapshots of such an animation:

[PLAIN]http://upload.wikimedia.org/wikipedia/en/6/6d/Three_frames.JPG[/CENTER]

Here's a demonstration

http://demonstrations.wolfram.com/LorentzTransformationForTwinParadox/​

 

[JDoolin]

Can you be specific about what "sense" you think you are "in" a given frame? Do you disagree that I can perform calculations from the perspective of a frame other than the one where I'm at rest, and I'll get all the same predictions about coordinate-independent facts (like what I am seeing at a particular instant, see below) as I would if I used my own rest frame?

All frames make identical predictions about local facts like what light rays are reaching a particular observer at the moment their own clock reads a particular time. Do you disagree? If not, then I don't see how it matters which frame you use to "calculate what light you are currently seeing". As for "where the events are that produced that light", does "where" mean the coordinates of those events? If so then the answer simply depends on what coordinate system I choose to use, and again I see no physical reason why I am "forced" to choose my own rest frame.

Two people traveling at different speeds that are co-located will see exactly the same events, but they will see them at different places. They will regard the events as having come from different distances and different amounts of times ago.

Even local effects will be like this, if the nearby objects are moving fast enough. It's sometimes all thrown in under the heading "aberration," or "Terrell rotation." These effects will be properly accounted for if you figure out the location of the events in your own reference frame.

 

[JDoolin]

I must strongly agree with dalespam. A somewhat silly example: suppose 'you' are a tentacled alien with eyes on the end of each tentacle. Now 'you' have at least a relevant frame for the end of each tentacle and for the brain which is in yet another location.

A frame does not exist in the real world. Any frame and any 'reasonable' coordinate system can correctly analyze any physical situation, and there is no reason to favor one over the other except for mathematical convenience.

Another example, that caused me confusion in another thread, are debates about the radiation in different frames. As certain comments from cesiumfrog made clear, what is really required (if using maxwell and classical fields only) is to model some antenna, and compute what actually happens given some (moving) charge and some (differently moving) antenna. This analysis can be done in any frame and must yield the same physical result: the antenna will or will not respond.

Though you have an alien with lots of eyes, each of those eyes can only face one direction. Each of those eyes can only have one velocity. In all likelyhood the beast would still draw its maps in the same way we do. With a single north, south, east, and west.

In our solar system with our Earth moving 30 km/s, think of that as a rapidity change of 2*10^-4. That is about equivalent to a rotation of 2*10^-4 radians, or .01^o. If I were to put two maps in front of you, and they were rotated by .01 degrees from each other, you would never notice the difference.

But when we're talking about our twins, Barbara and Alex, we are talking about much more significant rapidity changes, which should actually involve "Lorentz Boosting" the map.

 

[JesseM]

Two people traveling at different speeds that are co-located will see exactly the same events, but they will see them at different places.

I wasn't just talking about co-located observers. If an observer is far away from me, I can use my rest frame to calculations the different times that light from multiple events will reach him according to his own clock, and my calculations for the proper time on his clock that light from different events will reach him should match calculations done in his own rest frame.

They will regard the events as having come from different distances and different amounts of times ago.

By "see" do you mean visual appearances, or do you mean calculations of distances and times in his own rest frame? If the latter, then if you are trying to argue that identifying an observer's "perspective" with his rest frame is more than a matter of mere convention, then this is a completely circular argument! I would say that only if both observers choose to adopt the convention that the "regard" the distances and times of events by defining them in terms of their own rest frames will it be true that "they will regard the events as having come from different distances and different amounts of times ago", they could equally well adopt the convention that they will both use the same frame to do their calculations (and it needn't be either of their own rest frames), and in that case "they will regard the events as having come from the same distance and having happened the same amount of time ago". You haven't presented any arguments as to why they are forced to define their own perspective as what happens in their inertial rest frame--do you think they will be unable to perform measurements that determine the position and times of events in a frame other than their rest frame, or that their pencils will break when they try to perform calculations in some frame other than their rest frame?

Even local effects will be like this, if the nearby objects are moving fast enough. It's sometimes all thrown in under the heading "aberration," or "Terrell rotation." These effects will be properly accounted for if you figure out the location of the events in your own reference frame.

Aberration and Terrell rotation are visual effects only, having to do with which light rays hit your worldline at the same moment (for example, Terrell rotation has to do with the fact that the light rays from different parts of one object that hit your eye simultaneously were not actually emitted simultaneously in whatever inertial frame we are using to do calculations). Do you deny that all frames, not just your rest frame, will make the same predictions about which set of light rays will reach your eye simultaneously at a given point on your worldline, and thus they all make the same predictions about what you see visually at that moment?

 

[JDoolin]

I wasn't just talking about co-located observers. If an observer is far away from me, I can use my rest frame to calculations the different times that light from multiple events will reach him according to his own clock, and my calculations for the proper time on his clock that light from different events will reach him should match calculations done in his own rest frame.

By "see" do you mean visual appearances,

Yes

or do you mean calculations of distances and times in his own rest frame?

Yes.

If the latter, then if you are trying to argue that identifying an observer's "perspective" with his rest frame is more than a matter of mere convention, then this is a completely circular argument! I would say that only if both observers choose to adopt the convention that the "regard" the distances and times of events by defining them in terms of their own rest frames will it be true that "they will regard the events as having come from different distances and different amounts of times ago", they could equally well adopt the convention that they will both use the same frame to do their calculations (and it needn't be either of their own rest frames), and in that case "they will regard the events as having come from the same distance and having happened the same amount of time ago". You haven't presented any arguments as to why they are forced to define their own perspective as what happens in their inertial rest frame--do you think they will be unable to perform measurements that determine the position and times of events in a frame other than their rest frame,

Yes. (If it is a fast moving frame.)

or that their pencils will break when they try to perform calculations in some frame other than their rest frame?

It depends on how fast the objects are moving. If you happen to be in a system where another planet is going by at, say a rapidity=10. That planet will appear so suddenly in your view, and then be so time-dilated after it passes by; there would be absolutely no use whatsoever in defining your coordinates in terms of that planet's coordinate system.

I can tell you what it would look like: It would look like one big diagonal smear with each second on Earth stretching out for cosh(10)=11,013 seconds on the diagram. You might not break your pencil, but rounding errors would creep up very quickly because everything in your diagram for Earth would be just a smidgeon off the line x=c*t.

Mathematically, too many significant figures. Physically, just a useless diagram. We may not be forced to use our own reference frame, but we are forced to use a reference frame with a small relative rapidity to our own.

Aberration and Terrell rotation are visual effects only, having to do with which light rays hit your worldline at the same moment (for example, Terrell rotation has to do with the fact that the light rays from different parts of one object that hit your eye simultaneously were not actually emitted simultaneously in whatever inertial frame we are using to do calculations). Do you deny that all frames, not just your rest frame, will make the same predictions about which set of light rays will reach your eye simultaneously at a given point on your worldline, and thus they all make the same predictions about what you see visually at that moment?

No, I don't deny that. If two hypothetical observers are co-located, the same light would arrive at the same point at the same time. But the information has a different form.

 

[JDoolin]

No, I don't deny that. If two hypothetical observers are co-located, the same light would arrive at the same point at the same time. But the information has a different form.

I made a demo of this some time ago, though I didn't quite see all the implications at the time.

http://www.wiu.edu/users/jdd109/stuff/relativity/timetravel.swf

If you click on the right arrow key a couple of times you can see that the same light is reaching "Speedy T" and "Captain Green" at the same time. But they have different interpretations of that light.

The "actual" distances to the two space-stations is length contracted for "Speedy T," but the implications I didn't quite understand when I made the demo:

For the "observed" distance, you must locate the center of the circles that are currently reaching "Speedy T."

The image of space-station Blue will appear far-away, and approaching superluminally. Space-station red, will appear close-by, and receding slowly.

All images will appear stretched out as they approach, and then contract when they pass by.​

 

[JesseM]

Two people traveling at different speeds that are co-located will see exactly the same events, but they will see them at different places.

By "see" do you mean visual appearances,

Yes

or do you mean calculations of distances and times in his own rest frame?

Yes.

So you mean your statement to apply to both apparent visual distance and also to distance in each observer's rest frame? But in terms of apparent visual distance, it's not true that "two people traveling at different speeds that are co-located will see exactly the same events, but they will see them at different places"--if they are co-located, both will see exactly the same thing, so naturally the apparent visual distance of different objects (i.e. their visual size) and their visual arrangement relative to one another will be identical for both of the co-located observers at that instant.

-do you think they will be unable to perform measurements that determine the position and times of events in a frame other than their rest frame,

Yes. (If it is a fast moving frame.)

Why would the speed of the frame affect how hard it is to determine the position and time of an event? (which might be on the worldline of an object moving fast or slow relative to yourself)

It depends on how fast the objects are moving. If you happen to be in a system where another planet is going by at, say a rapidity=10. That planet will appear so suddenly in your view, and then be so time-dilated after it passes by; there would be absolutely no use whatsoever in defining your coordinates in terms of that planet's coordinate system.

What do you mean "no use"? The problem of the object appearing suddenly in your view is a visual issue which applies regardless of what coordinate system you use. I don't see why for any given object, whatever its visual appearance as it passes within range of your instruments, it should be harder to assign position and time coordinates in one imaginary coordinate grid than in another imaginary coordinate grid. Can you explain further, give a numerical example or something?

I can tell you what it would look like: It would look like one big diagonal smear

What would look like a diagonal smear? The visual appearance of the object passing by you, or your drawing of the object's worldline in a diagram which uses the frame moving rapidly relative to the Earth, or the drawing of the Earth's own worldline in a diagram in that frame?

with each second on Earth stretching out for cosh(10)=11,013 seconds on the diagram.

How does that make it hard to calculate the coordinates of any events? You just did the calculation yourself, showing that two events on the worldline of a clock on Earth which have 1 second of proper time between them must have occurred 11,013 seconds apart in the coordinate time of this frame. Piece of cake!

but rounding errors would creep up very quickly because everything in your diagram for Earth would be just a smidgeon off the line x=c*t.

What do "rounding errors" have to do with the speed of the Earth in this frame? Again, a frame is a purely imaginary thing, we can define the frame to be the one where the Earth is moving at some precisely specified velocity if we wish, rather than defining it in some other way and then trying to measure the speed of the Earth in this frame. And even if we do define the frame in some other way that requires us to measure the speed of the Earth in this frame, this is a practical issue, not a theoretical argument for why we must use a frame with low velocity relative to ourselves regardless of the precision of our measuring instruments.

 

[Dale]

Let's see if we're on the same page at all.

[URL]http://www.wiu.edu/users/jdd109/stuff/img/dolbygull.jpg​

[/URL]

Dolby and Gull have drawn lines of simultaneity in three different reference frames.

These lines of simultaneity are valid

• in Region P for Barbara's outbound trip,
• in Region F for Barbara's return trip,
• in Regions I and II the Lines of simultaneity are drawn for Alex's reference frame.

No, you are completely misunderstanding the figure and the point of the D&G paper. Those are the lines of simultaneity in Barbara's reference frame. This is a single frame which covers the whole spacetime from Barbara's perspective. The bends in the lines are because Barbara is a non-inertial observer, but they are all Barbara's lines of simultaneity. Alex's lines of simultaneity are not drawn on this figure, and note that Barbara's lines of simultaneity are 1-to-1 as required.

Now, that's how the lines of simultaneity are drawn, but if you look at the whole graph, and the way it is laid out on the page, the whole thing is actually drawn in Alex's reference frame. No real attempt is made to draw it in any of Barbara's frames.

Obviously. These are Barbara's lines of simultaneity plotted in Alex's frame; Barbara's lines of simultaneity in Barbara's frame would just be horizontal lines, no need to draw such a figure. For Alex's lines of simultaneity plotted in Barbara's frame see Figure 9. Note also the asymmetry between Barbara's path in Alex's frame and Alex's path in Barbara's frame by comparing figures 5 and 9.

 

[Dale]

Mathematically, too many significant figures. Physically, just a useless diagram. We may not be forced to use our own reference frame, but we are forced to use a reference frame with a small relative rapidity to our own.

This is completely wrong. In fact, on an almost daily basis, when people are designing experiments in particle accelerators they will use reference frames with very large rapidities relative to the lab frame. Your assertions are not only theoretically completely unfounded, they are directly contradicted from any practical standpoint also.

Tell me, in your own words, what does the first postulate of relativity mean in practice?

 

[PAllen]

An attempt at mutual understanding of these 'frame' issues:

1) Physical setup:

a) planet here, and two cameras instantaneously almost colocated there, one camera at small velocity relative to the planet, the other at huge velocity away from the planet.

b) a charge here and to two sets of iron filings instantaneously almost colocated there, one stationary relative to the charge, the other moving rapidly.

2) The physics is frame and coordinate independent.

a) One camera will take a normal picture of the planet, the other camera will take a very reddened picture. Both detect the same photons, but the difference in motion of the detectors (film or ccd) will cause a different result to be recorded.

b) One set of iron filings will sit inert, the other will line up with magnetic field resulting from the relative motion between detector and charge.

3) Complete freedom of choice in how to describe and calculate the physics. In (a) I can analyze everything from a coordinate system in which the planet is stationary at the origin, or either camera is stationary at the origin, or any other arbitrary coordinate system. Whichever I choose, if I do it correctly, I come to the exact same conclusions as to the physics. Same for situation (b).

In what sense is anything 'in a frame of reference' beyond linguistic sloppiness?

 

[Passionflower]

This is completely wrong. In fact, on an almost daily basis, when people are designing experiments in particle accelerators they will use reference frames with very large rapidities relative to the lab frame. Your assertions are not only theoretically completely unfounded, they are directly contradicted from any practical standpoint also.

Tell me, in your own words, what does the first postulate of relativity mean in practice?

Here is an example where high rapidity frames can actually be an advantage, as you need less time slices.

http://www.lbl.gov/cs/Archive/news080910.html

 

[JDoolin]

Here is an example where high rapidity frames can actually be an advantage, as you need less time slices.

http://www.lbl.gov/cs/Archive/news080910.html

Yes. When the events are close enough together in time and space, transferring to a high rapidity frame can be very useful. I was thinking of ordinary day-to-day life, like going to the store, or trying to find the directions on a map.

In these cases, by changing to a different reference frame, some events which are far apart in time and space will come closer together, while others which are close together in time and space will move far apart.

This may be exactly wha you are looking for if you are working with particle accelerators.

 

[JDoolin]

An attempt at mutual understanding of these 'frame' issues:

1) Physical setup:

a) planet here, and two cameras instantaneously almost colocated there, one camera at small velocity relative to the planet, the other at huge velocity away from the planet.

b) a charge here and to two sets of iron filings instantaneously almost colocated there, one stationary relative to the charge, the other moving rapidly.

2) The physics is frame and coordinate independent.

a) One camera will take a normal picture of the planet, the other camera will take a very reddened picture. Both detect the same photons, but the difference in motion of the detectors (film or ccd) will cause a different result to be recorded.

b) One set of iron filings will sit inert, the other will line up with magnetic field resulting from the relative motion between detector and charge.

3) Complete freedom of choice in how to describe and calculate the physics. In (a) I can analyze everything from a coordinate system in which the planet is stationary at the origin, or either camera is stationary at the origin, or any other arbitrary coordinate system. Whichever I choose, if I do it correctly, I come to the exact same conclusions as to the physics. Same for situation (b).

In what sense is anything 'in a frame of reference' beyond linguistic sloppiness?

You said it yourself.

You can analyze everything

(1) from a coordinate system in which the planet is stationary at the origin
(2) either camera is stationary at the origin
(3) any other arbitrary coordinate system​

Each of these are analysis "in a frame of reference."

In each of these frames, you DO get all the same events, and you DO get the same "space-time intervals" between the events, but you don't get the same locations or the same times. The distances are different; the times are different, the ANGLES are different.

Each analysis is in a single reference frame. And that makes these different frames physically significant.

Edit: I realized that this statement applies to Jesse's post as well:

So you mean your statement to apply to both apparent visual distance and also to distance in each observer's rest frame? But in terms of apparent visual distance, it's not true that "two people traveling at different speeds that are co-located will see exactly the same events, but they will see them at different places"--if they are co-located, both will see exactly the same thing, so naturally the apparent visual distance of different objects (i.e. their visual size) and their visual arrangement relative to one another will be identical for both of the co-located observers at that instant.

Again: In each of these frames, you DO get all the same events, and you DO get the same "space-time intervals" between the events, but you don't get the same locations or the same times. The distances are different; the times are different, the ANGLES are different.

 

[JDoolin]

What do you mean "no use"? The problem of the object appearing suddenly in your view is a visual issue which applies regardless of what coordinate system you use. I don't see why for any given object, whatever its visual appearance as it passes within range of your instruments, it should be harder to assign position and time coordinates in one imaginary coordinate grid than in another imaginary coordinate grid. Can you explain further, give a numerical example or something?


What would look like a diagonal smear? The visual appearance of the object passing by you, or your drawing of the object's worldline in a diagram which uses the frame moving rapidly relative to the Earth, or the drawing of the Earth's own worldline in a diagram in that frame?

How does that make it hard to calculate the coordinates of any events? You just did the calculation yourself, showing that two events on the worldline of a clock on Earth which have 1 second of proper time between them must have occurred 11,013 seconds apart in the coordinate time of this frame. Piece of cake!

What do "rounding errors" have to do with the speed of the Earth in this frame? Again, a frame is a purely imaginary thing, we can define the frame to be the one where the Earth is moving at some precisely specified velocity if we wish, rather than defining it in some other way and then trying to measure the speed of the Earth in this frame. And even if we do define the frame in some other way that requires us to measure the speed of the Earth in this frame, this is a practical issue, not a theoretical argument for why we must use a frame with low velocity relative to ourselves regardless of the precision of our measuring instruments.

Perhaps you're right.

Let's see. Imagine two 0-space-time intervals. One of them is a signal from the Earth to the moon, and another is a signal from your keyboard to your CPU. Let's just say these two beams happen to come from the same place at the same time, and are in exactly opposite directions.

It is possible using the Lorentz Transformations to find a frame where the distance between the earth-moon events is negligible, while the distance between the keyboard-CPU events is as far as you like. In such a frame, the Earth and moon would be length contracted to the point where the distances between cities would be negligible. This is where I though you would need many significant figures, but that would only happen if you addd on the x=v*t +(displacement). The v*t would give a very large number, while the lorentz-contracted displacement would give a very small number. (This is what I meant by the diagonal smear)

But I think you are right; there's not really any particular conceptual difficulty to adjusting to that; at least not too much more than adjusting to the notion of oncoming traffic.

 

[Passionflower]

Let's see. Imagine two 0-space-time intervals. One of them is a signal from the Earth to the moon, and another is a signal from your keyboard to your CPU.

I can understand that a light signal from the Earth to the Moon has a zero spacetime interval, but why do you say that a signal from the keyboard to the CPU has a zero spacetime interval?

It is possible using the Lorentz Transformations to find a frame where the distance between the earth-moon events is negligible, while the distance between the keyboard-CPU events is as far as you like.

You mean the proper time interval right?
The distance between two events is the same for all frames right?

 

[JDoolin]

I can understand that a light signal from the Earth to the Moon has a zero spacetime interval, but why do you say that a signal from the keyboard to the CPU has a zero spacetime interval?

Sorry. Use instead, the signal from your remote-control to your TV set.

You mean the proper time interval right?
The distance between two events is the same for all frames right?

The proper time interval (of time-like separated events) is the same as the space-time interval.

If you have two events that are connected by a photon, like the signal from Earth to moon, and the signal from my remote to my TV, those both have zero proper time between them.

But you can easily see that the distance to the TV and the distance to the moon are not zero, nor are they the same. But if they are in opposite directions, it is fairly trivial to come up with a Lorentz Transformation that would make the distances between the events the same.

Picture the Earth moving to the right very fast, and the moon following along behind it. The signal from the Earth to the moon barely has to move, because the moon catches up. On the other hand, the TV is moving really fast to the right, and it takes some time for the light from the remote to reach it.

Here, I attached a couple of diagrams of the same events in two different reference frames. I posted the source-code in my blog.

 

[Passionflower]

The proper time interval (of time-like separated events) is the same as the space-time interval.

Well yes of course but that is not what I was referring to.

If you have two events that are connected by a photon, like the signal from Earth to moon, and the signal from my remote to my TV, those both have zero proper time between them.

But you can easily see that the distance to the TV and the distance to the moon are not zero, nor are they the same. But if they are in opposite directions, it is fairly trivial to come up with a Lorentz Transformation that would make the distances between the events the same.

What I was referring to is your usage of "the distance between two events". When I think of the distance between two events I think of the spacetime distance, which of course is identical for all frames of reference. But it seems that you refer to spatial or temporal distance between two events from a frame of reference right? I think it is just a terminology issue.

By the way I really enjoy your pictures and graphs, please keep doing that :)

 

[Dale]

Yes. When the events are close enough together in time and space, transferring to a high rapidity frame can be very useful.

Yes, because the laws of physics are the same we can pick any frame that is useful or convenient. It seems like you now understand that, so I believe you are in agreement now.

 

[Dale]

Each of these are analysis "in a frame of reference."

Saying that an analysis is "in a frame of reference" is entirely different from saying that an object is "in a frame of reference". The former is just a colloquial way of saying that a particular frame was used during the analysis, the latter is nonsense. As long as you understand that there is nothing which requires us to use any specific frame then the rest of what we have been discussing follows.

 

[JDoolin]

Saying that an analysis is "in a frame of reference" is entirely different from saying that an object is "in a frame of reference". The former is just a colloquial way of saying that a particular frame was used during the analysis, the latter is nonsense. As long as you understand that there is nothing which requires us to use any specific frame then the rest of what we have been discussing follows.

There's still a rather vital nitpick remaining. If the "object" happens to be an "observer" such as Barbara, all of the data she collects to use for her analysis will come from whatever frame she is in.

Unless she's been dropping off probes along the way, there's no way for her to collect the data from any other frame except for the frame in which she is momentarily at rest.

You can analyze the data in whatever reference frame you like, but the nature of cameras and lab equipment requires you to collect the data in whatever reference frames are comoving with each piece of equipment.

 

[JDoolin]

Well yes of course but that is not what I was referring to.


What I was referring to is your usage of "the distance between two events". When I think of the distance between two events I think of the spacetime distance, which of course is identical for all frames of reference. But it seems that you refer to spatial or temporal distance between two events from a frame of reference right? I think it is just a terminology issue.

Might be terminology issue. I never use "distance" to mean space-time-interval.

Except for that mathematical fact that it is preserved under Lorentz Transformation, just like distance is preserved under rotation, they bear little in the way of conceptual similarity.

Coordinates move along circular paths in Euclidian space when the observer rotates. Events move along hyperbolic paths in spacetime when an observer accelerates.

Increasing a distance is like moving outward along concentric circles. But there's not really any analog for concentric hyperbolic arcs.

Also, after a rotation, the observer has the choice to stay put and rotate back the same way. But you can't stay put in spacetime. You have to move forward in time.

I'm probably just telling you what you already know. But I've always found the use of the word "distance" to describe s^2=c^2 t^2 - x^2 very confusing.

By the way I really enjoy your pictures and graphs, please keep doing that :)

Time permitting, whenever I think of something good, I will. As long as one person appreciates it, it's worth it. Thank you.

 

[Dale]

There's still a rather vital nitpick remaining. If the "object" happens to be an "observer" such as Barbara, all of the data she collects to use for her analysis will come from whatever frame she is in.

Unless she's been dropping off probes along the way, there's no way for her to collect the data from any other frame except for the frame in which she is momentarily at rest.

You can analyze the data in whatever reference frame you like, but the nature of cameras and lab equipment requires you to collect the data in whatever reference frames are comoving with each piece of equipment.

This is simply not true. All frames will agree on the predicted result of any measurement that Barbara makes.

You still seem to not understand the first postulate of relativity.

 

[JDoolin]

This is simply not true. All frames will agree on the predicted result of any measurement that Barbara makes.

You still seem to not understand the first postulate of relativity.

The first postulate of relativity is, according to Wikipedia:

The Principle of Relativity – The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems in uniform translatory motion relative to each other.

Do you think the laws being the same mean the measurements are the same?

What about momentum? In Barbara's frame, Barbara has no momentum. In other frames Barbara has momentum.

What about velocity? In Barbara's frame, Barbara has no velocity. In other frames Barbara has velocity.

Also, distances between events change, times between events change. The measurements are all different.

Only the laws are all the same. But when you collect the data, it's completely different data.

 

[Dale]

Do you think the laws being the same mean the measurements are the same?

Yes. The laws are what determine the measurements. The first implies the second.

What about momentum?

Momentum is not a measurement. Describe your physical device and procedure for measuring the momentum of some given object. That is what a measurement is, and all frames will agree on the result measured since all frames agree on the laws which govern the measuring device.

Not all frames will agree that the measured result actually represents the momentum of the object, but they will agree on the result of the measurement. That is required by the first postulate.

 

[JDoolin]

Yes. The laws are what determine the measurements. The first implies the second.

Momentum is not a measurement. Describe your physical device and procedure for measuring the momentum of some given object. That is what a measurement is, and all frames will agree on the result measured since all frames agree on the laws which govern the measuring device.

Not all frames will agree that the measured result actually represents the momentum of the object, but they will agree on the result of the measurement. That is required by the first postulate.

Okay, but you agree that the distances and times and velocities are different, right? They all disagree on results of measurement.

 

[Dale]

Okay, but you agree that the distances and times and velocities are different, right? They all disagree on results of measurement.

I only singled out momentum because it was the first one you mentioned, not because it was conceptually different from the others. The same thing I said about momentum applies to distances, times, and velocities. Describe the experimental set up of your measurement and all frames will agree on the result.