Benefits of time dilation / length contraction pairing?

[neopolitan]

Post #275

I did say I wanted to start again. I think I have said that a few times, but I don't want to trawl through old posts to show you that I have. So I will just repeat, I wish to start again.

Post #270

How about we take a pause for a bit while I fix what is wrong in #174 (as far as I can tell), then I make a temporal version of what is explained in the diagram in #264, I can repose the question from three paragraphs up and we can go from there?

Post #250

I've gone all the way back to the beginning so anything I have said in between to try to explain in your terms is defunct, so please try to start from here.

JesseM, Post #280

From post 243, here were your old definitions:

I don't what I can do more to make it plain. I am not trying to defend anything after #174 and prior to #250.

cheers,

neopolitan

 

[JesseM]

If you look at the context you'll see I only posted that quote from post 243 as a response to your comment "I don't recall ever meaning that, or writing it" and your comment that it "might well have been a typo" (The tone of this comment seemed dismissive, like you were saying I was the one who was confused when I pointed out that your new definition of x_B was different from the old one. You have repeatedly made comments on this thread suggesting that that the problem in our communication is all about me not trying hard enough to follow what you're saying, rather than acknowledging that your presentation might be confusing or inconsistent, so maybe you can understand why I'd be a little defensive). I was just pointing out that you had been using that definition previously, and it clearly wasn't a typo. Notice that the rest of my post was concerned with trying to understand your new definitions, so yes, I understood that you wanted to start from the beginning.

 

[neopolitan]

Sticking with the graphic approach, does http://www.geocities.com/neopolitonian/g2ev2_2.jpg" help? I moved the equations all to the next diagram in the sequence.

cheers,

neopolitan

Purely as an aside:

I did not mean to be dismissive, I actually held back from repeating that I didn't want to go over old things again, and in part that is because I have been plagued by typing issues. Each post just gets longer and more difficult to scope because it drags a lot of baggage, some intended and right (I think it is right), some intended at the time but now I can see is wrong, and quite a few things that were written in haste or in the middle of the night (like this section) and were just pure mistakes.

I apologise anyway, can we put it aside?

 

[JesseM]

Sticking with the graphic approach, does http://www.geocities.com/neopolitonian/g2ev2_2.jpg" help? I moved the equations all to the next diagram in the sequence.

A question about that: you say "the time it takes a photon to get from the YDE to B, according to B, is x'_B/c". But on the last diagram here you have x'_B = 4, are you saying it takes 4 seconds for the photon to get from the YDE to B? That wouldn't be right, because in B's frame the YDE occurs at x=10 (assuming the YDE still occurs at t=0 and x=8 in A's frame). I think you meant the time it takes a photon to get from E_B to B, where E_B is the meeting point of the top red line and the top green line in that last diagram (the point on the photon's worldline that's simultaneous with A and B being colocated in B's frame).

Likewise, you say "the time it takes a photon to get from the YDE to A, according to B is x_B/c." But in post 277 you said x_B was:

According to B, that distance (which is not when A and B are colocated), is x_B.

In context, I thought this made sense. The distance between B and the YDE, when? Well, B is an observer, or a body, or a frame, while the YDE is an event so "at the time of the YDE" has to be "when". I even state that this is not when A and B is colocated.

So, here you were saying x_B was the distance from B to the YDE at the time of the YDE in B's frame, which would be 10, so combining that with the above you're saying the time it takes a photon to get from the YDE to A, according to B is 10 seconds. But again that doesn't match with previous numbers, since the YDE occurs at t=-6 in B's frame, and the light reaches A at t=10 and x=-6 in B's frame, so the actual time for the photon to get from the YDE to A is 16 seconds. Again I think you may have put in the YDE when you really should have put in E_B, since the time for the light to get from E_B to A, according to B, is in fact 10 seconds.

Purely as an aside:

I did not mean to be dismissive, I actually held back from repeating that I didn't want to go over old things again, and in part that is because I have been plagued by typing issues. Each post just gets longer and more difficult to scope because it drags a lot of baggage, some intended and right (I think it is right), some intended at the time but now I can see is wrong, and quite a few things that were written in haste or in the middle of the night (like this section) and were just pure mistakes.

I apologise anyway, can we put it aside?

OK, sorry if I was being oversensitive, and yes I realize it easy for typos and other mistakes to happen in these long posts, so let's put it aside.

 

[neopolitan]

A question about that: you say "the time it takes a photon to get from the YDE to B, according to B, is x'_B/c". But on the last diagram here you have x'_B = 4, are you saying it takes 4 seconds for the photon to get from the YDE to B? That wouldn't be right, because in B's frame the YDE occurs at x=10 (assuming the YDE still occurs at t=0 and x=8 in A's frame). I think you meant the time it takes a photon to get from E_B to B, where E_B is the meeting point of the top red line and the top green line in that last diagram (the point on the photon's worldline that's simultaneous with A and B being colocated in B's frame).

Likewise, you say "the time it takes a photon to get from the YDE to A, according to B is x_B/c." But in post 277 you said x_B was:

So, here you were saying x_B was the distance from B to the YDE at the time of the YDE in B's frame, which would be 10, so combining that with the above you're saying the time it takes a photon to get from the YDE to A, according to B is 10 seconds. But again that doesn't match with previous numbers, since the YDE occurs at t=-6 in B's frame, and the light reaches A at t=10 and x=-6 in B's frame, so the actual time for the photon to get from the YDE to A is 16 seconds. Again I think you may have put in the YDE when you really should have put in E_B, since the time for the light to get from E_B to A, according to B, is in fact 10 seconds.

OK, sorry if I was being oversensitive, and yes I realize it easy for typos and other mistakes to happen in these long posts, so let's put it aside.

Diagrams http://www.geocities.com/neopolitonian/generality6_all_values.jpg" apply here.

My wording is poor.

If you look at the all values diagram, you can see that x'_B is a reflection of the time it takes the photon from YDE to get to B with the measurement started at colocation. Better said, perhaps, is "x'_B corresponds to the time between colocation and the photon from YDE hitting B". From that time, B can work out how distant the photon from YDE was at colocation - which is the event formerly known as E_b.

If you look at the all values diagram, you can see that x_B is the location of YDE at the time that YDE occurred, according to B.

That makes x'_B = 4 and x_B = 10.

I think you already understand this, but I'll make it explicit, the derivation does not call on the spacetime diagrams such as http://www.geocities.com/neopolitonian/generality6_all_values.jpg", we are using them so that we pin down what we are referring to and confirming that the values used in the derivations have appropriate physical meaning. But once the derivation was completed, the spacetime diagrams could be constructed.

cheers,

neopolitan

 

[neopolitan]

x'_B is the distance between B and the event on the worldline of the light from the YDE that's simultaneous with A&B being colocated in the B frame (this is the event that was formerly known as E_B--unless you have a way of defining x'_B without referring to this event, could we give it some label? We could stick with E_B or use some other label since you're no longer referring to the YDE as E_A). In this example it would be 4.

I'm a little reluctant to do this, because in my visualisation of this (an internal visualisation), A still considers that A is not in motion. Therefore while A considers that when A and B are colocated, A considers that B must think that YDE is closer than it is - if B considers that B is stationary - because the photon from it hits B a period of 4 later.

So A's conception of what B thinks is that the photon at colocation is at the other corner of the photon worldline-x'_B parallogram.

So, I think that when A is trying to work out x'_B, A will have the lower length in mind, rather than the upper length. (A will also have the x_A which is parallel with the x_A axis in mind.)

I think a similar thing happens when B is trying to work out x_A.

Do this words make any sense, or will I have to make a notation on an existing diagram?

cheers,

neopolitan

 

[JesseM]

I'm a little reluctant to do this, because in my visualisation of this (an internal visualisation), A still considers that A is not in motion. Therefore while A considers that when A and B are colocated, A considers that B must think that YDE is closer than it is - if B considers that B is stationary - because the photon from it hits B a period of 4 later.

But if A considers this, A is simply wrong about how things look in B's frame--in B's frame the YDE occurred farther from the position of colocation, and the reason this is compatible with B's clock only reading 4 when the photon hits it has to do with the relativity of simultaneity (B thinking YDE occurred much earlier than the time of colocation). It wouldn't make sense to use a wrong assumption of a derivation of a valid conclusion, so is this just another pedagogical comment, somehow? If so I think it's one that's likely to make things more confusing to people trying to follow your derivation, not less so.

So A's conception of what B thinks is that the photon at colocation is at the other corner of the photon worldline-x'_B parallogram.

By "other corner" you mean the one where the YDE occurs in the diagram, right? The corner that's opposite to the corner that marks the point where A and B pass next to one another?

So, I think that when A is trying to work out x'_B, A will have the lower length in mind, rather than the upper length.

I don't really understand the "So" here. What's the connection between A assuming (incorrectly) that B thinks the YDE occurred at the moment they were colocated, and A having the lower length in mind? If there were no disagreements about simultaneity then it wouldn't even look like a parallelogram, since the bottom corner of the parallelogram is defined as a point in spacetime that's colocated with the position of the photon when it passes B in A's frame, but simultaneous with the YDE in B's frame. So if A assumes B agrees with him about simultaneity A will draw this event as occurring at the same moment as colocation, at a position between the colocation event and the YDE, so the bottom red line would just be a horizontal line extending from the YDE to that point. But perhaps I am totally misunderstanding what you meant when you said "while A considers that when A and B are colocated, A considers that B must think that YDE is closer than it is", and you were not implying here that A was making the incorrect assumption that B agrees with A about simultaneity--if so please clarify.

 

[neopolitan]

But if A considers this, A is simply wrong about how things look in B's frame--in B's frame the YDE occurred farther from the position of colocation, and the reason this is compatible with B's clock only reading 4 when the photon hits it has to do with the relativity of simultaneity (B thinking YDE occurred much earlier than the time of colocation). It wouldn't make sense to use a wrong assumption of a derivation of a valid conclusion, so is this just another pedagogical comment, somehow? If so I think it's one that's likely to make things more confusing to people trying to follow your derivation, not less so.

It's really the same thing. Let's not get wrapped around the axles on it. Put it this way, I have a reason for not wanting to identify a separate event and label it E_B. I do understand that you do.

By "other corner" you mean the one where the YDE occurs in the diagram, right? The corner that's opposite to the corner that marks the point where A and B pass next to one another?

There are two black dots. One dot you want to label, I was talking about the one that is opposite to that.

I don't really understand the "So" here. What's the connection between A assuming (incorrectly) that B thinks the YDE occurred at the moment they were colocated, and A having the lower length in mind? If there were no disagreements about simultaneity then it wouldn't even look like a parallelogram, since the bottom corner of the parallelogram is defined as a point in spacetime that's colocated with the position of the photon when it passes B in A's frame, but simultaneous with the YDE in B's frame. So if A assumes B agrees with him about simultaneity A will draw this event as occurring at the same moment as colocation, at a position between the colocation event and the YDE, so the bottom red line would just be a horizontal line extending from the YDE to that point. But perhaps I am totally misunderstanding what you meant when you said "while A considers that when A and B are colocated, A considers that B must think that YDE is closer than it is", and you were not implying here that A was making the incorrect assumption that B agrees with A about simultaneity--if so please clarify.

It's probably better to use your event labelling. I agree.

I wanted to keep the lower x'_B as something significant because it is on the line joining the location of A simultaneous with YDE according to B (x_B) but I can see that perhaps, I should move the purple line there up to run along the x_B axis which would make it more consistent with x_A which is not offset from the x_A axis.

In that case, what I was worried about disappears entirely.

What about the post that was prior to the last you responded to?

cheers,

neopolitan

 

[JesseM]

It's really the same thing. Let's not get wrapped around the axles on it. Put it this way, I have a reason for not wanting to identify a separate event and label it E_B. I do understand that you do.

I just think it's good to label any event used in the definitions. You can define all the terms without referring to that event, but then you still need to use the black dot at the bottom of the parallelogram in your definitions, the one I have labeled E_C.

There are two black dots. One dot you want to label, I was talking about the one that is opposite to that.

Yes, that's the one I was calling E_C, defined as the event that is colocated with the event of the light hitting B in the A frame, and which is simultaneous with the YDE in the B frame.

Did you have any comments on my post 280? You only responded to the fact that I mentioned some of your old definitions at the beginning, but the rest of the post was an attempt to deal with your new definitions. As I said there, it seems like you currently want to define the terms this way:

So OK, I think based on the diagram I see what the definitions are (you don't show x_B in the diagram, but you explained that above), but please check to see if these are right:

x_A is the distance between YDE and A (in the A frame). In the example this would be 8.

x_B is the distance between the YDE and B (in the B frame). That distance is 10.

x'_A is the distance between YDE and the event of the light passing B (in the A frame). In this example it would be 5.

x'_B is the distance between B and the event on the worldline of the light from the YDE that's simultaneous with A&B being colocated in the B frame (this is the event that was formerly known as E_B--unless you have a way of defining x'_B without referring to this event, could we give it some label? We could stick with E_B or use some other label since you're no longer referring to the YDE as E_A). In this example it would be 4.

(Based on the diagram, x'_B could be defined in terms of either of the identical red lines, so I chose the top one since it was easier to state in words...if you wanted to use the bottom one, we could define another event E_C which was at the meeting point of the bottom green and red lines, it would be the event which is colocated in the A frame with the photon passing B and simultaneous in the B frame with the YDE, and then x'_B would be defined as the distance between E_C and the YDE.)

(based on your current comments, should I assume you would actually rather use the second paranthetical definition of x'_B above, where it's defined in terms of the bottom black dot on the parallelogram E_C and the YDE?)

Is this correct? If so, I pointed out there were some problems with justifying other equations if you defined the terms this way...the equation x_B = x'_B + vt'_B was easy to justify using your old definitions (which I realized were actually equivalent to your new ones in SR, but you need to already know the Lorentz transformation to show this equivalence), but I don't really see how you can justify it using the new definitions above. Likewise, the similarity between the equations x_B = (a factor times).x_A and x'_A = (a factor times).x'_B was much more apparent under the old definitions, with the new definitions there's no obvious way to show there is anything analogous about the quantities in the two equations.

What about the post that was prior to the last you responded to?

Sure:

Diagrams http://www.geocities.com/neopolitonian/generality6_all_values.jpg" apply here.

My wording is poor.

If you look at the all values diagram, you can see that x'_B is a reflection of the time it takes the photon from YDE to get to B with the measurement started at colocation. Better said, perhaps, is "x'_B corresponds to the time between colocation and the photon from YDE hitting B". From that time, B can work out how distant the photon from YDE was at colocation - which is the event formerly known as E_b.

For clarity I think it's good to define all coordinate intervals in terms of a pair of events...so x'_B can either be defined as the distance in the B frame between E_C (which I defined earlier) and the YDE, which would be the bottom red line in the parallelogram, or it can be defined as the distance between the event of A&B being colocated and the event E_B, which would be the top red line. Based on SR we can see these definitions are equivalent, but for the sake of a derivation we can't assume that, so I think it does make a difference which one we choose to use. Which of these two do you want to use as the definition of x'_B?

If you look at the all values diagram, you can see that x_B is the location of YDE at the time that YDE occurred, according to B.

Yes, and again based on SR we can see this is actually equivalent to the distance in the B frame between E_B and the event of the light passing A, which was your older definition of x_B (I can draw a diagram if it isn't clear why this should be true in general, but note that in our numerical example both would give the same value of 10). And again, without assuming SR to begin with I don't think there's any way to prove that these are equivalent, so it matters which one you choose as the definition. As I was saying in post 280, it's easy to see why the equation x_B = x'_B + vt'_B should be expected to hold using the old definitions, but I don't know if there's any way to justify this equation under the new ones.

I think you already understand this, but I'll make it explicit, the derivation does not call on the spacetime diagrams such as http://www.geocities.com/neopolitonian/generality6_all_values.jpg", we are using them so that we pin down what we are referring to and confirming that the values used in the derivations have appropriate physical meaning.

Yes, I understand, and this is exactly why I'm skeptical that some of the steps in the derivation are justifiable under the new definitions.

 

[neopolitan]

I just think it's good to label any event used in the definitions. You can define all the terms without referring to that event, but then you still need to use the black dot at the bottom of the parallelogram in your definitions, the one I have labeled E_C.

Yes, that's the one I was calling E_C, defined as the event that is colocated with the event of the light hitting B in the A frame, and which is simultaneous with the YDE in the B frame.

Noted.

Did you have any comments on my post 280? You only responded to the fact that I mentioned some of your old definitions at the beginning, but the rest of the post was an attempt to deal with your new definitions. As I said there, it seems like you currently want to define the terms this way:

(based on your current comments, should I assume you would actually rather use the second paranthetical definition of x'_B above, where it's defined in terms of the bottom black dot on the parallelogram E_C and the YDE?)

Is this correct? If so, I pointed out there were some problems with justifying other equations if you defined the terms this way...the equation x_B = x'_B + vt'_B was easy to justify using your old definitions (which I realized were actually equivalent to your new ones in SR, but you need to already know the Lorentz transformation to show this equivalence), but I don't really see how you can justify it using the new definitions above. Likewise, the similarity between the equations x_B = (a factor times).x_A and x'_A = (a factor times).x'_B was much more apparent under the old definitions, with the new definitions there's no obvious way to show there is anything analogous about the quantities in the two equations.

I did read them, but was taking them as "contaminated" since you used "x_B = x'_B + vt'_B", which was in sequence I had recently said I had to rework. I've done that, http://www.geocities.com/neopolitonian/g2ev2.htm".

Sorry about not responding to it before, but I thought it was pointless under the circumstances.

Sure:

For clarity I think it's good to define all coordinate intervals in terms of a pair of events...so x'_B can either be defined as the distance in the B frame between E_C (which I defined earlier) and the YDE, which would be the bottom red line in the parallelogram, or it can be defined as the distance between the event of A&B being colocated and the event E_B, which would be the top red line. Based on SR we can see these definitions are equivalent, but for the sake of a derivation we can't assume that, so I think it does make a difference which one we choose to use. Which of these two do you want to use as the definition of x'_B?

On reflection, I think I would have to go with colocation to E_B.

Yes, and again based on SR we can see this is actually equivalent to the distance in the B frame between E_B and the event of the light passing A, which was your older definition of x_B (I can draw a diagram if it isn't clear why this should be true in general, but note that in our numerical example both would give the same value of 10). And again, without assuming SR to begin with I don't think there's any way to prove that these are equivalent, so it matters which one you choose as the definition. As I was saying in post 280, it's easy to see why the equation x_B = x'_B + vt'_B should be expected to hold using the old definitions, but I don't know if there's any way to justify this equation under the new ones.

Looking specifically at the bottom of http://www.geocities.com/neopolitonian/g2ev2_2.jpg" hold?

You can see it together if you view http://www.geocities.com/neopolitonian/g2ev2.htm".

cheers,

neopolitan

 

[JesseM]

I did read them, but was taking them as "contaminated" since you used "x_B = x'_B + vt'_B", which was in sequence I had recently said I had to rework. I've done that, http://www.geocities.com/neopolitonian/g2ev2.htm".

Sorry about not responding to it before, but I thought it was pointless under the circumstances.

OK, didn't notice the change to that equation. Looking at the new equation x'_B = x_B - vt_B, I think the same basic argument holds. If we use the old definition of x_B where it refers to the distance between the event E_B and the event of the light passing A in the B frame (and also define t_B as the time of the light passing A), then it's fairly easy to understand why this equation works--if you rearrange it as x_B = x'_B + vt_B, then it's just a special case of the general equation x(t) = x'_B + vt, where x(t) represents the distance between A and the position of E_B as a function of time t, and where x'_B was the distance between A and E_B at time t=0. It's easy to see why this equation should hold since A is moving away from the position of E_B at speed v. And based on this general equation, the distance x_B at the time t_B when the light passes A would have to be x'_B + vt_B.

On the other hand, if we start out defining x_B as the distance between the YDE and B in B's frame, I don't see what argument you would use to justify the relation x'_B = x_B - vt_B if you don't already know from the Lorentz transformation that this definition of x_B is equivalent to the older one. If you think there is a justification for the equation which uses this definition of x_B rather than the older one, can you explain it?

Looking specifically at the bottom of http://www.geocities.com/neopolitonian/g2ev2_2.jpg" hold?

I'm unclear what the different dots represent in that first diagram. You have x'_B as the distance from the orange dot to the yellow dot in the right-hand drawing from B's perspective, and we know that x'_B can be defined either as the distance from the event of A&B being colocated to the E_B, or it can be defined as the distance from E_C (bottom dot on parallelogram) to the YDE. So does the orange dot represent E_C, or have you redefined the meaning of the yellow dot to mean E_B in this picture? If the former I can't figure out what the purple dot would be (what event to we know to be a distance of vt_B from E_C?), but if the latter then I suppose it's the position of A at the time t_B when the light reaches it. But in this case, I don't see why you label the distance from the purple dot to the yellow dot (which would really be E_B rather than the YDE) as x_B unless you are reverting to the old definition of x_B (distance between E_B and event of light passing A) as opposed to the newer one (distance between B and YDE).

 

[neopolitan]

OK, didn't notice the change to that equation. Looking at the new equation x'_B = x_B - vt_B, I think the same basic argument holds. If we use the old definition of x_B where it refers to the distance between the event E_B and the event of the light passing A in the B frame (and also define t_B as the time of the light passing A), then it's fairly easy to understand why this equation works--if you rearrange it as x_B = x'_B + vt_B, then it's just a special case of the general equation x(t) = x'_B + vt, where x(t) represents the distance between A and the position of E_B as a function of time t, and where x'_B was the distance between A and E_B at time t=0. It's easy to see why this equation should hold since A is moving away from the position of E_B at speed v. And based on this general equation, the distance x_B at the time t_B when the light passes A would have to be x'_B + vt_B.

On the other hand, if we start out defining x_B as the distance between the YDE and B in B's frame, I don't see what argument you would use to justify the relation x'_B = x_B - vt_B if you don't already know from the Lorentz transformation that this definition of x_B is equivalent to the older one. If you think there is a justification for the equation which uses this definition of x_B rather than the older one, can you explain it?

Right, remember, I said that the spacetime diagram is only (repeat only) to check the meanings of the various values.

As soon as you can agree that there are meanings, we can ditch the spacetime diagrams totally (at least figuratively) and work only from the Galilean boost like situation.

I think we both agree that it is unfair to use the spacetime diagrams in the derivation.

I'm unclear what the different dots represent in that first diagram. You have x'_B as the distance from the orange dot to the yellow dot in the right-hand drawing from B's perspective, and we know that x'_B can be defined either as the distance from the event of A&B being colocated to the E_B, or it can be defined as the distance from E_C (bottom dot on parallelogram) to the YDE. So does the orange dot represent E_C, or have you redefined the meaning of the yellow dot to mean E_B in this picture? If the former I can't figure out what the purple dot would be (what event to we know to be a distance of vt_B from E_C?), but if the latter then I suppose it's the position of A at the time t_B when the light reaches it. But in this case, I don't see why you label the distance from the purple dot to the yellow dot (which would really be E_B rather than the YDE) as x_B unless you are reverting to the old definition of x_B (distance between E_B and event of light passing A) as opposed to the newer one (distance between B and YDE).

It seems I am taking two positions, and I think I know why. Remember I said we were working from Galilean to Lorentz, so in the middle things are a little murky.

I think one definition of the distance makes more sense from one side (Galilean) and then, once you can draw the spacetime diagrams, another definition suddenly makes more sense.

Can you at least see that might be the case? Pinning down one definition certainly seems to be extremely difficult and this would explain why.

So, can we agree that there are a number of possible definitions for the values, and hypothesize that from one perspective one set of definitions makes sense, and from another perspective, another set of definitions makes sense?

The reason I say this is because I can understand arguments for both definitions of (for example) x_B, and my general approach is that "if two arguments are sound then possibly both are right from different viewpoints".

Can we agree to at least explore this avenue?

cheers,

neopolitan

Rather than having two posts, I add this:

On this diagram http://www.geocities.com/neopolitonian/g2ev2_2.jpg", I show all the values x' and x with subscripts A and B, plus vt'_A and -vt_B. This is starting from the Galilean perspective so:

x_A = distance from A to YDE at the time of YDE (simultaneous with colocation of A and B), according to A ( = 8)
x_B = distance from A to YDE at the time of YDE (not simultaneous with colocation of A and B), according to B ( = 10)
x'_A = distance from A to B when the photon from YDE passes B, according to A ( = 5)
x'_B = distance (at the time of colocation of A and B) from B to the photon which subsequently passes B ( = 4)
t'_A = time at which the photon passes B, according to A (measured from the time of colocation of A and B) ( = 5)
t_B = time at which the photon passes B, according to B (measured from the time of colocation of A and B) ( = 10)

Not in the diagram per se, but implied:

t_B = time between YDE and time that the photon from YDE passes A, according to A ( = 8)
t'_B = time between the colocation of A and B and the time that the photon from YDE passes B, according to B ( = 4)

Going from the the spacetime diagram http://www.geocities.com/neopolitonian/generality6_all_values.jpg":

x_A = separation between A and event E_A (YDE), according to A ( = 8)
x_B = separation between A and event E_A (YDE) at the time of YDE, according to B ( = 10)
x'_A = separation between B and the location of event E_A (YDE) when the photon from YDE passes B, according to A ( = 5)
x'_B = separation between B and the location of event E_B (the photon from YDE passes the x_B axis) when the photon from YDE passes B, according to B ( = 4)
t'_A = the time between colocation of A and B and the photon passing B, according to A ( = 5)
t_B = the time between when YDE occurred and when the photon passes A, according to B ( = 10)
t_A = the time between colocation of A and B and the photon passing A, according to A ( = 8)
t'_B = the time between colocation of A and B and the photon passing B, according to B ( = 4)


Not sure if you understand how much of a struggle it is to keep both viewpoints straight. I think I have it right, but there may be typos (late at night again).

The point is, it seems that I have may have been inconsistent, and that has bothered me but I think I have worked out why. If we talk about the http://www.geocities.com/neopolitonian/g2ev2.htm" page, it is "viewpoint 2" all the way.

In the next couple of days, I will head off on travel, so I will go quiet. I am not ignoring you, I will just be doing something other than bouncing the numbers 4, 5, 8 and 10 around in my head along with pretty coloured vectors. I will try to respond to any responsed to this, if I have enough time before I go.

cheers,

neopolitan

 

[JesseM]

It seems I am taking two positions, and I think I know why. Remember I said we were working from Galilean to Lorentz, so in the middle things are a little murky.

But you also said the Galilean stuff was just pedagogical, it's not part of the real substance of the derivation. Maybe by "Galilean" you mean the equation x_B = x'_B + vt_B, but as long as all these coordinates refer to the B frame, there's nothing specifically Galilean about this equation--in any coordinate system, if an object is moving away from some point it must be true that (distance at time t) = (distance at time 0) + (rate distance is changing)*t.

I think one definition of the distance makes more sense from one side (Galilean) and then, once you can draw the spacetime diagrams, another definition suddenly makes more sense.

Which one is which? Are you agreeing with me that defining x_B as the distance in the B frame between A and E_B at the moment the light reaches A is needed in order to justify x_B = x'_B + vt_B? I agree that if we later decide to plot things on a spacetime diagram based on the Lorentz transform, we can see that this is equivalent to the definition of x_B as the distance in the B frame from B to the YDE, although I don't really think either definition "makes more sense" than the other in a spacetime diagram, they both are reasonably simple to illustrate. But from the perspective of a step-by-step derivation where every new equation has to be justified only in terms of what came before, I do think it's necessary to use the first definition.

So, can we agree that there are a number of possible definitions for the values, and hypothesize that from one perspective one set of definitions makes sense, and from another perspective, another set of definitions makes sense?

Well, what criteria are you using to say whether a definition makes sense or not? I don't really understand why there would be a context where the second definition is clearly superior to the first definition, and in the context of a derivation it seems the first definition is the only one that makes sense if you want to justify x_B = x'_B + vt_B.

The reason I say this is because I can understand arguments for both definitions of (for example) x_B, and my general approach is that "if two arguments are sound then possibly both are right from different viewpoints".

Can we agree to at least explore this avenue?

I'm fine with using either definition if we're just talking about the consistency of the different values or analyzing their relationships in a spacetime diagram. As always, though, I'm not convinced it makes sense to use the second definition in an actual derivation.

On this diagram http://www.geocities.com/neopolitonian/g2ev2_2.jpg", I show all the values x' and x with subscripts A and B, plus vt'_A and -vt_B.

Could you define what events the different colored dots represent?

This is starting from the Galilean perspective so:

x_A = distance from A to YDE at the time of YDE (simultaneous with colocation of A and B), according to A ( = 8)
x_B = distance from A to YDE at the time of YDE (not simultaneous with colocation of A and B), according to B ( = 10)

That should be "distance from B to YDE at the time of YDE", I assume. Unless somehow "starting from a Galilean perspective" means B is assuming A agrees with him about simultaneity? I should tell you that I've never really understood your comments about starting from a Galilean perspective, you said it was just pedagogical so I thought it wasn't important to understand and moved on, but if you want to keep discussing it maybe you should try to explain in more detail what the concept is here, because I just find the whole thing totally confusing.

x'_A = distance from A to B when the photon from YDE passes B, according to A ( = 5)

I think that should be the distance from the YDE to B when the photon from YDE passes B, according to A.

x'_B = distance (at the time of colocation of A and B) from B to the photon which subsequently passes B ( = 4)
t'_A = time at which the photon passes B, according to A (measured from the time of colocation of A and B) ( = 5)
t_B = time at which the photon passes B, according to B (measured from the time of colocation of A and B) ( = 10)

That last one should be the time at which the photon passes A, according to B, right? In B's frame the photon passes B at t=4.

Not in the diagram per se, but implied:

t_B = time between YDE and time that the photon from YDE passes A, according to A ( = 8)

Since you defined t_B one line earlier, that should be t_A, right?

t'_B = time between the colocation of A and B and the time that the photon from YDE passes B, according to B ( = 4)

Going from the the spacetime diagram http://www.geocities.com/neopolitonian/generality6_all_values.jpg":

x_A = separation between A and event E_A (YDE), according to A ( = 8)
x_B = separation between A and event E_A (YDE) at the time of YDE, according to B ( = 10)

That should be separation between B and YDE at time of YDE.

x'_A = separation between B and the location of event E_A (YDE) when the photon from YDE passes B, according to A ( = 5)
x'_B = separation between B and the location of event E_B (the photon from YDE passes the x_B axis) when the photon from YDE passes B, according to B ( = 4)
t'_A = the time between colocation of A and B and the photon passing B, according to A ( = 5)
t_B = the time between when YDE occurred and when the photon passes A, according to B ( = 10)

Should be time between when YDE occurred (t=-6) and when the photon passed B (t=4), according to B. You have it drawn correctly in the diagram. The alternate definition would be the time between E_B and the event of the light passing A, which could also be depicted on the diagram if you wanted. (You'd just take that purple line and place the top end at the event of the light passing A, then the bottom end will naturally lie on B's x-axis which represents the set of all events at t=0 in the B frame, which is when E_B occurred. Likewise you could represent the alternate definition of x_B by moving the purple x_B line so the left end was on the event of the light passing A, and then if you drew an axis of constant x in the B frame which was parallel to B's time axis and which passed through E_B, the other end of the purple line would lie on this axis.)

t_A = the time between colocation of A and B and the photon passing A, according to A ( = 8)
t'_B = the time between colocation of A and B and the photon passing B, according to B ( = 4)

Not sure if you understand how much of a struggle it is to keep both viewpoints straight. I think I have it right, but there may be typos (late at night again).

No problem, let me know if you disagree with any of my suggested corrections. I'm confused by what you mean when you say "both viewpoints" though, since (assuming you agree with my corrections, maybe you won't) I don't actually see any differences in the two sets of definitions above. Maybe I'm missing something though.

The point is, it seems that I have may have been inconsistent, and that has bothered me but I think I have worked out why. If we talk about the http://www.geocities.com/neopolitonian/g2ev2.htm" page, it is "viewpoint 2" all the way.

On that first page, I take it that when you say "YDE" you are actually talking about different events depending on the context? You say x'_A/c is the time it takes the photon to get from the YDE event to B in A's frame (implying the YDE is E_A), but then you say x'_B/c is the time it takes the photon to get from the YDE to B in B's frame (implying YDE is E_B). Or are you in some sense having the two of them make the erroneous Galilean assumption that they agree about simultaneity?

 

[neopolitan]

But you also said the Galilean stuff was just pedagogical, it's not part of the real substance of the derivation. Maybe by "Galilean" you mean the equation x_B = x'_B + vt_B, but as long as all these coordinates refer to the B frame, there's nothing specifically Galilean about this equation--in any coordinate system, if an object is moving away from some point it must be true that (distance at time t) = (distance at time 0) + (rate distance is changing)*t.

Which one is which? Are you agreeing with me that defining x_B as the distance in the B frame between A and E_B at the moment the light reaches A is needed in order to justify x_B = x'_B + vt_B? I agree that if we later decide to plot things on a spacetime diagram based on the Lorentz transform, we can see that this is equivalent to the definition of x_B as the distance in the B frame from B to the YDE, although I don't really think either definition "makes more sense" than the other in a spacetime diagram, they both are reasonably simple to illustrate. But from the perspective of a step-by-step derivation where every new equation has to be justified only in terms of what came before, I do think it's necessary to use the first definition.

In answer to "which is which?" - I tried to detail all the values lower in my post. However, if there is an inconsistency between my words and the diagram, the diagram is the one to use.

Well, what criteria are you using to say whether a definition makes sense or not? I don't really understand why there would be a context where the second definition is clearly superior to the first definition, and in the context of a derivation it seems the first definition is the only one that makes sense if you want to justify x_B = x'_B + vt_B.

I'm fine with using either definition if we're just talking about the consistency of the different values or analyzing their relationships in a spacetime diagram. As always, though, I'm not convinced it makes sense to use the second definition in an actual derivation.

In the derivation only the first definitions would be used.

Could you define what events the different colored dots represent?

From left to right: photon passes A, A and B colocated, photon passes B and photon spawned.

That should be "distance from B to YDE at the time of YDE", I assume. Unless somehow "starting from a Galilean perspective" means B is assuming A agrees with him about simultaneity?

The diagram takes primacy over my words. Your words are right.

I should tell you that I've never really understood your comments about starting from a Galilean perspective, you said it was just pedagogical so I thought it wasn't important to understand and moved on, but if you want to keep discussing it maybe you should try to explain in more detail what the concept is here, because I just find the whole thing totally confusing.

It started off quite simple for me. Perhaps we should have tried suspending judgment on the definitions of the terms, done the derivation, then analysed the terms afterwards. That's probably totally alien to your mindset though.

Explaining in more detail will have to happen another day.

I think that should be the distance from the YDE to B when the photon from YDE passes B, according to A.

Yes. Again, the diagram takes primacy. This applies to all your similar comments. I've said before that I am very visual, the structure of this coding system doesn't help me at all.

Should be time between when YDE occurred (t=-6) and when the photon passed B (t=4), according to B. You have it drawn correctly in the diagram. The alternate definition would be the time between E_B and the event of the light passing A, which could also be depicted on the diagram if you wanted. (You'd just take that purple line and place the top end at the event of the light passing A, then the bottom end will naturally lie on B's x-axis which represents the set of all events at t=0 in the B frame, which is when E_B occurred. Likewise you could represent the alternate definition of x_B by moving the purple x_B line so the left end was on the event of the light passing A, and then if you drew an axis of constant x in the B frame which was parallel to B's time axis and which passed through E_B, the other end of the purple line would lie on this axis.)

No problem, let me know if you disagree with any of my suggested corrections. I'm confused by what you mean when you say "both viewpoints" though, since (assuming you agree with my corrections, maybe you won't) I don't actually see any differences in the two sets of definitions above. Maybe I'm missing something though.

Again, another day.

On that first page, I take it that when you say "YDE" you are actually talking about different events depending on the context? You say x'_A/c is the time it takes the photon to get from the YDE event to B in A's frame (implying the YDE is E_A), but then you say x'_B/c is the time it takes the photon to get from the YDE to B in B's frame (implying YDE is E_B). Or are you in some sense having the two of them make the erroneous Galilean assumption that they agree about simultaneity?

Dots defined above.

YDE is fixed, it spawns the photon. Photon passes B is fixed. A and B colocated is fixed. Photon passes A is fixed. By fixed I mean "fast, invariable", but not "corrected" and not "the same in all coordinate systems". What is the same in all coordinate systems

The lengths that are compared between frames are:

"colocation-photon passes B" (which is x'_B and x'_A), and

"colocation-photon passes A" (which is x_A and x_B).

Time's up for now.

cheers,

neopolitan

 

[neopolitan]

That should be "distance from B to YDE at the time of YDE", I assume. Unless somehow "starting from a Galilean perspective" means B is assuming A agrees with him about simultaneity? I should tell you that I've never really understood your comments about starting from a Galilean perspective, you said it was just pedagogical so I thought it wasn't important to understand and moved on, but if you want to keep discussing it maybe you should try to explain in more detail what the concept is here, because I just find the whole thing totally confusing.

It started off quite simple for me. Perhaps we should have tried suspending judgment on the definitions of the terms, done the derivation, then analysed the terms afterwards. That's probably totally alien to your mindset though.

Explaining in more detail will have to happen another day.

Try to put your brain into "pre-SR mode" which means you have to knowledge of a high school student who has paid enough attention to know that:

x' = x - vt

but you don't know anything more than that. That would give you the sort of knowledge that a person we are introducing to SR would have. Such a person won't have all the simultaneity issues you have, because they don't know enough to realize that there are simultaneity issues.

Then introduce the concept that if a photon is released from a distance of x away then it takes a period of t to reach you (remember you still don't know enough to realize there are simultaneity issues). Therefore:

x = ct and x' = x - vt

Now we know that the value t is not necessarily the same in both equations - but say we specifically want to know where the photon release location is in relation to an object moving at v away from us towards where the photon was released ... when the photon reaches us.

Then, we want to know how things look in the rest frame of that object. That is, how far from the photon release point are we when the photon reaches the object which is moving towards the photon release point (relative to us).

x' = ct' and x' = x - vt' or x = x' + vt'

We still know nothing about the relativity of simultaneity nor have any idea that the photon release location is not universally agreed. So we can try to make sense of what we have so far.

x' = x - vt and x = x' + vt'

so x' = (x' + vt') - vt

so vt' = vt which means t = t' which we know can't be right.

This is the very first step in the process. We have shown the student that just by thinking about a photon traveling past two observers in relative motion to each other, we prove that we need to have a better explanation than that given to us by Galileo and Newton.

I personally think that this is a very useful step, it engages the student's interest (at least if the student has a problem solving type of mindset) and shows that Einstein's relativity is necessary.

At this point it would probably be useful to discuss with the student the fact that whenever we measure the speed of light in an inertial frame, it is c - but note in the equations immediately above, c didn't come into it. (The x = c.t and x' = c.t' equations come into play in following steps.)

Can you understand the pedagogical process thus far?

No problem, let me know if you disagree with any of my suggested corrections. I'm confused by what you mean when you say "both viewpoints" though, since (assuming you agree with my corrections, maybe you won't) I don't actually see any differences in the two sets of definitions above. Maybe I'm missing something though.

Again, another day.

By "both viewpoints" I mean a viewpoint in which A considers A to be at rest and is considering how the universe looks from B's perspective:

x'_B = gamma.(x_A - v.t_A)
t'_B = gamma.(t_A - v.x_A/c²)

and a viewpoint in which B considers B to be at rest and is considering how the universe looks from A's perspective:

x_A = gamma.(x'_B - v.t'_B)
t_A = gamma.(t'_B - v.x'_B/c²)

Trying to keep it all straight in drawings, equations and words has been a bit of struggle even without the introduction of typos.

I'm hoping that these two responses go some way to giving you the answer to other questions you have posed (and perhaps negating some of the questions which are based on a misunderstanding or other uncertainty).

cheers,

neopolitan

 

[JesseM]

Just to let you know, right now I'm the one who's away on a trip, I'll get back to this sometime after I get home in about a week.

 

[neopolitan]

Just to let you know, right now I'm the one who's away on a trip, I'll get back to this sometime after I get home in about a week.

Enjoy your trip then

cheers,

neopolitan

 

[neopolitan]

From another thread:

Is it perhaps worthwhile to make sure everyone knows what proper time, proper length, coordinate time and coordinate length are?

http://en.wikipedia.org/wiki/Proper_time" - In relativity, proper time is time measured by a single clock between events that occur at the same place as the clock.

http://en.wikipedia.org/wiki/Proper_distance" - In relativistic physics, proper length is an invariant quantity which is the rod distance between spacelike-separated events in a frame of reference in which the events are simultaneous.

http://en.wikipedia.org/wiki/Coordinate_time" - In the theory of relativity, it is convenient to express results in terms of a spacetime coordinate system relative to an implied observer. An event is specified by one time coordinate and three spatial coordinates. The time measured by the time coordinate is referred to as coordinate time to distinguish it from proper time.

Coordinate distance is not described on wikipedia but we can extrapolate thus - In the theory of relativity, it is convenient to express results in terms of a spacetime coordinate system relative to an implied observer. An event is specified by one time coordinate and three spatial coordinates. The distance measured by the spatial coordinates can be referred to as coordinate distance to distinguish it from proper distance.

Interpreting all of those is the (relatively) difficult part.

Proper time - say you have an inertial clock, elapsed time on that clock is proper time. (t')

Proper length (distance) - say you have an inertial rod, the rest length of the rod (ie where the ends of the rod are simultaneous) is proper length (or proper distance between the ends of the rod). (L)

Coordinate time - we have an implied observer, the time on the observer's clock when events take place is coordinate time. (t)

Coordinate distance - we have an implied observer, the distance between the observer and an event is coordinate distance. A coordinate length would be the delta between two events, for a rod that would mean the ends of that rod. (L')

The relationship between coordinate time and proper time is given by:

$$t'=\frac{t}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$$

The relationship between coordinate length (or distance) and proper length (or distance) is given by:

$$L'=L.\sqrt{1-\frac{v^{2}}{c^{2}}}$$

Is this perhaps the answer to my original question?

I want to say "the odd thing is that the speed in question is given by proper distance over coordinate time" but I hesitate for two reasons. Firstly, perhaps it is not so odd after all and secondly, while I can get my head around "proper distance over coordinate time" it might not be completely kosher.

Just in case it is not a standard thing, I would see proper distance as the distance between the ends of a rod traversed by an observed body where the rod is at rest with respect to the observer. The times at which the observed body is colocated with the respective ends of the rod as measured on the clock of the observer is the coordinate time. So, the observed body traversed a distance of $$L = \Delta x$$ in a period of $$\Delta t$$, which is the speed of the observed body.

Thoughts?

cheers,

neopolitan

 

[JesseM]

Hi again, I'm back now so I'll return to our discussion:

Try to put your brain into "pre-SR mode" which means you have to knowledge of a high school student who has paid enough attention to know that:

x' = x - vt

but you don't know anything more than that. That would give you the sort of knowledge that a person we are introducing to SR would have. Such a person won't have all the simultaneity issues you have, because they don't know enough to realize that there are simultaneity issues.

Then introduce the concept that if a photon is released from a distance of x away then it takes a period of t to reach you (remember you still don't know enough to realize there are simultaneity issues). Therefore:

x = ct and x' = x - vt

But are you assuming both that x'=x-vt and that the light moves at c in both frames? As I'm sure you'd agree, these two assumptions aren't compatible, so is your pedagogical point just to show that they aren't compatible? If so, wouldn't it be a little easier to start from the Newtonian velocity addition equation w = v + u (where u is the speed of an object in the rest frame of observer A, and observer A is moving at speed v in the same direction in the frame of observer B, and we want to know the speed w of the original object in the frame of observer B)? This follows in a pretty direct way from x' = x - vt and it should in any case be familiar to anyone who's familiar with the most basic ideas of Newtonian frames.

Now we know that the value t is not necessarily the same in both equations

Why not? If the student knows x' = x - vt he should also know that this equation relates the coordinates of a single event x,t in one frame to the coordinates x',t' of the same event in the other frame, or else it relates the coordinate intervals between a single pair of events in one frame to the coordinate intervals between the same pair of events in the other frame...in either case t' = t. If you're talking about doing something different, like having x be the distance between where the photon was released and where it hit the unprimed observer as measured in the unprimed frame, while x' is the distance between where the photon was released and where it hit the primed observer in the primed frame, then the equation x' = x - vt should not be used.

but say we specifically want to know where the photon release location is in relation to an object moving at v away from us towards where the photon was released ... when the photon reaches us.

Then, we want to know how things look in the rest frame of that object. That is, how far from the photon release point are we when the photon reaches the object which is moving towards the photon release point (relative to us).

x' = ct' and x' = x - vt' or x = x' + vt'

We still know nothing about the relativity of simultaneity nor have any idea that the photon release location is not universally agreed.

Do x,t and x',t' represent the coordinates of the single event of the photon being released in each frame? If so, what do you mean by "nor have any idea that the photon release location is not universally agreed"? Even in basic Newtonian mechanics the same event can have different position coordinates in two frames, that's the whole point of x' = x - vt.

So we can try to make sense of what we have so far.

x' = x - vt and x = x' + vt'

so x' = (x' + vt') - vt

so vt' = vt which means t = t' which we know can't be right.

Again, what do t and t' represent so that the student knows t = t' can't be right?

By "both viewpoints" I mean a viewpoint in which A considers A to be at rest and is considering how the universe looks from B's perspective:

x'_B = gamma.(x_A - v.t_A)
t'_B = gamma.(t_A - v.x_A/c²)

and a viewpoint in which B considers B to be at rest and is considering how the universe looks from A's perspective:

x_A = gamma.(x'_B - v.t'_B)
t_A = gamma.(t'_B - v.x'_B/c²)

But how does the difference between these two perspectives related to the difference between this:

On this diagram http://www.geocities.com/neopolitonian/g2ev2_2.jpg", I show all the values x' and x with subscripts A and B, plus vt'_A and -vt_B. This is starting from the Galilean perspective so:

x_A = distance from A to YDE at the time of YDE (simultaneous with colocation of A and B), according to A ( = 8)
x_B = distance from A to YDE at the time of YDE (not simultaneous with colocation of A and B), according to B ( = 10)
x'_A = distance from A to B when the photon from YDE passes B, according to A ( = 5)
x'_B = distance (at the time of colocation of A and B) from B to the photon which subsequently passes B ( = 4)
t'_A = time at which the photon passes B, according to A (measured from the time of colocation of A and B) ( = 5)
t_B = time at which the photon passes B, according to B (measured from the time of colocation of A and B) ( = 10)

Not in the diagram per se, but implied:

t_B = time between YDE and time that the photon from YDE passes A, according to A ( = 8)
t'_B = time between the colocation of A and B and the time that the photon from YDE passes B, according to B ( = 4)

...and this?

Going from the the spacetime diagram http://www.geocities.com/neopolitonian/generality6_all_values.jpg":

x_A = separation between A and event E_A (YDE), according to A ( = 8)
x_B = separation between A and event E_A (YDE) at the time of YDE, according to B ( = 10)
x'_A = separation between B and the location of event E_A (YDE) when the photon from YDE passes B, according to A ( = 5)
x'_B = separation between B and the location of event E_B (the photon from YDE passes the x_B axis) when the photon from YDE passes B, according to B ( = 4)
t'_A = the time between colocation of A and B and the photon passing B, according to A ( = 5)
t_B = the time between when YDE occurred and when the photon passes A, according to B ( = 10)
t_A = the time between colocation of A and B and the photon passing A, according to A ( = 8)
t'_B = the time between colocation of A and B and the photon passing B, according to B ( = 4)Not sure if you understand how much of a struggle it is to keep both viewpoints straight. I think I have it right, but there may be typos (late at night again).

Or was I misunderstanding, and these two different ways of defining things aren't meant to map to the two viewpoints you were talking about?

 

[JesseM]

From another thread:

Is it perhaps worthwhile to make sure everyone knows what proper time, proper length, coordinate time and coordinate length are?

http://en.wikipedia.org/wiki/Proper_time - In relativity, proper time is time measured by a single clock between events that occur at the same place as the clock.

Proper length (distance) - In relativistic physics, proper length is an invariant quantity which is the rod distance between spacelike-separated events in a frame of reference in which the events are simultaneous.

Coordinate time - In the theory of relativity, it is convenient to express results in terms of a spacetime coordinate system relative to an implied observer. An event is specified by one time coordinate and three spatial coordinates. The time measured by the time coordinate is referred to as coordinate time to distinguish it from proper time.

Coordinate distance is not described on wikipedia but we can extrapolate thus - In the theory of relativity, it is convenient to express results in terms of a spacetime coordinate system relative to an implied observer. An event is specified by one time coordinate and three spatial coordinates. The distance measured by the spatial coordinates can be referred to as coordinate distance to distinguish it from proper distance.

Interpreting all of those is the (relatively) difficult part.

Proper time - say you have an inertial clock, elapsed time on that clock is proper time. (t')

Proper length (distance) - say you have an inertial rod, the rest length of the rod (ie where the ends of the rod are simultaneous) is proper length (or proper distance between the ends of the rod). (L)

Coordinate time - we have an implied observer, the time on the observer's clock when events take place is coordinate time. (t)

Coordinate distance - we have an implied observer, the distance between the observer and an event is coordinate distance. A coordinate length would be the delta between two events, for a rod that would mean the ends of that rod. (L')

The relationship between coordinate time and proper time is given by:

$$t'=\frac{t}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$$

The relationship between coordinate length (or distance) and proper length (or distance) is given by:

$$L'=L\sqrt{1-\frac{v^{2}}{c^{2}}}$$

Do you mean t' to be the proper time between a pair of events on the clock's own worldline as measured by that clock, while t is the coordinate time between those same events? If so you have the equation backwards, it should be:

$$t = \frac{t'}{\sqrt{1 - v^2/c^2}}$$

Of course the usual convention is to have unprimed t be proper time between events on the clock's worldline and primed t' be coordinate time between these same events in the frame where the clock is moving, so with that convention your equation above would be right.

Also, I'm confused by your definition of "coordinate distance"--are you talking about the delta-x' in the primed frame between a single pair of events which are simultaneous in the unprimed frame (so you're looking at the coordinate distance in the primed frame between events that are not simultaneous in the primed frame), or are you talking about the length in the primed frame of the same rod whose proper length you measured in the unprimed frame (with the understanding that 'length in the primed frame' means the coordinate distance between ends of the rod at a single instant in the primed frame)? Your length equation above only works under the second interpretation.

I want to say "the odd thing is that the speed in question is given by proper distance over coordinate time" but I hesitate for two reasons. Firstly, perhaps it is not so odd after all and secondly, while I can get my head around "proper distance over coordinate time" it might not be completely kosher.

v is defined in terms of the difference in position coordinate interval over difference in time coordinate interval in the unprimed frame for a pair of events on the worldline of an object at rest in the primed frame (like two events which occur at the origin of the primed coordinate system at different times).

Just in case it is not a standard thing, I would see proper distance as the distance between the ends of a rod traversed by an observed body where the rod is at rest with respect to the observer.

But of course this is the same as the coordinate distance between the event of the body passing one end and the event of the body passing the other end in this frame, since in the frame where the rod is at rest its proper length is equal to the coordinate distance between one end and the other end. Also, the term "proper distance" has a slightly different meaning than "proper length", since "proper length" refers to the length of some physical object like a rod in its own rest frame, while "proper distance" refers to taking a specific pair of spacelike-separated events and looking at the distance between them in the frame where they are simultaneous (which means if each event takes place on the end of a rod which is at rest in the frame where they are simultaneous, the proper distance between events is the same as the proper length of the rod).

 

[neopolitan]

It seems I overextended and messed up in the process. Now looking at it, I can't remember what I was thinking at the time, but the end result is certainly wrong when it comes to time.

The point remains that it would be helpful to clarify what proper time and coordinate time mean. It would just be good to do a better job of it than I did.

cheers,

neopolitan

 

[neopolitan]

But are you assuming both that x'=x-vt and that the light moves at c in both frames? As I'm sure you'd agree, these two assumptions aren't compatible, so is your pedagogical point just to show that they aren't compatible? If so, wouldn't it be a little easier to start from the Newtonian velocity addition equation w = v + u (where u is the speed of an object in the rest frame of observer A, and observer A is moving at speed v in the same direction in the frame of observer B, and we want to know the speed w of the original object in the frame of observer B)? This follows in a pretty direct way from x' = x - vt and it should in any case be familiar to anyone who's familiar with the most basic ideas of Newtonian frames.

Sure x'=x-vt and speed of light = c are compatible. You can only say they are incompatible if you have defined t to be something which makes them incompatible, which makes your question below a bit odd, because you are telling me you don't know what t is.

After a period of t (in the unprimed frame) the unprimed observer receives a photon. At t=0, that photon would have been a distance of x=ct away from the unprimed observer. At t=t, the primed observer, moving at v relative to the unprimed observer, is a distance of vt from the unprimed observer (assuming that at t=0 they were colocated, otherwise we just say the extra separation since t=0 is given by vt). According to the unprimed observer, the separation between where the primed observer is now, and where the photon was at t=0, is x'=x-vt.

Happy?

I said the student paid enough attention to know the galilean equation x'=x-vt. You can't introduce a new equation from Newton and say start there. You are welcome to do that as your own proposal, if you like.

Why not? If the student knows x' = x - vt he should also know that this equation relates the coordinates of a single event x,t in one frame to the coordinates x',t' of the same event in the other frame, or else it relates the coordinate intervals between a single pair of events in one frame to the coordinate intervals between the same pair of events in the other frame...in either case t' = t. If you're talking about doing something different, like having x be the distance between where the photon was released and where it hit the unprimed observer as measured in the unprimed frame, while x' is the distance between where the photon was released and where it hit the primed observer in the primed frame, then the equation x' = x - vt should not be used..

Actually, if a student knows about x' = x - vt, he or she should know about all the other conditions (for example, at t=0 the observers are colocated).

I explained x' above, but if you define it the way you have proposed - a way which doesn't really make sense - you are right, x' = x - vt should not be used.

Do x,t and x',t' represent the coordinates of the single event of the photon being released in each frame? If so, what do you mean by "nor have any idea that the photon release location is not universally agreed"? Even in basic Newtonian mechanics the same event can have different position coordinates in two frames, that's the whole point of x' = x - vt.

I just mean that we do not know about the rotation that happens with relativity, so as far as the student knows, if x' = x - vt then x = x' + vt'. (And in the very next paragraph of the original post, I show this can't be the case, so don't rush into knocking that straw man over.)

Again, what do t and t' represent so that the student knows t = t' can't be right?.

I had just explained in the original post that x'=ct' so you know that t' = x/c.

In the paragraphs before that, I explained what t is, specifically, x=ct so t=x/c.

I'm struggling to see what is confusing about this. Are you trying to jump forward, despite my request to try putting your brain into "pre-SR mode"?

But how does the difference between these two perspectives related to the difference between this:

...and this?

Or was I misunderstanding, and these two different ways of defining things aren't meant to map to the two viewpoints you were talking about?

There is at least one typo in what you quoted. I think you are confused anyway.

There are two perspectives in the equations I showed in the most recent post, there are two frames, one perspective is that one frame is at rest, the other perspective is that the other frame is at rest. I even wrote that. I am really not sure what your confusion is here.

Please try to go from the question posed by your statement "I'm confused by what you mean when you say "both viewpoints"", to the answer I provided where I specifically described the two viewpoints. Do you understand what I meant originally by "both viewpoints"? Does it help if I tell you that I meant viewpoint to have the same meaning as "perspective" in this context?

cheers,

neopolitan

 

[JesseM]

Sure x'=x-vt and speed of light = c are compatible.

It's not compatible with the speed of light being c in both frames--do you disagree?

You can only say they are incompatible if you have defined t to be something which makes them incompatible, which makes your question below a bit odd, because you are telling me you don't know what t is.

Not if you mean x'=x-vt to be part of the Galilei transformation where you are assigning coordinates to particular events and you also know that for any given event, t'=t. If you are suggesting that somehow the student wasn't even paying enough attention to understand the physical meaning of x'=x-vt, and just knows that the equation exists without knowing how it is actually used in the context of the Galilei transformation, then it seems like a rather bizarre pedagogical approach to cater to this one particular student with a very idiosyncratic misunderstanding as opposed to the typical student who can at least be expected to know the physical meaning of any equation he wants to use.

After a period of t (in the unprimed frame) the unprimed observer receives a photon.

A period of t between the photon emission event and the event of the photon passing the unprimed observer, presumably? In this case, if the student knows the Galilei transformation he'll also know that the period in the primed frame between these same two events is t'=t. Using this along with x'=x-vt, he'll find that the photon's distance/time in the primed frame was not c but rather c+v.

At t=0, that photon would have been a distance of x=ct away from the unprimed observer. At t=t, the primed observer, moving at v relative to the unprimed observer, is a distance of vt from the unprimed observer (assuming that at t=0 they were colocated, otherwise we just say the extra separation since t=0 is given by vt). According to the unprimed observer, the separation between where the primed observer is now, and where the photon was at t=0, is x'=x-vt.

I don't understand that last sentence, why would you say that the separation between where the primed observer is now and where the photon was at t=0 should be x'=x-vt? The equation x'=x-vt tells you the x' coordinate of an event with coordinates x,t in the unprimed frame, or the distance interval x' between a pair of events which have a distance and time interval of x and t in the unprimed frame, but you don't seem to be dealing with either type of question here. It's true of course that if you pick some fixed position x in the unprimed frame (like the position of the photon at t=0), and want to find the separation between the unprimed observer and that position at time t, then the answer will be x-vt, but it doesn't really make any sense to me to see this as an application of the Galilei transformation since you aren't even considering the primed frame here, and for that reason I also don't understand what it would mean to set this equal to x' if you're just calculating a separation in the unprimed frame. Note that it is also true that in SR if you had an observer moving at speed v (and located at the origin at t=0), and wanted to know the separation between this observer and some fixed position x at time t, then the answer would still be x-vt, in spite of the fact that the coordinate transformation equation x'=x-vt is wrong in SR. I think maybe you're getting confused by the superficial similarity between the Galilean coordinate transformation equation relating two different frames, namely x'=x-vt, and the equation for calculating the separation between an object moving at v and a fixed position x in the context of a single inertial frame, namely x-vt. The second does look like the right-hand side of the first but the physical meaning of what the equations are supposed to calculate is different.

Also, nowhere in the above paragraph do you calculate the distance/time for the light in the primed frame--again, when you said one of your assumptions was "speed of light = c" were you not talking about the assumption that the speed of light should be equal c in all inertial frames?

Actually, if a student knows about x' = x - vt, he or she should know about all the other conditions (for example, at t=0 the observers are colocated).

And one of the other conditions is that t'=t, yes?

I explained x' above, but if you define it the way you have proposed - a way which doesn't really make sense - you are right, x' = x - vt should not be used.

Why do you say "a way which doesn't really make sense"? Do you disagree that the standard interpretation of the Galilei transformation is that it either relates the coordinates of a single event in two different frames, or that it relates the distance and time intervals between a single pair of events in two different frames? If you're not addressing one of these questions then you shouldn't label whatever equations you use the "Galilei transformation".

Do x,t and x',t' represent the coordinates of the single event of the photon being released in each frame? If so, what do you mean by "nor have any idea that the photon release location is not universally agreed"? Even in basic Newtonian mechanics the same event can have different position coordinates in two frames, that's the whole point of x' = x - vt.

I just mean that we do not know about the rotation that happens with relativity, so as far as the student knows, if x' = x - vt then x = x' + vt'. (And in the very next paragraph of the original post, I show this can't be the case, so don't rush into knocking that straw man over.)

You didn't really answer my question at all, nor was I intending to debate the idea that x = x' + vt', I don't know how you got that from that question. My question was about the physical meaning of the symbols x, t, x', and t'. If you are using the Galilei transformation, these should either represent coordinates of a single event, or coordinate intervals between a single pair of events; if one of those is the case, please specify the event or events in question. If you are not using the symbols this way, then whatever you are doing cannot be seen as an application of the Galilei transformation, even if the equations you happen to use might look superficially similar like x-vt for the separation between an object moving at v and a fixed position x.

I had just explained in the original post that x'=ct' so you know that t' = x/c.

In the paragraphs before that, I explained what t is, specifically, x=ct so t=x/c.

Again, you're just giving equations without telling me their physical meaning in terms of specific events. When you write x=ct, do x and t represent the distance and time intervals in the unprimed frame's coordinates between the event of the photon being emitted and the event of the photon passing the unprimed observer? If so then we know by the Galilei transform that the distance between this same pair of events in the primed frame is x'=x-vt (and substituting x=ct back into this gives x'=ct-vt), and the time between this same pair of events in the primed frame is t'=t. But then you write x'=ct' which is incompatible with this.

I'm struggling to see what is confusing about this. Are you trying to jump forward, despite my request to try putting your brain into "pre-SR mode"?

No, I'm not talking about SR at all, just about the physical meaning of the Galilei transformation equations.

There is at least one typo in what you quoted. I think you are confused anyway.

There are two perspectives in the equations I showed in the most recent post, there are two frames, one perspective is that one frame is at rest, the other perspective is that the other frame is at rest. I even wrote that. I am really not sure what your confusion is here.

I don't understand how the two different blocks of equations I quoted correspond in any way to the two different rest frames of A and B. Each block of equations internally seems to contain both perspectives (for example, the first block defines t'_A as a time 'according to A' and t_B as a time 'according to B'), it's not like the first block shows only values calculated from the perspective of A's frame and the second shows only values calculated from the perspective of B's frame. So how does the difference between the two blocks relate to the difference between the two perspectives (A's rest frame and B's rest frame)? I don't see any connection at all.

 

[neopolitan]

I'm breaking this up, because it may get (even more) confusing.

The least important bit first - from my perspective.

I don't understand how the two different blocks of equations I quoted correspond in any way to the two different rest frames of A and B. Each block of equations internally seems to contain both perspectives (for example, the first block defines t'_A as a time 'according to A' and t_B as a time 'according to B'), it's not like the first block shows only values calculated from the perspective of A's frame and the second shows only values calculated from the perspective of B's frame. So how does the difference between the two blocks relate to the difference between the two perspectives (A's rest frame and B's rest frame)? I don't see any connection at all.

Each block of equations internally contains both perspectives, yes. Why each block should represent different perspectives as a whole, I have no idea. I can't really answer your question because it wasn't a position I was taking.

What I can say is that my original comment followed references to two diagrams (http://www.geocities.com/neopolitonian/g2ev2_2.jpg") which are two ways of looking at the same situation. I only introduced the second as part of a long discussion with you to show the physical meaning of the terms in a way that you would understand, and possibly accept. The first is what I had originally, and I would like to stick with that until (and if) we ever get to the point where we can progress further.

But the point is that both diagrams show the same thing, exactly the same thing, in a different way - different "viewpoints" (a pair of galilean-like diagrams and one spacetime diagram) on the same scenario which incorporates both perspectives (primed and unprimed).

The issue I was having at the time, was making sure that when I transitioned from one way at looking at the scenario to another, and back again, I didn't mess up with subscripts and prime notations.

cheers,

neopolitan

 

[JesseM]

Each block of equations internally contains both perspectives, yes. Why each block should represent different perspectives as a whole, I have no idea. I can't really answer your question because it wasn't a position I was taking.

OK, thanks. I wasn't assuming that the two blocks mapped to the two "perspectives" you talked about, I just wondered if that was the case (and I presented it as a question, saying 'But how does the difference between these two perspectives related to the difference between this [first block] ...and this? [second block] Or was I misunderstanding, and these two different ways of defining things aren't meant to map to the two viewpoints you were talking about?') You had presented the two blocks and then immediately said "Not sure if you understand how much of a struggle it is to keep both viewpoints straight", so it seemed natural to think that "both viewpoints" might refer to the difference between the two blocks.

What I can say is that my original comment followed references to two diagrams (http://www.geocities.com/neopolitonian/g2ev2_2.jpg") which are two ways of looking at the same situation. I only introduced the second as part of a long discussion with you to show the physical meaning of the terms in a way that you would understand, and possibly accept. The first is what I had originally, and I would like to stick with that until (and if) we ever get to the point where we can progress further.

If you'd like to stick to the first diagram, which I find more confusing, could you lay out specifically what each of the colored dots is supposed to represent? I asked about this earlier when I said:

I'm unclear what the different dots represent in that first diagram. You have x'_B as the distance from the orange dot to the yellow dot in the right-hand drawing from B's perspective, and we know that x'_B can be defined either as the distance from the event of A&B being colocated to the E_B, or it can be defined as the distance from E_C (bottom dot on parallelogram) to the YDE. So does the orange dot represent E_C, or have you redefined the meaning of the yellow dot to mean E_B in this picture? If the former I can't figure out what the purple dot would be (what event to we know to be a distance of vt_B from E_C?), but if the latter then I suppose it's the position of A at the time t_B when the light reaches it. But in this case, I don't see why you label the distance from the purple dot to the yellow dot (which would really be E_B rather than the YDE) as x_B unless you are reverting to the old definition of x_B (distance between E_B and event of light passing A) as opposed to the newer one (distance between B and YDE).

No need to address this right away if you want to deal with other issues first, just whenever you want to return the discussion to that first diagram.

 

[neopolitan]

If you'd like to stick to the first diagram, which I find more confusing, could you lay out specifically what each of the colored dots is supposed to represent?

The diagram in question is http://www.geocities.com/neopolitonian/g2ev2_2.jpg".

There are four dots:

Yellow Dot - location of an event associated with a photon, it could be the emission of a photon, or a colocated with a photon which just happened to be passing - we'd not be able to tell the difference. In the text above the diagram, I call this "the Yellow Dot Event" or YDE

Orange Dot - location of B and photon when B and photon are colocated

Green Dot - location of A when B and photon are colocated (according to A) - the logic is this:

in A's rest frame, A is at rest and B is in motion towards the YDE with speed of v. The time at which B and the photon are colocated is a period of t'_a after colocation (nominally t=t'=0). Therefore, in a period of t'_a, B must have moved a distance of vt'_a towards the YDE and the photon has traveled a distance of ct'_a towards B. Since the photon and B are colocated at this time, simple additon gives x_a = ct'_a + vt'_a = x'_a + vt'_a.

Purple Dot - location of A when A and photon are colocated (according to B) - the logic is similar to above

in B's rest frame, B is at rest and A is in motion away from the YDE with speed of v. The time at which A and the photon are colocated is a period of t_b after colocation (nominally t=t'=0). Therefore, in a period of t_b, A must have moved a distance of vt_b away from the YDE and the photon has traveled a distance of ct_b towards A. Since the photon and A are colocated at this time, simple additon gives x'_b = ct_b - vt_b = x_b + vt_b.

If I've done it right, "towards" is exchanged with "away from", A is exchanged with B and vice versa, a is exchanged with b and vice versa and primed is exchanged with unprimed and vice versa.

Note that in A's rest frame, the distance between the location of YDE and A does not change - therefore x_a does not change with time, but x'_a does (because x'_a is the distance between the location of YDE and B, according to A).

Similarly, note that in B's rest frame, the distance between the location of YDE and B does not change - therefore x'_b does not change with time, but x_b does (because x_b is the distance between the location of YDE and A, according to B).

Can you reply first with whether or not you understand this explanation. If you don't please explain specifically what it is that you don't understand. If you do understand and think it is wrong, then by all means explain where it is wrong, but first state that you understand. Thanks.

neopolitan

PS Please also answer only in terms of the diagram and this explanation, do you understand this explanation of the diagram I linked? Yes or no.

We can tie it up with the spacetime diagram and my earlier attempts to explain at a later date, if we get that far.

 

[JesseM]

The diagram in question is http://www.geocities.com/neopolitonian/g2ev2_2.jpg".

There are four dots:

Yellow Dot - location of an event associated with a photon, it could be the emission of a photon, or a colocated with a photon which just happened to be passing - we'd not be able to tell the difference. In the text above the diagram, I call this "the Yellow Dot Event" or YDE

But not just any event associated with the photon, right? Aren't you still assuming the YDE is the specific event on the photon's worldline that occurs at the same time as A&B being colocated in A's frame?

Orange Dot - location of B and photon when B and photon are colocated

Green Dot - location of A when B and photon are colocated (according to A)

OK, so this occurs at time t'_A in A's frame, which in the numerical example is equal to 5.

Purple Dot - location of A when A and photon are colocated (according to B) - the logic is similar to above

But is the meaning of the yellow dot unchanged in this right-hand diagram, or does the yellow dot now refer to the event on the photon's worldline that occurs at the same time as A&B being colocated in B's frame? Of course if we used Galilean frames this would be the same event as the one that occurred simultaneously with A&B being colocated in A's frame, but the Galilei transformation would be inconsistent with the actual numbers you gave for some of these quantities, and as I said it would also be inconsistent with the idea that the photon moves at c in both frames.

Note that in A's rest frame, the distance between the location of YDE and A does not change - therefore x_a does not change with time, but x'_a does (because x'_a is the distance between the location of YDE and B, according to A).

Isn't x'_A the distance between the YDE and a specific event on B's worldline indicated by the orange dot, namely the event of the photon passing B? If so it wouldn't change with time.

 

[neopolitan]

But not just any event associated with the photon, right? Aren't you still assuming the YDE is the specific event on the photon's worldline that occurs at the same time as A&B being colocated in A's frame?

Not really. I am saying that A and B colocated is one event and the YDE is another event. A certain time later a photon from that event passes B and then A, which both constitute events (photon colocated with B, photon colocated with A). Then I work with those.

Consider, A works out that, if t_a after colocation of A and a photon passes A, then at colocation of A and B that photon was c.t_a distant. This may have been the spawning of that photon - simultaneous with colocation of A and B in A's frame - (it doesn't have to be) and if it was, then it was a unique event (and even if it wasn't spawning of the photon, the photon's location is still a unique event) and the spacetime interval between A and B colocated and the photon's location simultaneous with colocation of A and B in A's frame is invariant. The same applies for B (ie thinking about the location of the photon simultaneous with the colocation of A and B, which could be the spawning of that photon but doesn't have to be).

Using a different coordinate system does not move the event.

But is the meaning of the yellow dot unchanged in this right-hand diagram, or does the yellow dot now refer to the event on the photon's worldline that occurs at the same time as A&B being colocated in B's frame? Of course if we used Galilean frames this would be the same event as the one that occurred simultaneously with A&B being colocated in A's frame, but the Galilei transformation would be inconsistent with the actual numbers you gave for some of these quantities, and as I said it would also be inconsistent with the idea that the photon moves at c in both frames.

I didn't give numbers so I don't know what you are talking about.

PS Please also answer only in terms of the diagram and this explanation, do you understand this explanation of the diagram I linked? Yes or no.

I repeat, YDE is an event which could be the spawning of a photon, A and B colocated is an event, photon colocated with B is an event, photon colocated with A is an event.

Try not to focus on simultaneity (it might be tough, since that seems to be your preferred avenue into relativity).

Think: YDE happens and colocation of A and B happens (not necessarily in that order, and not necessarily together), colocation of photon and B happens, colocation of photon and A happens.

There are two events colocated with A, ie colocation with B and colocation with the photon. That gives A a time, t_a and a distance (to photon when A and B were colocated) x_a. Using x'_a=ct'_a, A can also work out where and when B and the photon were colocated (in A's frame, if A were inclined to think in such terms) ... x'_a = x_a - vt'_a. In A's rest frame, the distance between where the photon was when A and B were colocated and A does not change. Note that it is this apparently unchanging distance that is the subject of one of my final equations (on a later drawing - http://www.geocities.com/neopolitonian/g2ev2_3.jpg"). This is silvered out because it is an aside. Just note it, I am not trying to prove it at this time.

The same goes for B (being careful with primes): B has a time t'_b and a distance (to photon when A and B were colocated) x'_b. Using x_b=ct_b, B can also work out where and when A and the photon will be colocated (in B's frame, if B were inclined to think in such terms) ... x_B = x'_b + vt_b. In B's rest frame, the distance between where the photon was when A and B were colocated and B does not change. Note that it is this apparently unchanging distance that is the subject of one of my final equations (on a later drawing).

Isn't x'_A the distance between the YDE and a specific event on B's worldline indicated by the orange dot, namely the event of the photon passing B? If so it wouldn't change with time.

I was unclear, I mean that as far as A is concerned, the separation between the location of the YDE and the location of A is fixed but the separation between the location of the YDE and the location of B is not fixed. My thinking, at that precise moment, was that x'_a = the separation (hence the x) between B and the YDE (hence the prime) according to A (hence the _a), which varies with time.

The selection of two specific times (photon passes B and photon passes A) is handy, but not essential. In the diagram the values apply for the moment depicted so x'_a is really x'_a(t'_a) for a very specific t'_a (when B and the photon are colocated according to A). There's nothing stopping you from looking at another time when B and the photon are not colocated and x'_a would still be the separation (x) between B and the location of YDE (') according to A (_a).

Does that make sense to you?

cheers,

neopolitan

 

[JesseM]

But not just any event associated with the photon, right? Aren't you still assuming the YDE is the specific event on the photon's worldline that occurs at the same time as A&B being colocated in A's frame?

Not really. I am saying that A and B colocated is one event and the YDE is another event. A certain time later a photon from that event passes B and then A, which both constitute events (photon colocated with B, photon colocated with A). Then I work with those.

Consider, A works out that, if t_a after colocation of A and a photon passes A, then at colocation of A and B that photon was c.t_a distant. This may have been the spawning of that photon - simultaneous with colocation of A and B in A's frame - (it doesn't have to be) and if it was, then it was a unique event (and even if it wasn't spawning of the photon, the photon's location is still a unique event)

I didn't say anything about the "spawning" of the photon, I just asked whether it was part of the definition of the YDE in the left-hand diagram that it is the specific event on the photon's worldline that is simultaneous with A&B being colocated as defined in A's frame. Can I take it from this reply that the answer is "yes"? Likewise, can I assume that it's part of the definition of the YDE in the right-hand diagram that it is the specific event on the photon's worldline that is simultaneous with A&B being colocated as defined in B's frame?

Using a different coordinate system does not move the event.

It does if the different coordinate systems disagree about simultaneity. If you use Galilean coordinate systems then they won't disagree about simultaneity so it'll be the same event, but then it is easy to show using the Galilei transform that the photon did not move at c in both frames. For example, say in A's frame the YDE occurred at x=ct_a, t=0, and the orange dot event occurred at x=vt'_a, t=t'_a, then if we assume the light was moving at c in this frame the relation between t_a and t'_a must be c = (ct_a - vt'_a)/t'_a, so multiplying both sides by t'_a gives ct'_a = ct_a - vt'_a which tells us that t_a = (ct'_a + vt'_a)/c. So, in A's frame the coordinates of the YDE are:

x = (ct'_a + vt'_a), t=0

and the coordinates of the orange dot are:

x = vt'_a, t = t'_a

Then if we want to find the coordinates of these same events in B's frame under the assumption that A and B's coordinates are related by the Galilei transformation, this would give the coordinates of the yellow dot event as:

x' = (ct'_a + vt'_a), t'=0

And the coordinates of the orange dot as:

x' = vt'_a - vt'_a = 0, t = t'_a

So in this case the light has traveled a distance of (ct'_a + vt'_a) in a time of t'_a, meaning its speed was (c + v) in this frame.

I didn't give numbers so I don't know what you are talking about.

Probably you do know what I'm talking about and are just trying to tell me that you want me to forget the numerical example that you had previously used to give values to symbols like x'_a and t_a (as in the blocks of text I quoted earlier). If so that's fine, but even without numbers, as long as we can assign abstract coordinates to the yellow dot and the orange dot as I did above, then it should be possible to show that the speed of light cannot be c in both frames if their coordinates are related by the Galilei transform.

Try not to focus on simultaneity (it might be tough, since that seems to be your preferred avenue into relativity).

Think: YDE happens and colocation of A and B happens (not necessarily in that order, and not necessarily together), colocation of photon and B happens, colocation of photon and A happens.

But you just said "Consider, A works out that, if t_a after colocation of A and a photon passes A, then at colocation of A and B that photon was c.t_a distant." Are you not assuming the YDE event is the same as the event of the photon being at a distance of c*t_a from A? If the YDE occurred "at colocation of A and B", presumably you mean at the same time that A and B were colocated, i.e. simultaneously with their being colocated.

There are two events colocated with A, ie colocation with B and colocation with the photon. That gives A a time, t_a

Gives A a time between what and what? I thought t_a represented the time between the YDE and the event of the photon being colocated with A (which is only the same as the time between the two events you mention if we assume the YDE is simultaneous with A and B being colocated), because I thought the YDE was supposed to occur at a distance of c*t_a from A. Is this incorrect?

and a distance (to photon when A and B were colocated) x_a.

But by talking about where the photon was "when A and B were colocated" you are using the concept of simultaneity. We don't have to get into the *relativity* of simultaneity of course, if we want to use the Galilei transform then simultaneity is non-relative. But in this case it's impossible that the light could move at c in both frames, as I keep saying.

I was unclear, I mean that as far as A is concerned, the separation between the location of the YDE and the location of A is fixed but the separation between the location of the YDE and the location of B is not fixed. My thinking, at that precise moment, was that x'_a = the separation (hence the x) between B and the YDE (hence the prime) according to A (hence the _a), which varies with time. The selection of two specific times (photon passes B and photon passes A) is handy, but not essential. In the diagram the values apply for the moment depicted so x'_a is really x'_a(t'_a) for a very specific t'_a (when B and the photon are colocated according to A).

OK, makes sense. You might consider changing the label next to the blue arrow from just x'_a to x'_a(t'_a), in order to make it consistent with the brown arrow whose label refers to the distance between A and B at the specific time t'_a.

 

[neopolitan]

I have been trying to get specific confirmation from you whether you understand the overall explanation and are nitpicking (or understand and feel that the explanation is wrong) or whether you still don't understand the explanation and are trying to get aspects explained so that you can understand (even if you may feel that the explanation is wrong). Is there any chance that you could address that in terms of the post where I specifically said:

Please also answer only in terms of the diagram and this explanation, do you understand this explanation of the diagram I linked? Yes or no

This quote indicates why I didn't understand your introduction of numbers.

Additionally, in the post that I gave the explanation of the diagram in question, #295, I got down to

so vt' = vt which means t = t' which we know can't be right

You keep banging on about the speed of light cannot be c in both frames. The thing is, the speed of light is c in all inertial frames, the initial framing of galilean relativity didn't specifically say it wasn't. What we find is that x' = x - vt and x = x' + vt' is not valid (which is pretty obvious, perhaps too obvious, we show the equations more because we use them in the next step and to help those for whom this fact is not so immediately obvious).

You are right if you mean that "the speed of light is c in all frames and x' = x - vt and x = x' + vt' and t' = t" is invalid. 100% But I never claim that.

I claim that "x' = x - vt and the speed of light is c is in all frames" is compatible, it's only a problem if you assert that "t' = t and x' in one frame = x' in another frame and x in one frame and x in another frame" is also valid. I introduce subscripts specifically because I know that this is not valid.

I didn't say anything about the "spawning" of the photon, I just asked whether it was part of the definition of the YDE in the left-hand diagram that it is the specific event on the photon's worldline that is simultaneous with A&B being colocated as defined in A's frame. Can I take it from this reply that the answer is "yes"? Likewise, can I assume that it's part of the definition of the YDE in the right-hand diagram that it is the specific event on the photon's worldline that is simultaneous with A&B being colocated as defined in B's frame?

The YDE is obviously key to you. I'm not totally fussed about where or when it is. There's a specific reason for this lack of concern.

First, the uncertainty about when and where the event is located comes later, once you get into the relativity of simultaneity.

Secondly, there are two sets of Lorentz transformations that you can arrive at, one pair from the perspective of A looking at B, and one pair from the perspective of B looking at A. We really only have to arrive at one pair.

I'm not worried that one pair might speaks about a YDE that is simultaneous in A's frame with A and B being colocated while the other speaks about a YDE that is simultaneous in B's frame with A and B being colocated. All I care about is whether the photon involved in each pair is the same photon, spawned by the same event.

Think about it. x_a is the separation between A and where the photon was at colocation of A and B in A's frame. x'_b is the separation between B and where the photon was at colocation of A and B in B's frame. Both are at rest in their own rest frames, so both consider that the other has a separation from that distant location that changes with time (x'_a and x_b respectively).

Taking just A's side of the story, A doesn't move, B does. The photon from YDE reaches A at a time t_a. That same photon passed B, and on B's clock at that time it said t'_b. According to A, that photon was at x_a when A and B were colocated. But according to B, that photon was at x'_b when A and B were colocated. The photon is the same. It's the photon that passes B and reaches A.

If you like, this not worring about the specific spacetime location of YDE is a little like a http://en.wikipedia.org/wiki/Lie-to-children" . If you want to focus heavily on the YDE and fix it in space and time, then I have to give you an overt "lie to children" and tell you it's the same event. Then, quite a bit later, we could go back, and show that the event that B thought was simultaneous with colocation of A and B was the not the same event that A thought was simultaneous with colocation of A and B. Which would be a useful introduction into the relativity of simultaneity.

The thing is, the average student being introduced to relativity would not be like you and want to know the precise spacetime location of the YDE. Can you understand that?

As an alternative, you could stop focusing on the YDE itself, and pay more attention to what I talked about in my last post which was the events colocated with each observer, from which the details of other events are extrapolated.

The events colocated with each observer are:

A: colocation with B, colocation of A and photon

B: colocation with A, colocation of B and photon

You asked:

Gives A a time between what and what? I thought t_a represented the time between the YDE and the event of the photon being colocated with A (which is only the same as the time between the two events you mention if we assume the YDE is simultaneous with A and B being colocated), because I thought the YDE was supposed to occur at a distance of c*t_a from A. Is this incorrect?

And said:

But by talking about where the photon was "when A and B were colocated" you are using the concept of simultaneity. We don't have to get into the *relativity* of simultaneity of course, if we want to use the Galilei transform then simultaneity is non-relative. But in this case it's impossible that the light could move at c in both frames, as I keep saying.

Addressing the second first, I'm not really using simultaneity. I'm using extrapolation.

If A and B are colocated and a photon passes A a period of t_a later, then A can extrapolate that the photon must have been at x_a=c.t_a when A and B were colocated.

If A and B are colocated and a photon passes B a period of t'_b later, then B can extrapolate that the photon must have been at x'_b=c.t'_b when A and B were colocated.

No explicit relativity of simultaneity.

And hopefully I have explained here why I am not fussed about the when and where of the YDE and what t_a is (between what and what).

Again, I very much want to get a feel whether you understand, but disagree or just don't understand. You might want to go back to the earlier explanation post https://www.physicsforums.com/showpost.php?p=2203097&postcount=295" with what I have subsequently tried to clarify.

cheers,

neopolitan

 

[JesseM]

I have been trying to get specific confirmation from you whether you understand the overall explanation and are nitpicking (or understand and feel that the explanation is wrong) or whether you still don't understand the explanation and are trying to get aspects explained so that you can understand (even if you may feel that the explanation is wrong). Is there any chance that you could address that in terms of the post where I specifically said:

Please also answer only in terms of the diagram and this explanation, do you understand this explanation of the diagram I linked? Yes or no

Didn't I do so in the purely symbolic notation above? Or do you disagree that in A's frame, the yellow dot event has coordinates x=ct_a and t=0, while the orange dot event has coordinates x=vt'_a and t=t'_a? (assuming we set the origin so the event of A and B being colocated has coordinates x=0 and t=0). If you think this is wrong, then I guess my answer would have to be "no", since in that case I don't really understand what t_a and t'_a are supposed to represent physically (I thought they were the time coordinates in A's frame of the photon passing A and B respectively).

Additionally, in the post that I gave the explanation of the diagram in question, #295, I got down to

so vt' = vt which means t = t' which we know can't be right

And as I said in my responses to that post, the meaning you were assigning these symbols was unclear to me...when you wrote "x = ct and x' = x - vt", what physically do x, t, and x' represent? This is especially confusing because x' = x - vt looks like the spatial component of the Galilei transform, a general equation that holds for arbitrary events that have coordinates x,t in one frame and x',t' in the other, whereas x = ct is clearly not a general relation that is supposed to apply to arbitrary events. If you want to refer to the positions and times of specific events as opposed to relationships that are supposed to hold between arbitrary sets of coordinates, it's really helpful if you put some subscripts to indicate this, and actually name in English the specific events they are the coordinates of (or the pairs of events that they represent distance and time intervals between). For example, if look at the following two events:

1. The event on the photon's worldline that occurs at t=0 in the frame of the unprimed observer, when he is colocated with the primed observer
2. The event of the photon being colocated with the unprimed observer

...then if x_a is used to denote the spatial coordinate of event #1 and t_a is used to denote the temporal coordinate of event #2, it would indeed be true that x_a = ct_a. However, since these are the coordinates of two different events, we cannot plug x_a and t_a in for x and t in the equation x' = x - vt to get the space coordinate of either event in the primed frame, since if you want the left side of this equation to be the primed space coordinate of a particular event, you have to plug in the x and t coordinates of that same single event in the right side.

On the other hand, the Galilei transform also gives us the equation dx' = dx - v*dt for the intervals between a pair of events, so if event #1 has coordinates (x=x_a, t=0) and event #2 has coordinates (x=0, t=t_a), then subtracting the coordinates of the first from the second gives dx = -x_a and dt = t_a. Then it would be valid to plug this value for dx and dt into the equation dx' = dx - v*dt to find the spatial interval in the primed frame between event #1 and event #2 above.

Either way, as I said in post #303, it only makes sense to use the Galilean equation x' = x - vt when you have a specific event that you know the x and t coordinates of in the unprimed frame, or a specific pair of events that you know the distance and time intervals between in the unprimed frame. The Galilei transformation equation x' = x - vt has no meaning outside of one of those specific contexts. You never addressed my post #303 so I don't know if you agree or disagree with this, if you disagree please say so.

You are right if you mean that "the speed of light is c in all frames and x' = x - vt and x = x' + vt' and t' = t" is invalid. 100% But I never claim that.

I claim that "x' = x - vt and the speed of light is c is in all frames" is compatible, it's only a problem if you assert that "t' = t and x' in one frame = x' in another frame and x in one frame and x in another frame" is also valid. I introduce subscripts specifically because I know that this is not valid.

Again, it would help if you would address post #303. What does the equation x' = x - vt mean if it isn't being written in the context of the full Galilei transformation? As I said in that post, it didn't really make sense to me when you wrote "According to the unprimed observer, the separation between where the primed observer is now, and where the photon was at t=0, is x'=x-vt", because the x' seemed superfluous here...you were just calculating the separation between the primed observer's position at time t and and the position "where the photon was at t=0", a calculation expressed entirely in terms of the unprimed frame, so the answer should just be x-vt, an equation that has nothing specifically to do with the Galilei transformation because it doesn't deal with multiple frames (the answer would still be x-vt in SR after all, something I also pointed out in post 303). Unless of course you were totally redefining the meaning of x' here, so that it no longer had jack squat to do with the coordinates of anything in the primed observer's own rest frame, but just was being used as a variable x'(t) to refer to the distance in the same unprimed frame between the primed observer and the position where the photon had been at t=0. But in this case it would be very strange to introduce the equation x'=x-vt without mentioning that the physical meaning of x' is totally different from what it means in the Galilei transformation which is the only context this equation would appear in physics books.

The YDE is obviously key to you. I'm not totally fussed about where or when it is. There's a specific reason for this lack of concern.

It's not the YDE specifically that's key to me, I just want to know the space and time coordinates of all three events (expressed in abstract rather than numerical terms is fine), otherwise the diagram and the terms don't seem very well-defined to me. In the left-hand diagram, do you agree or disagree that if the event of A and B being colocated is assigned coordinates x=0 and t=0, then x_a represents the position of the photon at t=0, t_a represents the time the photon passes A at x=0, and t'_a represents the time the photon passes B at x=vt'_a? That's all I want to know about the left-hand diagram.

First, the uncertainty about when and where the event is located comes later, once you get into the relativity of simultaneity.

I'm not talking about the relativity of simultaneity here, which involves multiple frames, just about whether the YDE is simultaneous with the event of A and B being colocated in any individual frame; as above, if they are colocated at t=0 in this frame, then does the YDE represent the event on the photon's worldline which also occurs at t=0? Or are you saying it would make no difference to you if we defined the YDE to be an event on the photon's worldline which occurred at some totally different time in this frame, say at t=-1000*t_a?

Secondly, there are two sets of Lorentz transformations that you can arrive at, one pair from the perspective of A looking at B, and one pair from the perspective of B looking at A. We really only have to arrive at one pair.

I never understand what your "looking at" terminology means, but presumably you refer to difference between a set of equations that takes as inputs the coordinates of an event in the A frame and gives as outputs the coordinates of the same event in the B frame, vs. a set of equations that takes B-coordinates as inputs and gives A-coordinates as outputs. Which of these sets corresponds in your terminology to A looking at B vs. B looking at A I'm not sure.

I'm not worried that one pair might speaks about a YDE that is simultaneous in A's frame with A and B being colocated while the other speaks about a YDE that is simultaneous in B's frame with A and B being colocated. All I care about is whether the photon involved in each pair is the same photon, spawned by the same event.

Huh? These equations wouldn't "speak" about any event in particular, they relate the coordinates of any arbitrary event in one frame to the coordinates of the same event in the other frame, but either way it is necessary that you have a specific physical event in mind. And if you're going to define terms like t_a or x_a in terms of relationships between specific events, you have to clearly specify what the events are, or else your terms aren't well-defined.

Think about it. x_a is the separation between A and where the photon was at colocation of A and B in A's frame.

OK, then what you've just said is that x_a is defined as the position coordinate of the event on the photon's worldline that is simultaneous with the colocation of A and B in A's frame.

x'_b is the separation between B and where the photon was at colocation of A and B in B's frame.

And here you've said that x'_b is defined as the position coordinate of the event on the photon's worldline that is simultaneous with the colocation of A and B in B's frame. These are perfectly good ways of defining x_a and x'_b in terms of coordinates of specific events, and that's all I was asking for. We don't have to worry (yet) about whether the event on the photon's worldline used in the first definition is identical to or different from the event o the photon's worldline used in the second definition.

Both are at rest in their own rest frames, so both consider that the other has a separation from that distant location that changes with time (x'_a and x_b respectively).

Sure (although again, if certain symbols are going to be variables as opposed to be constants, it would be helpful if you'd indicate them as such using notation like x'_a(t) and x_b(t)...or maybe it'd be x_b(t'), I dunno, this is another confusing aspect of your notation since you don't seem to follow the convention that unprimed terms always refer to coordinates in the first frame and primed terms always refer to coordinates in the second frame)

Taking just A's side of the story, A doesn't move, B does. The photon from YDE reaches A at a time t_a. That same photon passed B, and on B's clock at that time it said t'_b. According to A, that photon was at x_a when A and B were colocated. But according to B, that photon was at x'_b when A and B were colocated. The photon is the same. It's the photon that passes B and reaches A.

Sure, the photon is the same, but the event on the photon's worldline that occurred at a position of x=x_a in A's frame may or may not be the same event as the event on the photon's worldline that occurred at a position of x'=x'_b in B's frame. Terms like x_a must have well-defined physical definitions if we want to use them in a physics context.

If you like, this not worring about the specific spacetime location of YDE is a little like a http://en.wikipedia.org/wiki/Lie-to-children" . If you want to focus heavily on the YDE and fix it in space and time, then I have to give you an overt "lie to children" and tell you it's the same event.

You can't have a valid derivation that starts from a wrong premise, unless you're doing a proof by contradiction. In any case, the lie seems totally superfluous here. Why not have a yellow dot event in the left diagram that occurs at t=0 in the A frame, and a pink dot event in the right diagram that occurs at t'=0 in the B frame, and just not say anything one way or another about whether these two events are identical or different? What exactly would be lost?

The thing is, the average student being introduced to relativity would not be like you and want to know the precise spacetime location of the YDE. Can you understand that?

Anyone who's familiar with the use of coordinate systems at all (Galilean or otherwise) will want to know the coordinates of any event that's introduced, even if they are presented in abstract rather than numerical notation. As an example, how do you expect the student would understand that x_a (the space coordinate of the YDE in the A frame) should equal ct_a (where t_a is the time coordinate the the photon passes A) if they don't assume the photon was traveling at c and the YDE occurred at t=0?

Addressing the second first, I'm not really using simultaneity. I'm using extrapolation.

If A and B are colocated and a photon passes A a period of t_a later, then A can extrapolate that the photon must have been at x_a=c.t_a when A and B were colocated.

How does this contradict the idea that you're using simultaneity to define the YDE? If you define the YDE as where the photon was "when A and B were colocated", that's exactly equivalent to defining the YDE event as the point on the photon's worldline that's simultaneous with A and B being colocated--to say two events are simultaneous is just another way of saying one event happened when the other event did. The fact that you can then use this definition (along with the fact that the photon passed A at t_a, and the assumption that the light was traveling at c) to extrapolate the position of the YDE doesn't somehow invalidate the fact that simultaneity with A & B's colocation was key to the original definition.

No explicit relativity of simultaneity.

I just said that the YDE was defined in terms of simultaneity with A&B's colocation in each frame, I didn't say anything about the relativity of simultaneity. Again, simultaneity just means "at the same time coordinate", it's perfectly OK to use the word simultaneity in a discussion of Galilean frames when there is no relativity of simultaneity because all frames agree whether or not two events are simultaneous. I made this point in my last post too:

But by talking about where the photon was "when A and B were colocated" you are using the concept of simultaneity. We don't have to get into the *relativity* of simultaneity of course, if we want to use the Galilei transform then simultaneity is non-relative. But in this case it's impossible that the light could move at c in both frames, as I keep saying.

And hopefully I have explained here why I am not fussed about the when and where of the YDE and what t_a is (between what and what).

No, nothing you have said helps me to make any sense of what it could mean to have a well-defined problem where you introduce events without any notion of their coordinates, or introduce terms without knowing their physical meaning. If you're "not fussed" about the coordinates of the YDE or what t_a means, will it make no difference to your derivation if I secretly choose to assume the YDE occurred at t=-1000*t_a (still assuming that A and B were colocated at t=0), or that t_a refers to the time coordinate of A marking the 30th anniversary of the photon having passed him?

 

[neopolitan]

#303 strand

A common refrain, but your dividing strategy makes replying difficult. I did partially reply to #303. I may not have replied to something specific in #303 because I am not going to always give an individual reply to each of your paragraphs.

I did intend to reply to the totality of #303 in separate posts because there were two strands in there, which would lead to confusion (once you split everything into individual paragraphs, there is no longer a clear separation between those strands). I didn't get around to it but I do think I addressed most of what was in the first part of #303 in later posts.

Is there still a specific question in #303 which you need a specific answer to which I have not addressed since in response to a later question?

I have labelled this #303 strand.

cheers,

neopolitan

 

[neopolitan]

Or do you disagree that in A's frame, the yellow dot event has coordinates x=ct_a and t=0, while the orange dot event has coordinates x=vt'_a and t=t'_a? (assuming we set the origin so the event of A and B being colocated has coordinates x=0 and t=0).

I can live with that, yes. I don't really want to bring in a pink dot. There must be a mutually satisfactory way to avoid that.

These equations wouldn't "speak" about any event in particular, they relate the coordinates of any arbitrary event in one frame to the coordinates of the same event in the other frame, but either way it is necessary that you have a specific physical event in mind.

I want to focus in on this for the moment, if I may. Can we try to focus on one thing at a time?

For me, this is precisely what a good coordinate transformation equation would do. It would relate the coordinates of an arbritrary event in one frame to the coordinates of the same arbitrary event in another frame. I don't think that is a fault, it's an important feature of the coordinate transformation equation.

Do you agree that the event that results in (0,t_a) is arbitrary, if we chose a handy event (x_a=ct_a,0) as the initiating event and we haven't actually pinned down what numerical value t_a has?

We can certainly make it more general by selecting an initiating event which was not simultaneous with (0,0), but do you agree that that would be awkward?

Again, I really would like to focus on this, because it may be key. Can we do that?

cheers,

neopolitan

 

[JesseM]

A common refrain, but your dividing strategy makes replying difficult. I did reply to #303. I may not have replied to something specific in #303 because I am not going to always give an individual reply to each of your paragraphs.

You only responded to one incidental question in 303 about the relation between the two blocks of text and the "two perspectives" (which was just a continuation of a discussion about an incidental question I had asked in the second-to-last paragraph of 293), you didn't respond to the main body of the post which was part of the discussion about the equations you wrote out in 295 (I had responded to 295 in 299, then you responded to 299 in 302, and my 303 was in response to that).

I did intend to reply to #303 in separate posts because there were two strands in there, which would lead to confusion (once you split everything into individual paragraphs, there is no longer a clear separation between those strands). I didn't get around to it but I do think I addressed most of what was in the first part of #303 in later posts.

Is there still a specific question in #303 which you need a specific answer to which I have not addressed since in response to a later question?

Yes, the basic question I was focused on in 303 was whether you understood that the equation defining the separation between B and the position of the photon at t=0 as a function of time t in the A frame, namely x-vt, has only a superficial resemblance to the Galilei transformation equation, x'=x-vt, but their physical meanings are really quite different because the first is just calculating the distance between two things exclusively in A's frame with no reference to B's frame, while the fundamental purpose of the second is to relate the x,t coordinates of an event in the A frame to the x' coordinate of the same event in the B frame. So given this, I didn't understand why you had written "According to the unprimed observer, the separation between where the primed observer is now, and where the photon was at t=0, is x'=x-vt"...if you were just calculating the separation in the unprimed frame, that would just be x-vt, the x' doesn't make any sense there unless you have redefined x' to mean the separation in the unprimed frame, which would be very confusing and totally different from the meaning of the equation x'=x-vt in the Galilei transformation equation (and you said the student knew the equation x'=x-vt based on the fact that he 'paid enough attention' in class, so it would be pretty weird if what you really meant was that he knew the equation but totally misunderstood its physical meaning).

I had some other questions in post 303 about the meaning of the equations in your post 295, but I think I restated these questions a little more clearly in post 311 so maybe you could just respond to this section:

And as I said in my responses to that post, the meaning you were assigning these symbols was unclear to me...when you wrote "x = ct and x' = x - vt", what physically do x, t, and x' represent? This is especially confusing because x' = x - vt looks like the spatial component of the Galilei transform, a general equation that holds for arbitrary events that have coordinates x,t in one frame and x',t' in the other, whereas x = ct is clearly not a general relation that is supposed to apply to arbitrary events. If you want to refer to the positions and times of specific events as opposed to relationships that are supposed to hold between arbitrary sets of coordinates, it's really helpful if you put some subscripts to indicate this, and actually name in English the specific events they are the coordinates of (or the pairs of events that they represent distance and time intervals between). For example, if look at the following two events:

1. The event on the photon's worldline that occurs at t=0 in the frame of the unprimed observer, when he is colocated with the primed observer
2. The event of the photon being colocated with the unprimed observer

...then if x_a is used to denote the spatial coordinate of event #1 and t_a is used to denote the temporal coordinate of event #2, it would indeed be true that x_a = ct_a. However, since these are the coordinates of two different events, we cannot plug x_a and t_a in for x and t in the equation x' = x - vt to get the space coordinate of either event in the primed frame, since if you want the left side of this equation to be the primed space coordinate of a particular event, you have to plug in the x and t coordinates of that same single event in the right side.

On the other hand, the Galilei transform also gives us the equation dx' = dx - v*dt for the intervals between a pair of events, so if event #1 has coordinates (x=x_a, t=0) and event #2 has coordinates (x=0, t=t_a), then subtracting the coordinates of the first from the second gives dx = -x_a and dt = t_a. Then it would be valid to plug this value for dx and dt into the equation dx' = dx - v*dt to find the spatial interval in the primed frame between event #1 and event #2 above.

...or am I totally on the wrong track about the meaning of x'=x-vt, and is it not supposed to have the same meaning as in the Galilei transformation at all? As I suggested above, perhaps you are just redefining x' to mean the separation between B and the position of the YDE in the unprimed frame, so that every part of x'=x-vt deals with the unprimed frame and despite appearances it is not meant to be a coordinate transformation equation at all?

 

[JesseM]

I want to focus in on this for the moment, if I may. Can we try to focus on one thing at a time?

For me, this is precisely what a good coordinate transformation equation would do. It would relate the coordinates of an arbritrary event in one frame to the coordinates of the same arbitrary event in another frame. I don't think that is a fault, it's an important feature of the coordinate transformation equation.

Sure, that's just the standard meaning of what coordinate transformation equations are meant to do (although a coordinate transformation can also transform the intervals between an arbitrary pair of events in one frame to the intervals between the same pair of events in another frame).

Do you agree that the event that results in (0,t_a) is arbitrary, if we chose a handy event (x_a=ct_a,0) as the initiating event and we haven't actually pinned down what numerical value t_a has?

Yes, as I said I'm fine with defining events in abstract notation rather than numerical values.