Struggles with the geometrical analogy

[whatif]

{Moderators note: thread split from https://www.physicsforums.com/threa...dilation-and-the-twin-paradox-comments.842793 }

I am a novice about relativity, but I found this convoluted and very difficult to follow. I think the lengths in the different coordinate systems of the geometric analogue is just a matter of linear scaling with different base vectors. The lengths are identical in each frame when accounting for the scaling. As I understand it, the lengths and time intervals for special relativity are different, not as a result of arbitrary scaling but as a result of relative motion between of the inertial frames and the transformations are nonlinear.

 

[Orodruin]

I think the lengths in the different coordinate systems of the geometric analogue is just a matter of linear scaling with different base vectors.

Length has nothing to do with base vectors. It is invariant and does not depend on the coordinate system.

As I understand it, the lengths and time intervals for special relativity are different, not as a result of arbitrary scaling but as a result of relative motion between of the inertial frames and the transformations are nonlinear.

No, the Lorentz transformations are linear transformations. The entire point is that spacetime intervals do not depend on the coordinate system, just as little as the distance between you and your neighbour does not depend on the coordinate system.

 

[whatif]

the Lorentz transformations are linear transformations.

Sorry, yes they are.
​

Length has nothing to do with base vectors. It is invariant and does not depend on the coordinate system.

That was my point, except that that it applies in the same inertial reference frame. While the transformations are mathematically linear I understood that length varied according to relative motion of the frame in which it is measured.

 

[Ibix]

That was my point, except that that it applies in the same inertial reference frame. While the transformations are mathematically linear I understood that length varied according to relative motion of the frame in which it is measured.

You need to distinguish carefully between the two cases Orodruin was talking about in the article, and I don't think you are doing so. One case is Euclidean geometry, in which length is the same in all coordinate systems and time is no part of geometry. The other case is the Minkowski geometry that underlies relativity, which includes time as a dimension. In this case, a concept called "interval" is analogous to length in Euclidean geometry while the spatial length of an object is analogous to the difference in x coordinates between the ends of a rod in Euclidean geometry.

The interval is invariant between frames in relativity, but length is not. In Euclidean geometry, length is invariant and interval isn't a concept. The coordinate transforms between frames are linear in both cases.

 

[whatif]

You need to distinguish carefully between the two cases Orodruin was talking about in the article, and I don't think you are doing so.

I think that I understand the resolution of the twin paradox using a Minkowski diagram, but this geometrical analogue and explanation does not help me better understand time dilation. I would say that maybe it was not intended to if that was not stated as an aim.

Also, no analogue is perfect but it still seems to me that a significant flaw in the analogue is that the different coordinate systems come down to an arbitrary choice of axes but the transformations between the different moving frames is not about an arbitrary choice of axes; I think.

 

[Herman Trivilino]

I think the lengths in the different coordinate systems of the geometric analogue is just a matter of linear scaling with different base vectors.

Are you talking about the lengths of the line segments that are drawn on the spacetime diagrams?

 

[Ibix]

I think that I understand the resolution of the twin paradox using a Minkowski diagram, but this geometrical analogue and explanation does not help me better understand time dilation. I would say that maybe it was not intended to if that was not stated as an aim.

The ticks of a clock are regular marks on the timelike coordinate axis. If you are in motion with respect to me, your timelike coordinate axis is at an angle to mine. The spacing between your regular ticks, as measured by me, is the projection of the spacing you measure onto my coordinate axis. That's time dilation. The Euclidean analogy is two rulers at an angle to each other. Assuming the rulers cross at their zero mark, both rulers agree that the 1m mark on the other ruler is level with their 95cm mark (or whatever - depends on the angle). Minkowski geometry is just more mathematically interesting.

Also, no analogue is perfect but it still seems to me that a significant flaw in the analogue is that the different coordinate systems come down to an arbitrary choice of axes but the transformations between the different moving frames is not about an arbitrary choice of axes; I think.

Of course the choice of axes is arbitrary. How is an experiment affected if I observe it sutting still, or walking past, or flying by in a spaceship at 0.95c? Each of those options would naturally choose a different frame. The transform between any pair of frames is well defined, but no frame is right or wrong - the choice is completely arbitrary.

 

[Orodruin]

but it still seems to me that a significant flaw in the analogue is that the different coordinate systems come down to an arbitrary choice of axes but the transformations between the different moving frames is not about an arbitrary choice of axes; I think.

Then this is where your intuition fails you. It is exactly what it comes down to. A choice of inertial system is exactly analogous to an arbitrary choice of (mutually orthogonal) coordinate axes. The geometry of Minkowski space just implies that the corresponding transformation is a hyperbolic rotation rather than a normal rotation.

 

[Nugatory]

it still seems to me that a significant flaw in the analogue is that the different coordinate systems come down to an arbitrary choice of axes but the transformations between the different moving frames is not about an arbitrary choice of axes; I think.

It is completely arbitrary. There is no requirement that I do my calculations using the frame in which I am rest, and no matter which frame I use I will get the same results for anything and everything that happens - for example, the time on my wristwatch when light from some event reaches my eyes.

Often I choose to use the frame in which I am at rest, but that's not because that frame is special, it's because the problem at hand is most easily solved in that frame.

 

[whatif]

Are you talking about the lengths of the line segments that are drawn on the spacetime diagrams?

I was talking about the lengths of the projections of the line (object) onto the axes in the analogue, which was just about 2 D space.

 

[whatif]

Of course the choice of axes is arbitrary.

A choice of inertial system is exactly analogous to an arbitrary choice of (mutually orthogonal) coordinate axes.

It is completely arbitrary.

Only for the spacetime interval. As I see it, time is not independent of the location in space, but a restriction on the 3D location that can be occupied. In the 2D analogue there is no such interdependence/restriction. The article introduces the rotation of axes and ‘modified Pythagorean theorem’ without explanation, which, to me, reduces the idea to an arduous mathematical how to, rather than an insight.The twin paradox has a resolution when the twins reunite. However, so long as the twins separate in different inertial frames each is older than the other, forever in each's own frame. That, I imagine, is time dilation.

It is completely arbitrary. There is no requirement that I do my calculations using the frame in which I am rest.

Except that my rest frame is my experience.

 

[Dale]

Only for the spacetime interval.

The choice of reference frame or coordinates is not only arbitrary for the spacetime interval. It is also arbitrary for the expression of any physical law and for the prediction of the outcome of any physical measurement for any experiment. That is the meaning of the first postulate.

Except that my rest frame is my experience.

Not really. Your experience is much more closely related to your past light cone than your frame. And your past light cone is the same in all reference frames.

 

[whatif]

Your experience is much more closely related to your past light cone than your frame.

Rest frame was, perhaps, a bad choice of wording. What I meant was all frames in which I was stationary at the time, including non inertial frames of acceleration, which I guess correlates to the light cone. Yes/no?
To be more precise about the twin paradox. So long as each twin occupies a different inertial frame continuing to separate apart from the other's frame then each continues to age faster than the other. That, I think, exemplifies what requires insight; or just plane acceptance because it seems to be implicitly verified by experiment.

 

[Herman Trivilino]

I was talking about the lengths of the projections of the line (object) onto the axes in the analogue, which was just about 2 D space.

If you take the ratio of the projections onto the time axes that will give you the amount of time dilation. That ratio is not random, it depends on the angle that the time axes make with each other, which in turn depends on the relative speed.

 

[whatif]

If you take the ratio of the projections onto the time axes that will give you the amount of time dilation. That ratio is not random, it depends on the angle that the time axes make with each other, which in turn depends on the relative speed.

Reference https://www.physicsforums.com/threa...n-and-the-twin-paradox-comments.842793/page-3

Indeed, it is a mathematical construct that works. However, I suggest that the insight would be to understand how the experience of time and space for each occupier of different frames differs, which is counter intuitive. This explanation with the 2D analogue does not do it for me.

 

[Orodruin]

As I see it, time is not independent of the location in space, but a restriction on the 3D location that can be occupied. In the 2D analogue there is no such interdependence/restriction.

I am sorry, but this reads as word sallad.

Furthermore, position in space is frame dependent and therefore arbitrary.

What I meant was all frames in which I was stationary at the time, including non inertial frames of acceleration, which I guess correlates to the light cone. Yes/no?

No. What you experience has nothing to do with arbitrary coordinate systems. Convoluting and changing your statements is not going to change that.

So long as each twin occupies a different inertial frame continuing to separate apart from the other's frame then each continues to age faster than the other.

This again shows fundamental misunderstanding of SR. Inertial frames are not things to be ”occupied”. Anything that exists in one frame exists in all frames.

That, I think, exemplifies what requires insight; or just plane acceptance because it seems to be implicitly verified by experiment.

And the insight is that this is no stranger than the Euclidean analogue I mentioned. The only difference is that spacetime has a different geometry through the modified Pythagorean theorem. Why it has that geometry is a different question, but results from the SR postulates. The number of dimensions has nothing to do with this.

However, I suggest that the insight would be to understand how the experience of time and space for each occupier of different frames differs, which is counter intuitive.

Which you would understand if you read the insight. The insight is that, geometrically, it is completely analogous to the Euclidean case presented. Saying that it is not and instead banging your head on your own misunderstandings is not going to change this fact. You started by admitting that you are a novice in relativity, but you are stating things as if your own understanding must be the correct one rather than as if trying to understand what is going on. I do not think this will serve you well in the long run. Note that many of the people you are conversing with here are anything but novices when it comes to relativity.

I am not claiming that the Insight must be useful to everyone, but the fact of the matter is that the ideas behind time dilation are exactly analogous to the Euclidean case presented. It is also not intended to teach you why spacetime geometry is different, just about why this difference leads to some sign changes but otherwise time dilation is exactly the same effect as that discussed in the Euclidean setting.

Also note that inertial frames do not ”experience” time, they have a time coordinate, which is a different concept. The underlying thing to understand (without which you can never properly understand time dilation or any other effect in SR) is that simultaneity is relative, ie, events that are simultaneous in one frame are generally not in another.

 

[Ibix]

Only for the spacetime interval.

No. The choice of frame is completely arbitrary. That choice is also a definition of what "space" and "time" (in the sense of coordinate timel mean for this reference frame, so concepts such as length and duration follow from this. Length contraction and time dilation are the result of defining different things as the length of an object or its duration.

The Euclidean analogy to this is a rectangular table. Whether it's a long, narrow table or a short, wide one depends on an arbitrary choice of which direction we call "across" and which "along". That choice is arbitrary, and need not correspond to where you happen to be standing.

However, so long as the twins separate in different inertial frames each is older than the other, forever in each's own frame. That, I imagine, is time dilation.

More or less (although see my comment below on your "more precise" comment on the twin paradox). The important thing is that those answers depend on an arbitrary choice of what you want to call "at the same time as" one twin is aged 30. You can fairly easily construct a frame in which they are both always the same age. The difference with the twin paradox is that the twins reunite and, at that point, there must be a non-arbitrary answer to how old they both are.

Except that my rest frame is my experience.

Definitely not. Say a supernova appears in the sky tonight. Your experience is that a supernova appeared on December 13th 2018. That's all. If you pick a reference frame, you can then decide how far away the supernova was and when it happened. If you pick a different frame then you will come up with a different answer. But no choice of frame changes your personal experience of seeing a supernova and checking the calendar.

Frames have nothing to do with experience. They are all about interpretation of that experience, and multiple interpretations are possible.

What I meant was all frames in which I was stationary at the time, including non inertial frames of acceleration, which I guess correlates to the light cone. Yes/no?

No.

To be more precise about the twin paradox. So long as each twin occupies a different inertial frame continuing to separate apart from the other's frame then each continues to age faster than the other.

No. Nobody "occupies" a frame. As long as each twin remains moving inertially and chooses to use their inertial rest frame to interpret information about their twin, they will regard themselves as older. They can, at any time, adopt a different frame or just a different simultaneity convention and come up with a different answer.

 

[whatif]

I am sorry, but this reads as word sallad.
Reference https://www.physicsforums.com/threa...n-and-the-twin-paradox-comments.842793/page-3

I am sorry you feel that way.

you are stating things as if your own understanding must be the correct one rather than as if trying to understand what is going on
Reference https://www.physicsforums.com/threa...n-and-the-twin-paradox-comments.842793/page-3

That is not my intent.

The underlying thing to understand (without which you can never properly understand time dilation or any other effect in SR) is that simultaneity is relative
Reference https://www.physicsforums.com/threa...n-and-the-twin-paradox-comments.842793/page-3

That simultaneity is relative is something that I think I appreciate.

Your rap over my knuckles is acknowledged. However, it seems that an observer may chose a frame of reference to deem whether events are simultaneous and I dare to ask whether that is correct? I ask because if I measure two bolts of lightning in different locations to be simultaneous it seems that my experience of measuring simultaneity can be arbitrarily disregarded (and in a sense of using transformations to use a different frame of reference it can but to disregard my experience seems to involve its own convoluted rationale).

 

[Ibix]

? I ask because if I measure two bolts of lightning in different locations to be simultaneous it seems that my experience of measuring simultaneity can be arbitrarily disregarded

All frames will agree that you received light from the strikes simultaneously. All frames will agree that the flashes happened earlier than you saw them. Because they use different definitions of simultaneity they will not agree that the flashes happened simultaneously.

Your experience is not a frame. A frame is a tool for interpreting your experience.

 

[whatif]

All frames will agree that you received light from the strikes simultaneously. All frames will agree that the flashes happened earlier than you saw them. Because they use different definitions of simultaneity they will not agree that the flashes happened simultaneously.

Your experience is not a frame. A frame is a tool for interpreting your experience.

Which has always been my understanding. However, that implies that an observer and the reference frame for their experience is not arbitrary. And I am sorry that my fruit saladedy language is spurned but the frame innately used for my experience is not arbitrary and is a relationship that gives a sense of 'occupying' that frame.

 

[Orodruin]

However, that implies that an observer and the reference frame for their experience is not arbitrary.

Again, there is no "reference frame" for an observer's experience. It does not matter what frame you choose to label and interpret your events, your experience will be exactly the same.

 

[whatif]

Again, there is no "reference frame" for an observer's experience. It does not matter what frame you choose to label and interpret your events, your experience will be exactly the same.

Reference https://www.physicsforums.com/threa...n-and-the-twin-paradox-comments.842793/page-3

I think I see where you are coming from simply because the transformations when used will allow for all observers to agree about my experience of simultaneity when their experience is different from mine. To insist the reference frame that yields measured simultaneity by me has nothing to do with my experience takes its own kind of convoluted thinking, to my mind (and is an arbitrary abstract choice). But, OK you are the expert and I am not insisting that you are incorrect; just trying to learn.

 

[A.T.]

To insist the reference frame that yields measured simultaneity by me has nothing to do with my experience takes its own kind of convoluted thinking

Basing simultaneity on what you "experience" doesn’t even work in classical mechanics, due to signal delays. In Relativity, even accounting for signal delays, still leaves you with different simultaneity for frames in relative motion.

 

[whatif]

In Relativity, even accounting for signal delays, still leaves you with different simultaneity for frames in relative motion.

Fine, but having measured and accounted for signal delays simultaneity agrees with my measurement in a particular frame does it not? For other frames a transformation has to be performed.

 

[PeroK]

{Moderator’s note: this post refers to the original location of this discussion which was the insights discussion thread linked in the OP. The mentors agreed with this assessment and split the thread}

It seems to me that this sort of debate undermines the very idea of an "insight" - and, in this case, an insight by an expert professional physicist. This is not clarififying a point made in the insight but a separate analysis, challenging - to some extent at least - the basis for SR and confusing the issue with a beginner's misconceptions about SR. The place for those is a separate thread.

 

[A.T.]

Fine, but having measured and accounted for signal delays simultaneity agrees with my measurement in a particular frame does it not?

Not sure what you mean here. You define simultaneity based on the measurements and some convention, so it "agrees" with the measurements per definition.

 

[Herman Trivilino]

To insist the reference frame that yields measured simultaneity by me has nothing to do with my experience takes its own kind of convoluted thinking,

The reference frame doesn't yield measured simultaneity. Simultaneity is used to construct the reference frame.

 

[Herman Trivilino]

Fine, but having measured and accounted for signal delays simultaneity agrees with my measurement in a particular frame does it not?

You would have had to account for signal delays to establish simultaneity, so naturally you will be able to use that very same procedure to verify that it was done correctly.

For other frames a transformation has to be performed.

But if someone had carefully synchronized the clocks in any of those other frames, you'd have the same requirement imposed on your frame.

 

[Ibix]

Fine, but having measured and accounted for signal delays simultaneity agrees with my measurement in a particular frame does it not?

If your frame regards the actual emission events as equidistant from you, yes. The mere fact that you receive signals from events simultaneously tells you nothing about whether or not the events were simultaneous.

 

[Dale]

Rest frame was, perhaps, a bad choice of wording. What I meant was all frames in which I was stationary at the time, including non inertial frames of acceleration, which I guess correlates to the light cone. Yes/no?

No, not even remotely. Each frame includes all events in spacetime, including all future events. Surely you do not believe that you experience the future.

Similarly, even simultaneity is not something you experience. A star 1000 light years away may go supernova as you blow out candles on your birthday cake, but the star going supernova is not part of your experience because it is outside of your past light cone. However, a star 1000 light years away that went supernova 1000 years ago could be part of your birthday experience because it is in your past light cone. You know it did not happen simultaneously with blowing out your candles, but your experience was that the star went bright as the candles went dark. That is the light cone and it has nothing to do with frames or simultaneity.

This explanation with the 2D analogue does not do it for me.

That is fine. Not all analogies or explanations will do it for every person. You are certainly justified in your personal preference for any reason or for no reason at all. However, you need to be aware that the analogy is exceptionally solid. Your opinion-based objections are all legitimate, but so far your fact-based objections are all based on misunderstandings.

Personally, if you don’t like the analogy then I say you don’t need to use it. But the analogy is very common and powerful because of how accurate it is. It is far more exact than most other scientific analogies, so valid fact-based objections are relatively few and far between.