Q: "Time Dilation: Faster = Longer Wait?

[Ibix]

If there are no rockets firing, then whatever it is that is following the "different route", it can't be the traveling twin, or indeed any single object. The only word I can come up with for whatever it is that does follow the different route is "information". Perhaps that's not the best word, but we have to have some word for whatever it is that picks out the "route in spacetime" whose length is to be evaluated, in cases where no single object follows that route.

I don't disagree with this. I am just trying to argue that it is the whole route that matters, not just the corners. Darwin123 seems to me to be arguing the converse.

Perhaps I need a different example. Consider twins at rest at a space station. They leave together in identical rockets at velocity +v. At time t₁, one twin fires his motors and turns round, returning to the space station at velocity -v before braking to a relative stop. The other twin carries on until time t₂, when he also turns around and returns at -v before stopping at the space station. Both twins do identical accelerations, but it's easy to show that the difference in ages when they meet up is $\Delta t=2(t_2-t_1)(1-1/\gamma)$, which is zero only if they turn around at the same time or they don't travel at all. So acceleration isn't the only thing that matters. Both the amount of time between accelerations and the accelerations matter.

In the limited context of the classic twin paradox, you only need to know which twin accelerated to determine everything. So in this narrow circumstance, I agree one could argue that acceleration is the key. However, this isn't a useful view in general. In general, you need the complete history of both twins - i.e., their routes through spacetime.

Acceleration (or at least a frame change) is necessary for the worldlines to cross again. But it doesn't cause the age difference, any more than corners cause the triangle inequality. Different paths through a spacetime with a Minkowski geometry causes that.

 

[ghwellsjr]

And as I pointed out in that other thread, each of the IRF's that I drew in post #9 of this thread shows a different set of time-coordinates (and space coordinates) for each event which is illustrating the relativity of simultaneity. It also shows the relativity of time dilation. Neither of these are observable or measurable by the observers in the scenario, just like the one-way speed of light is not measurable. All three of these things are assigned by the definitions used in Special Relativity and after you assign them, then you can use the definitions and assignments to "read back" the same values you arbitrarily assigned to the events.

The essence of SR is that time is relative to the coordinate system or reference frame that you arbitrarily select. So is space. So is time dilation. So is simultaneity. Select a different coordinate system and all these characteristics change to different values. But what doesn't change are all the measurements and observations that each observer in the scenario makes. Each coordinate system preserves those measurements and observations. Maybe another way to say this is that the measurements and observations made at each event remain the same, even though the coordinates of each event take on different values in each reference frame.

Maybe you draw what you observe, measure, data, and I draw in my sketches why you observe and measure what you observe... (but if I understood you elsewhere, you seem not that much interested in what lays at the origin of the observations...?).
Let me put it to you this way. Say you have a forest full of trees... You can give me thousands of different coordinate systems with enless data lists of observations, from all over the place, and all plotted out in a different diagram. But as long as you do not tell me about the forest itself, I do not get it. This is what happens in SR discussions: data list talks. And where's the forest? (I'm glad that at least Bobc2 knows what the forest is in SR... and names it: 4D block universe)

Without a theory such as Special Relativity, it would be impossible to draw the data that we collect from our observations and measurements. All we would have are the lists of data. It would be and was very confusing until Einstein came along and presented his "simple and consistent theory" as he called it in the introduction of his 1905 paper.

So when I made the three separate diagrams in post #9, I used the same list of data from all observations and measurements with the aid of Einstein's definitions of what an Inertial Reference Frame (IRF) is. So the coordinates of each IRF are not part of the observations and measurements and the relationship between the coordinates and the measurements/observations are not recognizable to the observers. As I keep saying, how could they be? They change each time I use the Lorentz Transformation process to generate a new IRF and draw all the events with the new coordinate system.

You are right, I'm not interested in your opinion of "what lays at the origin of the observations" because I don't believe you or anyone else knows. In fact, I get the impression that you are promoting an idea that claims that it is no longer relative, once you see the forest for the trees. In any case, I have tried to understand the block universe concept that bobc2 and you are so fond of, but I find it so complicated that my eyes glaze over every time I see another one of those posts like #6 in the thread you linked to earlier. My opinion is that my diagrams and explanations would be easier for a newby to understand than the diagrams and explanations promoting the block universe. But, like I said in that other thread, we will have to get some feedback from various newbies to settle that issue.

In any case, I don't post comments about bobc2's or your diagrams saying that my way is "far more correct" or that there is "a danger of misinterpreting his diagrams" or implying that they are inherently incorrect or misleading. How would I know? I can't understand them.

 

[PeterDonis]

I don't disagree with this. I am just trying to argue that it is the whole route that matters, not just the corners. Darwin123 seems to me to be arguing the converse.

I think you're both right. You're right that the "corner" taken by itself is not sufficient to account for the actual differential aging that's observed; you need to look at the whole route, and by idealizing the "corner" to be instantaneous, you can idealize away any aging that actually occurs at the "corner", so that all of the actual aging has to be found by adding up aging over the segments of the route.

However, Darwin123 is right that except for the "corner", all motion involved is geodesic; so the "corner" is where you need to look if you want to find out why the "change in geodesics" occurred.

Similar remarks apply to the other scenarios you give: you have to look at the entire path through spacetime that each twin takes to get a final answer on relative aging, but if you want to understand why the paths are "crooked", why they're composed of segments of different geodesics instead of just one geodesic all the way, you need to look at the corners since that's where the "change in geodesics" happens.

 

[Vandam]

My opinion is that my diagrams and explanations would be easier for a newby to understand than the diagrams and explanations promoting the block universe.

Maybe it's enough to tell a newby that an observer sees a moving clock tick slower. And vice versa.
Everything is said by that.
Your time coordinates do not give any additional 'explanation' to it. But Block universe does.
Of course if you are not interestand in explanations then you have not to worry about that.

But I was worried about that when I came accros SR... I had to to understand the observations and maths. But again, if you are just happy putting numbers in Lorentz Tranformations... so be it.

 

[ghwellsjr]

My opinion is that my diagrams and explanations would be easier for a newby to understand than the diagrams and explanations promoting the block universe.

Maybe it's enough to tell a newby that an observer sees a moving clock tick slower. And vice versa.
Everything is said by that.

When did I ever say that? Don't you read what I write? I keep saying that an observer cannot see time dilation. I keep saying that an observer may see a moving clock ticking slower or faster than his own depending on the direction and orientation of their relative motion. I keep saying that Relativistic Doppler describes what observers see of moving clocks, not time dilation which is dependent on the selected Inertial Reference Frame (IRF) and always assigns the tick rate of moving clocks to be slower than the coordinate tick rate.

Your time coordinates do not give any additional 'explanation' to it.

They're not my time coordinates, they're Einstein's and that's what this forum is commited to teaching and helping newbies understand which is what I'm trying to do.

But Block universe does.

Like I said, I don't understand the block universe but if it promotes or teaches something that is at odds with SR, then it is not allowed here and I wouldn't be interested in it for that reason.

Of course if you are not interestand in explanations then you have not to worry about that.

But I was worried about that when I came accros SR... I had to to understand the observations and maths. But again, if you are just happy putting numbers in Lorentz Tranformations... so be it.

Yes, I find it fascinating that Einstein's concept of SR and using the Lorentz Transformation to understand relativity works the way it does. It's so simple once you understand it. I find that the biggest hurdle to helping newbies understand SR is getting rid of all the false notions they pick up from other sources, probably from those who don't understand SR themselves.

 

[bobc2]

How can you say that? I also just said:

But you don't need to analyze scenarios like this using Special Relativity. You can do it simply with a Relativistic Doppler Analysis which shows physically what each person actually observes and measures. But you have to discipline yourself and not ask about physical causes beyond what can actually be measured and observed so I doubt that that would be satisfying to you either.

That is an attitude generated back in the old Vienna Circle of philosophers, mathematicians and QM physicists. There are measurements and there are derivations. Most of our knowledge of physics is derived. Mostly, measurements performed do not directly measure the final quantity for which a "measurement" is said to have been made. The final "measurement" value is far more often than not a derived value. The speed of light was not directly measured. It was derived from measurements of distance and time. The masses of the elementary particles are not measured directly by a long shot. At NASA we used to measure the centripetal force applied to payloads under test by measuring the RPM of the centrifuge. When we measure voltage with a meter (with dial pointer) we are more directly measuring an angle of rotation of the pointer--beyond that the magnetic field strength associated with the meter movement coil, etc... Then we derive the voltage.

We could have a long discussion about examples of direct measurement of quantities in physics versus derived quanties. So, I think one has to be careful about minimizing the significance of derived "measurements" in physics. I think you make way too big a deal about the derived quanties leading to fundamental concepts in special relativity.

I don't understand your difficulty in interpreting Vandam's Loedel sketches. Many of my undergraduate students had a little trouble at first but caught on after spending some time really thinking about it. (No, I did not push block universe on them, but we did have interesting class discussions about some of the implications)

I'm really not trying to push the block universe concept here. However, I've documented in other threads the many notable physicists who embrace the concept (Paul Davies's book "About Time" is a good reference on the subject). You should not look upon it as a separate theory. It is a direct manifestation of Minkowski's geometric picture, i.e., Space-Time. Some people reject it, thinking it is a philosophical outlook. It is not. On the contrary, folks who reject it are doing it after they themselves bring in the field of philosophy--they are actually rejecting it on a philosophical basis, probably because they don't like some of the implications.

The implications--those are the cause of my own struggle with the concept. On the one hand I can't deny what Minkowski's Space-Time is showing us quite directly, but on the other hand I cannot quite make peace with it at a subjective level. I have too strong of a psychological sense of existing in a 3-D world that evolves with time. And I must give the disclaimer that I am a serious Christian and think a lot about the theological implications of foundational physics in that context.

So, my problem is that on the one hand I cannot refute the picture of the physical block universe that is manifest in special relativity, but on the other hand I can't handle it at the subjective level.

Vandam seems to have overcome those kinds of concerns--he just sticks to the facts and let's the chips fall where they may. So, for him the block universe is physical reality. (hope I haven't misrepresented Vandam). I always appreciate his comments and Loedel diagrams.

 

[ghwellsjr]

How can you say that? I also just said:

But you don't need to analyze scenarios like this using Special Relativity. You can do it simply with a Relativistic Doppler Analysis which shows physically what each person actually observes and measures. But you have to discipline yourself and not ask about physical causes beyond what can actually be measured and observed so I doubt that that would be satisfying to you either.

That is an attitude generated back in the old Vienna Circle of philosophers, mathematicians and QM physicists. There are measurements and there are derivations. Most of our knowledge of physics is derived. Mostly, measurements performed do not directly measure the final quantity for which a "measurement" is said to have been made. The final "measurement" value is far more often than not a derived value. The speed of light was not directly measured. It was derived from measurements of distance and time. The masses of the elementary particles are not measured directly by a long shot. At NASA we used to measure the centripetal force applied to payloads under test by measuring the RPM of the centrifuge. When we measure voltage with a meter (with dial pointer) we are more directly measuring an angle of rotation of the pointer--beyond that the magnetic field strength associated with the meter movement coil, etc... Then we derive the voltage.

We could have a long discussion about examples of direct measurement of quantities in physics versus derived quanties. So, I think one has to be careful about minimizing the significance of derived "measurements" in physics. I think you make way too big a deal about the derived quanties leading to fundamental concepts in special relativity.

How can you say that? I have on numerous occasions pointed out how to measure the round-trip speed of light using a single timing device colocated with a light source and a mirror some measured distance away. The determination of the speed of light is double the measured distance by the measured time interval.

But the unmeasurable one-way speed of light is defined in any Inertial Reference Frame to be equal to the measurable round trip speed of light.

But there is no way to measure or observe the time dilation of any clock. If you think there is then please describe how you would propose doing it.

For example, let's say you are moving in an IRF at some high speed. Your clock is time dilated but you can't tell, can you?

Or let's say you are observing a distant moving clock. All you can observe is the Relativistic Doppler shift which is independent of any reference frame but the propagation time of the image of the clock coming to you is dependent on the reference frame as well as the time dilation. Different frames trade off these two factors in such a way that the observation remains the same so you cannot observe or measure this trade off without which you cannot determine the time dilation. And unless you actually know what the distant clock is doing at any particular time, you can't even say what its speed is until the image of it gets to you.

I don't understand your difficulty in interpreting Vandam's Loedel sketches. Many of my undergraduate students had a little trouble at first but caught on after spending some time really thinking about it. (No, I did not push block universe on them, but we did have interesting class discussions about some of the implications)

I'm really not trying to push the block universe concept here. However, I've documented in other threads the many notable physicists who embrace the concept (Paul Davies's book "About Time" is a good reference on the subject). You should not look upon it as a separate theory. It is a direct manifestation of Minkowski's geometric picture, i.e., Space-Time. Some people reject it, thinking it is a philosophical outlook. It is not. On the contrary, folks who reject it are doing it after they themselves bring in the field of philosophy--they are actually rejecting it on a philosophical basis, probably because they don't like some of the implications.

The implications--those are the cause of my own struggle with the concept. On the one hand I can't deny what Minkowski's Space-Time is showing us quite directly, but on the other hand I cannot quite make peace with it at a subjective level. I have too strong of a psychological sense of existing in a 3-D world that evolves with time. And I must give the disclaimer that I am a serious Christian and think a lot about the theological implications of foundational physics in that context.

So, my problem is that on the one hand I cannot refute the picture of the physical block universe that is manifest in special relativity, but on the other hand I can't handle it at the subjective level.

Vandam seems to have overcome those kinds of concerns--he just sticks to the facts and let's the chips fall where they may. So, for him the block universe is physical reality. (hope I haven't misrepresented Vandam). I always appreciate his comments and Loedel diagrams.

Vandam's and your diagrams always put multiple coordinate systems into the same graph which makes it very difficult for me and I expect for a newby to grasp what is going on.

Look, as I said before, I don't complain that what you and Vandam are doing is wrong when you are presenting your explanations. Sometimes I have asked for clarification and understanding but I let you carry on without any hindrance from me. You just shouldn't complain about my explanations or insinuate that they are not accurate or not complete (unless you think they are in which case it would be helpful to specify exactly what the problem is).

 

[Vandam]

But there is no way to measure or observe the time dilation of any clock.

Again, how can you say you can not observe time dilation? Where do you get this?
I think we have to have a little converstion on what we both mean by 'observation'.

If you think there is then please describe how you would propose doing it.

Every time I or Bob gave you an explantion you tell me you do not understand it...

For example, let's say you are moving in an IRF at some high speed.

You never move in your IRF. Never.

Your clock is time dilated but you can't tell, can you?

In your own IRF your own time is never time dilated.
You do not understand relativity of simultaneity; that's the origin of time dilation!

Or let's say you are observing a distant moving clock. All you can observe is the Relativistic Doppler shift

I do not need Doppler shift to explain time dilation. Relativity of simultaneity suffices.

which is independent of any reference frame but the propagation time of the image of the clock coming to you is dependent on the reference frame as well as the time dilation. Different frames trade off these two factors in such a way that the observation remains the same so you cannot observe or measure this trade off without which you cannot determine the time dilation. And unless you actually know what the distant clock is doing at any particular time, you can't even say what its speed is until the image of it gets to you.

Vandam's and your diagrams always put multiple coordinate systems into the same graph which makes it very difficult for me and I expect for a newby to grasp what is going on.

Look, as I said before, I don't complain that what you and Vandam are doing is wrong when you are presenting your explanations. Sometimes I have asked for clarification and understanding but I let you carry on without any hindrance from me. You just shouldn't complain about my explanations or insinuate that they are not accurate or not complete
(unless you think they are in which case it would be helpful to specify exactly what the problem is).

I think we might disagree with what you mean with 'time-coordinates'. And you probably confuse 'time-coordinates' with the observation of 'observer independent clock indications'. I'll try to elaborate on this in a post sometime, when time allows me.

 

[ghwellsjr]

Vandam, I'm going to let others respond to your post. You obviously aren't going to accept anything I say. Maybe bobc2 can straighten you out since you seem to hold him in high esteem.

 

[Vandam]

You are right, I'm not interested in your opinion of "what lays at the origin of the observations" because I don't believe you or anyone else knows.

Ghwellsjr, I think you contradict yourself. Correct me if I am wrong.
SR talks about observers/observations. In order to understand SR we have to agree what observations mean, otherwise it is pointless to start dealing with observers and observations, or SR at all.
Just to make sure we understand each other as far as 'observation' is concerned;

You 'observe' (see) that an observer in the train sees a lightning hitting the front of the train because first there was lightning, then light travels to the observer, and then the light hits the observer's retinae. This is the meaning of 'to observe'. You see lightning hitting the front of the train because there is an observer independent event that's later observed by you (and other observers). There has/have to be event(s) to be observed and measured. Basic stuff. I guess you accept this.
Why then are you not interested in "what lays at the origin of the observations", i.e. the -observer independent- event: lightning hitting the front of the train?
You tell us about observation of events, but you refuse to tak about the events... Don't you contradict yourself?

SR is about observer independent events, which means: there are events out there (not part of your mind, be it physical -or mental, whatever) even before you observe them, otherwise there simply can not be an observation, nor observers.
If you refute this, then what are observations and observers in SR?

 

[ghwellsjr]

Vandam, I don't understand why you are asking me these questions when I have already specifically addressed these issues on this thread. Why don't my previous answers satisfy you?

The issue you and I are dealing with on this thread is whether or not time dilation can be observed and measured. I have said over and over again that it is dependent on the arbitrary frame of reference that you use to describe the scenario so how can it be measured by the observers in the scenario? Please go back and study my post #9 on this thread with regard to time dilation and tell me what observations or measurements the two observers can make that will enable them to determine the time dilations during each phase of the scenario and for each IRF.

 

[Vandam]

Really amazing, Ghwellsjr. Really.

 

[bobc2]

...The issue you and I are dealing with on this thread is whether or not time dilation can be observed and measured. I have said over and over again that it is dependent on the arbitrary frame of reference that you use to describe the scenario so how can it be measured by the observers in the scenario?...

ghwellsjr, I'm really not trying to be advesarial but am trying to understand your logic. Maybe we have a problem in the way we regard hyperplanes of simultaneity. Maybe if you could explain what significance you attach to this concept I would have a little better idea where you are coming from. Thanks.

 

[ghwellsjr]

...The issue you and I are dealing with on this thread is whether or not time dilation can be observed and measured. I have said over and over again that it is dependent on the arbitrary frame of reference that you use to describe the scenario so how can it be measured by the observers in the scenario?...

ghwellsjr, I'm really not trying to be advesarial but am trying to understand your logic. Maybe we have a problem in the way we regard hyperplanes of simultaneity. Maybe if you could explain what significance you attach to this concept I would have a little better idea where you are coming from. Thanks.

I have never used the term "hyperplanes of simultaneity" so now I guess I have to try to figure out what you mean by the term. If you go back to post #9 and look at the three graphs representing three different IRF's, each one of them is showing just one spatial dimension because, as is common in spacetime diagrams, we use the other dimension for time and we limit the activity in the scenario to just one dimension (usually referred to as the x-dimension) and we assume that the audience is familiar enough with this type of diagram that they know that the y- and z-dimensions are not shown but since nothing is happening at locations other than y=0 and z=0, we mentally recognize that when the graph shows a horizontal grid line, that is a line of simultaneity for a particular value of time which you look up at the left side of the graph and it means that all events along that horizontal line are simultaneous meaning they happen at the same time in that IRF. (I can't believe I'm explaining all this--nevertheless, I carry on.) Now since we don't show the y- and z- dimensions, we mentally realize that all the events that are simultaneous along that line are extrapolated out in those two extra dimensions so it is really a volume of simultaneity which I suppose is identical to your term hyperplane of simultaneity.

Now what's important is that two (or more) events that are simultaneous in one IRF (because they have the same value for their time coordinate) may not be simultaneous in another IRF as can be seen if you look at the three different graphs. I never really stopped to think in terms of a volume of simultaneity, assuming that that is what you mean by a hyperplane of simultaneity, but it is obviously the case although I would say it is so obvious that it doesn't need to be said.

Now if we wanted to show a two-dimensional scenario where the observers were moving around in both the x- and y-dimensions, we'd have a hard time putting that on a piece of paper but what we could do with today's technology is make an animation and present it as a movie. Each frame of the movie marks out a plane of simultaneity but the assumption is that it extends out into the z-dimension and so there really is a volume of simultaneity.

Does that communicate? Does it make sense to you? Is it in agreement with your concept of the hyperplane of simultaneity?

 

[bobc2]

ghwellsjr: I have never used the term "hyperplanes of simultaneity" so now I guess I have to try to figure out what you mean by the term. If you go back to post #9 and look at the three graphs representing three different IRF's, each one of them is showing just one spatial dimension because, as is common in spacetime diagrams, we use the other dimension for time and we limit the activity in the scenario to just one dimension (usually referred to as the x-dimension) and we assume that the audience is familiar enough with this type of diagram that they know that the y- and z-dimensions are not shown but since nothing is happening at locations other than y=0 and z=0, we mentally recognize that when the graph shows a horizontal grid line, that is a line of simultaneity for a particular value of time which you look up at the left side of the graph and it means that all events along that horizontal line are simultaneous meaning they happen at the same time in that IRF. (I can't believe I'm explaining all this--nevertheless, I carry on.) Now since we don't show the y- and z- dimensions, we mentally realize that all the events that are simultaneous along that line are extrapolated out in those two extra dimensions so it is really a volume of simultaneity which I suppose is identical to your term hyperplane of simultaneity.

Bobc2: Yes, we are on the same page here. Actually you do find the term “hyperplanes of simultaneity" in many places in the special relativity literature—and you have correctly figured out its meaning. I’m glad we have no problem reducing the analysis to the use of just two dimensions in our sketches.

ghwellsjr: Now what's important is that two (or more) events that are simultaneous in one IRF (because they have the same value for their time coordinate) may not be simultaneous in another IRF as can be seen if you look at the three different graphs. I never really stopped to think in terms of a volume of simultaneity, assuming that that is what you mean by a hyperplane of simultaneity, but it is obviously the case although I would say it is so obvious that it doesn't need to be said.

Bobc2: Yes, we are in perfect agreement on that. And when I use the term "hyperplanes of simultaneity" I also don't see a need to show all dimensions in the space-time diagrams.

ghwellsjr: Now if we wanted to show a two-dimensional scenario where the observers were moving around in both the x- and y-dimensions, we'd have a hard time putting that on a piece of paper…

Bobc2: But, that’s just what I’ve been trying to do with the space-time diagrams that include the various X1 axes for the different observers as well as the X4 axes. These axes are of course all identified using the velocities of the moving observers along with the Lorentz transformation. (see my first sketch below)

ghwellsjr: …but what we could do with today's technology is make an animation and present it as a movie. Each frame of the movie marks out a plane of simultaneity but the assumption is that it extends out into the z-dimension and so there really is a volume of simultaneity. Does that communicate? Does it make sense to you? Is it in agreement with your concept of the hyperplane of simultaneity?.

Bobc2: Yes, it certainly does. I have among my special relativity computer files examples of such an animation. And I’ve seen one posted on our forum here.

So, the sketch below illustrates how I show two different hyperplanes of simultaneity, blue and red, where two different observers are moving at the same speed in opposite directions with respect to the black inertial reference frame (the perpendicular coordinates representing X1 and X4 axes).

I have included the representation of a rod moving to the right with respect to the black frame, but the rod is at rest in the blue inertial frame. Thus, we see directly the length contraction aspect of special relativity. Blue sees the length of the rod as L0, whereas Red sees the rod length as L. And the reason I've used the symmetric space-time diagram (first introduced by Loedel of Mexico who received Einstein's blessing during their visit), is that it avoids the need to worry about the meaning of the line distances when comparing Blue and Red coordinates (you don't really need to be concerned with the hyperbolic calibration curves). This scheme was introduced to me in my first grad school special relativity course. My prof was fond of this means of communicating special relativity. I used it also later on when I was a physics instructor for undergrad physics and engineering students.

Of course it is easy to account for both X1 and X4 coordinates of Blue and Red using the Lorentz transformation hyperbolic calibration curves as shown below (the Red and Blue colors are reversed from the above sketch).

I was just trying to see if we are on the same page about the significance of these two different 3-D worlds (represented within the 4-dimensional space with just two coordinates) that blue and red occupy at points along their respective worldlines.

Finally, here is an interesting sketch, using the above concepts of hyperplanes of simultaneity to illustrate the motivation for the Block Universe model of special relativity. For now, I will spare you the pain of the addition of world lines of many different laser pulses (idealized in the diagrams as single photons). So, there is a scheme for deciphering the many laser light measurements that could be performed on signals transmitting back and forth and intersecting along the different world lines. To make the measurements more convincing you just add more observers at rest in the Blue inertial frame (collaborating results with any amount of data desired), and have matching Red observers participating in the experiment.

Perhaps I have not communicated these concepts well, or perhaps you understand the concept quite well and simply reject it. I just wanted to make sure I understood your thinking on these hyperplanes of simultaneity (X2 and X3 coordinates suppressed for clarity)
Maybe my basic questions are:

1) Do you accept the validity of the above sketches as correctly representing key aspects of special relativity (regardless of whether you attach any physical significance to it)?

2) Do you attach any physical significance to these hyperplanes of simultaneity?

3) What significance at all to the hyperplanes of simultaneity represented in the above space-time diagrams.

 

[K^2]

None of this makes absolutely any difference to anything ghwellsjr stated earlier. You don't need time in X' frame for every position x' to show time dilation. It is sufficient to show time along a single line of constant x_1-3. And world line of observer that is static in X' is just as good as any other. So proper time of observer static in X' is entirely sufficient to show time dilation. The fact that simultaneity lines are going to be different in X and X' is entirely irrelevant to this fact.

 

[PeterDonis]

Bobc2: But, that’s just what I’ve been trying to do with the space-time diagrams that include the various X1 axes for the different observers as well as the X4 axes.

If you were trying to represent motion in more than one spatial direction, you need to draw different diagrams. All of your diagrams only involve relative motion along one spatial direction, the X1 direction (what most people would just call the X direction). Different observers in different states of motion along that direction have different "X1" axes; but they all have the *same* Y and Z axes (or perhaps you would call them X2 and X3 axes) because none of them are moving at all in the Y and Z (or X2 and X3) spatial directions.

Your diagrams also show an X4 "direction", yes (which most people would call the "T" direction). But that's not a spatial direction.

 

[ghwellsjr]

ghwellsjr: I have never used the term "hyperplanes of simultaneity" so now I guess I have to try to figure out what you mean by the term. If you go back to post #9 and look at the three graphs representing three different IRF's, each one of them is showing just one spatial dimension because, as is common in spacetime diagrams, we use the other dimension for time and we limit the activity in the scenario to just one dimension (usually referred to as the x-dimension) and we assume that the audience is familiar enough with this type of diagram that they know that the y- and z-dimensions are not shown but since nothing is happening at locations other than y=0 and z=0, we mentally recognize that when the graph shows a horizontal grid line, that is a line of simultaneity for a particular value of time which you look up at the left side of the graph and it means that all events along that horizontal line are simultaneous meaning they happen at the same time in that IRF. (I can't believe I'm explaining all this--nevertheless, I carry on.) Now since we don't show the y- and z- dimensions, we mentally realize that all the events that are simultaneous along that line are extrapolated out in those two extra dimensions so it is really a volume of simultaneity which I suppose is identical to your term hyperplane of simultaneity.

Bobc2: Yes, we are on the same page here. Actually you do find the term “hyperplanes of simultaneity" in many places in the special relativity literature—and you have correctly figured out its meaning. I’m glad we have no problem reducing the analysis to the use of just two dimensions in our sketches.

But let's make it very clear that it's one dimension of space and one dimension of time. It's a 1-D scenario, agreed?

ghwellsjr: Now what's important is that two (or more) events that are simultaneous in one IRF (because they have the same value for their time coordinate) may not be simultaneous in another IRF as can be seen if you look at the three different graphs. I never really stopped to think in terms of a volume of simultaneity, assuming that that is what you mean by a hyperplane of simultaneity, but it is obviously the case although I would say it is so obvious that it doesn't need to be said.

Bobc2: Yes, we are in perfect agreement on that. And when I use the term "hyperplanes of simultaneity" I also don't see a need to show all dimensions in the space-time diagrams.

ghwellsjr: Now if we wanted to show a two-dimensional scenario where the observers were moving around in both the x- and y-dimensions, we'd have a hard time putting that on a piece of paper…

Bobc2: But, that’s just what I’ve been trying to do with the space-time diagrams that include the various X1 axes for the different observers as well as the X4 axes. These axes are of course all identified using the velocities of the moving observers along with the Lorentz transformation. (see my first sketch below)

Now maybe you can see why I made my previous comment. When I talk about a 2-D scenario, I specifically said two spatial dimensions, not one of space and one of time. I have never seen any of your diagrams that include anything more than X1 and X4. X4 is always the time dimension, correct? All your diagrams are for a 1-D scenario, not a 2-D scenario, agreed?

ghwellsjr: …but what we could do with today's technology is make an animation and present it as a movie. Each frame of the movie marks out a plane of simultaneity but the assumption is that it extends out into the z-dimension and so there really is a volume of simultaneity. Does that communicate? Does it make sense to you? Is it in agreement with your concept of the hyperplane of simultaneity?.

Bobc2: Yes, it certainly does. I have among my special relativity computer files examples of such an animation. And I’ve seen one posted on our forum here.

And would that be one that I posted?

So, the sketch below illustrates how I show two different hyperplanes of simultaneity, blue and red, where two different observers are moving at the same speed in opposite directions with respect to the black inertial reference frame (the perpendicular coordinates representing X1 and X4 axes).

I have included the representation of a rod moving to the right with respect to the black frame, but the rod is at rest in the blue inertial frame. Thus, we see directly the length contraction aspect of special relativity. Blue sees the length of the rod as L0, whereas Red sees the rod length as L. And the reason I've used the symmetric space-time diagram (first introduced by Loedel of Mexico who received Einstein's blessing during their visit), is that it avoids the need to worry about the meaning of the line distances when comparing Blue and Red coordinates (you don't really need to be concerned with the hyperbolic calibration curves). This scheme was introduced to me in my first grad school special relativity course. My prof was fond of this means of communicating special relativity. I used it also later on when I was a physics instructor for undergrad physics and engineering students.

Here's where my eyes glaze over. I'm not saying that there is anything wrong with your diagrams, I don't know, because I'm not motivated to learn about them. I don't see the attraction for them. They don't communicate anything that can't be communicated in a series of simple graphs like the ones I presented in post #9. Do you think they communicate something more than several IRF type graphs?

Of course it is easy to account for both X1 and X4 coordinates of Blue and Red using the Lorentz transformation hyperbolic calibration curves as shown below (the Red and Blue colors are reversed from the above sketch).

Maybe it's easy for you but not for me.

I can understand how these kinds of graphs would be important a hundred years ago but nowadays, we can let our computers take care of all the computations.

I was just trying to see if we are on the same page about the significance of these two different 3-D worlds (represented within the 4-dimensional space with just two coordinates) that blue and red occupy at points along their respective worldlines.

We probably aren't on the same page, especially if you see eye-to-eye with Vandam, because he thinks the three separate IRF plots hide or mask information that is evident on the kinds of diagrams you make. Do you share his opinion?

I don't hand-draw my plots. I use a computer and once I set up a scenario, the computer draws the first plot in the same IRF that I entered the scenario into. Then I enter a speed parameter that creates a new plot using the Lorentz Transformation. I repeat for the third plot. So I know that there is no more information in the second and third plots (or as many others as I want to make) than there is in the first one.

My question to you is: would it be possible to have a computer take the scenario the way I set it up for the first IRF and then instead of transforming to an IRF at a different speed, could it generate one of your diagrams that combines the information from three simple IRF graphs?

There is one piece of information that can be gleaned from watching the computer redraw the graphs for the different IRF's that you would not see from anyone of them and that is it makes it obvious which characteristics are frame invariant and which are not but aside from that, no new insight or conclusions can be obtained simply by presenting the same information in different IRF's or in one of your (or Vandam's) diagrams that combine the information from multiple IRF's. Do you agree with this assessment?

Finally, here is an interesting sketch, using the above concepts of hyperplanes of simultaneity to illustrate the motivation for the Block Universe model of special relativity. For now, I will spare you the pain of the addition of world lines of many different laser pulses (idealized in the diagrams as single photons). So, there is a scheme for deciphering the many laser light measurements that could be performed on signals transmitting back and forth and intersecting along the different world lines. To make the measurements more convincing you just add more observers at rest in the Blue inertial frame (collaborating results with any amount of data desired), and have matching Red observers participating in the experiment.

My hat's off to anyone that can make sense out of these diagrams, let alone, draw them.

Perhaps I have not communicated these concepts well, or perhaps you understand the concept quite well and simply reject it. I just wanted to make sure I understood your thinking on these hyperplanes of simultaneity (X2 and X3 coordinates suppressed for clarity)

If by "concept", you mean your diagrams, then you can fault the student--not the teacher. But if you mean, as you posed the question to me earlier, the concept of simultaneity, then I understand it quite well. It's simply all the events that have the same time coordinate in any given IRF.

Maybe my basic questions are:

1) Do you accept the validity of the above sketches as correctly representing key aspects of special relativity (regardless of whether you attach any physical significance to it)?

Since others accept their validity, then I will accept their opinion.

Do you accept the validity of graphs like the ones on page #9 as being exactly equivalent to your diagrams?

2) Do you attach any physical significance to these hyperplanes of simultaneity?

No, not in your diagrams or in the type that I draw.

Do you attach any physical significance to the origin of an IRF?

3) What significance at all to the hyperplanes of simultaneity represented in the above space-time diagrams.

IRF's are man-made constructs. If they exist physically in nature, we have no way of determining that. It's like asking for the absolute rest state of the ether. Even we believe, like Lorentz that such an ether exists, we still would prefer Einstein's Special Relativity over a Lorentz Ether Theory because the Transformation process allows us to make any IRF just as valid as the one and only ether IRF.

All of Special Relativity, not just issues of simultaneity are very important in our understanding of the world. Without it, we would still be floundering around searching for that illusive ether. Without it, we would not have the simple and consistent means of interpreting the data from our measurements. One of the most important tenets of SR is that there is no preferred reference frame. It appears to me that you and Vandam want to get rid of all reference frames in favor of some super interpretation that incorporates several reference frames all at the same time. One of the other important tenets of SR is that you don't conflate coordinates from two or more reference frames which is what I see you and Vandam doing.

One last question: what does any of this have to do with the issue of whether time dilation is observable or measurable by the observers in the scenario?

 

[Vandam]

Here's where my eyes glaze over. I'm not saying that there is anything wrong with your diagrams, I don't know, because I'm not motivated to learn about them.

Then I would stop telling they are useless.

I don't see the attraction for them.

Of course, as long as you are not motivated to learn about them, you will never apprectiate what they offer (block universe).

They don't communicate anything that can't be communicated in a series of simple graphs like the ones I presented in post #9.

We definitely do not agree about that.

Do you think they communicate something more than several IRF type graphs?

Yes. Block universe. But for me 'relativity of simultaneous events' suffices. The problem is that mathermatics do not read this from their calculators. What does 'different time coordinates for on event' mean? Of course you have to look at the greater picture to understand this. I get back to ther forest analogy. Whatever coordinate system you choose to measure the space and time coordinates of the trees, that is in fact irrelevant of the 'real' position of the trees in the forest (= what is out there to be observed from a coordinate system). Again, there is nothing wrong with your different timecoordinate charts, thousands of mathematicians can juggle with the Lorentz transformations, and in essence they do not have to worry about anything else. But they miss the broader picture, but because that will not change anything to their calculations they consider it superfluous.

Maybe it's easy for you but not for me.

Of course you should first take the effort to learn about diagrams. No offence, but maybe you simply do not have the conceptual ability to read 4D diagrams. I personally can not read a piano partiture (scores?), my mind just doesn't get it. Nothing wrong with that, but I will never get on a forum and pretend they are superfulous and without any meaning for understanding music.

I can understand how these kinds of graphs would be important a hundred years ago but nowadays, we can let our computers take care of all the computations.

Sigh.

We probably aren't on the same page, especially if you see eye-to-eye with Vandam, because he thinks the three separate IRF plots hide or mask information that is evident on the kinds of diagrams you make. Do you share his opinion?

I don't hand-draw my plots. I use a computer and once I set up a scenario, the computer draws the first plot in the same IRF that I entered the scenario into. Then I enter a speed parameter that creates a new plot using the Lorentz Transformation. I repeat for the third plot. So I know that there is no more information in the second and third plots (or as many others as I want to make) than there is in the first one.

My question to you is: would it be possible to have a computer take the scenario the way I set it up for the first IRF and then instead of transforming to an IRF at a different speed, could it generate one of your diagrams that combines the information from three simple IRF graphs?

Of course it could! Peace of cake. You put in the relative speed and hop there is the drawing. (Unfortunately I am not a cumputer programmer)

There is one piece of information that can be gleaned from watching the computer redraw the graphs for the different IRF's that you would not see from anyone of them and that is it makes it obvious which characteristics are frame invariant and which are not but aside from that, no new insight or conclusions can be obtained simply by presenting the same information in different IRF's or in one of your (or Vandam's) diagrams that combine the information from multiple IRF's. Do you agree with this assessment?

My hat's off to anyone that can make sense out of these diagrams, let alone, draw them.

Thanks. I hope you will soon be one of them. There is nothing difficult to these diagrams.

If by "concept", you mean your diagrams, then you can fault the student--not the teacher. But if you mean, as you posed the question to me earlier, the concept of simultaneity, then I understand it quite well. It's simply all the events that have the same time coordinate in any given IRF.

Yes. So far the mathematics. Numbers. And what do your numbers stand for? Think about the forest.
Of course you can say: "I do not care what time coordinates are. They are figures, and that's all what I need...". Sigh.

Since others accept their validity, then I will accept their opinion.

Do you accept the validity of graphs like the ones on page #9 as being exactly equivalent to your diagrams?

It depends what you mean with equivalent. Are the 2D sections through a house equivalent with the 3D house?

No, not in your diagrams or in the type that I draw.

Then you probably have a problem with 'observer independent events'.

Do you attach any physical significance to the origin of an IRF?

Do you mean the (0,0) coordinate? Let me get back to the forest. Is the the spot from where you measure the distance between the trees a physical spot. Yes I guess. You can put that spot anywhere in the forest, that will not change (alter) the structure of the forest.

IRF's are man-made constructs.

In the sense: they depend on the obsever. Observer dependent. the way you measure the forest is observer dependent. But it would be wrong to state that the forest is a 'man-made construct'!

If they exist physically in nature, we have no way of determining that. It's like asking for the absolute rest state of the ether. Even we believe, like Lorentz that such an ether exists, we still would prefer Einstein's Special Relativity over a Lorentz Ether Theory because the Transformation process allows us to make any IRF just as valid as the one and only ether IRF.

All of Special Relativity, not just issues of simultaneity are very important in our understanding of the world. Without it, we would still be floundering around searching for that illusive ether. Without it, we would not have the simple and consistent means of interpreting the data from our measurements. One of the most important tenets of SR is that there is no preferred reference frame. It appears to me that you and Vandam want to get rid of all reference frames in favor of some super interpretation that incorporates several reference frames all at the same time. One of the other important tenets of SR is that you don't conflate coordinates from two or more reference frames which is what I see you and Vandam doing.

What you say is: different observers measure, observe, but you refute anything that is there to be measured. Therefore I asked in my other post what then you mean with 'obvervation' in SR.
https://www.physicsforums.com/showpost.php?p=4197377&postcount=45http://

One last question: what does any of this have to do with the issue of whether time dilation is observable or measurable by the observers in the scenario?

Because we have to agree on what you mean with 'observation/measure'. You mean probably: the coordinates you measure in the forest. I mean: comparing observer independent time indications on the clocks (trees...).

 

[K^2]

Vandam, your graphs do not introduce any extra information contained in extra dimensions. You are both working with 2D sections. The ONLY extra information you provide is that of simultaneity, which is irrelevant to discussion.

 

[Vandam]

Vandam, your graphs do not introduce any extra information contained in extra dimensions. You are both working with 2D sections. The ONLY extra information you provide is that of simultaneity, which is irrelevant to discussion.

Simultaneity is irrelevant?? The plot thickens...
Do you know what is Special Relativity all about?
Relativity of simultaneity!
Ever read Einstein's 1905 paper?
Or his train gedanken experiment? Relativity of simultaneity is the core of Special Relativity.
Talking about observations is O.K., but you have to grasp the relativity of simultaneity or you don't understand SR. Sure, you can say that an event 'lightning hits the front of the train' gets different timescoordinates depending of the observer, but again: we have to agree what you mean with timecoordinates. And then I refer back to my previous post. Keeping on saying it's is not relevant only proves you didn't get the essence of Special Relativity: relativity of simultaneity.

 

[ghwellsjr]

One last question: what does any of this have to do with the issue of whether time dilation is observable or measurable by the observers in the scenario?

Because we have to agree on what you mean with 'observation/measure'. You mean probably: the coordinates you measure in the forest. I mean: comparing observer independent time indications on the clocks (trees...).

No, I don't mean the coordinates. Those are arbitrarily assigned by the selected IRF and change when a new one is selected. I mean for example, the observations by observers of the other ones clock which is handled by the Relativistic Doppler analysis and doesn't change with each new reference frame and doesn't assign a Time Dilation value to any clock. Since you have already rejected the Doppler analysis as being relevant in this discussion, I have no idea what you mean by "observer independent time indications on the clocks". I would express it as "independent observer observations of time indications on the clocks" which is what is used in the Doppler analysis. I know you will claim that this is because I refuse to grasp the notions of the block universe but I can rely on what others have said who do understand it, that it is irrelevant.

 

[Vandam]

The relativistic Doppler effect is pure relativity of simultaneity.
Leo Sartori draws a Loedel spacediagram of the doppler scenario in his book 'Understanding Relativity' page 161.
I can find no reference to that drawing on the net. And because you are probably not really interested in such a diagram (?) I am not too motivated to copy and post it here now... (I suffer shortage of time now...)

 

[Mentz114]

.. the essence of Special Relativity:[is] relativity of simultaneity.

This is a very blinkered view. SR is based on the two principles, the clock postulate ( and possibly some other postulates). Time dilation and relativity of simultaneity can be deduced from the aformentioned principles etc. RoS is not the essence of SR, it is a deduction ( and a rather obvious one ).

 

[K^2]

Simultaneity is irrelevant?? The plot thickens...
Do you know what is Special Relativity all about?
Relativity of simultaneity!

Topic was time dilation. Time dilation does not require discussion of simultaneity across multiple coordinate systems. I have my X coordinate system. I've written down (t, x) of the rocket in my coordinate system. I've taken dt/d$\small \tau$ in my frame. I got the time dilation. That's it.

Yes, when at time t, I claim that rocket's proper time is $\small \tau$ from the start, the man on the rocket, having experienced amount of time $\small \tau$ from the start will think of my time t as something that's yet to happen. So when I compare time dilation in two different frames, I need to consider simultaneous events as according to whom.

But this is getting pretty far from original topic. ghwellsjr's original plots give correct positions and proper times of rockets in each of the coordinate systems. To get time dilation in a particular system, all you need to look at is proper time of each rocket at given time t as defined by the coordinate system choice. All the information you need to derive time dilation is already on these graphs. Introducing constant time slices for each of the participants is absolutely unnecessary.

 

[Vandam]

This is a very blinkered view. SR is based on the two principles, the clock postulate ( and possibly some other postulates). Time dilation and relativity of simultaneity can be deduced from the aformentioned principles etc. RoS is not the essence of SR, it is a deduction ( and a rather obvious one ).

Clock postulate?
RoS not the esssence but a deduction?
I really think you have some homework to do.

 

[Mentz114]

Clock postulate?
RoS not the esssence but a deduction?
I really think you have some homework to do.

Nah, I'm fine.

For me the essence of relativity is the way EM is relativistically invariant and the fact that identifying the invariant proper interval with the time recorded on a clock eliminates clock paradoxes.

I suppose you'll say those things depend on RoS, but you'd be wrong.

 

[K^2]

RoS not the esssence but a deduction?

It is not one of the postulates, therefore, it is a deduction.

 

[Vandam]

Clock postulate?
RoS not the esssence but a deduction?
I really think you have some homework to do.

It is not one of the postulates, therefore, it is a deduction.

You are correct. I did take a bit of a shortcut there. Too much in a hurry.
There is no clock postulate either.
The clock synchro, time coordinates and RoS are a deduction of the constant light speed postulate.
But that takes us nowhere in this thread.

I have to read the OP again and Ghwellsjr's posts... Maybe the point I want to make can better be explained in another thread.
So I bail out for a moment.

 

[bobc2]

ghwellsjr, you have been considering your graphics to represent just one frame of reference. I'm thinking that your sketch actually implies three sets of coordinates, and you have used the Lorentz transformations to assign values to the time dimensions (X4 = ct) of the other two time coordinates. You haven't labled your coordinate time axes, so I've added in the labels for your three time coordinates in sketch a) below. Sketch b) just explicitly includes the X1 coordinate axes for the three sets of coordinates used in your presentation. The X1 axes are easy to identify since we know that in any frame the photon of light worldline must bisect the angle between X1 and X4. That assures that the speed of of light will be the same in all frames and the coordinate systems will all be in conformance with Einstein's postulate asserting the laws of physics are the same for all frames. The numbers on the coordinates in your presentation make it clear that you have done a good job of applying the Lorentz transformations between the various sets of coordinate systems.

I'm not trying to be critical of your presentation at all, because you have prepared it to minimize the information needed in order to focus on the point you were getting across about the different time increments along the different X4 (=ct) axes. And you do not wish to clutter up your graphs with any more detail than necessary to get your point across.

 

[K^2]

ghwellsjr, you have been considering your graphics to represent just one frame of reference. I'm thinking that your sketch actually implies three sets of coordinates, and you have used the Lorentz transformations to assign values to the time dimensions (X4 = ct) of the other two time coordinates.

Wrong. His diagrams only show one set of coordinates each. The other world lines only have proper time marked along them.

You CAN chose a coordinate system where proper time of a given object corresponds to time coordinate of the system, but you don't have to do that to discuss time dilation.

Your plots of additional coordinate systems are not wrong, but they are outside of the scope of the initial discussion, and are absolutely unnecessary for discussion of time dilation.

 

[Vandam]

Bob,
Last night I went late to bed because when I started reading from the beginning of the thread I immediately got stuck when I got to ghwellsjrs post. I think he started switching the A and B stationary and traveler, and then started using 21 months instead of 20,78 (24 / 1,1547). I do not know why because the opening post mentioned 2 years. Anyway, I got though that. After this little hickup it took me another 20 minutes to realize his drawings are NO space time diagrams at all. They are just time charts taken in one IRF all the way through. So K^2'last post is indeed correct.

But here is why you and I got mistaken: in fact there IS one chart of the three (in his post #9) that can indeed work as a full spacetime diagram (Minkowski), and that's the one you selected and marked up. Unfortunately you made the same 'mistake' as I did (on one of his charts in another post: https://www.physicsforums.com/showpost.php?p=4189020&postcount=35): you add the X1 ax. On that chart it does work, but Ghwellsjr doesn't understand what it (the ax) does there because his diagrams are time charts in one IRF only. Period. It took me nearly a sleepless night to get there.

His charts are correct, but of course they miss the complete space and time picture.
Furthermore the 3 charts insinuate the dilation occurs because of the space stretching between the dots on a worldline. But -as I see it- the lines in his charts are no worldlines..., just plotting timecoordinates.
A Loedel diagram could show him there is no stretching of dotspacing involved, but because he has 3 observers a Loedel diagram can not handle that.
I can only make a Minkowski for the three observers, but there he will again say that there is stretching of the dots.

I also have to admit I thought I was posting on that other thread of two opposite direction traveling spaceships. There it does make sense to show the simultaneity lines etc to explain time dilation. (But it didn't make sense to him)
But now on this tread I suddenly realized that his charts are no spacetime diagrams, and because here the two observers meet again there is indeed no need to get space axes involved, I guess.
So I think Ghwellsjr can here get away with it by the skin of his teeth.

I will drop a sketch to reformulate what I/we tried to get across.

 

[Dale]

His charts are correct

So why bother with the rest of this conversation? You may like your charts better, but you recognize that there isn't anything actually wrong with his.

So to me it seems like you are arguing over trivialities like font choices and colors. So what?

 

[ghwellsjr]

ghwellsjr, you have been considering your graphics to represent just one frame of reference.

Yes, that's because in Special Relativity, a given scenario is presented in the context of just one IRF, or if it isn't, calculations are made to eventually get it into a single IRF. Otherwise, it will be ambiguous and impossible to analyze.

I'm thinking that your sketch actually implies three sets of coordinates, and you have used the Lorentz transformations to assign values to the time dimensions (X4 = ct) of the other two time coordinates.

I start with the IRF presented by the OP and make a graph corresponding to that IRF. Now in that graph, I calculate the spacing of the dots for each observer, including the stationary one, using the time dilation equation (not the Lorentz Transform). So for each observer/clock, I use the speed assigned by the OP to calculate gamma and gamma times speed. Then I space the dots along the time axis according to gamma and space the dots along the distance axis according to gamma times speed. The Lorentz Transformation is not used to create the data for the original scenario in its IRF, only the Time Dilation factor (gamma) and the speed are used.

You should not think of the blue vertical line with the blue dots as being associated with the time coordinate of the IRF anymore than for the other observer/clock. In another scenario, an OP might not have any observer/clock at rest in the IRF and so there would be no vertical line with dots in the defining IRF.

You haven't labled your coordinate time axes, so I've added in the labels for your three time coordinates in sketch a) below.

As K^2 pointed out, there is only one set of coordinates, clearly labeled and marked and providing grid lines so that the coordinates of any event can be easily determined. What you are calling coordinate time axes are not axes at all, they simply show how the Proper Time of each observer/clock advances as a function of the clearly labeled coordinate time

I could have numbered the dots to make it easier to see what time is on each clock but that would have been more work for me so I leave it up to the viewer to count the dots if they care what the Proper Time is at any point in the diagram.

The whole purpose of this exercise is to show that Time Dilation is the ratio of accumulated Coordinate Time to accumulated Proper Time and that it changes with each IRF but still everything comes out the same for anything that the observers/clocks can see, observe and measure.

Since you want to talk only in terms of coordinate time, what is your definition of Time Dilation?

Sketch b) just explicitly includes the X1 coordinate axes for the three sets of coordinates used in your presentation. The X1 axes are easy to identify since we know that in any frame the photon of light worldline must bisect the angle between X1 and X4. That assures that the speed of of light will be the same in all frames and the coordinate systems will all be in conformance with Einstein's postulate asserting the laws of physics are the same for all frames. The numbers on the coordinates in your presentation make it clear that you have done a good job of applying the Lorentz transformations between the various sets of coordinate systems.

If you want to see how the speed of light remains c with respect to the IRF even after you transform to a different IRF, I will show you a bunch of graphs illustrating this. Keep in mind once I set up the original scenario, I merely put in a speed parameter to get each of these different graphs. It is the super simple Lorentz Transformation calculation done on all the points (events) of the original graph that creates each new graph. (I do have to do a little more work to limit the scope of each graph to the significant area.) Since I can only upload three graphs in each post, I will continue this in a second post.

First. a repeat of the original scenario with a flash of light sent out by both observers each month according to their own clocks. The thicker yellow lines are sent out by the blue observer and the thinner black lines are sent out by the black observer.

You can note that during the first part of the scenario, each observer sees the other ones clock advancing by the same amount. For example, after 19 months for each observer, they are seeing the other observer at 11 months.

Similarly, during the last part of the scenario, each observer sees the other ones clock advancing by the same (but different than before) amount. For example, between blue's Proper Time going from 41 to 48 months which is 7 months, he sees 12 new flashes coming from the black observer and for the black observer between the coordinates of about 38 and 46, his clock advances by 7 months and he sees 12 new flashes coming from the blue observer.

Now for the next two graphs transformed at 0.5c and -0.5c:

If you care to count out how each observer sees the other ones time progressing just like I described earlier, you can count out the dots to see that it doesn't make any difference what IRF we use, the same information is present in all of them.

Continued on next post...

 

[ghwellsjr]

Now I want to show three more IRF's. The first two are at a transformed speed where the speed of the two observers is identical for the first part of the trip (0.268) and for the last part of the trip (-0.268).

Note that since the speeds of the two observers are the same in these two IRF's for a part of the trip, their Time Dilations are also the same. Can you see that?

Now for one more IRF at an arbitrary random transformed speed of (0.35c) just to show that it doesn't have to be associated with anything in particular that is happening in the scenario and yet all the same information is present.

Each observer still sees everything identically to what they see in any other IRF. All measurements are identical. All observations are identical. But the Time Dilations are all different but still follow the same definition of being the ratio of accumulated Coordinate Time to accumulated Proper Time.

I'm not trying to be critical of your presentation at all, because you have prepared it to minimize the information needed in order to focus on the point you were getting across about the different time increments along the different X4 (=ct) axes. And you do not wish to clutter up your graphs with any more detail than necessary to get your point across.

I appreciate your congenial attitude but you should understand that I'm not trying to minimize the information--it's already minimized. There is no more information from which to minimize.

All you are doing is heaping the same information presented in different ways onto the same graph and thinking that it is more information and then you think that I'm trying to minimize the information when I don't do that.

I could, if I really wanted to, develop a computer program that would allow me to transform a scenario into another IRF but instead of presenting the coordinates in a normal square pattern (like on graph paper), I could distort the axes so that the physical locations of the events would remain in the same physical places as in the original scenario and then overlay the two plots so that you don't see the events move to new locations but instead see the axes with their labels and grid lines in different locations. That's all that is done in a conventional Minkowski diagram except that usually the grid lines are eliminated forcing the viewer to mentally establish their locations. I just don't see any advantage in doing that.