Benefits of time dilation / length contraction pairing?

[neopolitan]

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

There are four values involved (which is a fair indication to the astute student that there might be two events and therefore is already laying the ground work for introducing the relativity of simultaneity). The values which are primed are as per the Galilean boost. The subscripted values mean "according to ..." (as clarified in post https://www.physicsforums.com/showpost.php?p=2166309&postcount=175").

I didn't take your E and E' as prescriptive but rather descriptive since I had already written twice by that time that the subscripted values mean "according to ..." and that the values which are primed are as per the Galilean boost (which I was in the process of trying to explain).

If you want to refer to the events (x_A,t_A) according to A and (x_B,t_B) according to B via a priming notation, go ahead, but it's not really sensible. Personally, I would refer to the events as E_A and E_B. Remember also I was using primes in the same sense as they are used in the Galilean boost, where there are no events to speak of, primed or unprimed.

Try comparing http://www.geocities.com/neopolitonian/uniquespacetimelocation.jpg").

Until I know for sure what you are calling the primed E and what you are calling the unprimed E, since you do seem to jump around a lot, I'd prefer you use either "event at (x_A,t_A) according to A" and "the event at (x'_B,t'_B) according to B" or E_A and E_B. Until then, I am not totally convinced that you aren't talking about completely different events.

Driving the point home even further, I specify explicitly that in http://www.geocities.com/neopolitonian/uniquespacetimelocation.jpg", E_A is "Photon when A is colocated with B, according to A" and "E_B is "Photon when A is colocated with B, according to B".

Note the consistency with the "acccording to ..." notation.

cheers,

neopolitan

 

[JesseM]

I didn't take your E and E' as prescriptive but rather descriptive since I had already written twice by that time that the subscripted values mean "according to ..." and that the values which are primed are as per the Galilean boost (which I was in the process of trying to explain).

I don't understand what you mean by "prescriptive rather than descriptive". Could you please just write out explicitly what each of the following refer to?

x_A refers to position coordinate of ___ (or the distance between ___ and ___) in the A frame
x_B refers to position coordinate of ___ (or the distance between ___ and ___) in the B frame
x'_A refers to position coordinate of ___ (or the distance between ___ and ___) in the A frame
x'_B refers to position coordinate of ___ (or the distance between ___ and ___) in the B frame

I put in those parentheses because it seems that at times these symbols refer to actual coordinates of individual events in SR, while at other times they refer to distance intervals between events rather than coordinates of individual events.

If you want to refer to the events (x_A,t_A) according to A and (x_B,t_B) according to B via a priming notation, go ahead, but it's not really sensible. Personally, I would refer to the events as E_A and E_B.

But then if I see notation like x_A, there's nothing in the notation itself that tells me whether it refers to the position in A's frame of event E_A or the position in A's frame of event E_B? If you don't want to use the prime to tell me which event is being referred to, could you add some other indication? Like instead of calling the events E_A and E_B you could call them E₁ and E₂, then x_A1 would refer to the position coordinate of event E₁ in frame A, x_B2 would refer to the position coordinate of event E₂ in frame B, and so forth.

Remember also I was using primes in the same sense as they are used in the Galilean boost, where there are no events to speak of, primed or unprimed.

Why do you say that? All coordinate transformations are showing you the coordinates of a single event in two different frames. For example, if an event has coordinates (x,t) in the unprimed frame, then according to the Galilei transformation (which is what is meant by the 'Galilean boost') in the primed coordinate system it has coordinates (x',t') given by:

x' = x - vt
t' = t

Try comparing http://www.geocities.com/neopolitonian/uniquespacetimelocation.jpg").

But the second diagram is just a weird transformation (which I think would be horrendously complicated to write the equations for) that has nothing to do with the Galilei transformation, and was based on your misinterpretation of what I meant when I said instantaneous communication wasn't necessary in the Galilei transformation, so what's the relevance?

Until I know for sure what you are calling the primed E and what you are calling the unprimed E, since you do seem to jump around a lot

I didn't specify because I assumed you were using primed vs. unprimed to refer to the two events, since you said I was correct when I said you were actually distinguishing between the two events using primed and unprimed (calling one event E and the other E', say), and I didn't know which was supposed to be which in your notation.

I'd prefer you use either "event at (x_A,t_A) according to A" and "the event at (x_B,t_B) according to B" or E_A and E_B. Until then, I am not totally convinced that you aren't talking about completely different events.

Of course I was talking about different events! We already established that you were discussing two different events, you showed them in this diagram. Are you saying the above notation which you "prefer" is not talking about two different events? If so, which of the events are you talking about with that notion, the one that was simultaneous with colocation in A's frame, or the one that was simultaneous with colocation in B's frame?

Driving the point home even further, I specify explicitly that in http://www.geocities.com/neopolitonian/uniquespacetimelocation.jpg", E_A is "Photon when A is colocated with B, according to A" and "E_B is "Photon when A is colocated with B, according to B".

OK, then please tell me what notation you would use for each of the following:

1. position and time coordinates of E_A in A's frame
2. position and time coordinates of E_A in B's frame
1. position and time coordinates of E_B in A's frame
2. position and time coordinates of E_B in B's frame

Note the consistency with the "acccording to ..." notation.

Yes, I understood that subscripts referred to which frame we were using, that wasn't the issue.

 

[JesseM]

But the second diagram is just a weird transformation (which I think would be horrendously complicated to write the equations for) that has nothing to do with the Galilei transformation, and was based on your misinterpretation of what I meant when I said instantaneous communication wasn't necessary in the Galilei transformation, so what's the relevance?

By the way, I spent a bit of time figuring out what the transformation equations implied by that diagram would actually be, turns out it's not so complicated, in units where c=1 I believe it'd be:

x_B = [1/(1 - v^2)]*(x_A - vt_A)
t_B = [1/(1 - v^2)]*(t_A - vx_A)

So, almost like the Lorentz transformation but with x_B and t_B multiplied by gamma. And the reverse transformation would just be:

x_A = x_B + vt_B
t_A = t_B + vx_B

 

[neopolitan]

I don't understand what you mean by "prescriptive rather than descriptive". Could you please just write out explicitly what each of the following refer to?

Descriptive - thou could if thou wanted
Prescriptive - thou shalt

Sure you can call the events what you want. E' and E. E₁ and E₂, E and F. Bonny and Clyde. Frank and Ernest.

You didn't initially say not only must I use your nomenclature but I also must use your definitions.

x_A refers to position coordinate of event E_A (or the distance between the origin of the x_A axis and Event E_A) in the A frame
x_B refers to position coordinate of Event E_B (or the distance between the origin of the x_B axis and Event E_B) in the B frame
x'_A refers to the distance between the location of B at t_A and the location of Event E_A in the A frame
x'_B refers to the distance between the location of A at t'_B and the location of Event E_B in the B frame

I put in those parentheses because it seems that at times these symbols refer to actual coordinates of individual events in SR, while at other times they refer to distance intervals between events rather than coordinates of individual events.

I've filled it in, in red.

As you can see, yes, two of them represent both a coordinate and a distance interval between two events. Note that the origin of either axis at a specific time is "an event" since it's both a time and a location, so really any coordinate really represents an interval between two events.

But then if I see notation like x_A, there's nothing in the notation itself that tells me whether it refers to the position in A's frame of event E_A or the position in A's frame of event E_B? If you don't want to use the prime to tell me which event is being referred to, could you add some other indication? Like instead of calling the events E_A and E_B you could call them E₁ and E₂, then x_A1 would refer to the position coordinate of event E₁ in frame A, x_B2 would refer to the position coordinate of event E₂ in frame B, and so forth.

You could prime it as well, if you like. But let's not.

E_Aa = (x_A,t_A) when viewed by A. E_Ab is the same event when viewed by B.
E_Bb = (x'_B,t'_B) when viewed by B. E_Ba is the same event when viewed by A.

Why do you say that? All coordinate transformations are showing you the coordinates of a single event in two different frames. For example, if an event has coordinates (x,t) in the unprimed frame, then according to the Galilei transformation (which is what is meant by the 'Galilean boost') in the primed coordinate system it has coordinates (x',t') given by:

x' = x - vt
t' = t

I'm going to have to take your word for that. I've only ever seen the Galilean boost talked of in terms of locations, except where SR is what is really being discussed. Events seem to be introduced in SR.

That may be just because t'=t in Galilean relativity, so it is safe to discuss just locations, but underneath, back when it was written Galileo was thinking in terms of events. (I just don't think so, I think he thought in terms of something like "vehicle moving towards shed, at t=t, what is the distance interval between vehicle and shed". Thinking in terms of events is a consequence of working backwards from SR.)

But the second diagram is just a weird transformation (which I think would be horrendously complicated to write the equations for) that has nothing to do with the Galilei transformation, and was based on your misinterpretation of what I meant when I said instantaneous communication wasn't necessary in the Galilei transformation, so what's the relevance?

You just couldn't help yourself, could you?

I repeat what I said when I directed you to that diagram "Try to restrict your response to how the events were named".

Instead, you totally ignored it. It boggles the mind.

I didn't specify because I assumed you were using primed vs. unprimed to refer to the two events, since you said I was correct when I said you were actually distinguishing between the two events using primed and unprimed (calling one event E and the other E', say), and I didn't know which was supposed to be which in your notation.

You were wrong. You wanted to prime one of the events. I'd specifically avoid priming one of the events because it is misleading.

Of course I was talking about different events! We already established that you were discussing two different events, you showed them in this diagram. Are you saying the above notation which you "prefer" is not talking about two different events? If so, which of the events are you talking about with that notion, the one that was simultaneous with colocation in A's frame, or the one that was simultaneous with colocation in B's frame?

"Completely" different events. Are you talking about completely different events? I was talking about two events, were you talking about two completely different events which are not the events that I was talking about?

You say "of course", but I am not so sure since you point back to the two events that I am talking about.

OK, then please tell me what notation you would use for each of the following (coordinates from my diagram http://www.geocities.com/neopolitonian/uniquespacetimelocation.jpg"):

1. position and time coordinates of E_A in A's frame - (x_A , 0) (8,0)
2. position and time coordinates of E_A in B's frame - (gamma.x_A , gamma.(-v.c².x_A)) (10,-6)
1. position and time coordinates of E_B in A's frame - (gamma.x'_B , gamma.(+v.c².x'_B)) (5,3)
2. position and time coordinates of E_B in B's frame - (x'_B , 0) (4,0)

Getting the code right for writing this up is much more difficult than working out what I actually need to write.

cheers,

neopolitan

 

[neopolitan]

By the way, I spent a bit of time figuring out what the transformation equations implied by that diagram would actually be, turns out it's not so complicated, in units where c=1 I believe it'd be:

x_B = [1/(1 - v^2)]*(x_A - vt_A)
t_B = [1/(1 - v^2)]*(t_A - vx_A)

So, almost like the Lorentz transformation but with x_B and t_B multiplied by gamma. And the reverse transformation would just be:

x_A = x_B + vt_B
t_A = t_B + vx_B

Yeah, I did the same thing. But there were some c's in my answer.

 

[JesseM]

Descriptive - thou could if thou wanted
Prescriptive - thou shalt

Sure you can call the events what you want. E' and E. E₁ and E₂, E and F. Bonny and Clyde. Frank and Ernest.

You didn't initially say not only must I use your nomenclature but I also must use your definitions.

I take it you didn't understand that my question was specifically about me trying to understand what you meant by different symbols? If you understood that, you would understand how confusing it would be to answer "correct" if what you really meant was "well, that's not what I meant by primed and unprimed, but you're free to use that notation if you like".

I've filled it in, in red.

OK, thanks. You seem to have no notation to represent the position coordinate of event E_A in the B frame or of E_B in the A frame, but perhaps that's not relevant to your derivation?

You could prime it as well, if you like. But let's not.

E_Aa = (x_A,t_A) when viewed by A. E_Ab is the same event when viewed by B.
E_Bb = (x'_B,t'_B) when viewed by B. E_Ba is the same event when viewed by A.

We can do it that way if you like. I was thinking that the different subscripts for E represented different physical events--that's the convention I normally see in relativity problems of this kind--but if you want them to refer to an event and its coordinates that's fine too. Again you seem to have no notation to represent the x and t coordinates associated with E_Ab or E_Ba, but perhaps you don't need it.

You just couldn't help yourself, could you?

I repeat what I said when I directed you to that diagram "Try to restrict your response to how the events were named".

Instead, you totally ignored it. It boggles the mind.

Jeez, relax, I just didn't catch the significance of that phrase, I didn't intentionally "ignore" it. There's a lot of stuff I write that you misunderstand too, I don't complain about how mind-boggling it is that you would fail to get everything I say.

You were wrong. You wanted to prime one of the events. I'd specifically avoid priming one of the events because it is misleading.

No, I asked a question about how you were using the notation: "was I in fact mistaken about your notation, and you were actually distinguishing between the two events using primed and unprimed (calling one event E and the other E', say)"? I didn't "want" to use that notation, I was just asking if that's what you meant when you put primes on some coordinates and not others, because I was trying to follow your equations and I didn't understand what that notation meant...you answered "correct" to my question. Can you see why I was confused?

"Completely" different events. Are you talking about completely different events? I was talking about two events, were you talking about two completely different events which are not the events that I was talking about?

I think I understand what you mean now, but it wasn't obvious from the context--the word "completely" does not naturally imply "two events different from the two events I am talking about" as opposed to "two different events, as opposed to a single physical event with different coordinates in different frames". I thought you were suggesting that x_A and x_B were supposed to refer to the coordinates of a single event in the A frame vs. the B frame.

Anyway, now that you've given a clear definition of your spatial terms:

x_A refers to position coordinate of event E_A (or the distance between the origin of the x_A axis and Event E_A) in the A frame
x_B refers to position coordinate of Event E_B (or the distance between the origin of the x_B axis and Event E_B) in the B frame
x'_A refers to the distance between the location of B at t_A and the location of Event E_A in the A frame
x'_B refers to the distance between the location of A at t'_B and the location of Event E_B in the B frame

...can I ask for definitions of the temporal terms too? Comparing to the above I would initially think t_A is the time coordinate of E_A in the A frame, and t_B is the time coordinate of E_B in the B frame, but that would mean t_A = 0 and t_B = 0, which probably isn't what you meant. So are t_A and t_B just two arbitrary time-coordinates in each frame? And what's the difference between them and t'_A and t'_B?

 

[neopolitan]

I take it you didn't understand that my question was specifically about me trying to understand what you meant by different symbols? If you understood that, you would understand how confusing it would be to answer "correct" if what you really meant was "well, that's not what I meant by primed and unprimed, but you're free to use that notation if you like".

I wasn't saying "correct" to the primes. It didn't figure highly. But I think we are past that now.

OK, thanks. You seem to have no notation to represent the position coordinate of event E_A in the B frame or of E_B in the A frame, but perhaps that's not relevant to your derivation?

Nope, not necessary.

We can do it that way if you like. I was thinking that the different subscripts for E represented different physical events--that's the convention I normally see in relativity problems of this kind--but if you want them to refer to an event and its coordinates that's fine too. Again you seem to have no notation to represent the x and t coordinates associated with E_Ab or E_Ba, but perhaps you don't need it.

Jeez, relax, I just didn't catch the significance of that phrase, I didn't intentionally "ignore" it. There's a lot of stuff I write that you misunderstand too, I don't complain about how mind-boggling it is that you would fail to get everything I say.

I specifically put the phrase in there because I thought that (without it) you would go off on a tangent. But you did anyway. I was actually more irritated about the "completely different" thing. Oh well, such is life.

No, I asked a question about how you were using the notation: "was I in fact mistaken about your notation, and you were actually distinguishing between the two events using primed and unprimed (calling one event E and the other E', say)"? I didn't "want" to use that notation, I was just asking if that's what you meant when you put primes on some coordinates and not others, because I was trying to follow your equations and I didn't understand what that notation meant...you answered "correct" to my question. Can you see why I was confused?

Yes. We were focussed on different things. What I was saying correct to was not what you thought I was saying correct to. As I said, I think we are past this now.

I think I understand what you mean now, but it wasn't obvious from the context--the word "completely" does not naturally imply "two events different from the two events I am talking about" as opposed to "two different events, as opposed to a single physical event with different coordinates in different frames". I thought you were suggesting that x_A and x_B were supposed to refer to the coordinates of a single event in the A frame vs. the B frame.

Anyway, now that you've given a clear definition of your spatial terms:

...can I ask for definitions of the temporal terms too? Comparing to the above I would initially think t_A is the time coordinate of E_A in the A frame, and t_B is the time coordinate of E_B in the B frame, but that would mean t_A = 0 and t_B = 0, which probably isn't what you meant. So are t_A and t_B just two arbitrary time-coordinates in each frame? And what's the difference between them and t'_A and t'_B?

You could ask for definitions of the temporal terms, but for the purposes of the derivation, you don't need them.

Since x=ct and x'=ct', you just divide through by c.

x'=gamma.(x-vt) =>
x'/c = gamma.(x/c-vt/c) =>
t' = gamma.(t - v.(x/c)/c) =>
t' = gamma.(t - vx/c²)

But if you comprehend how you get to the spatial Lorentz transformation, doing the same for time is not difficult - if you must derive it separately.

Is it possible that you could look at the post where I introduced this derivation and see if you can work it through, now you know what refers to what?

You asked a while back why the yellow E was still one event rather than two. I answered it my own sort of way along with some other questions, but perhaps you didn't think that I answered.

For each of A and B, E is one event (E_A for A, and E_B for B). Both A and B will think that E according to the other is another event (E_Ba and E_Ab).

cheers,

neopolitan

 

[phyti]

Neo:

Since you seem to have a handle on space-time drawings, and they do tend to simplify for you and the reader, here is an extension to the example you are working. Hope it helps.
In the 3rd case, the viewer is always the rest frame.
View attachment neo problem0425.doc

 

[neopolitan]

Neo:

Since you seem to have a handle on space-time drawings, and they do tend to simplify for you and the reader, here is an extension to the example you are working. Hope it helps.
In the 3rd case, the viewer is always the rest frame.
View attachment 18619

Thanks phyti, I have pretty much always had something like that in mind.

I did wonder if E1 is misplaced on the second two. It sits on the x_A axis at first, then moves to the x_B which at first glance can't be right.

I think I know what you mean though, that E1 is simultaneous with the colocation of A and B, according to the viewer of the graph (for whom it is square) and at first that is someone at rest with A and then it is someone at rest with B. Which means that there are two separate E1s. Am I right?

In terms of the discussion which precedes this, then E1 is both E_Aa and E_Bb (although the assumption that E_A and E_B are causally linked is absent).

cheers,

neopolitan

 

[JesseM]

You could ask for definitions of the temporal terms, but for the purposes of the derivation, you don't need them.

Since x=ct and x'=ct', you just divide through by c.

x'=gamma.(x-vt) =>
x'/c = gamma.(x/c-vt/c) =>
t' = gamma.(t - v.(x/c)/c) =>
t' = gamma.(t - vx/c²)

But if you comprehend how you get to the spatial Lorentz transformation, doing the same for time is not difficult - if you must derive it separately.

Is it possible that you could look at the post where I introduced this derivation and see if you can work it through, now you know what refers to what?

You asked a while back why the yellow E was still one event rather than two. I answered it my own sort of way along with some other questions, but perhaps you didn't think that I answered.

For each of A and B, E is one event (E_A for A, and E_B for B). Both A and B will think that E according to the other is another event (E_Ba and E_Ab).

cheers,

neopolitan

OK, going back to the third drawing in post 174:

Question #1: Why do both A and B conclude the same factor G will appear in the equations x_A = G*x_B and x'_B = G*x'_A? The two equations have fairly different physical meanings...the first means:

(position coordinate of E_A in A's frame) = G*(position coordinate of E_B in B's frame)

But the second means:

(distance between position of A at t'_B and position of E_B, in the B frame) = G*(distance between position of B at t'_A and position of E_A, in the A frame)

Can we go through a numerical example so I can try to understand what additional assumptions you might be making that lead you to think the factor relating these would be the same?

Let's suppose in A's frame B has a velocity of 0.6c. Suppose in A's frame the event E_A has position x_A = 16 light-seconds. In this case the light will pass B at a position of 6 light-seconds and a time of 10 seconds in A's frame. Since B's clock is running slow by a factor of 0.8 in this frame, the light must hit B when B's clock reads 8 seconds, so E_B must occur at position x_B = 8 light-seconds in B's frame. So here, G=2.

Now, in the A frame the distance between B and the position of E_A as a function of time must be (16 - 0.6*t'_A). And in the B frame the distance between A and the position of E_B as a function of time must be (8 + 0.6*t'_B). Obviously we can't pick an arbitrary t'_A and t'_B and know that these two expressions will always be related by the same G-factor of 2. So do you specifically want the relation between t'_A and t'_B to be such that the two expressions are related by a factor of 2, i.e.

(8 + 0.6*t'_B) = 2*(16 - 0.6*t'_A) ?

If so we could solve for t'_B as a function of t'_A

t'_B = 40 - 2*t'_A

For example, if we pick t'_A = 3, then t'_B = 34. Did you intend this sort of relation between the two primed times, or am I misunderstanding? If I am, it really would be helpful if you'd explain a little more about what the primed times are supposed to mean physically...

 

[neopolitan]

OK, going back to the third drawing in post 174:

Question #1: Why do both A and B conclude the same factor G will appear in the equations x_A = G*x_B and x'_B = G*x'_A? The two equations have fairly different physical meanings...the first means:

(position coordinate of E_A in A's frame) = G*(position coordinate of E_B in B's frame)

But the second means:

(distance between position of A at t'_B and position of E_B, in the B frame) = G*(distance between position of B at t'_A and position of E_A, in the A frame)

Can we go through a numerical example so I can try to understand what additional assumptions you might be making that lead you to think the factor relating these would be the same?

Let's suppose in A's frame B has a velocity of 0.6c. Suppose in A's frame the event E_A has position x_A = 16 light-seconds. In this case the light will pass B at a position of 6 light-seconds and a time of 10 seconds in A's frame. Since B's clock is running slow by a factor of 0.8 in this frame, the light must hit B when B's clock reads 8 seconds, so E_B must occur at position x_B = 8 light-seconds in B's frame. So here, G=2.

Now, in the A frame the distance between B and the position of E_A as a function of time must be (16 - 0.6*t'_A). And in the B frame the distance between A and the position of E_B as a function of time must be (8 + 0.6*t'_B). Obviously we can't pick an arbitrary t'_A and t'_B and know that these two expressions will always be related by the same G-factor of 2. So do you specifically want the relation between t'_A and t'_B to be such that the two expressions are related by a factor of 2, i.e.

(8 + 0.6*t'_B) = 2*(16 - 0.6*t'_A) ?

If so we could solve for t'_B as a function of t'_A

t'_B = 40 - 2*t'_A

For example, if we pick t'_A = 3, then t'_B = 34. Did you intend this sort of relation between the two primed times, or am I misunderstanding? If I am, it really would be helpful if you'd explain a little more about what the primed times are supposed to mean physically...

I was all ready to get irritated when I realized that I had not put labels the third diagram (the diagram that it is taken from does have labels, which would have prevented some problems) and further, I slipped up with primes https://www.physicsforums.com/showpost.php?p=2173445&postcount=214" when defining what all the x and x' values meant. I'll start again.

Remember I said that primes are "as per Galilean boosts"? In other words,

x'=x-vt

Part of the consequence of that is that x'_B is B talking about B and x_A is A talking about A.

Then there is plenty of scope for confusion about what x_B and x'_A mean.

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


So:

x_A is the distance between A and event E_A according to A ( <- position coordinate)
(value in the example we have been discussing: 8)

x'_A is the distance between B and event E_A at t=5 according to A
(value in the example we have been discussing: 5)

x'_B is the distance between B and event E_B according to B ( <- position coordinate)
(value in the example we have been discussiing: 4)

x_B is the distance between A and event E_B at t=-10 according to B
(value in the example we have been discussing: 10)

Therefore (after you work it all through):

x_B = x_A.gamma (= 4*1.25 = 5 )
and
x'_A = x'_B.gamma (= 8*1.25 = 10 )


I should not have cut and pasted, I should have written it all out from first principles.

I apologise for any confusion caused.


Remember, the derivation allows you go back and show that events E_A and E_B are not the same. But it does not initially assume it.

cheers,

neopolitan

 

[JesseM]

x_A is the distance between A and event E_A according to A ( <- position coordinate)
(value in the example we have been discussing: 8)

OK, it was actually 16 in my example but this works fine too.

x'_A is the distance between B and event E_A at t=5 according to A
(value in the example we have been discussing: 5)

In this case the distance between B and E_A as a function of time t_A in A's frame is 8 - 0.6*t_A, so that works.

x'_B is the distance between B and event E_B according to B ( <- position coordinate)
(value in the example we have been discussiing: 4)

x_B is the distance between A and event E_B at t=-10 according to B
(value in the example we have been discussing: 10)

And here the distance between A and E_B as a function of time t'_B in B's frame is 4 + 0.6*t'_B...I guess that should be t'_B=10 rather that t'_B=-10, right?

Therefore (after you work it all through):

x_B = x_A.gamma (= 4*1.25 = 5 )
and
x'_A = x'_B.gamma (= 8*1.25 = 10 )

But this only works because of the two particular times you chose for t_A and t'_B, right? You could just as easily pick a pair of t-values such that the factor was gamma^2 or anything else, as well as a pair such that the factors in the two equations were different. Do these times have any physical significance other than that they were chosen to make these two equations include a gamma factor?

Also, in equation 3 from the third diagram you wrote x'_B + vt'_B = x'_B + vx'_B/c, so it looks like you were making the substitution t'_B = x'_B/c, but that doesn't work with the above numbers where t'_B = 10 seconds and x'_B/c = 4 seconds.

 

[neopolitan]

I'm clearly going to have to go back to the beginning with a separate diagram and show you all the relationships as I build the derivation.

I need to do that because I only used a flat 1 dimensional diagram, and it doesn't easily translate to a 1+1 diagram.

You can preempt me, if you like, by looking at https://www.physicsforums.com/attachment.php?attachmentid=18548&d=1240077258" and thinking about how they could be melded.

cheers,

neopolitan

 

[neopolitan]

But this only works because of the two particular times you chose for t_A and t'_B, right? You could just as easily pick a pair of t-values such that the factor was gamma^2 or anything else, as well as a pair such that the factors in the two equations were different. Do these times have any physical significance other than that they were chosen to make these two equations include a gamma factor?

I'm obviously tiring.

x_A is the distance from the origin of the x_A axis to event E_A.

So why the hell did I have x_A=4? Fixing it:

x_B = x_A.gamma (= 8*1.25 = 10 )
and
x'_A = x'_B.gamma (= 4*1.25 = 5 )

x_A = t_A.c = 8
x'_B = t'_B.c = 4

Then we have to remember that x'=x-vt only works because t=t'

Otherwise, x'=x-vt' so, in this case (my mess up again, we really should start from the beginning):

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

where t'_A is when the photon from E_A passes B (eg t=5) according to A and vt_B is when the photon from E_B passes A according to B (eg t=10).

Summarising:

x_A=8 ... t_A=8
x'_A=5 ... t'_A=5

x'_B=4 ... x'_B=4
x_B=10 ... t_B=10

x_B = x_A.gamma (= 8*1.25 = 10 )
x'_A = x'_B.gamma (= 4*1.25 = 5 )

x'_A=x_A - vt'_A = 8 - 0.6*5 = 5
x_B=x'_B + vt_B = 4 + 0.6*10 = 10

which means it all works.

I very very much hope this has been written out correctly this time, I even printed it out and checked it. I will still do the diagram, because I think it might help even though the diagram is only something you can construct retrospectively.

cheers,

neopolitan

 

[JesseM]

where t'_A is when the photon from E_A passes B (eg t=5) according to A and vt_B is when the photon from E_B passes A according to B (eg t=10).

Ah, OK, then that answers my question about the physical meaning of the time coordinates, thanks.

So, x_B = x_A * gamma can be written as:

(distance between A and E_B at the time t_B when the photon from E_B reaches A, all in the B frame) = (distance between A and E_A in the A frame) * gamma

And x'_A = x'_B * gamma can be written as:

(distance between B and E_A at the time t'_A when the photon from E_A reaches B, all in the A frame) = (distance between B and E_B in the B frame) * gamma

So, this does seem to work in SR (although I'm having trouble conceptualizing why it must work). I guess my next question is, if you don't assume SR from the start, what starting assumptions do you use to derive the fact that these equations should include the same factor G relating the two quantities in each equation?

Also, since the equations in diagram 3 from post 174 are x_A = x_B*G and x'_B = x'_A*G, I guess that means the G factor is actually 1/gamma? Or you could just rewrite the equations in that diagram as x_B = x_A*G and x'_A = x'_B*G if you prefer, that way G would still end up being equal to gamma.

x_A = t_A.c = 8
x'_B = t'_B.c = 4

OK, so that means t_A is the time for the photon to get from E_A to A in the A frame, and t'_B is the time for the photon to get from E_B to B in the B frame.

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

These equations are actually different from what's written in equation 3 and 4 of the diagram, here you have t'_A where the diagram has t_A, and t_B where the diagram has t'_B. I think the revised versions make sense given what you say above, for example the first revised equation would mean:

(distance between B and E_A at the time t'_A when the photon from E_A reaches B, all in the A frame) = (distance between A and E_A in the A frame) - v*(time t'_A when the photon from E_A reaches B as seen in A frame)

...which does make sense, since B started at the same distance from E_A as A did, and then at any later time t, B has moved closer to the position of E_A by an amount vt (all as seen in A frame). But with this revised version x'_A=x_A - vt'_A, you can't substitute x_A/c in for t'_A as you did in the diagram's version with t_A right? So doesn't that mean the final part of equation 4 in the diagram is no longer valid? And likewise with the final part of equation 3?

I guess you would instead want to substitute x'_A/c in for t'_A, giving x_A = x'_A + vx'_A/c = x'_A*(1 + v/c), an altered version of equation 4. Likewise for x_B=x'_B + vt_B you could substitute x_B/c for t_B, giving x'_B = x_B - vx_B/c = x_B*(1 - v/c), an altered version of equation 3. Then with the rewritten equations x_B = x_A*G and x'_A = x'_B*G (the ones I suggested earlier so G would still end up being equal to gamma) you'd be able to plug in and write x_B = G*x'_A*(1 + v/c) along with x'_A = G*x_B*(1 - v/c), which would be altered versions of your equations 6 and 5.

 

[JesseM]

So, to continue with the slightly altered version of your derivation, we can combine the equations x_B = G*x'_A*(1 + v/c) along with x'_A = G*x_B*(1 - v/c) to get x_B = G*(G*x_B*(1 - v/c))*(1 + v/c) = G^2*x_B*(1 - v^2/c^2). So divide both sides bey x_B to get 1 = G^2*(1 - v^2/c^2), or G^2 = 1/(1 - v^2/c^2), showing G is the familiar gamma factor. We can plug that back into x_B = x_A*G to get x_B = gamma*x_A. Now I think you'd want to combine that with another revised equation, x'_A=x_A - vt'_A or equivalently x_A = x'_A + vt'_A, and then we get x_B = gamma*(x'_A + vt'_A). Now, you would say this is a derivation of the spatial portion of the Lorentz transformation, correct? The problem I see here is that when you actually examine the meaning of the symbols, they don't seem to be all coordinates of a specific event or pair of events as in the Lorentz transformation--x_B is the difference in the B frame between the spatial coordinates of the event E_B and the event of the photon from E_B hitting A, but x'_A is the difference in the A frame between the spatial coordinates of a different pair of events, namely the event E_A and the event of the photon from E_A hitting B (t'_A is the difference between the time coordinates of the same pair of events as x'_A). So unless I've gotten the equations wrong or the symbols mixed up, it seems that any resemblance to the spatial Lorentz transformation equation here is coincidental, this equation doesn't have the same physical meaning at all, and in fact it would only be applicable to the specific definitions of the symbols in terms of the particular physical scenario you described, rather than the Lorentz equation delta-x_B = gamma*(delta-x_A + v*delta-t_A) which is applicable to the difference in spatial coordinates and difference in time coordinates of two arbitrary events (events that may not lie on the worldline of a photon for example), but with the understanding that delta-x_B and delta-x_A are the difference in spatial coordinates between the same pair of events in two frames.

 

[neopolitan]

So, this does seem to work in SR (although I'm having trouble conceptualizing why it must work). I guess my next question is, if you don't assume SR from the start, what starting assumptions do you use to derive the fact that these equations should include the same factor G relating the two quantities in each equation?

Galilean invariance

Also, since the equations in diagram 3 from post 174 are x_A = x_B*G and x'_B = x'_A*G, I guess that means the G factor is actually 1/gamma?

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

and

t'_A = x'_A/c
t_B = x_B/c

so

x'_A=x_A - vx'_A/c
x_B=x'_B + vx_B/c

so

x_A=x'_A + vx'_A/c = x'_A * (1 + v/c )
x'_B=x_B - vx_B/c = x_B * (1 - v/c )

and

x_B = x_A.G
x'_A = x'_B.G

so

x_B = x_A.G = G * x'_A * (1 + v/c )

and

x'_A = x'_B.G = G * x_B * (1 - v/c )

so

x_B = G * ( G * x_B * (1 - v/c )
) * (1 + v/c )

so

1 = G² (1 - v²/c²)

so

G = gamma - if you put G on the other sides of the first equations you come across it in, then you end up with G = 1/gamma (which is what you originally asked and the answer is yes), but you always end up with:

x_B = x_A.gamma
x'_A = x'_B.gamma

then taking the next step:

x_B=x'_B + vt_B

so

x'_B=x_B - vt_B

andx'_A = (x_B - vt_B).gamma

which is your spatial Lorentz transformation

then dividing through by c

t'_A = (t_B - vt_B/c).gamma

and since t_B = x_B/c then

t'_A = (t_B - v.x_B/c²).gamma

which is your temporal Lorentz transformation

These equations are actually different from what's written in equation 3 and 4 of the diagram, here you have t'_A where the diagram has t_A, and t_B where the diagram has t'_B. I think the revised versions make sense given what you say above, for example the first revised equation would mean:

(distance between B and E_A at the time t'_A when the photon from E_A reaches B, all in the A frame) = (distance between A and E_A in the A frame) - v*(time t'_A when the photon from E_A reaches B as seen in A frame)

...which does make sense, since B started at the same distance from E_A as A did, and then at any later time t B has moved an additional vt from E_A (all as seen in A frame). But with this revised version x'_A=x_A - vt'_A, you can't substitute x_A/c in for t'_A as you did in the diagram's version with t_A right? So doesn't that mean the final part of equation 4 in the diagram is no longer valid? And likewise with the final part of equation 3?

Sort of answered above, these supplant the diagrams and any errors therein.

cheers,

neopolitan

 

[neopolitan]

So, to continue with the slightly altered version of your derivation, we can combine the equations x_B = G*x'_A*(1 + v/c) along with x'_A = G*x_B*(1 - v/c) to get x_B = G*(G*x_B*(1 - v/c))*(1 + v/c) = G^2*x_B*(1 - v^2/c^2). So divide both sides bey x_B to get 1 = G^2*(1 - v^2/c^2), or G^2 = 1/(1 - v^2/c^2), showing G is the familiar gamma factor. We can plug that back into x_B = x_A*G to get x_B = gamma*x_A. Now I think you'd want to combine that with another revised equation, x'_A=x_A - vt'_A or equivalently x_A = x'_A + vt'_A, and then we get x_B = gamma*(x'_A + vt'_A). Now, you would say this is a derivation of the spatial portion of the Lorentz transformation, correct? The problem I see here is that when you actually examine the meaning of the symbols, they don't seem to be all coordinates of a specific event or pair of events as in the Lorentz transformation--x_B is the difference in the B frame between the spatial coordinates of the event E_B and the event of the photon from E_B hitting A, but x'_A is the difference in the A frame between the spatial coordinates of a different pair of events, namely the event E_A and the event of the photon from E_A hitting B (t'_A is the difference between the time coordinates of the same pair of events as x'_A). So unless I've gotten the equations wrong or the symbols mixed up, it seems that any resemblance to the spatial Lorentz transformation equation here is coincidental, this equation doesn't have the same physical meaning at all, and in fact it would only be applicable to the specific definitions of the symbols in terms of the particular physical scenario you described, rather than the Lorentz equation delta-x_B = gamma*(delta-x_A + v*delta-t_A) which is applicable to the difference in spatial coordinates and difference in time coordinates of two arbitrary events (events that may not lie on the worldline of a photon for example), but with the understanding that delta-x_B and delta-x_A are the difference in spatial coordinates between the same pair of events in two frames.

What I've shown you so far is how to end up with:

A talking about the location of event E_A as B sees it, modified by gamma. E_A can be any event even two events which do not share the same world line which you can then find the delta between.

and

B talking about the location of event E_B as A sees it, modified by gamma. E_B can be any event and even two events which do not share the same world line which you can then find the delta between.

If you want to call this the Lorentz boost to distinguish it from the Lorentz Transformation, since it comes from the Galilean boost, then you are welcome to.

It's such a simple thing compared to what you have to go through to work out the matrix versions, perhaps it is better to call it a boost and note that it is a 100% accurate approximation of the Lorentz Transformation limited to two dimensions but that you can apply the same logic to all three spatial dimensions to work out what happens in y and z if v is not limited.

cheers,

neopolitan

(The derivation of the temporal Lorentz transformation does seem dodgy, I accept that, it just happens to work. I understand your concern. Note, however, that I did say at one point something along the lines of "Once you understand how to derive the spatial Lorentz transformation, it is not difficult to do the same for the temporal". Would you be more happy if you could see that I can derive the temporal Lorentz transformation so that I have find the spatial component of any event, followed by the temporal component of any event. If I was showing students, I think I would have to show the temporal derivation as well.)

 

[JesseM]

What you end up with is:

A talking about event E_A as B sees it, modified by gamma. E_A can be any event even two events which do not share the same world line which you can then find the delta between.

and

B talking about event E_B as A sees it, modified by gamma. E_B can be any event and even two events which do not share the same world line which you can then find the delta between.

The problem is that when we actually look at the meaning of the symbols in your equations, they are not talking about the coordinates of the same event (or the coordinate intervals between the same pair of events) in two different frames. And there is nothing in your derivation to show it can be generalized to such a case, everything about your derivation referred to specific events on the worldline of a light ray that were chosen based on simultaneity with A&B being colocated.

For example, you say x'_A = (x_B - vt_B).gamma is the spatial Lorentz transform, but it just isn't, not with the stated meaning of the symbols--the left side deals with the distance between two events (E_A, photon hitting B), while the right side deals with the distance and time between a different pair of events (E_B, photon hitting A). It so happens because of the symmetry of the problem that the distance and time between the first pair should be the same as the distance and time between the second pair in any given frame, but how do you prove that the equation would still work if we changed the meaning of the symbols so both sides were referring to the same pair of events, events which didn't necessarily have to lie on the worldline of a light ray? To derive the Lorentz transform you need an equation where the physical meaning of the symbols is the same, not just an equation that has the same form.

It's such a simple thing compared to what you have to go through to work out the matrix versions

If you want a relatively simple derivation of the 1+1 dimensional Lorentz transform that sticks to basic algebra, see my post #14 on this thread.

 

[neopolitan]

The problem is that when we actually look at the meaning of the symbols in your equations, they are not talking about the coordinates of the same event (or the coordinate intervals between the same pair of events) in two different frames. And there is nothing in your derivation to show it can be generalized to such a case, everything about your derivation referred to specific events on the worldline of a light ray that were chosen based on simultaneity with A&B being colocated.

For example, you say x'_A = (x_B - vt_B).gamma is the spatial Lorentz transform, but it just isn't, not with the stated meaning of the symbols--the left side deals with the distance between two events (E_A, photon hitting B), while the right side deals with the distance and time between a different pair of events (E_B, photon hitting A). How do you prove that the equation would still work if we changed the meaning of the symbols so both sides were referring to the same pair of events? To derive the Lorentz transform you need an equation where the physical meaning of the symbols is the same, not just an equation that has the same form.

If you want a relatively simple derivation of the 1+1 dimensional Lorentz transform that sticks to basic algebra, see my post #14 on this thread.

I modified my post. Perhaps you might want to modify your reply in light of that, then I can delete this.

 

[JesseM]

What I've shown you so far is how to end up with:

A talking about the location of event E_A as B sees it, modified by gamma. E_A can be any event even two events which do not share the same world line which you can then find the delta between.

and

B talking about the location of event E_B as A sees it, modified by gamma. E_B can be any event and even two events which do not share the same world line which you can then find the delta between.

Are you still talking about the equation x'_A = (x_B - vt_B).gamma though? If so I don't see how your description fits, x'_A on the left side isn't the spatial coordinates of any individual event in A's frame, rather it's the interval (x-coordinate A assigns to event E_A) - (x-coordinate A assigns to photon hitting B). Likewise x_B on the right side is the interval (x-coordinate B assigns to E_B) - (x-coordinate B assigns to photon hitting A), and t_B does the same but for time coordinates. Also, you say "E_B can be any event" but you don't justify that, your derivation of this equation made use of some facts that were specifically related to the fact that these are events on the worldline of a light ray, like the substitution t_B = x_B/c.

 

[neopolitan]

Are you still talking about the equation x'_A = (x_B - vt_B).gamma though? If so I don't see how your description fits, x'_A on the left side isn't the spatial coordinates of any individual event in A's frame, rather it's the interval (x-coordinate A assigns to event E_A) - (x-coordinate A assigns to photon hitting B). Likewise x_B on the right side is the interval (x-coordinate B assigns to E_B) - (x-coordinate B assigns to photon hitting A), and t_B does the same but for time coordinates. Also, you say "E_B can be any event" but you don't justify that, your derivation of this equation made use of some facts that were specifically related to the fact that these are events on the worldline of a light ray, like the substitution t_B = x_B/c.

Yes, my E_A and E_B are causally linked and right at the beginning, if you remember, I talked about one photon which could have been emitted anywhere so long as it subsequently passes B and then A.

This derivation is based on the fact that the event emits a photon which is subsequently detected, otherwise A and B know nothing about the event and quite possibly could never know anything about it.

cheers,

neopolitan

 

[JesseM]

Yes, my E_A and E_B are causally linked and right at the beginning, if you remember, I talked about one photon which could have been emitted anywhere so long as it subsequently passes B and then A.

This derivation is based on the fact that the event emits a photon which is subsequently detected, otherwise A and B know nothing about the event and quite possibly could never know anything about it.

The emission event doesn't even enter into any of your equations, the photon could have been traveling along the same path for infinite time, it wouldn't make a difference. And if you agree your derivation involves a very particular choice of four events to look at, then do you agree that you aren't really deriving the Lorentz transformation? The Lorentz transform deals with the coordinates of a single arbitrary event in two frames (or the coordinate intervals between a single arbitrary pair of events in two frames), whereas your equation shows a relation between two different pairs of events that have particular properties that you defined for them (like the fact that all four events have a light-like separation from one another). It so happens that because of a symmetry in the way you defined the events, the spatial and temporal separation between the first pair is guaranteed to be the same as the spatial and temporal separation between the second pair (try drawing both the 'E_A/photon hitting B' pair and the 'E_B/photon hitting A' pair on the same graph and hopefully you'll see what I mean), which is why your relation looks just like the Lorentz transform equation dealing with a single pair of events, but your derivation wouldn't generalize to an arbitrarily-chosen pair of events, in particular a pair that had a time-like or space-like separation.

 

[neopolitan]

The emission event doesn't even enter into any of your equations, the photon could have been traveling along the same path for infinite time, it wouldn't make a difference. And if you agree your derivation involves a very particular choice of four events to look at, then do you agree that you aren't really deriving the Lorentz transformation? The Lorentz transform deals with the coordinates of a single arbitrary event in two frames (or the coordinate intervals between a single arbitrary pair of events in two frames), whereas your equation shows a relation between two different pairs of events that have particular properties that you defined for them (like the fact that all four events have a light-like separation from one another).

You really are going to have to make up your mind.

I think the Lorentz transform is one event described in two frames. Do you agree?

Or do you want the Lorentz transform to be coordinate intervals between a pair of events described in two frames?

Or do you want it to be both?

I don't know where you are getting four events from.

It so happens that because of a symmetry in the way you defined the events, the spatial and temporal separation between the first pair is guaranteed to be the same as the spatial and temporal separation between the second pair (try drawing both the 'E_A/photon hitting B' pair and the 'E_B/photon hitting A' pair on the same graph and hopefully you'll see what I mean), which is why your relation looks just like the Lorentz transform equation dealing with a single pair of events, but your derivation wouldn't generalize to an arbitrarily-chosen pair of events, in particular a pair that had a time-like or space-like separation.

I think you don't understand, yes.

Take one event. Any event, E, any where any time.

Take a single observer, A, any where any time.

Now that we have a single observer and a single event (we don't stuff around by changing anything), there is a single distance interval between the spatial location of event E and the spatial location of the observer and a single time interval between the time location of event E and any specific time reading for the observer.

The difference is that x=0 makes sense, but really t=0 doesn't make much sense, except perhaps if you start when the observer is born/created/assigned the task. But if you pick any time for A and call it t=0, then there is a single time interval between the event (0,0) and the event E.

So there is a single spacetime interval between the event (0,0) and the event E (irrespective of how that spacetime interval is measured, ie what coordinate system you use to measure it - even if the coordinate system of A makes the most sense).

Do we agree on this?

I'm not going further than this. Please agree or disagree on this, then I can proceed. But if we disagree on this, or I have a fundamental misunderstanding, then we have to sort that out first.

cheers,

neopolitan

 

[JesseM]

You really are going to have to make up your mind.

I think the Lorentz transform is one event described in two frames. Do you agree?

Or do you want the Lorentz transform to be coordinate intervals between a pair of events described in two frames?

Or do you want it to be both?

The Lorentz transform can be written both ways. For example, the spatial part can be written as

x' = gamma*(x - vt)

where (x,t) are the coordinates of a single event in the unprimed frame and x' is the spatial coordinate of the same event in the primed frame, or as

delta-x' = gamma*(delta-x - v*delta-t)

where delta-x and delta-t are the coordinate intervals in the unprimed frame between a pair of events, and delta-x' is the spatial coordinate interval between the same pair of events in the primed frame. It's easy to derive the second equation from the first.

I don't know where you are getting four events from.

I mentioned the four events in your equation in all three of my previous posts. For example, in post 229:

you say x'_A = (x_B - vt_B).gamma is the spatial Lorentz transform, but it just isn't, not with the stated meaning of the symbols--the left side deals with the distance between two events (E_A, photon hitting B), while the right side deals with the distance and time between a different pair of events (E_B, photon hitting A).

The first parentheses is one pair of events, the one x'_A gives the coordinate distance between in frame A according to previous definitions, while the second parentheses is a different pair of events, the one x_B and t_B give the coordinate intervals between in frame B. So your equation relates one pair of events on the left side to a different pair of events on the right side.

I think you don't understand, yes.

Take one event. Any event, E, any where any time.

Take a single observer, A, any where any time.

Now that we have a single observer and a single event (we don't stuff around by changing anything), there is a single distance interval between the spatial location of event E and the spatial location of the observer and a single time interval between the time location of event E and any specific time reading for the observer.

The difference is that x=0 makes sense, but really t=0 doesn't make much sense, except perhaps if you start when the observer is born/created/assigned the task. But if you pick any time for A and call it t=0, then there is a single time interval between the event (0,0) and the event E.

So there is a single spacetime interval between the event (0,0) and the event E (irrespective of how that spacetime interval is measured, ie what coordinate system you use to measure it - even if the coordinate system of A makes the most sense).

Do we agree on this?

Sure, but the Lorentz transformation doesn't directly calculate the invariant spacetime interval x^2 - c^2*t^2. It says if you know the coordinates of E are (x,t) in one frame, then it can tell you the corresponding coordinates (x',t') of the same event E in another frame. Likewise, if you have two arbitrary events E1 and E2 and you know the coordinate distance delta-x and coordinate time delta-t between them in one frame, then it can tell you the delta-x' and delta-t' for the same specific pair E1 and E2 in another frame. Your equation does neither of these things.

 

[neopolitan]

I mentioned the four events in your equation in all three of my previous posts. For example, in post 229:

you say x'_A = (x_B - vt_B).gamma is the spatial Lorentz transform, but it just isn't, not with the stated meaning of the symbols--the left side deals with the distance between two events (E_A, photon hitting B), while the right side deals with the distance and time between a different pair of events (E_B, photon hitting A).

The first parentheses is one pair of events, the one x'_A gives the coordinate distance between in frame A according to previous definitions, while the second parentheses is a different pair of events, the one x_B and t_B give the coordinate intervals between in frame B. So your equation relates one pair of events on the left side to a different pair of events on the right side.

You're not confused here Jesse, you are simply wrong. There are two relevant events in terms of the derivation, photon crosses the x_A axis (at t=0 according to A and t=-6 according to B) and photon crosses the x_B axis (at t=3 according to A and t=0 according to B).

Agreed that both can use another two events to work out where on their own x-axis that crossing took place, but otherwise those events (photon passes B and photon passes A) are irrelevant (in this part of the derivation I really don't care when A and B work out where a photon from event E_A passed their x axis).

But do you at least agree that all four events could involve the same photon? (I think you do because you were complaining about them all being on the same world line.)

Sure.

I light of that I present a semi humorous pair of drawings.

cheers,

neopolitan

 

[JesseM]

You're not confused here Jesse, you are simply wrong. There are two relevant events in terms of the derivation, photon crosses the x_A axis (at t=0 according to A and t=-6 according to B) and photon crosses the x_B axis (at t=3 according to A and t=0 according to B).

Are you claiming that E_A and E_B don't also figure into the definition of x'_A and x_B? Here was how you defined those terms in post #221:

x'_A is the distance between B and event E_A at t=5 according to A
(value in the example we have been discussing: 5)

x_B is the distance between A and event E_B at t=-10 according to B
(value in the example we have been discussing: 10)

Note that t=5 in the A frame was the time when the light hit B according to the A frame, so according to the above x'_A is the distance in the A frame between B and E_A at the moment the light hits B (i.e. the distance in the A frame between the event E_A and the event of light hitting B)

Likewise, t=10 (which is what I assume you mean to write in the second line rather than t=-10) in the B frame was the time when the light hit A according to the B frame (which is also how you defined t_B earlier), so according to the above x_B is the distance in the B frame between A and E_B at the moment the light hit A (i.e. the distance in the B frame between the event E_B and the event of light hitting A)

So unless you've revised your definitions of the terms x'_A and x_B, it appears that the first refers to the distance between two events (E_A, light hits B) and the second refers to the distance between two different events (E_B, light hits A). If you think I am misunderstanding something here, please explain.

But do you at least agree that all four events could involve the same photon? (I think you do because you were complaining about them all being on the same world line.)

Yes, I agree with that--where I write "photon hits A" or "photon hits B" it shouldn't be taken to imply the photon is absorbed, perhaps it would be better to write "photon passes" instead.

 

[neopolitan]

Going back as far as post #212 (when I filled in forms for you), I can't see anywhere where I have spoken about photons have hit A and B and given notation for those events.

You brought those events up for the first time in post #229. I followed up #229 with #230 in which I said "I modified my post (#228). Perhaps you might want to modify your reply in light of that, then I can delete this."

You didn't modify #229 so I have subsequently ignored it thinking that #231 which referred to the modified #228 replaced #229.

After that I did talk about the photon being subsequently detected and detection was always part of the scenario, but I never formally described any event which was the photon passing either A or B. (Those events would be (0,8) and (0,4) in the frame of the observer being passed.)

I don't think you did either.

cheers,

neopolitan

 

[JesseM]

Going back as far as post #212 (when I filled in forms for you), I can't see anywhere where I have spoken about photons have hit A and B and given notation for those events.

In post #224 you said:

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

where t'_A is when the photon from E_A passes B (eg t=5) according to A and vt_B is when the photon from E_B passes A according to B (eg t=10).

x_A was the coordinate position of E_A in the A frame, and in this frame B is moving towards E_A at velocity v starting at the origin, so naturally the distance between B and the position of E_A as a function of time in this frame is x_A - vt. If you define t = t'_A as the time when the photon passes B as seen in the A frame, that means x_A - vt'_A must be the distance in this frame between B and E_A at the moment the photon passes B, i.e. the distance in the A frame between the event of the photon passing B and the event E_A. Do you disagree?

You brought those events up for the first time in post #229. I followed up #229 with #230 in which I said "I modified my post (#228). Perhaps you might want to modify your reply in light of that, then I can delete this."

Was the modification to post 228 supposed to modify any of the definitions of yours which I quoted above?

After that I did talk about the photon being subsequently detected and detection was always part of the scenario, but I never formally described any event which was the photon passing either A or B. (Those events would be (0,8) and (0,4) in the frame of the observer being passed.)

t'_A and t_B referred to the time in one observer's frame when the other observer was being passed by the photon, and they occurred at t=5 and t=10 which did figure into your calculations (remember in the second-to-last paragraph of my post #222 where I pointed out that the relations x_B = x_A*gamma and x'_A = x'_B*gamma only included the gamma factor because you chose that particular pair of times, and I asked why you had chosen them...your definitions of t'_A and t_B in post #224 in terms of photons passing observers was what I took as an answer to that question).

 

[neopolitan]

Are you claiming that E_A and E_B don't also figure into the definition of x'_A and x_B?

Have I said that? The photon passing B and then A don't really figure as events in my derivation, otherwise I would have drawn them in as events - we know that a photon at 8ls away from A traveling towards A will pass A 8s later. But the event I am talking about is the photon being 8ls away from A, E_A -the photon passing A 8s later is a consequence of that event as well as being an event in its own right.

It's just that I am focussed on the former (consequence) rather then the latter (a separate event in its own right).

Note that t=5 in the A frame was the time when the light hit B according to the A frame, so according to the above x'_A is the distance in the A frame between B and E_A at the moment the light hits B (i.e. the distance in the A frame between the event E_A and the event of light hitting B)

Likewise, t=10 (which is what I assume you mean to write in the second line rather than t=-10) in the B frame was the time when the light hit A according to the B frame (which is also how you defined t_B earlier), so according to the above x_B is the distance in the B frame between A and E_B at the moment the light hit A (i.e. the distance in the B frame between the event E_B and the event of light hitting A)

So unless you've revised your definitions of the terms x'_A and x_B, it appears that the first refers to the distance between two events (E_A, light hits B) and the second refers to the distance between two different events (E_B, light hits A). If you think I am misunderstanding something here, please explain.

Step 1, ask if something that seems odd is me screwing up.

Step 2, wait for a response, it's not as if I am making you wait.

Step 3, work from the correction, if there is indeed one.

The value t=-10 is correct. When the photon passes E_A, then according to B, t=-10. When the photon passes A, then according to B, t=10. Your incorrect correction is right, since you redefined what I was saying.

Note very very carefully, I said in #224:

vt_B is when the photon from E_B passes A according to B (eg t=10)

and in #221:

x_B is the distance between A and event E_B at t=-10 according to B

I didn't put a subscript on t=-10. It's not an error.

Yes, I agree with that--where I write "photon hits A" or "photon hits B" it shouldn't be taken to imply the photon is absorbed, perhaps it would be better to write "photon passes" instead.

I know what you mean, I have tried to use "passes", but if I say "hits" I mean "passes". I mentioned "thought experiment magic" before and I fully expect you to have the right to use the same magic in your descriptions, so a photon which hits B recovers immediately and continues on its way to A with no delay or other ill effects.

Any comments on the diagrams? They were semi-humorous in the sense that they were semi-serious.

cheers,

neopolitan

 

[neopolitan]

x_A was the coordinate position of E_A in the A frame, and in this frame B is moving towards E_A at velocity v starting at the origin, so naturally the distance between B and the position of E_A as a function of time in this frame is x_A - vt. If you define t = t'_A as the time when the photon passes B as seen in the A frame, that means x_A - vt'_A must be the distance in this frame between B and E_A at the moment the photon passes B, i.e. the distance in the A frame between the event of the photon passing B and the event E_A. Do you disagree?

No, I agree.

Was the modification to post 228 supposed to modify any of the definitions of yours which I quoted above?

No.

t'_A and t_B referred to the time in one observer's frame when the other observer was being passed by the photon, and they occurred at t=5 and t=10 which did figure into your calculations (remember in the second-to-last paragraph of my post #222 where I pointed out that the relations x_B = x_A*gamma and x'_A = x'_B*gamma only included the gamma factor because you chose that particular pair of times, and I asked why you had chosen them...your definitions of t'_A and t_B in post #224 in terms of photons passing observers was what I took as an answer to that question).

They are consequences of the placements of the photon at events E_A and E_B, like I said before, I didn't consider the passing A and B by the photon to be significant events in their own right.

It's like falling down the top of the stairs (top step being JesseM's reference step) - the event "tripping" is important but really, all the little events that follow (Jesse M hits the third step down, JesseM hits the fourth and fifth step down, JesseM starts rolling and hits all steps until step 13) are consequences of the initial trip. Perhaps you could mention one other important event, trying to stop falling by reaching for the banister and failing, but once you are really falling, the end result is pretty much all scripted.

If I made step 12 a special step (it's the one I hide all my gold under, so it is my reference step) would I be obliged to refer to JesseM hits step 12 as a special event?

In the same sense, although I know that the photon eventually passes both B and A, I don't feel obliged to refer to them as special events.

cheers,

neopolitan

 

[JesseM]

Have I said that? The photon passing B and then A don't really figure as events in my derivation, otherwise I would have drawn them in as events - we know that a photon at 8ls away from A traveling towards A will pass A 8s later. But the event I am talking about is the photon being 8ls away from A, E_A -the photon passing A 8s later is a consequence of that event as well as being an event in its own right.

It's just that I am focussed on the former (consequence) rather then the latter (a separate event in its own right).

Regardless of whether you were "focussed" on them, what I was saying was that if we look at the physical meaning of x'_A and x_B as you defined them, there doesn't seem to be any way to define them that doesn't involve those events of the photons passing A and B. If you can think of a rigorous definition of x'_A and x_B that make no mention of these events (and don't refer to other coordinates which themselves are defined in terms of these events), then please explain.

The value t=-10 is correct. When the photon passes E_A, then according to B, t=-10. When the photon passes A, then according to B, t=10. Your incorrect correction is right, since you redefined what I was saying.

Note very very carefully, I said in #224:

vt_B is when the photon from E_B passes A according to B (eg t=10)

And was it an error that you wrote vt_B there as opposed to t_B? vt_B would be a distance rather than a time.

and in #221:

x_B is the distance between A and event E_B at t=-10 according to B

OK, but that would imply x_B = 2, which doesn't fit with what you wrote elsewhere. After all, in B's frame A was moving in the -x direction at 0.6c, so at t=-10, A would have been at position x=+6, while the event E_B occurred at x=+4 in B's frame.

I didn't put a subscript on t=-10. It's not an error.

When you say you "didn't put a subscript", you mean t=-10 is distinct from t_B which is t=10 according to the definition above (if we remove the v), right? But then it seems you have offered two distinct definitions of x_B, one of which makes use of t=-10 and one of which makes use of t_B = 10:

x_B is the distance between A and event E_B at t=-10 according to B

x_B=x'_B + vt_B

These two definitions would be equivalent if you had written t=10 in the first, which was part of why I assumed it was a mistake. But if it's not a mistake, the definitions appear to be incompatible--as I said, the first would seem to imply x_B = 2, while the second implies x_B = 4 + 0.6*10 = 10

Any comments on the diagrams? They were semi-humorous in the sense that they were semi-serious.

I couldn't quite follow the point you were making, but it seemed like you were saying my objection was that the red event might not lie on the light ray that crossed through E_A and E_B...if so that wasn't really my objection, I realize that you can always draw a new light ray which goes through any arbitrary event, and define a new E_A and E_B in terms of where this ray crosses the x-axes of A and B's frames. But even if we assume our "arbitrary event" is along this ray, my problem is that none of the coordinates you defined--x_A, x'_A, x_B, x'_B--have anything to do with that event specifically as opposed to any of an infinite number of other possible events along the same ray, all the events on this ray would yield the same values for those coordinates that you defined. So, the relation between these coordinates doesn't really demonstrate anything about how the coordinates of the event itself in each frame will be related to one another, it only tells us about relations between coordinates of events involved in the definitions of x_A and your other coordinates.

 

[neopolitan]

Regardless of whether you were "focussed" on them, what I was saying was that if we look at the physical meaning of x'_A and x_B as you defined them, there doesn't seem to be any way to define them that doesn't involve those events of the photons passing A and B. If you can think of a rigorous definition of x'_A and x_B that make no mention of these events (and don't refer to other coordinates which themselves are defined in terms of these events), then please explain.

Do you disagree that the photon passing A at a specific time (in either coordinate frame) is a consequence of that photon being at E_A at another specific time and having a specific direction? I agree that the photon being at that spacetime location leads to the photon passing B and then A. I've never come close to denying that.

Just as much as event E_A has a specific spacetime location, so too does the photon pass B and A (constituting two new events in your parlance). But which comes first?

I'm giving priority to the first consequential event in each frame (either E_A or E_B, not the detection of the event(s) (a detection which is in itself a new event in your parlance).

It gets more complicated if we use the B frame, because if the photon is spawned by E_B then it never passed the x_A axis and so there was no location of the photon simultaneous with the colocation of A and B, according to A. That's why I started with the idea of a photon which just passes the x axes, and call those passings events E_A and E_B.

And was it an error that you wrote vt_B there as opposed to t_B? vt_B would be a distance rather than a time.

Yeah, I've been having all sorts of problems with cutting and pasting code. Delete the v.

OK, but that would imply x_B = 2, which doesn't fit with what you wrote elsewhere. After all, in B's frame A was moving in the -x direction at 0.6c, so at t=-10, A would have been at position x=+6, while the event E_B occurred at x=+4 in B's frame.

I can only refer you back to posts #227 and #224.

x_A is the distance between the origin of the x_A axis and E_A, according to A, which is 8.

x'_B is the distance between the origin of the x_B axis and E_B, according to B, which is 4.

t_A is the time it takes a photon to travel from event E_A to the origin of the x_A axis, according to A, which is 8.

t'_B is the time it takes a photon to travel from event E_B to the origin of the x_B axis, according to B, which is 4.

t'_A is the time it takes a photon to travel from event E_A and pass the t_B axis (and hence B), according to A, which is 5.

t_B is the time it takes a photon to travel from event E_B and pass the t_A axis (and hence B), according to B, which is 10.

x'_A is the distance beween B and event E_A when the photon passes B (which is, I stress, just a consequence of the spacetime location of event E_A), according to A, which is 5.

x_B is the distance beween A and event E_B when the photon passes A (which is, I stress, just a consequence of the spacetime location of event E_B), according to B, which is 10.

When you say you "didn't put a subscript", you mean t=-10 is distinct from t_B which is t=10 according to the definition above (if we remove the v), right?

Yes, t=-10 is not shown anywhere on the diagram, but if you took the t_B axis and relocated it so it crossed E_B and then followed it down until it crossed the t_A axis, then the that crossing would be t = -10 on the t_B axis.

I couldn't quite follow the point you were making, but it seemed like you were saying my objection was that the red event might not lie on the light ray that crossed through E_A and E_B...if so that wasn't really my objection, I realize that you can always draw a new light ray which goes through any arbitrary event, and define a new E_A and E_B in terms of where this ray crosses the x-axes of A and B's frames. But even if we assume our "arbitrary event" is along this ray, my problem is that none of the coordinates you defined--x_A, x'_A, x_B, x'_B--have anything to do with that event specifically as opposed to any of an infinite number of other possible events along the same ray, all the events on this ray would yield the same values for those coordinates that you defined. So, the relation between these coordinates doesn't really demonstrate anything about how the coordinates of the event itself in each frame will be related to one another, it only tells us about relations between coordinates of events involved in the definitions of x_A and your other coordinates.

Pick any event, relocate (conceptually) your axes, and you can work out x_B in terms of x_A by seeing where a photon from the event which now lies on the x_A axis crosses the x_B axis.

Do a similar thing with the t axes and you can work out t_B in terms of t_A.

Perhaps I have confused you. I talked about an event that happens anywhere on the world line defined by E_A and E_B. Really, I only want to talk about one "real" event which I purposely shift my axes to align up so that the event is on the x_A axis for the purposes of deriving the spatial Lorentz transformation.

To do the same with the temporal Lorentz transformation you can shift the axes so that the event is on the t_A axis. When does (or did) a photon which crosses the t_A axis at t_A cross the t_B axis?

The relationship will actually be the same (just shifted) as the relationship between the two events you want to add to the mix, photon passing B and photon passing A.

t'_A = (t_B - v.x_B/c²).gamma

5 = ( 10 - 0.6*10 ) * 1.25 = 4 * 1.25 = 5

or (noting v in the other direction)

t_B = (t'_B + v.x'_B/c²).gamma

10 = ( 5 + 0.6*5 ) * 1.25 = 8 * 1.25 = 10

If this doesn't help then, without some animation, I really wonder if there is any way to get this through to you.

I don't really have the facilities here to do animation. If there is anyone following this thread who understands what I am trying to explain and can do animation, then perhaps you could help by showing the temporal relocation of the x_A and x_B axes and the spatial relocation of the t_B axis to align with any event that JesseM would like to choose.

For example, the event in http://www.geocities.com/neopolitonian/uniquespacetimelocation.jpg" is nominally:

(x_A,t_A)=(8,0).

Say we chose an event:

(x_A,t_A)=(5,-4)

the relocation would make this event be:

(x_A,t_A+4)=(5,0)

The animation I am thinking of is the three axes in question sliding down four to align with the new event. Is that possible?

cheers,

neopolitan

(There may be some cut and paste, or failure to subscript errors in here. I am really getting tired, physically and intellectually tired, of explaining something that seems quite obvious to me, but clearly isn't obvious to everybody, or perhaps anybody else. And the more I write, the more chances there are that something I write is not perfect.)

 

[JesseM]

Do you disagree that the photon passing A at a specific time (in either coordinate frame) is a consequence of that photon being at E_A at another specific time and having a specific direction? I agree that the photon being at that spacetime location leads to the photon passing B and then A. I've never come close to denying that.

I don't disagree with you in causal terms, but I'm not talking about causality, I'm just talking about the definition of terms. Since E_B happens at a later time than E_A you could equally well say that the photon passing through the point E_B is a "consequence" of it having been at E_A, but there's no need to refer to two events in the definition of x'_B which just refers to the coordinate position of the photon at time 0 in the B frame (and this event is of course E_B). In contrast, your definitions of x'_A and x_B was in terms of the distance between two events, and you don't have any other way to define the meaning of these terms. So, your equation x'_A=gamma*(x_B - vt_B) is not physically equivalent to the Lorentz transform, despite the fact that it looks the same if you forget about the definitions of the terms.

and in #221:

x_B is the distance between A and event E_B at t=-10 according to B

OK, but that would imply x_B = 2, which doesn't fit with what you wrote elsewhere. After all, in B's frame A was moving in the -x direction at 0.6c, so at t=-10, A would have been at position x=+6, while the event E_B occurred at x=+4 in B's frame.

I can only refer you back to posts #227 and #224.

x_A is the distance between the origin of the x_A axis and E_A, according to A, which is 8.

x'_B is the distance between the origin of the x_B axis and E_B, according to B, which is 4.

t_A is the time it takes a photon to travel from event E_A to the origin of the x_A axis, according to A, which is 8.

t'_B is the time it takes a photon to travel from event E_B to the origin of the x_B axis, according to B, which is 4.

t'_A is the time it takes a photon to travel from event E_A and pass the t_B axis (and hence B), according to A, which is 5.

t_B is the time it takes a photon to travel from event E_B and pass the t_A axis (and hence B), according to B, which is 10.

x'_A is the distance beween B and event E_A when the photon passes B (which is, I stress, just a consequence of the spacetime location of event E_A), according to A, which is 5.

x_B is the distance beween A and event E_B when the photon passes A (which is, I stress, just a consequence of the spacetime location of event E_B), according to B, which is 10.

How does referring me back to these definitions (which I don't object to) answer my question about your comment in post #221, where you defined x_B in a different way, not in terms of the distance between A and E_B at t_B=10 as above, but rather in terms of the distance between A and E_B at t=-10?

Yes, t=-10 is not shown anywhere on the diagram, but if you took the t_B axis and relocated it so it crossed E_B and then followed it down until it crossed the t_A axis, then the that crossing would be t = -10 on the t_B axis.

"Relocated it"? Your definition in post #221 didn't say anything about such a relocation. Also, are you talking about shifting the point in spacetime that you label the crossing point of A&B (in which case you'd have to change which point you call E_B and E_A), or are you talking about keeping that point the same but having B's time axis no longer pass through it, so it's as if A and B are actual physical observers who cross at some point, but B is using a coordinate system where he's at rest but not located at x=0?

I couldn't quite follow the point you were making, but it seemed like you were saying my objection was that the red event might not lie on the light ray that crossed through E_A and E_B...if so that wasn't really my objection, I realize that you can always draw a new light ray which goes through any arbitrary event, and define a new E_A and E_B in terms of where this ray crosses the x-axes of A and B's frames. But even if we assume our "arbitrary event" is along this ray, my problem is that none of the coordinates you defined--x_A, x'_A, x_B, x'_B--have anything to do with that event specifically as opposed to any of an infinite number of other possible events along the same ray, all the events on this ray would yield the same values for those coordinates that you defined. So, the relation between these coordinates doesn't really demonstrate anything about how the coordinates of the event itself in each frame will be related to one another, it only tells us about relations between coordinates of events involved in the definitions of x_A and your other coordinates.

Pick any event, relocate (conceptually) your axes, and you can work out x_B in terms of x_A by seeing where a photon from the event which now lies on the x_A axis crosses the x_B axis.

Do a similar thing with the t axes and you can work out t_B in terms of t_A.

Perhaps I have confused you. I talked about an event that happens anywhere on the world line defined by E_A and E_B. Really, I only want to talk about one "real" event which I purposely shift my axes to align up so that the event is on the x_A axis for the purposes of deriving the spatial Lorentz transformation.

OK, I don't think you said before that you wanted to relocate the axes so to ensure that the event lies on the x_A axis (in which case the event would be the new E_A). But then you haven't really proved the general Lorentz transformation which says that events at arbitrary coordinates (x,t) in one frame and (x',t') in the other are related by x'=gamma*(x - vt), you've only shown that if you pick a pair of coordinate systems such that the event lies on the x-axis of one of the frames, then something like this relation holds. And I say "something like" because your equation does not actually relate the coordinates of the individual event E_A in the A frame with the coordinates of the same individual event in the B frame, rather it relates the interval (E_A, photon passing B) in the A frame to the interval (E_B, photon passing A) in the B frame.

 

[neopolitan]

I don't disagree with you in causal terms, but I'm not talking about causality, I'm just talking about the definition of terms. Since E_B happens at a later time than E_A you could equally well say that the photon passing through the point E_B is a "consequence" of it having been at E_A, but there's no need to refer to two events in the definition of x'_B which just refers to the coordinate position of the photon at time 0 in the B frame (and this event is of course E_B). In contrast, your definitions of x'_A and x_B was in terms of the distance between two events, and you don't have any other way to define the meaning of these terms. So, your equation x'_A=gamma*(x_B - vt_B) is not physically equivalent to the Lorentz transform, despite the fact that it looks the same if you forget about the definitions of the terms.

I'm glad that you point out that E_B is causally linked to E_A. I thought you had grasped that (perhaps not consciously) the whole time.

Think like this, if you can. According B, B is stationary, so the distance between B and the location of E_A never changes, correct? So the distance between B and the location of E_A at any time, according to B, is invariant (Lorentz invariant but B doesn't need to say that).

When A and B are colocated, t_B = 0 and 4 time units later B passes a photon, so B "knows" that the separation between B and the photon when A and B were colocated was 4 space units. Correct?

If the photon was spawned by E_A it will pass E_B, so a photon spawned by E_A is indistinguishable from a photon spawned by _B. Correct?

My equations reflect this. How I can word that so that it makes you happy, I don't know.

What I do know is that somehow I have single handedly come up with a way to derive equations which are indistinguishable from the Lorentz transformations. Not sure what I should call them though, since if I tell people I have derived these new equations, they will tell me "No, that is just a recasting of the Lorentz transformations". I'm pretty damn sure that if I started off like that, saying I had new equations which just look like Lorentz transformations, you would be telling me that they are not new, they actually are the Lorentz transformations recast. But that's ok, I've come up with new equations. I'm happy with that.

How does referring me back to these definitions (which I don't object to) answer my question about your comment in post #221, where you defined x_B in a different way, not in terms of the distance between A and E_B at t_B=10 as above, but rather in terms of the distance between A and E_B at t=-10?

Because in post #224 I said what I had written (in post #224) supersedes what had come earlier. I thought if I did it all again, you could work from that, rather than going to something that is superseded.

"Relocated it"? Your definition in post #221 didn't say anything about such a relocation. Also, are you talking about shifting the point in spacetime that you label the crossing point of A&B (in which case you'd have to change which point you call E_B and E_A), or are you talking about keeping that point the same but having B's time axis no longer pass through it, so it's as if A and B are actual physical observers who cross at some point, but B is using a coordinate system where he's at rest but not located at x=0?

In post #221 I had no inkling of the lengths I would have to go to try to get you to understand. But anyway, #221 is superseded.

OK, I don't think you said before that you wanted to relocate the axes so to ensure that the event lies on the x_A axis (in which case the event would be the new E_A). But then you haven't really proved the general Lorentz transformation which says that events at arbitrary coordinates (x,t) in one frame and (x',t') in the other are related by x'=gamma*(x - vt), you've only shown that if you pick a pair of coordinate systems such that the event lies on the x-axis of one of the frames, then something like this relation holds. And I say "something like" because your equation does not actually relate the coordinates of the individual event E_A in the A frame with the coordinates of the same individual event in the B frame, rather it relates the interval (E_A, photon passing B) in the A frame to the interval (E_B, photon passing A) in the B frame.

Again, I never thought I would have to go to such lengths.

I'm going to hope you get an idea of what I am getting at, and later I will try to do a be all and end all diagram (but only from one perspective) that will help you understand.

cheers,

neopolitan