Benefits of time dilation / length contraction pairing?

[neopolitan]

Nevertheless, the situation I am thinking about is analogous to the diagram with a red and green photon worldline, but with one photon which could have come from anywhere (along its path) and any time. Two observers are initially colocated, one travels away at v, and the photon passes one observer then the other one.

The observers consider themselves to be at rest and the other to be moving. Both assume that the event that spawned the photon occurred when they were colocated with the other observer.

Why would they both assume that? You agree that one of them must be objectively right and the other objectively wrong if they each assume the event occurred simultaneously with their being colocated according to their own rest frame's definition of simultaneity, right

You are right, they don't need to assume that the event that spawned the photon occurred when they were collocated with the other observer.

This was very poorly worded (particularly the words I have made red) and I apologise for the confusion it created.

It is better to say that when the photon passes B at t_B' in my later diagram (my poor phrasing quoted above was in post #168, diagrams were post #174), B could assume that one of the possible spawning events could have been at ct_B' when A and B were collocated.

When the photon passes A at t_A, A could assume that one of the possible spawning events could have been at ct_A when A and B were collocated.

I do believe that neither A nor B are wrong in being able to make these assumptions. Being objectively right never comes into it.

Where I wrote "Both assume", I should have written "Both could assume". It made sense to me at the time

cheers,

neopolitan

 

[neopolitan]

While awaiting a response on https://www.physicsforums.com/showpost.php?p=2165684&postcount=174". That diagram only showed length contraction (LC).

I have developed three more diagrams, one for time dilation (TD), one for spatial analogue for time dilation (SAFTD) and one for temporal analogue for length contraction (TAFLC).

In each of the diagrams I have tried to be consistent with the use of colours. I hope that the sheets (light blue and light orange) are easily understood. Trying to explain briefly, depending on what is being shown:

the blue sheets have a spatial of L or a temporal width of 1 tick in the A frame
the orange sheets have a spatial width of L or a temporal width of 1 tick in the B frame.

The darker blue bar shows either a length L or the period between two consecutive ticks in the A frame. The darker orange bar shows either a length L or the period between two consecutive ticks in the B frame.

The green bars show measurements by A of a length or period in the B frame.
The purple bars show measurements by B of a length or period in the A frame.

For completeness, I will repost JesseM's diagram (which contains some of the information, but not all).

cheers,

neopolitan

It seems that I can't post diagrams at the moment. So, I have posted them all on one web page http://www.geocities.com/neopolitonian/index.htm".

I'm aware that only a few may understand the background to the diagrams. Therefore, I thought I might try to explain a little.

Length contraction manifests in situations such as when two rods of the same rest length pass each other. Both will observe the other to be contracted.

Time dilation manifests with in situations such as when you have two identical clocks where one is in motion relative to the other. Time dilation is not so easily observed. Part of this is because photons received from a clock moving away from you will be spaced out due to the movement of the clock away from you, even before taking into account relativity. The combined effect of standard doppler and relativity is "relativistic doppler" and in the diagrams the overall effect is to double the period between "moving" ticks compared to "stationary" ticks where v=0.6c. A pure doppler effect for one observer assumed to be stationary with the other observer is in motion at v=0.6c would be: (c+0.6c)/(c+0) = 1.6. To make relativistic doppler, we apply a gamma of 1.25 to get 1.6*1.25 = 2.

To understand spatial analogue for time dilation (SAFTD) you really need to look at the diagram. Note that the period between ticks is stretched out for B relative to A. This is the effect of time dilation. The length of L in B's frame is stretched out in A's frame in a similar fashion.

As for the temporal analogue for length contraction (TAFLC), this is similar to the idea of two rods passing each other. Imagine there are no doppler considerations, in the time it takes the clock in A's frame to tick once, according to A, the clock in B's frame has not had enough time to tick. In other words, according to A, 1 tick in A = less than 1 tick in B. This means that the clock in A will tick more, so more time will elapse in A than in B.

Clear as mud?

 

[JesseM]

It seems that I can't post diagrams at the moment. So, I have posted them all on one web page http://www.geocities.com/neopolitonian/index.htm".

The labels on the time dilation diagram appear to be backwards. You label the purple segment the time dilation of B looking at A, but that appears to be the time dilation that A sees when it looks at B's clock--it's the difference in A's time coordinates between two events on the worldline of B's clock that are separated by 1 second in B's frame (and the height of the purple segment is greater than 1 second in A's frame, hence the 'dilation'). Likewise, the green bar seems to show the time in B's frame between two ticks of a clock at rest in A's frame, so that should be "time dilation of B looking at A" rather than "A looking at B". Unless I'm misunderstanding what "looking at" is supposed to mean (I interpreted 'A looking at B' to mean how B's clock behaved in A's frame).

 

[JesseM]

Some clarifications to the last post (done in the early hours of the morning).

G to E 01 - (Galileo to Einstein) - shows the standard Galilean boost. There is an assumption of instantaneous transfer of information (or god-like powers to see everything at once).

Why is instantaneous transfer of information relevant? Even in a Galilean universe it could be true that information has limited speed (for example, the fastest information transfer might be vibrations in the aether which travel at c in the aether frame). The coordinates an observer assigns to an event are done in retrospect, once I have already received information about an event. For example, if an event happens 12 light-seconds away from me at t=0, then if you have instantaneous transfer of information I'll learn about the event at t=0, while if information only travels at c I won't learn about it until t=12; but in the latter case I'll take into account the speed of the signal and backdate the event to t=0, so the coordinates are the same either way.

G to E 02 - shows what happens when you remove the assumption of instantaneous transfer of information. E is an "event". I stress that it could be something that causes the emission of a photon, or it just could be an event along the path of the photon. A and B see the same photon (thought experiment magic).

In this diagram, I take it t refers to the time in A's frame the light reached A, and t' refers to the time in B's frame the light reached B? If so it also seems that x refers to the position of the photon at t=0 in A's frame, while x' refers to the position of the same photon at t'=0 in B's frame (because of the relativity of simultaneity these must refer to different events on the photon's worldline). So in each frame you're calculating the distance and time between a totally different pair of events, correct?

G to E 03 - shows the reconciliation, with subscripts to show "according to ..."

In this process, it becomes explicit what sort of physical things your x, t, x', t' and, if you like, L and L' refer to.

It still isn't really clear to me...is A still measuring the distance and time between an event #1 on the photon's worldline (the event on the photon's worldline which in A's frame is simultaneous with A and B being colocated) and the event of the photon passing A, while B is still measuring the distance and time between a different event #2 on the photon's worldline (the event on the photon's worldline which in B's frame is simultaneous with A and B being colocated) and the event of the photon passing B? Or are the assumptions supposed to be different in this diagram? (or did I misunderstand the assumptions of the previous diagram?) If I'm getting the assumptions wrong, can you try to explain in clear terms what two events A is measuring the distance x and time t between, and likewise what two events B is measuring the distance x' and time t' between?

 

[neopolitan]

The labels on the time dilation diagram appear to be backwards. You label the purple segment the time dilation of B looking at A, but that appears to be the time dilation that A sees when it looks at B's clock--it's the difference in A's time coordinates between two events on the worldline of B's clock that are separated by 1 second in B's frame (and the height of the purple segment is greater than 1 second in A's frame, hence the 'dilation'). Likewise, the green bar seems to show the time in B's frame between two ticks of a clock at rest in A's frame, so that should be "time dilation of B looking at A" rather than "A looking at B". Unless I'm misunderstanding what "looking at" is supposed to mean (I interpreted 'A looking at B' to mean how B's clock behaved in A's frame).

Time dilation continues to be a problem.

What the diagram tried to show is a value seen from the A frame (from the A frame, ticks are all colocal) such that t_B = gamma.t_A. Since someone said a while back:

The usual convention is that the unprimed t represents the time interval between two events on the worldline of a clock as measured in the clock's rest frame (so both events happen at the same spatial location in the unprimed frame, and t will be equal to the time interval as measured by the clock itself), whereas the primed t' represents the time interval between the same two events in a frame where the clock is moving (so the events happen at different locations in the primed frame).

That means that, according to A, t_B = t' and t_A = t, therefore:

t' = gamma.t

which is time dilation.

I could learn to like time dilation more if the stretching of the time axis in B's frame is time dilation and/or I am allowed to explain something like this:

Time dilation manifests with in situations such as when you have two identical clocks where one is in motion relative to the other. Time dilation is not so easily observed. Part of this is because photons received from a clock moving away from you will be spaced out due to the movement of the clock away from you, even before taking into account relativity. The combined effect of standard doppler and relativity is "relativistic doppler" and in the diagrams the overall effect is to double the period between "moving" ticks compared to "stationary" ticks where v=0.6c. A pure doppler effect for one observer assumed to be stationary with the other observer is in motion at v=0.6c would be: (c+0.6c)/(c+0) = 1.6. To make relativistic doppler, we apply a gamma of 1.25 to get 1.6*1.25 = 2.

Application of the gamma of 1.25 is time dilation and the stretching of the t_B axis.

This would mean that the darker orange line is time dilation for A and the dark blue line is time dilation for B.

I'm happy with that.

But you can see that time dilation is such a totally different thing to Length contraction, which we have covered before, and that in the SAFTD diagrams the same would apply, the darker orange line would be SAFTD for A and the dark blue line would be SAFTD for B.

Are you happy with that?

What you can't really do though, is measure these values directly. Even to show it on the diagram is difficult, I would have to animate it or show a rotation around the origin of the axis (which may introduce problems). You can measure the time dilation indirectly in either of the ways described above (relativistic doppler or some variant of the twin's experiment).

You might say they have already tested it and showed that it works, and I would say "yes, they have done so, indirectly". What they have not done is something equivalent to directly measuring length contraction which could be done by simply passing two rods past each other - and both would measure the other to be shorter.

If you can come up with an experiment in which both observers can directly observe the other be time dilated, I'd be curious to know about it. I think that relativistic doppler is as close as you can get.

I've posted updated TD and SAFTD diagrams (Attempt 2) http://www.geocities.com/neopolitonian/index.htm".

cheers,

neopolitan

 

[neopolitan]

Why is instantaneous transfer of information relevant? Even in a Galilean universe it could be true that information has limited speed (for example, the fastest information transfer might be vibrations in the aether which travel at c in the aether frame). The coordinates an observer assigns to an event are done in retrospect, once I have already received information about an event. For example, if an event happens 12 light-seconds away from me at t=0, then if you have instantaneous transfer of information I'll learn about the event at t=0, while if information only travels at c I won't learn about it until t=12; but in the latter case I'll take into account the speed of the signal and backdate the event to t=0, so the coordinates are the same either way.

You want to reintroduce an aether?

I have never actually seen Galilean relativity done that way, but then I have never seen time brought into it at all. Galilean relativity seems to be based on a snapshot. It is certainly based on absolute space (http://en.wikipedia.org/wiki/Galilean_invariance" ).

Really, I am just going from the Galilean boost to Lorentz Transformations though. That boost is given by x'=x-vt. Do we at least agree on that?

The Galilean assumption, in terms of my diagram, is that B is moving with an absolute velocity of v towards a location E which is a distance of x from A and, at a time t, the distance from B to E is x' = x - vt. This means that when t=0, A and B were colocated. Do we agree on that?

In Galilean relativity, at t, A has not moved, B is moving with a velocity of v and is located vt closer to E than A is. Do we agree on that?

In Galilean relativity, we could have an event at E, (x,t) in A's frame and (x',t) in B's frame. Do we agree on that?

In Galilean relativity, we could have an event at E, (x,t) in A's frame and (x',t') in B's frame, because time is absolute and t=t'. Do we agree on that?

In Galilean relativity, if B is told that E is currently x' away, and B has observed that A has been moving away at -vt, then B will calculate that A-E is currently x = x' + vt . Do we agree on that?

Do we further agree that if an event took place at (x,t) in A's frame in Special Relativity and even in a more careful analysis of Galilean relativity, that neither A nor B would know about it until a photon from the event is received?

If x = ct, in Galilean relativity, when A receives the photon at 2t, x' = x - 2vt. Do we agree that if we now talk about where a photon from the same event (x,t) hits B, this is not x' as calculated above?

I guess I could agree that Galilean relativity is based on either absolute space (ie there's an aether frame) or instantaneous transmission of information. Can you agree that it is one or the other? In a paper I put together on this, I actually had a few assumption including preferred frame and instantaneous transmission of information, I can see that I should put them as "and/or".

Can you see that if information is transmitted instantaneously and an event takes place at (x,t) in the A frame, then in the A frame that event will be detected by A at (0,t) and B at (vt,t)? And in the B frame, the event was at (x',t), B detects is at (0,t) and A at (-vt,t) where x'=x-vt? And can you see that these can all be related by the Galilean boosts?

(Because of LET, I wonder if it actually works with just an aether frame. I'd have to put more time into, and I am running out of time rapidly.)

In this diagram G to E 02, I take it t refers to the time in A's frame the light reached A, and t' refers to the time in B's frame the light reached B? If so it also seems that x refers to the position of the photon at t=0 in A's frame, while x' refers to the position of the same photon at t'=0 in B's frame (because of the relativity of simultaneity these must refer to different events on the photon's worldline). So in each frame you're calculating the distance and time between a totally different pair of events, correct?

One photon. One event spawning the photon. Two events where the photon passes B, then A. One event when A and B were colocated and t=0 and t'=0.

A thinks that at colocation, the photon was located at x=ct.

B thinks that at colocation, the photon was located at x'=ct'.

What is the relationship between x' and x, and t and t'?

Does that help?

What we do know is that, irrespective of coordinate system, when A and B were colocated, the photon had one unique location. Correct?

It still isn't really clear to me...is A still measuring the distance and time between an event #1 on the photon's worldline (the event on the photon's worldline which in A's frame is simultaneous with A and B being colocated) and the event of the photon passing A, while B is still measuring the distance and time between a different event #2 on the photon's worldline (the event on the photon's worldline which in B's frame is simultaneous with A and B being colocated) and the event of the photon passing B? Or are the assumptions supposed to be different in this diagram? (or did I misunderstand the assumptions of the previous diagram?) If I'm getting the assumptions wrong, can you try to explain in clear terms what two events A is measuring the distance x and time t between, and likewise what two events B is measuring the distance x' and time t' between?

Hopefully the above helped. My time is up for now.

cheers,

neopolitan

 

[JesseM]

The labels on the time dilation diagram appear to be backwards. You label the purple segment the time dilation of B looking at A, but that appears to be the time dilation that A sees when it looks at B's clock--it's the difference in A's time coordinates between two events on the worldline of B's clock that are separated by 1 second in B's frame (and the height of the purple segment is greater than 1 second in A's frame, hence the 'dilation'). Likewise, the green bar seems to show the time in B's frame between two ticks of a clock at rest in A's frame, so that should be "time dilation of B looking at A" rather than "A looking at B". Unless I'm misunderstanding what "looking at" is supposed to mean (I interpreted 'A looking at B' to mean how B's clock behaved in A's frame).

Time dilation continues to be a problem.

What the diagram tried to show is a value seen from the A frame (from the A frame, ticks are all colocal) such that t_B = gamma.t_A.

A value of what, though? Which two events are you supposed to be measuring the time between in both frames? The orange bar which projects through the purple line crosses the events 1 and 2 on B's worldline, so I assumed the purple bar was showing the time in A's frame between events 1 and 2 on B's worldline which have a separation of 1 second in B's frame. But if t_A represents the time between this pair of events in A's frame and t_B represents the between the same pair in B's frame, then t_A = 1.25 and t_B = 1, which means t_A = gamma * t_B, the reverse of what you write above.

Maybe the two events you're thinking of are not 1 and 2 on B's worldline, but just the events which actually lie at the endpoints of the purple line? But then the bar is confusing because it doesn't help you show the time between these events in B's frame, you should instead draw a slanted bar (like the light blue one) whose top and bottom cross the events at the top and bottom of the purple line, and then see where this slanted bar intersects B's time axis, which would give you t_B for these two events.

Since someone said a while back:

The usual convention is that the unprimed t represents the time interval between two events on the worldline of a clock as measured in the clock's rest frame (so both events happen at the same spatial location in the unprimed frame, and t will be equal to the time interval as measured by the clock itself), whereas the primed t' represents the time interval between the same two events in a frame where the clock is moving (so the events happen at different locations in the primed frame).

Right, and going by the interpretation that the purple line is supposed to show the time between 1 and 2 on B's worldline, then since these events are colocal in B's frame but not in A's frame, B should be unprimed and A should be primed.

That means that, according to A, t_B = t' and t_A = t

Well, not if your events are 1 and 2 on B's worldline. And why do you say "according to A"? Once you have picked the two events you want to measure the time between, according to my description of the primed vs. unprimed convention both observers will agree on which frame should be primed and which should be unprimed, the unprimed is always the frame where those particular events are colocated. But I'm not sure you're clear on the fact that we always have to have a fixed idea of what two events we're talking about in advance before we can use the time dilation equation, which is comparing the time-interval between those specific events in both frames (one of which is the frame where they are colocated, which is normally labeled as the unprimed frame).

I could learn to like time dilation more if the stretching of the time axis in B's frame is time dilation and/or I am allowed to explain something like this:

Time dilation manifests with in situations such as when you have two identical clocks where one is in motion relative to the other. Time dilation is not so easily observed. Part of this is because photons received from a clock moving away from you will be spaced out due to the movement of the clock away from you, even before taking into account relativity. The combined effect of standard doppler and relativity is "relativistic doppler" and in the diagrams the overall effect is to double the period between "moving" ticks compared to "stationary" ticks where v=0.6c. A pure doppler effect for one observer assumed to be stationary with the other observer is in motion at v=0.6c would be: (c+0.6c)/(c+0) = 1.6. To make relativistic doppler, we apply a gamma of 1.25 to get 1.6*1.25 = 2.

True, if "pure Doppler" is taken to mean Doppler shift when we assume no time dilation, you just multiply that by gamma to get the relativistic Doppler equation (for period rather than frequency as in your calculation).

Application of the gamma of 1.25 is time dilation and the stretching of the t_B axis.

I don't follow. I'd say that application of gamma is because if the clock is emitting signals once every T_emit according to its own readings, then in your frame the time interval between signal emissions is gamma*T_emit and its distance from you increases by v*gamma*T_emit in this time, so each successive signal has to cross this extra distance which takes an extra time of v*gamma*T_emit/c in your frame, so instead of receiving the signals once every gamma*T_emit (the 'actual' time between emissions in your frame), you only receive them once every gamma*T_emit + v*gamma*T_emit/c = gamma*T_emit*(1 + v/c). So if T_obs is the time between observing successive signals from the moving clock, we get the relativistic Doppler equation T_obs = gamma*T_emit*(1 + v/c), whereas the Galilean Doppler equation would just be T_obs = T_emit*(1 + v/c).

This would mean that the darker orange line is time dilation for A

The orange line is slanted in A's frame, so what do you mean by "time dilation for A" if this line goes in a mix of space and time directions in A's frame? Are you talking about the purely vertical distance between event 2 at the bottom of the orange line and event 3 at the top? If so your diagram doesn't actually show anything corresponding to this vertical distance, it'd be more clear if you drew a horizontal bar whose top and bottom edge cross these two events, so you could see the times this bar's top and bottom intersect the t_A axis (which would represent the dilated time in A's frame between events 2 and 3 on B's worldline that are colocated in B's frame). Alternatively you could just relocate the darker orange line so it goes between events 1 and 2 on B's worldline, since you already have a light orange horizontal bar there.

But you can see that time dilation is such a totally different thing to Length contraction, which we have covered before, and that in the SAFTD diagrams the same would apply, the darker orange line would be SAFTD for A and the dark blue line would be SAFTD for B.

I would say that the horizontal length of the orange bar in the SAFTD diagram represents the spatial distance in A's frame between events 1 and 4 on the t_B axis, which are simultaneous in B's frame, and that's exactly the idea of the SAFTD...in other words, if we write SAFTD as dx' = dx * gamma, then here dx is the distance in B's frame between events 1 and 4, and dx' is the distance in A's frame between events 1 and 4 (which can also be represented by the green line). As long as that matches what you're saying above, this diagram makes sense to me.

What you can't really do though, is measure these values directly. Even to show it on the diagram is difficult

Isn't showing where the appropriate bars cross the appropriate axis "showing it on the diagram"? For example, in the SAFTD diagram, you can see the dark orange line has a length of 3 in B's coordinate system (since it goes from x=1 to x=4 on B's space axis), but you can also see that the vertical light orange bar whose edges go through the two ends of the dark orange line intersects A's x-axis at two points that are a distance of 3*1.25 = 3.75 apart (which represents the distance in A's frame between the events on the ends of the dark orange line).

 

[neopolitan]

A value of what, though? Which two events are you supposed to be measuring the time between in both frames? The orange bar which projects through the purple line crosses the events 1 and 2 on B's worldline, so I assumed the purple bar was showing the time in A's frame between events 1 and 2 on B's worldline which have a separation of 1 second in B's frame. But if t_A represents the time between this pair of events in A's frame and t_B represents the between the same pair in B's frame, then t_A = 1.25 and t_B = 1, which means t_A = gamma * t_B, the reverse of what you write above.

The light coloured sheets are simultaneity planes, if you like:
The light orange sheet is 1 tick.gamma in the A frame, spanning 1 tick in the B frame.
The light blue sheet is 1 tick.gamma in the B frame, spanning 1 tick in the A frame.

If you must have events, the light orange sheet shows all the events which are simultaneous in the A frame with two consecutive ticks of B's clock, and all the events between. In the B frame, the two consecutive ticks are 1 tick apart and not simultaneous with all the other events which constitute the boundaries of the light orange sheet. In the A frame, the two consecutive ticks are 1 tick.gamma.

The light blue sheet shows all the events which are simultaneous in the B frame with two consecutive ticks of A's clock, and all the events between. (And so on.)

Maybe the two events you're thinking of are not 1 and 2 on B's worldline, but just the events which actually lie at the endpoints of the purple line? But then the bar is confusing because it doesn't help you show the time between these events in B's frame, you should instead draw a slanted bar (like the light blue one) whose top and bottom cross the events at the top and bottom of the purple line, and then see where this slanted bar intersects B's time axis, which would give you t_B for these two events.

The purple line just shows the time between the events which are simultaneous in the A frame on clock which is at rest with A but not colocated. Any point along the sheet will do.

t_B for these two events is already shown on the t_B axis. That's why the light orange sheet spans two consecutive ticks on the t_B axis.

But I'm not sure you're clear on the fact that we always have to have a fixed idea of what two events we're talking about in advance before we can use the time dilation equation, which is comparing the time-interval between those specific events in both frames (one of which is the frame where they are colocated, which is normally labeled as the unprimed frame).

We don't really need two specific events. We can use simultaneity sheets. It might not be immediately obvious, but you can do it.

I don't follow. I'd say that application of gamma is because if the clock is emitting signals once every T_emit according to its own readings, then in your frame the time interval between signal emissions is gamma*T_emit and its distance from you increases by v*gamma*T_emit in this time, so each successive signal has to cross this extra distance which takes an extra time of v*gamma*T_emit/c in your frame, so instead of receiving the signals once every gamma*T_emit (the 'actual' time between emissions in your frame), you only receive them once every gamma*T_emit + v*gamma*T_emit/c = gamma*T_emit*(1 + v/c). So if T_obs is the time between observing successive signals from the moving clock, we get the relativistic Doppler equation T_obs = gamma*T_emit*(1 + v/c), whereas the Galilean Doppler equation would just be T_obs = T_emit*(1 + v/c).

Yep, like I said. Except I said it more quickly, because I thought you would understand. In my words:

A pure doppler effect for one observer assumed to be stationary with the other observer is in motion at v=0.6c would be: (c+0.6c)/(c+0) = 1.6. To make relativistic doppler, we apply a gamma of 1.25 to get 1.6*1.25 = 2.

The orange line is slanted in A's frame, so what do you mean by "time dilation for A" if this line goes in a mix of space and time directions in A's frame? Are you talking about the purely vertical distance between event 2 at the bottom of the orange line and event 3 at the top? If so your diagram doesn't actually show anything corresponding to this vertical distance, it'd be more clear if you drew a horizontal bar whose top and bottom edge cross these two events, so you could see the times this bar's top and bottom intersect the t_A axis (which would represent the dilated time in A's frame between events 2 and 3 on B's worldline that are colocated in B's frame). Alternatively you could just relocate the darker orange line so it goes between events 1 and 2 on B's worldline, since you already have a light orange horizontal bar there.

I would say that the horizontal length of the orange bar in the SAFTD diagram represents the spatial distance in A's frame between events 1 and 4 on the t_B axis, which are simultaneous in B's frame, and that's exactly the idea of the SAFTD...in other words, if we write SAFTD as dx' = dx * gamma, then here dx is the distance in B's frame between events 1 and 4, and dx' is the distance in A's frame between events 1 and 4 (which can also be represented by the green line). As long as that matches what you're saying above, this diagram makes sense to me.

Isn't showing where the appropriate bars cross the appropriate axis "showing it on the diagram"? For example, in the SAFTD diagram, you can see the dark orange line has a length of 3 in B's coordinate system (since it goes from x=1 to x=4 on B's space axis), but you can also see that the vertical light orange bar whose edges go through the two ends of the dark orange line intersects A's x-axis at two points that are a distance of 3*1.25 = 3.75 apart (which represents the distance in A's frame between the events on the ends of the dark orange line).

You are splitting stuff up and ignoring context again.

I said it was difficult to draw, didn't I?

Imagine that the t_B axis is a wiper fixed at the origin. Move it so it is parallel with the t_A axis. The t_B is not only rotated, but also stretched. Do you recall anyone saying that?

Keep in mind that if you draw 1-second ticks along A's time axis and 1-light second ticks along A's space axis, then also draw similar ticks along B's space and time axis, then when drawn from the perspective of A's frame, B's space and time axes are not merely rotated versions of A's, in A's frame the distances (in the diagram) between ticks on B's axes are greater than the distances between ticks on A's axes.

The extent to which the t_B axis is stretched is given by time dilation.

cheers,

neopolitan

 

[JesseM]

The light coloured sheets are simultaneity planes, if you like:
The light orange sheet is 1 tick.gamma in the A frame, spanning 1 tick in the B frame.
The light blue sheet is 1 tick.gamma in the B frame, spanning 1 tick in the A frame.

Right, that's what I figured.

If you must have events, the light orange sheet shows all the events which are simultaneous in the A frame with two consecutive ticks of B's clock, and all the events between.

Yes, and those two consecutive ticks are events 1 and 2 on B's worldline, as I suggested. In this case t_A for these events is 1.25 and t_B is 1.

The light blue sheet shows all the events which are simultaneous in the B frame with two consecutive ticks of A's clock, and all the events between. (And so on.)

Yes, here I assume the two events would be on either end of the dark blue line, the events labeled 2 and 3 on A's time axis. In A's frame t_A for these events is 1, while in B's frame t_B for these events is 1.25. Can I assume it's this pair of events, and the time between them in both frames, that you were referring to when you said "What the diagram tried to show is a value seen from the A frame (from the A frame, ticks are all colocal) such that t_B = gamma.t_A"? But in this case the time dilation occurs in B's frame, so I still don't really understand why you labeled the green line "time dilation A looking at B" rather than "B looking at A".

The purple line just shows the time between the events which are simultaneous in the A frame on clock which is at rest with A but not colocated.

They're not colocated with the clock at x=0 in the A frame, but the two events are colocated with one another in A's frame.

t_B for these two events is already shown on the t_B axis. That's why the light orange sheet spans two consecutive ticks on the t_B axis.

But the light orange sheet shows the events on B's t-axis that are simultaneous with the events at either end of the purple line in A's frame, not in B's frame. The event at the bottom of the purple line does not occur at t=1 in B's frame, and the event at the top of the frame does not occur at t=2 in B's frame. Nowhere in the diagram do you show points on B's axis that are simultaneous with the events at the top and bottom of the purple line, so the diagram just doesn't show t_B for those particular events.

Yep, like I said. Except I said it more quickly, because I thought you would understand. In my words:

A pure doppler effect for one observer assumed to be stationary with the other observer is in motion at v=0.6c would be: (c+0.6c)/(c+0) = 1.6. To make relativistic doppler, we apply a gamma of 1.25 to get 1.6*1.25 = 2.

Yes, and I did follow that part. What I didn't follow was the subsequent statement "Application of the gamma of 1.25 is time dilation and the stretching of the t_B axis." What does it mean to consider "stretching of the t_B axis" as something separate from time dilation? You can see that in my derivation of the relativistic Doppler effect I didn't consider any effect from "stretching of the t_B axis", I just made use of time dilation.

This would mean that the darker orange line is time dilation for A

The orange line is slanted in A's frame, so what do you mean by "time dilation for A" if this line goes in a mix of space and time directions in A's frame? Are you talking about the purely vertical distance between event 2 at the bottom of the orange line and event 3 at the top? If so your diagram doesn't actually show anything corresponding to this vertical distance, it'd be more clear if you drew a horizontal bar whose top and bottom edge cross these two events, so you could see the times this bar's top and bottom intersect the tA axis (which would represent the dilated time in A's frame between events 2 and 3 on B's worldline that are colocated in B's frame). Alternatively you could just relocate the darker orange line so it goes between events 1 and 2 on B's worldline, since you already have a light orange horizontal bar there.

But you can see that time dilation is such a totally different thing to Length contraction, which we have covered before, and that in the SAFTD diagrams the same would apply, the darker orange line would be SAFTD for A and the dark blue line would be SAFTD for B.

I would say that the horizontal length of the orange bar in the SAFTD diagram represents the spatial distance in A's frame between events 1 and 4 on the tB axis, which are simultaneous in B's frame, and that's exactly the idea of the SAFTD...in other words, if we write SAFTD as dx' = dx * gamma, then here dx is the distance in B's frame between events 1 and 4, and dx' is the distance in A's frame between events 1 and 4 (which can also be represented by the green line). As long as that matches what you're saying above, this diagram makes sense to me.

What you can't really do though, is measure these values directly. Even to show it on the diagram is difficult

Isn't showing where the appropriate bars cross the appropriate axis "showing it on the diagram"? For example, in the SAFTD diagram, you can see the dark orange line has a length of 3 in B's coordinate system (since it goes from x=1 to x=4 on B's space axis), but you can also see that the vertical light orange bar whose edges go through the two ends of the dark orange line intersects A's x-axis at two points that are a distance of 3*1.25 = 3.75 apart (which represents the distance in A's frame between the events on the ends of the dark orange line).

You are splitting stuff up and ignoring context again.

I said it was difficult to draw, didn't I?

Imagine that the t_B axis is a wiper fixed at the origin. Move it so it is parallel with the t_A axis. The t_B is not only rotated, but also stretched. Do you recall anyone saying that?

Yes, I understand what you mean by "stretching", but that wasn't an issue I was asking in any of the questions I asked above, if you think stretching is relevant to my questions you need to explain why. If you think I'm ignoring context, it would help if you would address the specific questions in a way that explains what context I'm missing for that particular question. I asked a bunch of questions there, so if you want we can concentrate on just one:

This would mean that the darker orange line is time dilation for A

The orange line is slanted in A's frame, so what do you mean by "time dilation for A" if this line goes in a mix of space and time directions in A's frame? Are you talking about the purely vertical distance between event 2 at the bottom of the orange line and event 3 at the top?

Answering these questions would help me understand what you mean by "the darker orange line is time dilation for A", I didn't see any context there that would help me understand this (though perhaps your last comment below shines some light on what you meant).

The extent to which the t_B axis is stretched is given by time dilation.

The axis is actually stretched by more than gamma, if you're referring to the diagonal distance on the diagram between a pair of ticks on B's time axis as compared with the vertical distance between a pair of ticks on A's time axis. If you draw it so that the vertical distance between ticks 1 and 2 on A's time axis is 1 centimeter, and then you look at ticks 1 and 2 on B's slanted time axis, then the purely vertical distance between these points is gamma*1 centimeter = 1.25 centimeter, while the horizontal distance is 0.6*1.25 centimeter = 0.75 centimeter, so the diagonal distance between these points must be sqrt(1.25^2 + 0.75^2) = 1.45774 cm. So, you can see that the extent of the stretching in the diagram is not given by the gamma factor that appears in the time dilation equation.

 

[neopolitan]

Yes, here I assume the two events would be on either end of the dark blue line, the events labeled 2 and 3 on A's time axis. In A's frame t_A for these events is 1, while in B's frame t_B for these events is 1.25. Can I assume it's this pair of events, and the time between them in both frames, that you were referring to when you said "What the diagram tried to show is a value seen from the A frame (from the A frame, ticks are all colocal) such that t_B = gamma.t_A"? But in this case the time dilation occurs in B's frame, so I still don't really understand why you labeled the green line "time dilation A looking at B" rather than "B looking at A".

I find time dilation really awkward. It seems I am not alone.

Can you step back, momentarily and just follow this and see if it is right?

Two events on the t_A axis are consecutive ticks of a clock, so t_A = 1 tick.

These two events are simultaneous in the B frame with two events on the t_B axis which are separated by t_B = 1 tick.gamma.

Assuming a gamma of 1.25, if t_A = 1s then t_B = 1.25s

Is that right?

Now the unprimed frame is that in which ticks are colocal. The primed frame in that in which ticks are not colocal.

If we take the A frame as our reference point, the A frame is the frame in which ticks are colocal and the B frame is the frame in which ticks are not colocal.

Therefore, t_A = t and t_B = t'.

Therefore t' = t_B = t_A.gamma = t.gamma, or

t' = t.gamma

Which is your time dilation equation from the A frame, considering the B frame, or "A looking at B".

They're not colocated with the clock at x=0 in the A frame, but the two events are colocated with one another in A's frame.

I thought that was staggeringly obvious, but yes.

But the light orange sheet shows the events on B's t-axis that are simultaneous with the events at either end of the purple line in A's frame, not in B's frame. The event at the bottom of the purple line does not occur at t=1 in B's frame, and the event at the top of the frame does not occur at t=2 in B's frame. Nowhere in the diagram do you show points on B's axis that are simultaneous with the events at the top and bottom of the purple line, so the diagram just doesn't show t_B for those particular events.

The purple line spans two colocated events in the A frame which are simultaneous in the A frame with two consecutive ticks of the B clock. Note that if this is an issue for you with the purple line, it should be an issue for you with the green line and any issue you have with the green line should also be an issue for you with the purple line.

Did you previously understand that? Perhaps this might clarify:

The purple/green line spans two colocated events in the A/B frame which are simultaneous in the A/B frame with two consecutive ticks of the B/A clock.

Yes, and I did follow that part. What I didn't follow was the subsequent statement "Application of the gamma of 1.25 is time dilation and the stretching of the t_B axis." What does it mean to consider "stretching of the t_B axis" as something separate from time dilation? You can see that in my derivation of the relativistic Doppler effect I didn't consider any effect from "stretching of the t_B axis", I just made use of time dilation.

Ok, this makes more sense now. I was wondering where the confusion was.

Application of the gamma of 1.25 is time dilation and the stretching of the t_B axis is also representative of time dilation. Maybe that doesn't help right now, but see my response below.

Yes, I understand what you mean by "stretching", but that wasn't an issue I was asking in any of the questions I asked above, if you think stretching is relevant to my questions you need to explain why. If you think I'm ignoring context, it would help if you would address the specific questions in a way that explains what context I'm missing for that particular question. I asked a bunch of questions there, so if you want we can concentrate on just one:

Answering these questions would help me understand what you mean by "the darker orange line is time dilation for A", I didn't see any context there that would help me understand this (though perhaps your last comment below shines some light on what you meant).

The axis is actually stretched by more than gamma, if you're referring to the diagonal distance on the diagram between a pair of ticks on B's time axis as compared with the vertical distance between a pair of ticks on A's time axis. If you draw it so that the vertical distance between ticks 1 and 2 on A's time axis is 1 centimeter, and then you look at ticks 1 and 2 on B's slanted time axis, then the purely vertical distance between these points is gamma*1 centimeter = 1.25 centimeter, while the horizontal distance is 0.6*1.25 centimeter = 0.75 centimeter, so the diagonal distance between these points must be sqrt(1.25^2 + 0.75^2) = 1.45774 cm. So, you can see that the extent of the stretching in the diagram is not given by the gamma factor that appears in the time dilation equation.

I know you had a specific question, but it is sort of off track, so perhaps answering it will cause more confusion that trying to address the core issue.

In your last paragraph above, you make an incorrect assumption. Note that I didn't say that time dilation accounts for stretching of the t_A that would be required to match the t_B axis.

I fully admit that I wasn't being entirely clear, but I did say:

The extent to which the t_B axis is stretched is given by time dilation.

I have done another couple of diagrams, showing Galilean relativity. These diagrams kill two birds with one stone, since they hopefully show that maybe I was right about instantaneous transmission of information being an issue - keeping in mind that the Galilean boost is:

x' = x - vt
(not x' = x - vt')

They are http://www.geocities.com/neopolitonian/gal_rel_JesseM.jpg".

That aside, look at the separation between the ticks of the B clock in Galilean relativity (either diagram). If the t_B axis is imagined as a wiper and swung around to line up with the t_A axis then the ticks are further apart. Now go back to the original time dilation diagram and keep in mind the comment about the t_B axis being stretched.

cheers,

neopolitan

PS This was written yesterday when the system wasn't letting me post because of a disc failure. I think I finished editing it, but it is possible that it is not quite the finished product I wanted it to be.

 

[JesseM]

I find time dilation really awkward. It seems I am not alone.

Can you step back, momentarily and just follow this and see if it is right?

Two events on the t_A axis are consecutive ticks of a clock, so t_A = 1 tick.

These two events are simultaneous in the B frame with two events on the t_B axis which are separated by t_B = 1 tick.gamma.

Assuming a gamma of 1.25, if t_A = 1s then t_B = 1.25s

Is that right?

Yeah, all that is right. But if you're drawing things from the perspective of the A frame, you may find it less awkward to think about ticks on a clock which is moving in the A frame, and the time between them in the A frame (switching all the A's and B's in your explanation above). If two events on the B worldline are separated by 1 second in B's frame, then the time between these events in A's frame--which is just the vertical distance between the events in the diagram--is 1.25 seconds.

Now the unprimed frame is that in which ticks are colocal. The primed frame in that in which ticks are not colocal.

If we take the A frame as our reference point, the A frame is the frame in which ticks are colocal and the B frame is the frame in which ticks are not colocal.

Therefore, t_A = t and t_B = t'.

Therefore t' = t_B = t_A.gamma = t.gamma, or

t' = t.gamma

Which is your time dilation equation from the A frame, considering the B frame, or "A looking at B".

Your equations are right but I still just don't get the phrasing...why do you say the time dilation is "from the A frame" when the actual dilation (greater amount of time) occurs in the B frame in your example? You're taking two events on A's worldline which have a separation of 1 second in A's frame, then using that to figure out the time between these same events in the B frame, namely 1.25 seconds. So aren't you figuring out what B observing when it's "looking at" A's clock, and observing that A's ticks seem to be dilated by a factor of 1.25?

Like I said, I think if you're going to draw things in A's frame, it's much more natural to think about time dilation if you pick two events on B's worldline and then understand the dilated time to be the vertical distance between these events in the diagram of A's frame (like events 1 and 2 on B's time axis in your diagram, where the vertical distance is shown by the purple line). In this case t_B is the frame where the events are colocated and t_A is the frame where they're not, so you still have the equation t' = t*gamma, but now t is the time interval in B's frame and t' is the time interval in A's frame. It's this that I would call "the time dilation in A's frame", since A's time t' is dilated relative to B's time t, although the meaning of such words is ambiguous and perhaps you find it more natural to define them differently.

The purple line spans two colocated events in the A frame which are simultaneous in the A frame with two consecutive ticks of the B clock.

If that's the intended meaning of the purple line that's fine with me, although visually I think it would be more clear that the purple line is supposed to relate to those specific ticks (ticks 1 and 2) on B's worldline if the dark orange line went between those ticks rather than two other ticks. In that case the relation between the vertical purple line and the slanted dark orange line in the time dilation illustration would be exactly analogous to the relation between the horizontal green line and the slanted dark orange line in the SAFTD illustration. But strangely you label the horizontal green line "A looking at B" while you label the vertical purple line "B looking at A"--whatever "looking at" is supposed to mean, you're using it inconsistently here, because the SAFTD is supposed to be exactly analogous to time dilation but with the roles of space and time reversed.

Note that if this is an issue for you with the purple line, it should be an issue for you with the green line and any issue you have with the green line should also be an issue for you with the purple line.

But in the case of the green line, the diagram makes it clear it's supposed to relate to the same to events spanned by the dark blue line.

I know you had a specific question, but it is sort of off track, so perhaps answering it will cause more confusion that trying to address the core issue.

In your last paragraph above, you make an incorrect assumption. Note that I didn't say that time dilation accounts for stretching of the t_A that would be required to match the t_B axis.

I fully admit that I wasn't being entirely clear, but I did say:

The extent to which the tB axis is stretched is given by time dilation.

I done another couple of diagrams, showing Galilean relativity. These diagrams kill two birds with one stone, since they hopefully show that maybe I was right about instantaneous transmission of information being an issue - keeping in mind that the Galilean boost is:

x' = x - vt
(not x' = x - vt')

They are http://www.geocities.com/neopolitonian/gal_rel_JesseM.jpg".

I think you need some other label for the first diagram besides Galilean relativity where t does not equal t', since in the Galilean boost (i.e. the Galilei transformation) t does equal t', and in the first diagram x' does not equal x - vt either (except in the special case of events along B's time axis of x'=0), so the first diagram would represent some other coordinate transformation that is neither Galilean nor Lorentzian. The second diagram does correctly show two coordinate systems related by the Galilei transformation though. But going back to the issue of what you meant by "The extent to which the t_B axis is stretched is given by time dilation", are you saying that since the diagonal distance between ticks on B's axes already appear visually stretched in these diagrams (though of course the vertical distance between ticks on B's time axis is the same as the vertical distance between ticks on A's time axis in both diagrams, and the horizontal distance between ticks on B's space axis is the same as the horizontal distance between ticks on A's space axis in the first diagram), we can apply gamma to this preexisting stretching to get the amount of diagonal stretching seen on the diagram of the Lorentz transform?

 

[neopolitan]

Yeah, all that is right. But if you're drawing things from the perspective of the A frame, you may find it less awkward to think about ticks on a clock which is moving in the A frame, and the time between them in the A frame (switching all the A's and B's in your explanation above). If two events on the B worldline are separated by 1 second in B's frame, then the time between these events in A's frame--which is just the vertical distance between the events in the diagram--is 1.25 seconds.

The B clock is moving in the A frame. So ticks on the t_B axis are "ticks on a clock which is moving in the A frame" which should make you happy with http://www.geocities.com/neopolitonian/TDv2.JPG".

Your equations are right but I still just don't get the phrasing...why do you say the time dilation is "from the A frame" when the actual dilation (greater amount of time) occurs in the B frame in your example? You're taking two events on A's worldline which have a separation of 1 second in A's frame, then using that to figure out the time between these same events in the B frame, namely 1.25 seconds. So aren't you figuring out what B observing when it's "looking at" A's clock, and observing that A's ticks seem to be dilated by a factor of 1.25?

Sort of, yes. But that is the confusing thing with time dilation. In which frame is the clock moving? Work from the A frame, in which the A clock is at rest, which means that t_A = t_unprimed and t_B = t_primed, or more simply t_A = t and t_B = t'.

You want to me go from B's frame, so that means the B clock is the one at rest and the A clock is in motion, which means that t_B = t and t_A = t'. Then, if you want me to use the same sheet as I was using (in this http://www.geocities.com/neopolitonian/TD.JPG"), ie the light blue one which:

when it intersects the t_A axis spans 1 tick, therefore t' = 1 tick
and
when it intersects the t_B axis spans 1 tick.gamma, therefore t = 1 tick.gamma

therefore:

t=t'.gamma

Which is inverse time dilation and is not what I am trying to show. I don't think you want me to use the blue sheet as it is.

I think you want me to use the orange sheet or something like it as you go on to explain ...

Like I said, I think if you're going to draw things in A's frame, it's much more natural to think about time dilation if you pick two events on B's worldline and then understand the dilated time to be the vertical distance between these events in the diagram of A's frame (like events 1 and 2 on B's time axis in your diagram, where the vertical distance is shown by the purple line). In this case t_B is the frame where the events are colocated and t_A is the frame where they're not, so you still have the equation t' = t*gamma, but now t is the time interval in B's frame and t' is the time interval in A's frame. It's this that I would call "the time dilation in A's frame", since A's time t' is dilated relative to B's time t, although the meaning of such words is ambiguous and perhaps you find it more natural to define them differently.

But isn't that just what the light orange sheet shows?

Do you still not understand that the light orange sheet is the skewed rotation of the light blue sheet and the purple line is the skewed rotation of the green line, and that if you recast the whole thing so that the t_B and x_B axes were square, then you would have a (sort of) mirror image of this diagram? (Mirrored in the sense that the t^A axis would have a negative slope, not mirrored in the sense that the green photon worldlines would not also be mirrored.)

I'm asking this straight out, because I just don't understand what your problem is. You seem to be saying "you are doing this completely wrong with the blue sheet, look do it this way" and then you go and do what I have done with the orange sheet, but to do that you have to change all the terminology used with the blue sheet, and lo and behold, you have the terminology that I used with the orange sheet.

Try this. Start off the way you like it. Use the orange sheet. Then, try to see how it would look if you flipped frames - and how you would draw it on the same diagram. If it is different from the way it looks on my blue sheet, then we have something to talk about. Otherwise, I really think you are at cross purposes.

If that's the intended meaning of the purple line that's fine with me, although visually I think it would be more clear that the purple line is supposed to relate to those specific ticks (ticks 1 and 2) on B's worldline if the dark orange line went between those ticks rather than two other ticks. In that case the relation between the vertical purple line and the slanted dark orange line in the time dilation illustration would be exactly analogous to the relation between the horizontal green line and the slanted dark orange line in the SAFTD illustration. But strangely you label the horizontal green line "A looking at B" while you label the vertical purple line "B looking at A"--whatever "looking at" is supposed to mean, you're using it inconsistently here, because the SAFTD is supposed to be exactly analogous to time dilation but with the roles of space and time reversed.

My error. It's fixed.

I think you need some other label for the first diagram besides Galilean relativity where t does not equal t', since in the Galilean boost (i.e. the Galilei transformation) t does equal t', and in the first diagram x' does not equal x - vt either (except in the special case of events along B's time axis of x'=0), so the first diagram would represent some other coordinate transformation that is neither Galilean nor Lorentzian. The second diagram does correctly show two coordinate systems related by the Galilei transformation though.

Jesse, it's my whole point ie "in the Galilean boost (i.e. the Galilei transformation) t does equal t'". That diagram shows Galilean relativity as you tried to tell me it would work, even if information is not transmitted instantaneously. That's why that diagram's file is called "gal_rel_JesseM.jpg".

Note that in the diagram I don't have x' = x - vt. I have x' = x - vt'. Given that, do you agree that t' does not equal t?

But going back to the issue of what you meant by "The extent to which the t_B axis is stretched is given by time dilation", are you saying that since the diagonal distance between ticks on B's axes already appear visually stretched in these diagrams (though of course the vertical distance between ticks on B's time axis is the same as the vertical distance between ticks on A's time axis in both diagrams, and the horizontal distance between ticks on B's space axis is the same as the horizontal distance between ticks on A's space axis in the first diagram), we can apply gamma to this preexisting stretching to get the amount of diagonal stretching seen on the diagram of the Lorentz transform?

Yes (it is the stretching shown in diagrams TD, LC, SAFTD and TAFLC which was calculated using the Lorentz transformations).

cheers,

neopolitan

 

[JesseM]

Yeah, all that is right. But if you're drawing things from the perspective of the A frame, you may find it less awkward to think about ticks on a clock which is moving in the A frame

The B clock is moving in the A frame.

But you weren't talking about ticks on the B clock when you said "Two events on the t_A axis are consecutive ticks of a clock, so t_A = 1 tick", you were talking about ticks on the A clock, and figuring out how far apart they were in the B frame. There's nothing wrong with doing that of course, I was just saying that if you want to think about time dilation in a diagram drawn from the A frame perspective, it's conceptually easier to think about two ticks on the B clock which is moving in this frame, and then the dilated time between these ticks in the A frame is just the vertical distance between where the ticks are drawn in the diagram.

So ticks on the t_B axis are "ticks on a clock which is moving in the A frame" which should make you happy with http://www.geocities.com/neopolitonian/TDv2.JPG".

Yes, I'm quite happy with that diagram (although to make it perfect you'd need to extend the bottom of the light orange and light blue bars to make them line up better with the bottom of the dark orange line segment and the dark blue line segment...not really important because I understand the intent though).

Your equations are right but I still just don't get the phrasing...why do you say the time dilation is "from the A frame" when the actual dilation (greater amount of time) occurs in the B frame in your example? You're taking two events on A's worldline which have a separation of 1 second in A's frame, then using that to figure out the time between these same events in the B frame, namely 1.25 seconds. So aren't you figuring out what B observing when it's "looking at" A's clock, and observing that A's ticks seem to be dilated by a factor of 1.25?

Sort of, yes. But that is the confusing thing with time dilation. In which frame is the clock moving? Work from the A frame, in which the A clock is at rest, which means that t_A = t_unprimed and t_B = t_primed, or more simply t_A = t and t_B = t'.

Well, only if you're using t_A to represent the time in the A frame between two events on A's own clock. I was suggesting you use t_A to represent the time in the A frame between two events on B's clock, in which case t_A = t_primed.

You want to me go from B's frame

No I don't, I want you to use two events on B's clock but figure out the coordinate time between these events in A's frame. This is the most natural way to think about time dilation--whatever frame you're using (A in this case), it tells you how much the time between events on a clock moving relative to that frame is dilated in that frame, if you know how much proper time the clock itself measured between them. In this case, if there are two events on B's clock which the clock itself measured as being separated by 1 second, in A's frame these same two events are separated by a dilated coordinate time of 1.25 seconds.

It's always simplest to think about time dilation in terms of a relation between a proper time on a clock and a coordinate time in a frame where the clock is moving, I think. If you try to think about two different clocks at rest in different frames, and lines of simultaneity connecting events on one clock with events on another, it gets way more confusing.

so that means the B clock is the one at rest and the A clock is in motion, which means that t_B = t and t_A = t'. Then, if you want me to use the same sheet as I was using (in this http://www.geocities.com/neopolitonian/TD.JPG"), ie the light blue one which:

when it intersects the t_A axis spans 1 tick, therefore t' = 1 tick
and
when it intersects the t_B axis spans 1 tick.gamma, therefore t = 1 tick.gamma

If you're using the events at either end of the dark blue line segment, then these events are colocated in the A frame where the time between them is 1 second, so that should be the unprimed frame, while the primed frame should be the B frame where they are not colocated and the coordinate time between them is 1*gamma (the height of the light blue sheet in the B frame).

therefore:

t=t'.gamma

Which is inverse time dilation and is not what I am trying to show. I don't think you want me to use the blue sheet as it is.

See above, you weren't sticking to the convention that the unprimed frame is the one where the events are colocated.

I think you want me to use the orange sheet or something like it as you go on to explain ...

Like I said, I think if you're going to draw things in A's frame, it's much more natural to think about time dilation if you pick two events on B's worldline and then understand the dilated time to be the vertical distance between these events in the diagram of A's frame (like events 1 and 2 on B's time axis in your diagram, where the vertical distance is shown by the purple line). In this case tB is the frame where the events are colocated and tA is the frame where they're not, so you still have the equation t' = t*gamma, but now t is the time interval in B's frame and t' is the time interval in A's frame. It's this that I would call "the time dilation in A's frame", since A's time t' is dilated relative to B's time t, although the meaning of such words is ambiguous and perhaps you find it more natural to define them differently.

But isn't that just what the light orange sheet shows?

Sure, but in the above paragraph I was responding to the verbal analysis in your previous post, which was specifically about two events on the worldline of A's clock and the time between them in A's frame and B's frame. Again, there was nothing wrong with this analysis, but I was just suggesting time dilation is easier to understand intuitively if you pick one frame (frame A in your diagram) and then consider events on a clock which is moving in that frame, the time dilation equation relating the proper time as measured by the clock to the coordinate time as measured in the frame you're using (which in this frame will just be the vertical distance between the two events).

Do you still not understand that the light orange sheet is the skewed rotation of the light blue sheet and the purple line is the skewed rotation of the green line

Of course I understand that.

I'm asking this straight out, because I just don't understand what your problem is. You seem to be saying "you are doing this completely wrong with the blue sheet, look do it this way" and then you go and do what I have done with the orange sheet, but to do that you have to change all the terminology used with the blue sheet, and lo and behold, you have the terminology that I used with the orange sheet.

Where did I say you were "completely wrong" about anything? The main issue I brought up was that I thought the words you used to describe segments, like "A looking at B", seemed backwards to me from how I would naturally interpret them (and also inconsistent with how you used the same words in the SAFTD diagram). The other minor issue I brought up was that if the light orange bar was showing the time in A's frame between events 1 and 2 on B's clock, the diagram would be more clear if the dark orange line segment went between events 1 and 2, as you have it in the new diagram here.

Jesse, it's my whole point ie "in the Galilean boost (i.e. the Galilei transformation) t does equal t'". That diagram shows Galilean relativity as you tried to tell me it would work, even if information is not transmitted instantaneously. That's why that diagram's file is called "gal_rel_JesseM.jpg".

You misunderstood me, my point about the speed of information transmission had nothing to do with suggesting a new type of coordinate transformation different from the standard Galilei transformation. My point was that I didn't understand why you thought the standard Galilei transformation implied instantaneous information transmission in the first place! I was saying that it would still make sense to use the standard Galilei transformation in a universe with basically Newtonian laws but where information transmission had some finite upper limit--why wouldn't it? As long as the laws of physics are such that moving clocks don't slow down and moving rulers don't shrink, then if different observers construct their coordinate systems using a lattice of inertial rulers and clocks at rest relative to themselves, and different observers use a method of synchronizing clocks that won't cause different frames to disagree about simultaneity (like bringing the clocks together to a single location and synchronizing them there, then moving them apart to their respective positions in the lattice, which unlike in relativity won't cause them to get-out-of-sync because there is no time dilation associated with movement), then the resulting coordinate systems will be related by the standard Galilei transformation. Nothing here depends on whether information is transmitted instantaneously or at finite speed, it's an irrelevant issue.

Note that in the diagram I don't have x' = x - vt. I have x' = x - vt'. Given that, do you agree that t' does not equal t?

Yeah, but in my last post I wasn't disagreeing with you that t doesn't equal t' in your first diagram, I was just saying you shouldn't use the term Galilean relativity where t does not equal t' to refer to it because this coordinate transformation isn't "Galilean" at all.

 

[neopolitan]

Yes, I'm quite happy with that diagram (although to make it perfect you'd need to extend the bottom of the light orange and light blue bars to make them line up better with the bottom of the dark orange line segment and the dark blue line segment...not really important because I understand the intent though).

Then you effectively end up with what I started with. I thought I would have to tell you that what is already is what I started off with, even without your suggested modification.

Take the light orange sheet from http://www.geocities.com/neopolitonian/TDv2.JPG", make it wider to span consecutive ticks on the t_B axis.

Move the light orange sheet down so it spans t_B=1 to t_B=2

Then put a purple bar on the t_A axis where the light orange sheet crosses it. Then to make it visually clearer (so you don't have bars lying over each other), move the purple bar to x_A=6.

Then take the light blue sheet and make it wider to span consecutive ticks on the t_A axis and move it down slight so it spans t_A=2 to t_A=3.

Then put a green bar on the t_B axis where the light blue sheet crosses it. Then to make it visually clearer, move the green bar along the light blue sheet until it aligns with x_B=2.

Can you see now?

If you are happy with http://www.geocities.com/neopolitonian/TDv2.JPG". Because they are the same.

Unless you can see this, then we will have to agree to disagree.

You misunderstood me, my point about the speed of information transmission had nothing to do with suggesting a new type of coordinate transformation different from the standard Galilei transformation. My point was that I didn't understand why you thought the standard Galilei transformation implied instantaneous information transmission in the first place! I was saying that it would still make sense to use the standard Galilei transformation in a universe with basically Newtonian laws but where information transmission had some finite upper limit--why wouldn't it? As long as the laws of physics are such that moving clocks don't slow down and moving rulers don't shrink, then if different observers construct their coordinate systems using a lattice of inertial rulers and clocks at rest relative to themselves, and different observers use a method of synchronizing clocks that won't cause different frames to disagree about simultaneity (like bringing the clocks together to a single location and synchronizing them there, then moving them apart to their respective positions in the lattice, which unlike in relativity won't cause them to get-out-of-sync because there is no time dilation associated with movement), then the resulting coordinate systems will be related by the standard Galilei transformation. Nothing here depends on whether information is transmitted instantaneously or at finite speed, it's an irrelevant issue.

Perhaps we disagree about what is relevant and irrelevant.

To see the equation x' = x - vt, in "my" Galilean relativity, which we seem to both agree is Galilean relativity rather than the other one which I ascribed to you, it requires either:

instant understanding of where A is and B and the event is at 5 ticks, rather than waiting until 8 ticks when a photon from E reaches A then working backwards to see what happened when that same photon passed B to get a similar equation,

a god like observer (but really that god like observer sees everything instantly, so you are back at square one) and the god like observer implies a preferred frame (part of the perks of being a god).

Yeah, but in my last post I wasn't disagreeing with you that t doesn't equal t' in your first diagram, I was just saying you shouldn't use the term Galilean relativity where t does not equal t' to refer to it because this coordinate transformation isn't "Galilean" at all.

I don't what you were going on about a few posts ago then (post #179). Perhaps you can show a diagram in which you have an aether frame which works with Galilean relativity and you get the right equation - and you show it complete with photons and no hint of instantaneous transmission of information.

cheers,

neopolitan

 

[JesseM]

Then you effectively end up with what I started with. I thought I would have to tell you that what is already is what I started off with, even without your suggested modification.

Take the light orange sheet from http://www.geocities.com/neopolitonian/TDv2.JPG", make it wider to span consecutive ticks on the t_B axis.

Wait, so it was intentional that the light orange bar doesn't already span from one end of the dark orange line segment to the other? In that case I'm not very happy with the diagram, I thought that was just an accident of your drawing, which is why I said "(although to make it perfect you'd need to extend the bottom of the light orange and light blue bars to make them line up better with the bottom of the dark orange line segment and the dark blue line segment...not really important because I understand the intent though)". I thought the point of the light orange bar was that its vertical height showed the dilated time in the A frame between the two events on the dark orange bar which had a separation of 1 second on B's clock.

Move the light orange sheet down so it spans t_B=1 to t_B=2

Then put a purple bar on the t_A axis where the light orange sheet crosses it. Then to make it visually clearer (so you don't have bars lying over each other), move the purple bar to x_A=6.

But if you did that, can you also move the dark orange line segment down, so that it spans the same two ticks that are spanned by the light orange bar? Remember, that was my only visual criticism of your original diagram, that it was confusing that the dark orange line segment spanned two different ticks than the light orange bar, so the light orange bar wasn't clearly indicated as illustrating the dilated time between the two ends of the dark orange line segment (and hence the purple line segment wasn't clearly indicated as illustrating this either).

Then take the light blue sheet and make it wider to span consecutive ticks on the t_A axis and move it down slight so it spans t_A=2 to t_A=3.

Again, this is what I thought your intention was, for the light blue slanted bar to span from one end of the dark blue line segment to the other. If not then again I take back what I said about being happy with the diagram, the point of each light-colored bar should be to show the dilated time in one frame between points on the worldline of a clock which is at rest in the other frame, since again it is easiest to conceptualize time dilation as relating the proper time between events on a clock to the coordinate time between the same events in the frame where the clock is moving.

Then put a green bar on the t_B axis where the light blue sheet crosses it. Then to make it visually clearer, move the green bar along the light blue sheet until it aligns with x_B=2.

Can you see now?

Can I see what, exactly? You talk as though there was something about the original diagram I didn't "see". Again, my main criticism of the original was just about the "A looking at B" terminology, and the other minor criticism was that I thought the dark orange line segment should be moved down so it was spanned by the light orange bar, just as the light blue bar spanned the dark blue line segment (that way each bar is clearly showing the time interval in one frame between events on the worldline of a clock at rest in the other frame)

Perhaps we disagree about what is relevant and irrelevant.

To see the equation x' = x - vt, in "my" Galilean relativity, which we seem to both agree is Galilean relativity rather than the other one which I ascribed to you, it requires either:

instant understanding of where A is and B and the event is at 5 ticks, rather than waiting until 8 ticks when a photon from E reaches A then working backwards to see what happened when that same photon passed B to get a similar equation,

a god like observer (but really that god like observer sees everything instantly, so you are back at square one) and the god like observer implies a preferred frame (part of the perks of being a god).

I really don't understand why you think standard Galilean relativity "requires" this, since the Galilie transformation is about relating the coordinates that two observers assign to events, not about when they "know" the events occurred. The same is true of the Lorentz transform! In SR if an event E occurs 10 light-seconds away from my position and I assign it the coordinate time t=5, same as the coordinate time of my own clock reading 5 seconds, you understand that doesn't mean I actually instantaneously know about the event as soon as my clock reads 5 seconds, right? The light from this event can't actually reach me until t=15, so I won't know it occurred until then. But if I use a lattice of rulers and synchronized clocks to define my coordinate system, then when the light does reach me at t=15, I'll be able to see that the event occurred next to the 10 light-second mark on my rulers, and that the clock sitting at that mark (which is synchronized with mine) read 5 seconds as the event happened. So, in retrospect I assign this event the position coordinate x=10 and time coordinate t=5, even though I myself was totally oblivious to the existence of the event prior to t=15.

Well, it would be exactly the same in a Galilean universe with a finite speed of information transmission. Each observer would define their own coordinate system using a network of rulers and synchronized clocks at rest relative to themselves, and the coordinates of distant events would be determined in retrospect based on what ruler-marking and clock-reading the event was seen to be next to when it had happened. The difference in Galilean relativity would be that 1) one observer's rulers wouldn't be measured to be shrunk relative to another observer's if the first was moving relative to the second, 2) one observer's clock wouldn't be measured to be slowed down relative to another's if the first was moving relative to the second, and 3) different observers wouldn't disagree about simultaneity, because they could all agree to synchronize their own clocks by bringing them together to a common location and synchronizing them there before moving them to their respective positions on the lattice. As long as 1-3 are true, then naturally the coordinates that different observers assign to the same event will be related by the Galilei transform.

I don't what you were going on about a few posts ago then (post #179). Perhaps you can show a diagram in which you have an aether frame which works with Galilean relativity and you get the right equation - and you show it complete with photons and no hint of instantaneous transmission of information.

Since the coordinates of different observers would be related by the Galilei transform, different coordinate axes would be drawn just as in this diagram. If you want to show a finite speed of information transmission in this diagram, just draw light cones emanating from events. The time that an observer learns about a given event will be represented by the time his worldline crosses into the future light cone of the event, which will be later than the time the event actually occurred--just as in SR. The only difference with SR is that if light moves at the same speed in both directions in one preferred frame, so both sides of the light cone are at the same angle from the horizontal in the diagram for this frame, then in other frames light would move at different speeds in different directions so the two sides of the light cone would have different angles if you drew things from the perspective of other frames.

 

[neopolitan]

Wait, so it was intentional that the light orange bar doesn't already span from one end of the dark orange line segment to the other?

We are both bright enough to know that it is just dividing through by gamma, right? It doesn't change the equations.

We do know that don't we?

Remember, that was my only visual criticism of your original diagram, that it was confusing that the dark orange line segment spanned two different ticks than the light orange bar, so the light orange bar wasn't clearly indicated as illustrating the dilated time between the two ends of the dark orange line segment (and hence the purple line segment wasn't clearly indicated as illustrating this either).

Can I see what, exactly? You talk as though there was something about the original diagram I didn't "see". Again, my main criticism of the original was just about the "A looking at B" terminology, and the other minor criticism was that I thought the dark orange line segment should be moved down so it was spanned by the light orange bar, just as the light blue bar spanned the dark blue line segment (that way each bar is clearly showing the time interval in one frame between events on the worldline of a clock at rest in the other frame)

So if, in the first time dilation diagram, if I moved the dark orange bar down slightly so that it spanned a specific pair of events, you'd be happier?

It wasn't overly important to me, the axis bars were just showing a tick, not any specific tick. But I see what you mean.

The odd thing is that you are happy with the same terminology as I have in the original diagram being used the second. http://www.geocities.com/neopolitonian/TD.JPG".

I need to think more about the Galilean - Lorentz thing, but I am basically talked around.

What leaves me a little stumped is ... it worked. So, I need to see what it is that makes it work.

cheers,

neopolitan

 

[JesseM]

We are both bright enough to know that it is just dividing through by gamma, right? It doesn't change the equations.

When you say "it is just dividing through by gamma", what does "it" refer to? What quantity are you dividing by gamma to get what other quantity? Do you mean that if we look at the horizontal orange bar spanned between the top of the dark orange slanted line segment, then if you take the height of the orange bar and divide by gamma, you get the proper time between the ends of the dark orange segment? If so I agree of course, but if that's not what you were referring to could you clarify?

So if, in the first time dilation diagram, if I moved the dark orange bar down slightly so that it spanned a specific pair of events, you'd be happier?

By "dark orange bar" you mean the slanted line segment rather than the thick horizontal orange bar, right? And yeah, if you moved the line segment down so that its top and bottom coincided with the upper and lower edges of the horizontal orange bar, I think the visual relation between them would be clearer.

The odd thing is that you are happy with the same terminology as I have in the original diagram being used the second. http://www.geocities.com/neopolitonian/TD.JPG".

I admit I didn't actually think about the terminology in the second one, I was just saying I was happy with what was being shown visually.

I need to think more about the Galilean - Lorentz thing, but I am basically talked around.

What leaves me a little stumped is ... it worked. So, I need to see what it is that makes it work.

When you say "it worked", as before I'm not quite sure what the "it" refers to...did you do some sort of calculation to check what I was saying?

 

[JesseM]

Responding also to your older post #181:

You want to reintroduce an aether?

Not in real life, but then I don't want to reintroduce Galilei-symmetric physics in real life either, I was just making the point that the Galilei transformation is not incompatible with a finite speed of information transmission. But I think we cleared this up in the most recent posts.

Really, I am just going from the Galilean boost to Lorentz Transformations though. That boost is given by x'=x-vt. Do we at least agree on that?

Yes, that's the spatial component of the boost, and of course the temporal part is t'=t.

The Galilean assumption, in terms of my diagram, is that B is moving with an absolute velocity of v towards a location E which is a distance of x from A and, at a time t, the distance from B to E is x' = x - vt. This means that when t=0, A and B were colocated. Do we agree on that?

I assume x' means the distance of B from E, so that's fine.

In Galilean relativity, at t, A has not moved, B is moving with a velocity of v and is located vt closer to E than A is. Do we agree on that?

A has not moved in the A frame, although it has moved in the B frame (there is no need to assume absolute space in Galilean relativity, although of course many classical physicists did believe in absolute space). Of course in the B frame it's E that's moving towards B, but either way, it's true that at time t (or t') B will be vt (or vt') closer to E than A is.

In Galilean relativity, we could have an event at E, (x,t) in A's frame and (x',t) in B's frame. Do we agree on that?

Yes, and the Galilei transformation is telling us that if an event has coordinates x,t in A's frame, it has coordinates x'=x-vt, t'=t in B's frame.

In Galilean relativity, we could have an event at E, (x,t) in A's frame and (x',t') in B's frame, because time is absolute and t=t'. Do we agree on that?

The first part just looks like a repeat of the previous paragraph, but why do you say "because time is absolute and t=t' "? It's certainly true that t=t', but I don't see how the fact that an event has coordinates x,t in A's frame and x',t' in B's frame is "because" of this.

In Galilean relativity, if B is told that E is currently x' away, and B has observed that A has been moving away at -vt, then B will calculate that A-E is currently x = x' + vt . Do we agree on that?

Yes, if an event occurs at time t' on E's worldline at position x' in B's frame, then the Galilei transformation tells us that the same event must occur at position x = x' + vt' in A's frame, and since t'=t this is also at position x = x' + vt.

Do we further agree that if an event took place at (x,t) in A's frame in Special Relativity and even in a more careful analysis of Galilean relativity, that neither A nor B would know about it until a photon from the event is received?

Yes, but in SR the time-coordinate they learned about the event would be different than tht time-coordinate they retroactively assigned to the event itself, as discussed before.

If x = ct, in Galilean relativity, when A receives the photon at 2t, x' = x - 2vt. Do we agree that if we now talk about where a photon from the same event (x,t) hits B, this is not x' as calculated above?

But x' = x - vt relates the coordinates of a single event in two frames. The event of a photon hitting B is different from the event of a photon hitting A, so we would not expect the x,t coordinates of the photon hitting A and the x',t' coordinates of the photon hitting B to be related by the Galilei transformation.

I guess I could agree that Galilean relativity is based on either absolute space (ie there's an aether frame) or instantaneous transmission of information. Can you agree that it is one or the other?

I don't see why, can you explain? If by "aether frame" you just mean there's one frame where the upper limit on the speed of information is the same in both directions, while in other frames it's different in one direction than the other, then I'd agree with that, but I'm not sure that's what you meant.

Can you see that if information is transmitted instantaneously and an event takes place at (x,t) in the A frame, then in the A frame that event will be detected by A at (0,t) and B at (vt,t)? And in the B frame, the event was at (x',t), B detects is at (0,t) and A at (-vt,t) where x'=x-vt? And can you see that these can all be related by the Galilean boosts?

Sure, but this is trivially true since the coordinates of any event in two different frames are related by the Galilei boosts if you're dealing with Galilean frames, that's the very definition of what the Galilei transformation is supposed to do! If information is transmitted at c in the -x direction of A's frame (which means it's transmitted at c+v in the -x' direction of B's frame, based on Galilean velocity addition), then we can also figure out when A and B will receive the information in both frames, and the coordinates of each event in the two frames are also related by the Galilei transformation.

In this diagram G to E 02, I take it t refers to the time in A's frame the light reached A, and t' refers to the time in B's frame the light reached B? If so it also seems that x refers to the position of the photon at t=0 in A's frame, while x' refers to the position of the same photon at t'=0 in B's frame (because of the relativity of simultaneity these must refer to different events on the photon's worldline). So in each frame you're calculating the distance and time between a totally different pair of events, correct?

One photon. One event spawning the photon. Two events where the photon passes B, then A. One event when A and B were colocated and t=0 and t'=0.

A thinks that at colocation, the photon was located at x=ct.

B thinks that at colocation, the photon was located at x'=ct'.

What is the relationship between x' and x, and t and t'?

But x and x' refer to the coordinates of two different events on the photon's worldline in this case, at least if we're dealing with relativity (and we must be if both observers say the photon is traveling at c). The event E_A on the photon's worldline that is simultaneous in A's frame with A&B being colocated is different from the event E_B on the photon's worldline that is simultaneous in B's frame with A&B being colocated, due to the relativity of simultaneity.

Do you agree that x refers to the position coordinate (in frame A) of one event E_A and x' refers to the position coordinate (in frame B) of a different event E_B? And do you agree that in the spatial component of the Lorentz transform, x' = gamma*(x - vt), x and x' are supposed to refer to the position coordinates of a single event in two different frames?

What we do know is that, irrespective of coordinate system, when A and B were colocated, the photon had one unique location. Correct?

Not if simultaneity is relative! And if you want to say that both observers measure the speed of photons to be c regardless of direction, it has to be (as demonstrated by Einstein's train/embankment thought-experiment).

 

[neopolitan]

When you say "it is just dividing through by gamma", what does "it" refer to? What quantity are you dividing by gamma to get what other quantity? Do you mean that if we look at the horizontal orange bar spanned between the top of the dark orange slanted line segment, then if you take the height of the orange bar and divide by gamma, you get the proper time between the ends of the dark orange segment? If so I agree of course, but if that's not what you were referring to could you clarify?

The only real difference between TD.JPG and TDv2.JPG is that the light blue and light orange sheets are wider in one than the other.

The relationship between the widths of those sheets remains the same and the relationship between the width of the sheets and what they span on the t-axes remains the same.

The factor relating the sheets in TD.JPG and the sheets in TDv2.JPG is gamma, ie the TD.JPG sheets are gamma wider than those in TDv2.JPG. The dark green bar in TD.JPG is gamma longer than the dark orange bar in both. The purple bar in TD.JPG is gamma longer than the dark blue bar in both.

By "dark orange bar" you mean the slanted line segment rather than the thick horizontal orange bar, right? And yeah, if you moved the line segment down so that its top and bottom coincided with the upper and lower edges of the horizontal orange bar, I think the visual relation between them would be clearer.

Fixed. A similar issue in TAFLC is also fixed.

When you say "it worked", as before I'm not quite sure what the "it" refers to...did you do some sort of calculation to check what I was saying?

"It" refers to the derivation I showed for going from Galilean boosts to Lorentz Transformations, which are in part based on removing the assumption of instantaneous transmission of information. It was done back https://www.physicsforums.com/showpost.php?p=2165684&postcount=174".

cheers,

neopolitan

 

[neopolitan]

The first part just looks like a repeat of the previous paragraph, but why do you say "because time is absolute and t=t' "? It's certainly true that t=t', but I don't see how the fact that an event has coordinates x,t in A's frame and x',t' in B's frame is "because" of this.

It's not a repeat, because in the second I say (x,t) in A and (x',t') in B, whereas in the first it is (x',t) (unprimed t) in B. I could reduce " because time is absolute and t=t' " to "because t=t' ". Putting it all back in context, a common refrain:

In Galilean relativity, we could have an event at E, (x,t) in A's frame and (x',t) in B's frame.

If we had an event at E which is (x,t) in A's frame and (x',t) in B's frame, then we would have an event at E which is at (x,t) in A's frame and (x',t') in B's frame, because t=t'.​

Is that better? I'm not sure what reason you have for t=t' other than time is absolute. But in any event, we agree that t=t' so the reason for it is immaterial.

I don't see why, can you explain? If by "aether frame" you just mean there's one frame where the upper limit on the speed of information is the same in both directions, while in other frames it's different in one direction than the other, then I'd agree with that, but I'm not sure that's what you meant.

That's basically what I meant. I guess if you apply http://en.wikipedia.org/wiki/Galilean_invariance" and include the speed of light as a fundamental law of physics (which by its inclusion in such things as the Planck equations, it certainly seems to be), then can't have the aether and different measured values for the speed of light, which means that for the Galilean boosts to work, you have to look for something else. I took instantaneous transmission of information.

It seems odd to have Galilean relativity without proper Galilean invariance (although omission of the speed of light as a fundamental law of physics was a forgivable lapse at the time).

One photon. One event spawning the photon. Two events where the photon passes B, then A. One event when A and B were colocated and t=0 and t'=0.

A thinks that at colocation, the photon was located at x=ct.

B thinks that at colocation, the photon was located at x'=ct'.

What is the relationship between x' and x, and t and t'?

But x and x' refer to the coordinates of two different events on the photon's worldline in this case, at least if we're dealing with relativity (and we must be if both observers say the photon is traveling at c). The event E_A on the photon's worldline that is simultaneous in A's frame with A&B being colocated is different from the event E_B on the photon's worldline that is simultaneous in B's frame with A&B being colocated, due to the relativity of simultaneity.

Do you agree that x refers to the position coordinate (in frame A) of one event E_A and x' refers to the position coordinate (in frame B) of a different event E_B? And do you agree that in the spatial component of the Lorentz transform, x' = gamma*(x - vt), x and x' are supposed to refer to the position coordinates of a single event in two different frames?

I think you've misread.

I'll try again, and highlight what I think you have misread:

One photon (there is only one photon, which by thought experiment magic could be detected by both B and then A).

One event spawning the photon (there's only one photon, so there is only one event, so this is really tautological, but despite that you still want to have two events related to the photon, perhaps I have misread you, but you certainly are not on the same page as me on this one).

Two events where the photon passes B, then A (these are two separate events, which I agree with you on, assuming that you agree).

One event when A and B were colocated and t=0 and t'=0 (when A and B are colocated, that is one event, their colocation is the event I am talking about, I am not talking about any other event at any other time or location. To drive it in, this event is (x=0,t=0) and (x'=0, t'=0) and irrespective of their coordinate systems, this is still a unique event to each of them).

A thinks that at colocation, the photon was located at x=ct.

B thinks that at colocation, the photon was located at x'=ct'.

What is the relationship between x' and x, and t and t'?

What we do know is that, irrespective of coordinate system, when A and B were colocated, the photon had one unique location. Correct?

Not if simultaneity is relative! And if you want to say that both observers measure the speed of photons to be c regardless of direction, it has to be (as demonstrated by Einstein's train/embankment thought-experiment).

Again, I think you have misread, or perhap misunderstood.

In any coordinate system (say A's), a photon will have a unique spacetime location when A and B are colocated. In another coordinate system (say B's), the same photon will have a unique spacetime location when A and B are colocated. Therefore, when B is colocated with A is it true that:
1. that one unique photon is in different spacetime locations or
2. A and B ascribe different coordinates to the one unique spacetime location (which, I must stress, doesn't have to be any "true" or "objectively correct" spacetime location)?

I vote for #1. You either misread or misunderstood or vote for #2.

cheers,

neopolitan

 

[JesseM]

It's not a repeat, because in the second I say (x,t) in A and (x',t') in B, whereas in the first it is (x',t) (unprimed t) in B.

Ah, I missed that it was unprimed.

I'm not sure what reason you have for t=t' other than time is absolute.

Just to show the full coordinate transformation I suppose, same reason they write y'=y and z'=z in the 3D version of the Lorentz transformation.

It seems odd to have Galilean relativity without proper Galilean invariance (although omission of the speed of light as a fundamental law of physics was a forgivable lapse at the time).

True, if you have a finite speed of information transmission which is different in different frames, then if this finite speed is built into the fundamental laws of physics (rather than being a consequence of a physical substance like aether which just happens to be at rest in one particular frame, but which could in principle be moved) then the fundamental laws are not really Galilei-invariant.

But x and x' refer to the coordinates of two different events on the photon's worldline in this case, at least if we're dealing with relativity (and we must be if both observers say the photon is traveling at c). The event E_A on the photon's worldline that is simultaneous in A's frame with A&B being colocated is different from the event E_B on the photon's worldline that is simultaneous in B's frame with A&B being colocated, due to the relativity of simultaneity.

Do you agree that x refers to the position coordinate (in frame A) of one event E_A and x' refers to the position coordinate (in frame B) of a different event E_B? And do you agree that in the spatial component of the Lorentz transform, x' = gamma*(x - vt), x and x' are supposed to refer to the position coordinates of a single event in two different frames?

I think you've misread.

I'll try again, and highlight what I think you have misread:

One photon (there is only one photon, which by thought experiment magic could be detected by both B and then A).

One event spawning the photon (there's only one photon, so there is only one event, so this is really tautological, but despite that you still want to have two events related to the photon, perhaps I have misread you, but you certainly are not on the same page as me on this one).

Two events where the photon passes B, then A (these are two separate events, which I agree with you on, assuming that you agree).

One event when A and B were colocated and t=0 and t'=0 (when A and B are colocated, that is one event, their colocation is the event I am talking about, I am not talking about any other event at any other time or location. To drive it in, this event is (x=0,t=0) and (x'=0, t'=0) and irrespective of their coordinate systems, this is still a unique event to each of them).

A thinks that at colocation, the photon was located at x=ct.

B thinks that at colocation, the photon was located at x'=ct'.

What is the relationship between x' and x, and t and t'?

That doesn't really clarify the issue I had. When you said "A thinks that at colocation, the photon was located at x=ct" and "B thinks that at colocation, the photon was located at x'=ct'", are you implying that one is objectively right and one is objectively wrong? I interpreted "thinks" to just mean "A is defining x and t to be the position and time coordinates (in the A frame) of the event E_A on the photon's worldline that occurs simultaneously with A&B being colocated (in the A frame)", and likewise "B is defining x' and t' to be the position and time coordinates (in the B frame) of the event E_B on the photon's worldline that occurs simultaneously with A&B being colocated (in the B frame)". Since A and B must naturally disagree about simultaneity if they both measure the photon traveling at c, E_A and E_B would be two different events on the photon's worldline (with one or both happening after the emission event).

If that's not what you meant, could you describe in specific terms what event you are saying should be assigned coordinates x and t in the A frame, and what event you are saying should be assigned coordiantes x' and t' in the B frame? It wouldn't make sense for you to be talking about the emission event in both cases, since if the emission event happens "at colocation" in one frame, it does not happen at the moment of colocation in the other frame.

Again, I think you have misread, or perhap misunderstood.

In any coordinate system (say A's), a photon will have a unique spacetime location when A and B are colocated. In another coordinate system (say B's), the same photon will have a unique spacetime location when A and B are colocated. Therefore, when B is colocated with A is it true that:
1. that one unique photon is in different spacetime locations or
2. A and B ascribe different coordinates to the one unique spacetime location (which, I must stress, doesn't have to be any "true" or "objectively correct" spacetime location)?

I vote for #1. You either misread or misunderstood or vote for #2.

Why do you say that? My interpretation was that they were both talking about two physically separate events ('physically separate' meaning different points in spacetime) on the photon's worldline, E_A and E_B, so that sounds like #1 unless I'm misunderstanding what you're saying there. In your way of speaking, is saying "when A&B pass, one unique object X is in different spacetime locations" equivalent to "according to A's definition of simultaneity, when A&B pass this is simultaneous with one event on the worldline of a unique object X, but according to B's definition of simultaneity, when A&B pass this is simultaneous with a different event (at a different spacetime location) on the worldline of the same unique object X"? For example, if two distant observers A and B wanted to know my age at the moment they passed each other, and in A's frame the passing was simultaneous with my turning 20 but in B's frame the passing was simultaneous with my turning 30, would you sum this up by saying "when A&B pass there is one unique Jesse at two different spacetime locations (and thus two different ages)"?

 

[neopolitan]

That doesn't really clarify the issue I had. When you said "A thinks that at colocation, the photon was located at x=ct" and "B thinks that at colocation, the photon was located at x'=ct'", are you implying that one is objectively right and one is objectively wrong? I interpreted "thinks" to just mean "A is defining x and t to be the position and time coordinates (in the A frame) of the event E_A on the photon's worldline that occurs simultaneously with A&B being colocated (in the A frame)", and likewise "B is defining x' and t' to be the position and time coordinates (in the B frame) of the event E_B on the photon's worldline that occurs simultaneously with A&B being colocated (in the B frame)". Since A and B must naturally disagree about simultaneity if they both measure the photon traveling at c, E_A and E_B would be two different events on the photon's worldline (with one or both happening after the emission event).

If that's not what you meant, could you describe in specific terms what event you are saying should be assigned coordinates x and t in the A frame, and what event you are saying should be assigned coordiantes x' and t' in the B frame? It wouldn't make sense for you to be talking about the emission event in both cases, since if the emission event happens "at colocation" in one frame, it does not happen at the moment of colocation in the other frame.

Why do you say that? My interpretation was that they were both talking about two physically separate events ('physically separate' meaning different points in spacetime) on the photon's worldline, E_A and E_B, so that sounds like #1 unless I'm misunderstanding what you're saying there. In your way of speaking, is saying "when A&B pass, one unique object X is in different spacetime locations" equivalent to "according to A's definition of simultaneity, when A&B pass this is simultaneous with one event on the worldline of a unique object X, but according to B's definition of simultaneity, when A&B pass this is simultaneous with a different event (at a different spacetime location) on the worldline of the same unique object X"? For example, if two distant observers A and B wanted to know my age at the moment they passed each other, and in A's frame the passing was simultaneous with my turning 20 but in B's frame the passing was simultaneous with my turning 30, would you sum this up by saying "when A&B pass there is one unique Jesse at two different spacetime locations (and thus two different ages)"?

I'm such a visual sort of fellow. Another diagram, http://www.geocities.com/neopolitonian/uniquespacetimelocation.jpg".

The photon has a unique spacetime location both times. Do we agree on that?

It's just that for the first event (it's the first event in both frames), the photon is at (8,0) according to A and (10,-6) according to B and for the second event, the photon is at (5,3) according to A and (4,0) according to B.

But even though each event is described differently, both events describe one unique location of the photon. A and B just disagree about which event is simultaneous with their colocation with the other.

Do we agree on this?

cheers,

neopolitan

 

[JesseM]

I'm such a visual sort of fellow. Another diagram, http://www.geocities.com/neopolitonian/uniquespacetimelocation.jpg".

The photon has a unique spacetime location both times. Do we agree on that?

It's just that for the first event (it's the first event in both frames), the photon is at (8,0) according to A and (10,-6) according to B and for the second event, the photon is at (5,3) according to A and (4,0) according to B.

But even though each event is described differently, both events describe one unique location of the photon. A and B just disagree about which event is simultaneous with their colocation with the other.

Do we agree on this?

Yes, the two arrows on the diagram are pointing to exactly the events I was thinking of when I referred to E_A and E_B.

 

[neopolitan]

Yes, the two arrows on the diagram are pointing to exactly the events I was thinking of when I referred to E_A and E_B.

Do you agree that they are both unique events? Irrespective of what coordinates A and B might assign to them and whether or not A and B might disagree about which is simultaneous with their colocation (unless they do some Lorentz Transformations, of course).

 

[neopolitan]

I don't really want to smother the previous post even if I am pretty sure that you will agree, but I've been wondering what caused the misunderstanding which led to it.

I think it was that in #181, I wrote in haste "What we do know is that, irrespective of coordinate system, when A and B were colocated, the photon had one unique location."

I was in a real rush at the time, and it seems that although the comment made sense to me at the time I now see I should have been more careful - some words were in my head but didn't get converted to pixels.

Let me rephrase "What we do know is that, irrespective of what coordinate system you chose to use, when A and B were colocated, the photon had one unique location."

It's still not perfect, since I should add that you must chose only one coordinate system, either A or B's but not both since A and B have different ideas about what is simultaneous with the colocation event.

Anyway, it seems you did misunderstand but, given the poor wording, it was hardly your fault.

cheers,

neopolitan

 

[JesseM]

Do you agree that they are both unique events?

Yes, and that was kind of my point, I was confused about how you intended to use these two different events E_A and E_B to derive the Lorentz transform when the Lorentz transform relates the coordinates that two different frames assign to a single event. I can't really follow what's going on in the second diagram from post #174, and hence I can't follow the third diagram either, could you walk me through them a bit? In the second diagram when you say "According to A, there is one event E" are you referring to the event I've called E_A? In that diagram, do x and t refer to the coordinates that A assigns to E_A while x' and t' refer to the coordinates that B assigns to E_B? Why would A "tentatively" believe that x' = x - vt if x' refers to the position coordinate of an entirely different event E_B from the event E_A that x and t refer to? Is it because they don't realize the events are distinct because they're still thinking in terms of the Galilei transform where simultaneity is absolute? But if that's the case, why do they both assume the light from the event was traveling at c to reach them, given that this is impossible under the Galilei transform?

And in the third diagram, do x_A and x_B refer to the position coordinates in A's frame of E_A and E_B respectively, while x'_A and x'_B refer to the position coordinates of these two events in B's frame? If so why do you still show only a single yellow event E in this diagram?

 

[neopolitan]

Yes, and that was kind of my point, I was confused about how you intended to use these two different events E_A and E_B to derive the Lorentz transform when the Lorentz transform relates the coordinates that two different frames assign to a single event. I can't really follow what's going on in the second diagram from post #174, and hence I can't follow the third diagram either, could you walk me through them a bit? In the second diagram when you say "According to A, there is one event E" are you referring to the event I've called E_A? In that diagram, do x and t refer to the coordinates that A assigns to E_A while x' and t' refer to the coordinates that B assigns to E_B? Why would A "tentatively" believe that x' = x - vt if x' refers to the position coordinate of an entirely different event E_B from the event E_A that x and t refer to? Is it because they don't realize the events are distinct because they're still thinking in terms of the Galilei transform where simultaneity is absolute? But if that's the case, why do they both assume the light from the event was traveling at c to reach them, given that this is impossible under the Galilei transform?

And in the third diagram, do x_A and x_B refer to the position coordinates in A's frame of E_A and E_B respectively, while x'_A and x'_B refer to the position coordinates of these two events in B's frame? If so why do you still show only a single yellow event E in this diagram?

Because in its original form, the derivation is a staged process starting with the Galilean boost (Galilei transform) and ending up at the Lorentz transformations. In hind sight, you can see clearly that so long as event E is simultaneous with when A and B are colocated, then event E is actually two events along the photon's worldline. But in the Galilean boost, event E isn't two events.

The same goes for the question in the previous paragraph, except a little reversed, once both assume that information/light gets to them at c, then they find that the Galilean boost is invalid, except as an approximation for where v << c.

So it is step 1, show the Galilean boost. Step 2, introduce the speed of light considerations (ensuring that it is Galilean invariant). Step 3, introduce Galiliean invariance more fully. In the process of doing Steps 2 and 3, it is shown that the Galilean boost is invalid. Step 4, derive the Lorentz transformations.

Then you can do what you seem to want to do, if you like: Step 5, go back and show that event E shown in the diagrams is actually describing two events E and thereby introduce the relativity of simultaneity.

Our approaches seem fundamentally different, if we were talking about how to make a robot, you might start with instructing the class to study gyroscopes to explain how they would be used to keep the robot upright because the robot is top heavy whereas I'd be telling them to how to smelt metal. Your explanation assumes you already know how the robot is built (top heavy), whereas I am starting from close to the ground up (I didn't start right back at "climb down from the trees" or "first evolve into multicellular into lifeform"). Similarly, you seem to want me to assume relatively advanced prior knowledge on the part of the student in order to show a derivation of the Lorentz transform (ie, that the Galilean boost is invalid and that the relativity of simultaneity applies).

cheers,

neopolitan

 

[JesseM]

Our approaches seem fundamentally different, if we were talking about how to make a robot, you might start with instructing the class to study gyroscopes to explain how they would be used to keep the robot upright because the robot is top heavy whereas I'd be telling them to how to smelt metal. Your explanation assumes you already know how the robot is built (top heavy), whereas I am starting from close to the ground up (I didn't start right back at "climb down from the trees" or "first evolve into multicellular into lifeform"). Similarly, you seem to want me to assume relatively advanced prior knowledge on the part of the student in order to show a derivation of the Lorentz transform (ie, that the Galilean boost is invalid and that the relativity of simultaneity applies).

Well, the normal derivation of the Lorentz transform just starts from the idea that we want a transformation where the two postulates of SR are true, and sees what conclusions can be derived from these postulates alone. That's how any "derivation" in math or physics is supposed to work, you pick some starting assumptions and then show in a step by step way what follows from the assumptions. From what you're saying it sounds like the first two diagrams are not really part of your "derivation", but are just part of a sort of pedagogical strategy of showing why we can't continue to use the Galilei transformation if we want the speed of light to be constant in all frames (which also follows in a more direct way from the Galilean velocity addition formula)--if you had made clear that these diagrams weren't part of the actual derivation I wouldn't have been confused, I have no problem with making some pedagogical points before delving into a derivation (for example, one might also point out the context of why Einstein wanted to postulate that the speed of light be the same for all observers, which would involve talking a little about Maxwell's laws and the Michelson-Morley results).

So let's skip the first two diagrams and go directly to number 3. I wrote:

And in the third diagram, do x_A and x_B refer to the position coordinates in A's frame of E_A and E_B respectively, while x'_A and x'_B refer to the position coordinates of these two events in B's frame? If so why do you still show only a single yellow event E in this diagram?

Your response addressed the last sentence but not the previous one, can I assume that's because my interpretation of the meaning of those symbols matched your intentions? If so I don't really understand the next step...if x_A and x_B both refer to the position coordinates in A's frame of the two different events, what does this have to do with A concluding that "B measures distance and/or time oddly"? Or was I in fact mistaken about your notation, and you were actually distinguishing between the two events using primed and unprimed (calling one event E and the other E', say), so that x_A represents the position in A's frame of the unprimed event E while x_B represents the position in B's frame of the same unprimed event E?

 

[neopolitan]

Or was I in fact mistaken about your notation, and you were actually distinguishing between the two events using primed and unprimed (calling one event E and the other E', say), so that x_A represents the position in A's frame of the unprimed event E while x_B represents the position in B's frame of the same unprimed event E?

Correct.

There are four values involved (which is a fair indication to the astute student that there might be two events and therefore is already laying the ground work for introducing the relativity of simultaneity). The values which are primed are as per the Galilean boost. The subscripted values mean "according to ..." (as clarified in post https://www.physicsforums.com/showpost.php?p=2166309&postcount=175"). I didn't go into depth about what the primes meant, perhaps I should have.

Anyway, we have:

x'_A = x_A - vt_A
x_B = x'_B + vt'_B

which work, even in SR. The primes relate to measurements between B and E and the unprimes relate to measurements between A and E.

Is it starting to make more sense?

cheers,

neopolitan

PS Something that I was leading to with an earlier post, but I got distracted from:

Do you now agree that Galilean relativity assumes either instantaneous transmission of information (totally ignoring light) or a variant speed of light (which leads to an internal contradiction since Galilean invariance is a component of Galilean relativity)? While you may still not agree with the former, do you agree with this catch-all phrasing?

"Galilean relativity includes an invalid assumption about the transmission of information and light"

(Where the invalid assumption might be either of the above, but doesn't specifically have to be one or the other.)

 

[neopolitan]

(for example, one might also point out the context of why Einstein wanted to postulate that the speed of light be the same for all observers, which would involve talking a little about Maxwell's laws and the Michelson-Morley results).

I was actually trying to go from Galileo to Einstein in one fell swoop. I think you can do it via Galilean invariance, but I do accept that it might be unfair since the invariance of the speed of light didn't get shown until a long time after.

I remain curious about when we had enough information to arrive at SR, I was thinking that it was around Galileo's time. There was an experiment proposed by one of Galileo's contempories as early as 1629 to http://en.wikipedia.org/wiki/Speed_of_light#Measurements_of_the_speed_of_light".

Galileo said he had done the experiment and wikipedia has conflicting reports of the results (one article says the speed he came up with was in the order of 1000 times greater than the speed of sound - same link as above - and another said the results indicated that the speed of light was http://en.wikipedia.org/wiki/Galileo_Galilei#Physics"). When the Accademia del Cimento carried out the experiment in 1667, no delay was detected. Robert Hooke explained the negative results by saying that they don't necessarily indicate that the speed of light is infinite, but could just be extremely high.

The point being that it is not unreasonable to posit that when Galileo formulated his boosts, an assumption therein was not that there is an aether frame in which the speed of light is a constant but rather that the speed of light is infinite, and light (and thus information) is transmitted instantaneously.

The last nail in the coffin of infinite speed of light was apparently driven home by James Bradley in 1728, eighty years after Galileo's death.

So, I suppose, you could say that there was not quite enough information available in Galileo's time to arrive at SR, but there was in 1728 - 160 years before Michelson and Morely conducted their experiment and more than a century before the birth of Maxwell.

cheers,

neopolitan

 

[JesseM]

Or was I in fact mistaken about your notation, and you were actually distinguishing between the two events using primed and unprimed (calling one event E and the other E', say), so that x_A represents the position in A's frame of the unprimed event E while x_B represents the position in B's frame of the same unprimed event E?

Correct.

There are four values involved (which is a fair indication to the astute student that there might be two events and therefore is already laying the ground work for introducing the relativity of simultaneity). The values which are primed are as per the Galilean boost. The subscripted values mean "according to ..."

Why do you say "as per the Galilean boost"? According to the definition you called "correct" above, primed and unprimed just refer to two different events. If we use the same subscript, like x'_A and x_A, then we are exclusively talking about a single frame, the rest frame of A, no boosts are involved at all.

x'_A = x_A - vt_A
x_B = x'_B + vt'_B

which work, even in SR. The primes relate to measurements between B and E and the unprimes relate to measurements between A and E.

I don't think it does work in SR, given the definitions above. Let's suppose in A's frame B has a velocity of 0.6c. Suppose the unprimed event is the one that's simultaneous with A&B passing each other in A's frame, so in A's frame t_A = 0, and let's say this unprimed event occurs at position x_A = 16 light-seconds. In this case the light will pass B at a position of 6 light-seconds and a time of 10 seconds in A's frame. Since B's clock is running slow by a factor of 0.8 in this frame, the light must hit B when B's clock reads 8 seconds, so the primed event which occurs at t'_B = 0 in B's frame must occur at position x'_B = 8 light-seconds in B's frame. Using the Lorentz transformation we can find the coordinates of the primed event in the A frame:

x'_A = 0.8*(x'_B + v*t'_B) = 0.8*(8 + 0.6*0) = 6.4
t'_A = 0.8*(t'_B + v*x'_B/c^2) = 0.8*(0 + 0.6*8) = 3.84

But in this case your first equation x'_A = x_A - vt_A doesn't work, since 6.4 doesn't equal 16 - 0.6*0.

Maybe instead you want the primed event to be the one that's simultaneous with A&B passing each other in A's frame...in this case let's use the same numbers and say t'_A = 0 and x'_A = 16, and similarly t_B = 0 and x_B = 8. Now we can figure out the coordinates of the primed event in the B frame:

x'_B = 0.8*(x'_A - v*t'_A) = 0.8*(16 - 0.6*0) = 12.8
t'_B = 0.8*(t'_A - v*x'_A) = 0.8*(0 - 0.6*16) = 7.68

So in this case your second equation x_B = x'_B + vt'_B doesn't work because 8 is not equal to 12.8 + 0.6*7.68.

Do you now agree that Galilean relativity assumes either instantaneous transmission of information (totally ignoring light) or a variant speed of light (which leads to an internal contradiction since Galilean invariance is a component of Galilean relativity)? While you may still not agree with the former, do you agree with this catch-all phrasing?

Yes, I'd agree with that. However, we can imagine a possible set of laws in which the Galilei transformation is still the most "natural" one for inertial observers to use despite the fact that the laws of physics are not all Galilei-symmetric, because the behavior of rulers and clocks is Galilei-symmetric so coordinate systems constructed out of such rulers and clocks will be related to one another by the Galilei transformation. In fact, this is exactly how 19th century physicists who believed in a preferred aether frame assumed things would work, at least prior to Michelson-Morley.

 

[neopolitan]

According to A, at any time an event happens (x_A,t_A), how far has B travelled?

I think that distance is vt_A, according to A. Therefore, using the simple Galilean boost, the distance between B and the event according to A, at that time according to A is x'_A = x_A - vt_A.

Do you understand that? Or have I made a mistake?

cheers,

neopolitan

 

[JesseM]

According to A, at any time an event happens (x_A,t_A), how far has B travelled?

I think that distance is vt_A, according to A.

Yes.

Therefore, using the simple Galilean boost, the distance between B and the event according to A, at that time according to A is x'_A = x_A - vt_A.

But none of your symbols were supposed to represent the distance between B and the event in A's frame. x_A referred to the distance between A and the unprimed event in A's frame, x_B referred to the distance between B and the unprimed event in B's frame, x'_A referred to the distance between A and the primed event in A's frame, and x'_B referred to the distance between B and the primed event in B's frame. Or were you mistaken when you said that my earlier statement about the notation was "correct"? If so can you please explain specifically what each of these symbols is supposed to refer to?

 

[neopolitan]

Or were you mistaken when you said that my earlier statement about the notation was "correct"?

Or was I in fact mistaken about your notation, and you were actually distinguishing between the two events using primed and unprimed (calling one event E and the other E', say), so that x_A represents the position in A's frame of the unprimed event E while x_B represents the position in B's frame of the same unprimed event E?

Correct.

I stand by that.

<snip> can you please explain specifically what each of these symbols is supposed to refer to?

First, before I go into lots of detail, can you confirm that you agree that this is correct:

A and B separate at v and an event or events take place closer to B than A, such that the direction that the event lies is notionally taken as the positive direction. In other words, according to A, B has a velocity of +v and according to B, A has a velocity of -v.

If according to A, an event happens at (x_A,t_A), then according to A B has traveled vt_A, therefore according to A, the distance between B and where the event took place according to A, can at that time be given by x'_A = x_A - vt_A.

And, if according to B, an event happens at (x'_B,t'_B), then according to B A has traveled -vt'_B, therefore according to B, the distance between A and where the event took place according to B, can at that time be given by x_B = x'_B + vt'_B.

If it's wrong, I'd like to know what assumptions you are making that make it wrong (which I think you will find are not in accordance with the assumptions that I have obviously not made sufficiently clear).

cheers,

neopolitan

 

[JesseM]

Or was I in fact mistaken about your notation, and you were actually distinguishing between the two events using primed and unprimed (calling one event E and the other E', say), so that xA represents the position in A's frame of the unprimed event E while xB represents the position in B's frame of the same unprimed event E?

Correct.

I stand by that.

OK, I guess I didn't state it explicitly, but I thought it was obvious that if primed and unprimed was just supposed to distinguish between the two events E and E'--since you agreed with my statement "you were actually distinguishing between the two events using primed and unprimed (calling one event E and the other E', say)"--then if you also agreed that "x_A represents the position in A's frame of the unprimed event E while x_B represents the position in B's frame of the same unprimed event E", then naturally a corollary would be that "x'_A represents the position in A's frame of the primed event E' while x'_B represents the position in B's frame of the same primed event E' ". Is that not correct?

First, before I go into lots of detail, can you confirm that you agree that this is correct:

A and B separate at v and an event or events take place closer to B than A, such that the direction that the event lies is notionally taken as the positive direction. In other words, according to A, B has a velocity of +v and according to B, A has a velocity of -v.

If according to A, an event happens at (x_A,t_A)

So based on my statement "x_A represents the position in A's frame of the unprimed event E", I assume this still refers specifically to the position and time of the unprimed event E in A's frame?

then according to A B has traveled vt_A, therefore according to A, the distance between B and where the event took place according to A, can at that time be given by x'_A = x_A - vt_A.

Yes, if you use the notation x'_A to mean "the distance between B and the unprimed event E at the moment E occurred, all in A's frame." But this would seem to be inconsistent with the statement earlier that primed and unprimed was to distinguish between the two events: "you were actually distinguishing between the two events using primed and unprimed (calling one event E and the other E', say)".

And, if according to B, an event happens at (x'_B,t'_B)

Which event? The unprimed event E or the primed event E'?

then according to B A has traveled -vt'_B, therefore according to B, the distance between A and where the event took place according to B, can at that time be given by x_B = x'_B + vt'_B.

Sure, if x_B represents the distance between A and the event in B's frame. But this seems to contradict my statement "x_B represents the position in B's frame of the same unprimed event E" which you said was "correct" earlier.