Stopped Clock Paradox; Relativity Down for the Count

[Carleas]

James, I take it you took that equation from the Wikipedia page on time dilation. However, after presenting the equation, it is clarified that

∆t is the time interval between two co-local events (i.e. happening at the same place) for an observer in some inertial frame (e.g. ticks on his clock)

That's not what's happening here, so that's not the equation we should be using. In fact, we are looking for the coordinates of the flashers in the stations frame of reference (call it (x',t')) given their coordinates in the train's frame (x,t). Specifically, we want the time coordinate.

... due to the timing devices being all set to read the same at the point of interest with respect to BOTH frames, "Flash time".

Isn't this just what we're trying to decide? You can't stipulate the conclusion. My point is that it is impossible to set them to flash at the same time at when the clocks pass in both frames: the phrase "the same time" is frame-relative.

If you want to use that diagram AFTER the "Flash point"...

The diagram plots the entire exchange, both before and after, equally well. Could you expand on this point?

Note that at T0, P0, the time READINGS are set to be different intentionally between the station and the train items so that at flash time, there will be no dt, difference in time readings.

This is one of two scenarios, which I called "run #2" a few posts ago. It's possible to set to the flashers beforehand so that they will be simultaneous for the station clock, but in that case they are not simultaneous for the train clock.

The relative positioning of the flashers to the train clock; x, x', do not vary thus there is no time dilation or delta time related to those items. Make a diagram showing the relative motion between the two flashers and you should see them on the same x and y angle along with the train clock, merely displaced fore and aft in train clock's x positioning but identical timeline (y).

First, I'd like to suggest that we use the standard notation that the variables in one frame of reference be plain (x,y,z,t) and those in the other be primed (x',y',z',t'). Accepting a convention will help us avoid confusion. In that case, the locations of the flashers x1 and x2 do not vary in the frame of the train, but t1 and t1', and t2 and t2' do vary.
The clocks will have parallel timelines for both frames of reference, however their location in time, their 't' coordinate, will vary between the frames of reference. In one, frame, they will hit y=0 at the same time, i.e. they will both be on the x axis at the same time. But in the 'squished' frame, they will be before and after 0. Here are a couple new diagrams to illustrate this:

Train.jpg (22.8 KiB) Viewed 521 times

This is the train's frame of reference. The clocks are set to be simultaneous with respect to the train in this scenario (again, this decision is arbitrary, there are two scenarios, and each will have a lack of simultaneity in one frame). At time zero (the x axis), the flashers flash, the light travels to the clock at the origin (x=0) and the clock stops.

Station.jpg (52.03 KiB) Viewed 521 times

This is the frame of reference of the station. I left it skewed to illustrate that the skewing doesn't change the facts within it when it is skewed, it is still a perfectly good Cartesian plane. Here, the flashers flash early and late; the red lines indicate the approximate value of t when each flashes (that the red line hits the green circle is due solely to the selection of values for v and x).

These two diagrams are useful just for separating out the two frames of reference. They more clearly convey the information contained in one of my earlier graphs, but in two separate graphs. They also make it easier to see that the skewed coordinate plane is still a full coordinate plane. It is the plane of the stations frame of reference, and must be used to evaluate t relative to the station.

[Mad Man P]

Can you guys please accelerate so we can all get synchronised again?

I'm really feeling bad for Carleas here, he's going through allot of truble in his posts with the diagrams and the illustrations.

Let's just agree that what Jimmy is refering to as SR is a bad theory that indeed does run into a paradox here, just as Jimmy says!
Let's agree that what we refer to as SR is a DIFFERENT theory, and that it does not run into a paradox here.

And then eithe leave it at that or go LOOK UP which theory is being refered to as SR by physicists these days.

All this frustration would stop and we would all be in perfect agreement!

We have no no disagreement with Jimmy, GIVEN we were working with the theory he refers to as SR.
We're just each insisting that what the other refers to as SR isn't "really" SR... semantics.

(and now my predictions have all come true... and you may all refer to me as Mad Man Prophet! )

[James S Saint]

James, I take it you took that equation from the Wikipedia page on time dilation. However, after presenting the equation, it is clarified that

∆t is the time interval between two co-local events (i.e. happening at the same place) for an observer in some inertial frame (e.g. ticks on his clock)

That's not what's happening here, so that's not the equation we should be using. In fact, we are looking for the coordinates of the flashers in the stations frame of reference (call it (x',t')) given their coordinates in the train's frame (x,t). Specifically, we want the time coordinate.

You are right. We don't need ∆t ("dt"), just as I said. We need "tf", a time dilation factor component.

... due to the timing devices being all set to read the same at the point of interest with respect to BOTH frames, "Flash time".

Isn't this just what we're trying to decide? You can't stipulate the conclusion. My point is that it is impossible to set them to flash at the same time at when the clocks pass in both frames: the phrase "the same time" is frame-relative.

It was stipulated that they must flash together at P1. Answer 1 proposed how to ensure that. We are now deciding if that is possible to accomplish.

Relative Motion - Station Frame 2.jpg (109.35 KiB) Viewed 497 times

Relative Motion - Train Frame 3.jpg (90.05 KiB) Viewed 497 times

I believe those are the diagrams involved.

Where the train timers are set must be horizontal in the diagram parallel to the x axis because they are set all together at the "same time". On the x axis, the flashers must be centered about the train clock, representing an "x" and "-x" position from the train clock.

[James S Saint]

Let's agree that what we refer to as SR is a DIFFERENT theory

After all, it IS relative.

[Mad Man P]

After all, it IS relative.

That which we call a rose, by any other name....

[Calrid]

It seems to me James has two choices he can either accept that simultaneity is not a conserved event or he can accept that anything of the sort is a paradox. This isn't an argument at all. James just doesn't understand the theory and everything he says just further demonstrates that he can't grasp that there might not be something paradoxical about one frame of reference being different from another when clocks are set up to achieve relativistic adjusted times. As I said ages ago his axioms are false the Lorentz transforms are the correct axioms. GIGO.

Garbage In Garbage OUT.

You remind me of a poster on another forum who has spent nearly 5 years trying to convince everyone that there is absolute time because energy cannot be destroyed and space is eternal. Ultimately if you can't explain why something is a paradox without making fundamental mistakes, no one is going to take you seriously. In fact the more you mangle the theory the less people will give you the benefit of the doubt.

It's possible to adjust a clock without it reflecting the time the frame of reference is actually experiencing. all the clocks will eventually reach 4 o' clock but it is simply not true that the actual photon strikes will be simultaneous unless all things are equal according to relativity. In order to disprove that you would need to set up an experiment that didn't assume that a paradox was in fact just a natural result of a theory and was entirely consistent with real world experiment without the universe disappearing up its own ass. What needs to be relatively shifted is your ability to understand the non addition of velocities in frames of reference and the implications to time and distance therein, not the theory itself which is currently fairly untouchable. In as much as any theory can be said to be.

"All science is but one experiment from death."

Me just then, unashamedly semi-plagiarising Neils Bohr.

[Mad Man P]

This isn't an argument at all. James just doesn't understand the theory

He's just refusing to call it SR, from what I can tell...

[Calrid]

This isn't an argument at all. James just doesn't understand the theory

He's just refusing to call it SR, from what I can tell...

Absolutely the argument isn't is SR a paradox, its why don't I understand why it isn't. Either James is a supra genius, who's skills are beyond Einstein, Bohr and Feynman, the more humble posters here and science in general, or he's just wrong. I'm not saying he's wrong because of that, his inelegant and faulty axioms have convinced me of that. But I would like to suggest he's no more a supra genius than I am, I think I can get away with that without too much trouble.

Sometimes you are just wrong. There's a fine line between genius and crackpot. Experiment will out.

[Carleas]

James, thank you for providing diagrams, this provides a lot of insight into how you are arriving at a paradox.

Unfortunately, your diagrams must be incorrect. The light lines (the red and blue lines in your diagram) have different slopes, so as you've drawn it, the speed of light is not even constant in a single frame of reference! Furthermore, as you have it, the train is traveling faster than some of the rays of light emitted by the flashers!

The slope of all light lines must be equal, and is conventionally set to 1 because it makes it easier to overlay one coordinate plane on top of the other. This means that, since the speed of any object must be less than c, the slope must always be greater than 1 (the slope is t/x, and v=x/t, so larger slope means smaller velocity).

The diagram shows time versus position. That means that if less distance is covered over the same amount of time (less run per unit rise), the object in question is moving slower. Look at the red line coming out of B in the first diagram. This line should represent light, but its slope is greater than that of the train: it moves less distance in the same amount of time than the train does. But then look at the blue line: its slope is less than the slope of the train's worldline, meaning it is traveling faster than the train. So you have two particles of light traveling at different velocities, and a train traveling faster than one and slower than the other.

The same sorts of problems exist in your second diagram. These cannot be the correct diagrams; the information they convey represents numerous inconsistent assumptions about the speed of light. If that is what is causing the paradox, it is easy to identify improper assumptions at work and eliminate them to resolve the paradox.

[Farsight]

There is no paradox. The situation is very simple, as we can see if we replace the two photons hitting the clock with two spaceships which crash into each other. It doesn't matter how the observers are moving, and it doesn't matter how fast they deem the spaceships to be moving, or when they deem the spaceships started moving, or which one they deem to have moved first. The two spaceships crash into each other, everybody sees it happen, and that's that.

Calrid, who's this poster on another forum?

[James S Saint]

James, thank you for providing diagrams, this provides a lot of insight into how you are arriving at a paradox.

Unfortunately, your diagrams must be incorrect. The light lines (the red and blue lines in your diagram) have different slopes, so as you've drawn it, the speed of light is not even constant in a single frame of reference! Furthermore, as you have it, the train is traveling faster than some of the rays of light emitted by the flashers!

Oh okay. I didn't pretend to be familiar with your diagram conventions. Due to an earlier comment, I was thinking that perhaps you were using a light circle convention extending from the flash point. But if the axes are to maintain their concept throughout, then certainly it won't do to have the train exceeding the speed of light. That would most terribly upset Penelope... but then again, that thought might keep Whiplash further away from the tracks.

Relative Motion - Station Frame 2.jpg (104.98 KiB) Viewed 430 times

Relative Motion - Train Frame 3.jpg (99.93 KiB) Viewed 460 times

As you noted in your earlier diagrams, it gets tough to keep things on the diagram if anything gets very realistic. But are those better? As you can see, they don't really change the issue. I was asking earlier of what information they were supposed to be conveying, because to me they don't really say any more than we already knew. But is there more they are to show?

[Carleas]

I believe these are correct space time diagrams, but they are for two separate scenarios, run #1 and run #2. In run #1, represented by your first diagram, the flashers are set to be synchronized in the station's frame of reference (both rest on the horizontal line t=0). However, note that they do not rest on the t'=0 line of the train's frame of reference. In the first diagram, the train's t'=0 line (not shown) would be a line whose slope is the multiplicative inverse of the train's speed. Because it is sloped, the flashers would be below and above it rather than on it.

In the second diagram, you represent run #2, where the flashers are synchronized with respect to the train. Taking the trains frame as the primed frame for consistency, the station's t=0 line (not shown) is a line whose slope is the multiplicative inverse of the station's velocity (with respect to the train). Here, the lights would flash simultaneously for train, because they are both on the train's t'=0 line, but they would not flash simultaneously for the station, whose zero line is not shown.

These diagrams each show both frames of reference. The differences between them are 1) that the station is (arbitrarily) taken as the right-angle frame in one and the train is in the other (this change has no mathematical consequences), and 2) the flashers are set to flash simultaneously with respect to the station in one and the train in the other. They do not depict the same event. The best way to see this is to draw in the t'=0 line for the train in the first diagram, and then calculate where the flashers are when they flash relative to that line (eyeballing will be enough to see that t'≠0), then look at where they flash in the second diagram (where it is clear that the flashers flash at t'=0).

[James S Saint]

I believe these are correct space time diagrams, but they are for two separate scenarios, run #1 and run #2.

Emmm... no. ..except in the train frame diagram, the station is accelerating rather than the train. That changes who gets the "tf" added to their clocks, but all else is the same. They are not two different events.

In run #1, represented by your first diagram, the flashers are set to be synchronized in the station's frame of reference (both rest on the horizontal line t=0).

All events on a horizontal line happen at the same moment, regardless of who's time frame it might be.

However, note that they do not rest on the t'=0 line of the train's frame of reference.

The "t' " line is in the Train Frame reference diagram as the vertical axis (the one you are calling run#2). On the station diagram, t' is shown stretched out along the train clock path. You have to stretch it to indicate the consumption of "tf" as time advances.

In the first diagram, the train's t'=0 line (not shown) would be a line whose slope is the multiplicative inverse of the train's speed. Because it is sloped, the flashers would be below and above it rather than on it.

You cannot lay one time frame on top of another and draw conclusions by mixing them. That is like translating every other word of the Bible into Hebrew and then saying because Hebrew is read backwards, we have to read each sentence by taking one word from the back then the next word from the front. You have to deal with only one frame at a time. A skewed alternate frame line that is on top of the station's frame, does not relate at all to time or distance. That is why they make multiple diagrams. If you try to see both frames at once, you will get a little of one mixed with a little of the other. And that seems to be what you are actually doing.

Can we agree on;
1) Both flashers were set to match the train clock while the train was at rest?
2) Both flashers were accelerated equally and thus traveled the same distance before they flashed?

These diagrams each show both frames of reference. The differences between them are

1) that the station is (arbitrarily) taken as the right-angle frame in one and the train is in the other (this change has no mathematical consequences), and
2) the flashers are set to flash simultaneously with respect to the station in one and the train in the other.

Emm.. no. The flashers are set to match the train and trigger when all 3 read "T0+tt". But for the station clock to read "T0+tt" (4:00) at that same moment that the train items do, the train items are each given an identical "tf" component. So at P0, the station clock and the train items do not agree in either frame. No matter which frame, the train clock and the flashers always agree.

The best way to see this is to draw in the t'=0 line for the train in the first diagram, and then calculate where the flashers are when they flash relative to that line (eyeballing will be enough to see that t'≠0), then look at where they flash in the second diagram (where it is clear that the flashers flash at t'=0).

I think that "t' " line that you are imagining is where you are tricking yourself. That t' line IS the train reference frame time line and the flashers are all set to match the train clock at the initial moment "T0" at "P0. That is true for either frame, just as shown.

You seem to be proposing that while the station time stood still, the train items were set one at a time waiting between each setting. It doesn't make sense. In ALL frames, a single moment is a single moment for everyone in any frame.

[Carleas]

James, your post reveals a fundamental misunderstanding of spacetime diagrams. The entire point of these diagrams is that they show both frames of reference.

Simple.jpg (42.47 KiB) Viewed 436 times

This is a stripped down version of the diagrams. The (blue) line x'=0 shows an object moving with respect to the (black) x,t coordinate plane. But with the line t'=0 it also defines another coordinate plane about the (yellow) light line that represents the frame of reference of a person traveling along x'=0. In this second coordinate frame, a stationary body is one whose worldline has a slope equal to the slope of x'=0. The light line bisects the first quadrant in both systems, because in relativity light travels at the same speed (and so has the same slope) in all frame of reference.

The (x',t') coordinate system is shown below:

xprimetprime.jpg (154.55 KiB) Viewed 436 times

Though coordinate planes are usually drawn with x and y at a right angle to each other, the choice is arbitrary, and any line can be drawn with equal accuracy on a skewed plane (this is always the case, not just in relativity). The reason it's possible to put both coordinate planes into the same diagram is because there is a constant transformation between the two, defined by the Lorenz equations, and because we know that in both frames the light line will have an equal slope. In this second illustration, you can see again that, for example, a stationary object at x'=1, t'=0 will only be stationary if it is still at x'=1 when t'=2 (i.e. its change in distance per change in time is zero). Though the line appears to have a slope, the speed is zero because the whole coordinate plane is skewed. If you follow x'=1 (the first light blue line to the right of the more vertical dark blue line), it will remain one unit away from x'=0, so its change in distance is zero, and so the ratio x'/t', which is its velocity, is also zero.

It doesn't make sense. In ALL frames, a single moment is a single moment for everyone in any frame.

Welcome to relativity. A 'single moment' is only a single moment at one location. If the two moments are separated by distance, they are also separated by time. Because we're talking about two events separated in space, we cannot assume that they will be simultaneous for everyone in every frame.

[James S Saint]

James, your post reveals a fundamental misunderstanding of spacetime diagrams. The entire point of these diagrams is that they show both frames of reference.

Not surprising, but interesting. I was about to say the same concerning you.

I am pretty certain finally, that I can see what you are seeing and thinking. You are misusing your tool, but don't believe that you are and certainly have no reason to take my word for it. But as I wonder how to crack that plank in your eye, I can easily hear your probable responses to anything I would say. It would be a bit like trying to get fly paper off of your hand while reading the instructions that you are holding with the other hand.

We need to stick with one diagram only, although laying another upon it temporarily might be needed for clarification. And I think that your mistake involves you rotating a frame so as to view a different frame. To create a different frame, you can stretch or skew either time, distance, or both, but you do not rotate.

Rather than try to bring me into your illusion, sense you believe that you understand both your diagram as well as mine, how about we start with my Station Frame and you add items or lines to it until it fits your version as well.

Now the first thing I would want to see is the diagram show the very first beginning movement of both flashers. In my Station Frame, I show that they begin their movement at T0, P0 horizontal. All items are at T0, P0 (+/- for the flashers) and the Station Frame is the only frame at that moment. All train items begin to move in unison from there.

But even from the train's frame, they all begin to move from that same position and time. Aboard the train, they are all just beginning to leave P0, T0. So at the very beginning of the scenario, there really is only one frame and the train frame must slowly begin to alter from that position and time.

So according to whatever frame you would like to add to my Station Frame diagram, could you maybe copy my pic and draw on it ;

A) where you see the flasher's F and B starting up? Remember that they all must be at T0, P0 according to the station frame just before they begin to move.

Each travel line must be identical in length for all train items because they all move together, so ;

B) where would the 3 train item lines lead to and end within the "tt" time of the station frame. If they are each at the same angle and are the same length, the station frame should see them all still parallel to where ever they were when they started.

Welcome to relativity. A 'single moment' is only a single moment at one location. If the two moments are separated by distance, they are also separated by time. Because we're talking about two events separated in space, we cannot assume that they will be simultaneous for everyone in every frame.

Well, welcome to your illusion. We are only talking about one event. You think we are talking about two. Where were they all immediately as they started up and at what time, purveyed by the train or the station since they were both "still" at startup?

[PhysBang]

Well, welcome to your illusion. We are only talking about one event. You think we are talking about two. Where were they all immediately as they started up and at what time, purveyed by the train or the station since they were both "still" at startup?

The illusion here is all yours. You imagine that both ends of the train beginning to move is one event. Every part of the train has it's own event when it starts to move.

[Carleas]

James, I don't know why you think I'm rotating the frame of reference. In every diagram where I've illustrated both coordinate planes, they have been skewed. If they were rotated, they would look like this:

Baddiagram.jpg (29.85 KiB) Viewed 405 times

We can both agree that this is the wrong way to translate between the frames, if only because if the frames are rotated in this way, the slope of the light line cannot be constant in both frames.
______________________________________________________________
It isn't necessary to include the initial acceleration in the diagram, because what we're actually concerned with is two inertial reference frames in motion relative to each other. Even if the train started and stopped and traveled backwards and stopped and traveled forward again, we can safely ignore it as long as we're still asking about a time when the two clocks are in inertial reference frames with constant velocity relative to each other. Acceleration would look something like this:

Acceleration.jpg (18.59 KiB) Viewed 405 times

The bottom quarter could be cut off this graph and the math of what happens in the top three quarters would be unaffected.
________________________________________________________________________
Here's a depiction of the train's frame of reference, with the flashers flashing simultaneously in that frame.

Trainsframe.jpg (40.52 KiB) Viewed 405 times

When we overlay the station's coordinate system (see the next diagram), we can see that if both flashers are flashing at t'=0 (the tain's frame), they can't possibly be flashing at t=0 (the station's frame). t'=0 is slanted with respect to t=0; since both are straight lines, they can only intersect at a single point. In this diagram, that point is where the two clocks line up. If we draw in the train when each flasher goes off, we see where the other end of the train is in the the station's reference frame when each flasher goes off (red lines).

bothframes.jpg (75.52 KiB) Viewed 405 times

The lines representing the train in the station's frame of reference must be parallel to the station's coordinate plane, and so cannot be parallel to the train's coordinate plane. What happens for the train at t'=0 happens for the station at some different, non-zero time for each flasher.

We are only talking about one event. You think we are talking about two.

I'm talking about two runs, i.e. two different experimental setups. That is to be expected if there is no universal simultaneity: if we can't make it simultaneous for both frames in the same run, then there is one run in which it is simultaneous for one frame and one where it is simultaneous for the other.

But PhysBang's point is a good one: "Every part of the train has it's own event when it starts to move." The flashers aren't co-local, so they are different events. They may be temporally connected in one frame, but because they are separate events they don't need to be connected temporally in all frames (and they won't be in any two frames in motion relative to each other).

[James S Saint]

Is this just a delay tactic?

So according to whatever frame you would like to add to my Station Frame diagram, could you maybe copy my pic and draw on it ;

A) where you see the flasher's F and B starting up? Remember that they all must be at T0, P0 according to the station frame just before they begin to move.

Each travel line must be identical in length for all train items because they all move together, so ;

B) where would the 3 train item lines lead to and end within the "tt" time of the station frame. If they are each at the same angle and are the same length, the station frame should see them all still parallel to where ever they were when they started.

[Carleas]

As I said, the period of acceleration is irrelevant to the state of affairs when the clocks are aligned.

[James S Saint]

As I said, the period of acceleration is irrelevant to the state of affairs when the clocks are aligned.

I didn't ask about acceleration (nor concerned about it: assume instantaneous acceleration). I asked of position; "Where" are the flashers at the very beginning moment of the scenario in your additional frame? They should be located in the same position as the beginning of the Station Frame because at that moment there is only one frame.

Given that, then where would the 2 lines representing the flashers lead to and how long will they be before the flashing occurs. They must be the same length because they are treated identically and thus travel the same distance.

[Calrid]

As an side from this strange thread instantaneous acceleration is a debatable concept. If one could achieve light speed instantaneously then in theory as no time had passed then objects of mass would be able to travel at light speed. Obviously that's why it's somewhat dubious to assume instantaneous without specifying what you mean exactly. Mind you most physics concepts you have to be very careful to specify a coherent terminology or you can get lost in space. Crackpots then seep into the gaps (not referring to anyone here) and light speed becomes possible for space ships and so on and time travel and all the other hypothetical guff kinda gets added in in the usual ad hoc way.

Best way to explain instantaneous is probably to say at t>0 speed = a or whatever speed/acceleration it instantaneously is said to achieve.

[Carleas]

Where? They're at some arbitrary starting point that doesn't play in any equation that determines their relative positions or times. They must be the same length in the Train's frame of reference, but not in other frames of reference. Remember, length in these diagrams is not a measure of distance, but is the hypotenuse of a triangle of height t and width x.

Nor does it matter how long it is before the flashing occurs. When the flashing occurs, the flashers are time shifted with respect to each other from the station's frame of reference.

What you're really looking for a coordinate system that maps the non-inertial frame of the train's acceleration to the inertial frame of the station. There is no such mapping, because the relationship between the two stations is nonlinear.

How does this relate to the question of whether the flashers are simultaneous? Will you respond to my earlier post (to which you alleged I was employing a "delay tactic", though I don't see how I could possibly benefit from slowing this discussion down)?

[James S Saint]

Where? They're at some arbitrary starting point that doesn't play in any equation that determines their relative positions or times. They must be the same length in the Train's frame of reference, but not in other frames of reference. Remember, length in these diagrams is not a measure of distance, but is the hypotenuse of a triangle of height t and width x.

Actually the length if along an axis displays the value seen by the displayed frame. The length at an angle displays the time or distance dilation. Note in my Station diagram there is a "tt" length directly vertical and a "tt+tf" length sloped between where the train began at T0, P0 and where the flashes occur at P1.

How can any point involving either axis, position or time, be "arbitrary"??? In effect, that is saying that "I want them to end up here, so I'll just start them ... emm.. lets say over here." And in this case, you WANT them to end up being at different times for the station so you "arbitrarily" start them where ever you like to cause that end result. That isn't Science or logic. It is pathos.

Your lines showing the travel of the flashers MUST start at the same location as the Station frame, again, because at that start moment, there is only one frame.

Nor does it matter how long it is before the flashing occurs.

Then what is the problem in showing it? The major point is that it is the same length for both flashers. At the beginning the flashers were in sync with the station and each other. The flashers will get time dilated together and thus no longer be in sync with the station, but still in perfect sync with each other.

When the flashing occurs, the flashers are time shifted with respect to each other from the station's frame of reference.

But that is your assertion. You can't merely assume your asserted result and then proclaim that your diagram shows it. Your diagram is supposed to display how it got there, else what good is it? I could just say, "no this diagram here is the real diagram" and provide anything I wanted. Any proposed new frame diagram, must begin at the same positions as the inertial frame and then show the slopes involved relating the two frames (the dilations). That is the only reason there is any slope at all.

What you're really looking for a coordinate system that maps the non-inertial frame of the train's acceleration to the inertial frame of the station.

That is exactly what my Station frame is showing. Why else have such diagrams?

There is no such mapping, because the relationship between the two stations is nonlinear.

Non-linearity wouldn't make it un-displayable. It would merely make the slopes into curves. But for a constant dilation factor, there are no non-linear equations involved.

How does this relate to the question of whether the flashers are simultaneous? Will you respond to my earlier post (to which you alleged I was employing a "delay tactic", though I don't see how I could possibly benefit from slowing this discussion down)?

In presenting your diagram, you are merely asserting that "it will be like this". But any diagram that is going to reveal the truth is going to be generated by some rationale, not arbitrarily splashed on top of the other frame at your whim. There has to be a reason for each point being where the diagram proclaims it to be.

You say that the flashes will not be simultaneous. Yet at the origin, they were synchronized to the Station. A proper diagram will display how they got un-synchronized (if they did).

By showing that they were synchronized at origin and traveled the same length and time as each other, they should still be synchronized with each other, even though both equally different than the station. The slope of the lines and their length is what is telling you how much different they would be from the station.

The issue that you seem to be avoiding and proclaiming irrelevant is the only real issue of this discussion.

[PhysBang]

You say that the flashes will not be simultaneous. Yet at the origin, they were synchronized to the Station. A proper diagram will display how they got un-synchronized (if they did).

This is not something that can be shown on a diagram. Special relativity lays out the prerequisites for having correct spacetime diagrams if one wants to preserve the ability to translate from one diagram to another without rewriting the laws of physics and while keeping the speed of light the same in all frames. These two things together entail that clocks that are synchronized on one frame are not in another. The times that things happen according to physical laws must change between reference frames or else different reference frames will either produce different events or the speed of light will not be constant. This is why, if we use SR, we must use the Lorentz transformations to translate between reference frames.

We can image that we can make clocks for our flashers that will stay synchronized with the clocks in the station frame, but this means that these flasher clocks will drift away from the time of the frame that we use for calculating the physics of that frame.

[James S Saint]

There are no equations in Science that express the idea that a clock merely being in a different location will automatically have a different time dilation merely due to it being located off to the right rather than the left.

What you guys are saying is that if NASA were to do the exact same time dilation experiment with orbiting clocks, they would get entirely different readings now because the Earth is always in a different location. Where is your Science evidence that such has ever occurred or even that any Science formula suggests that?

The Lorentz transformations, as pointed out by Calrid days ago, change nothing. They are what the motion diagrams are made from. No one in Science has ever suggested what you are thinking.

This discussion isn't about the difference between me and Science, but the difference between you and Science. Show me anything in science that says that time dilation is dependent on location rather than the changing in location relative to the changing in time.

There are only 2 frames involved. I have shown both frames. The x and y axis are one frame while the sloped lines are the other frame, in both diagrams. This mystical frame that you guys think displays the flashers going off in different times is pure contrived imagination having nothing to do with Lorentz, Einstein, or anyone of Science whom I am aware.

It appears that the idea got into your heads that a different location, rather a different changing in location, constitutes a different dilation. You have merely misunderstood what "they" have said. Show me anything to indicate otherwise.