Benefits of time dilation / length contraction pairing?

[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

 

[JesseM]

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.

Yes, this has always been obvious to me, but I don't see the relevance.

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).

It's certainly invariant in B's frame, but if you think it's "Lorentz invariant" you misunderstand the meaning of the term. "Lorentz invariant" means "invariant under the Lorentz transform", i.e. something which is the same in all inertial frames, like the invariant interval ds^2 = dx^2 - c^2*dt^2. The distance between B and the location of E_A is not something that's the same in every frame (in fact in most frames it's not constant with time), so it's not Lorentz invariant.

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?

Sure.

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.

I'm not sure how you think this is relevant. Yes, obviously E_A, E_B, the event of the photon passing B, and the event of the photon passing A all lie on the worldline of a single photon. This doesn't change the fact that x'_A is defined as the difference in position between the first and third event in the A frame, while x_B is defined as the difference in position between the second and fourth event in the B frame, and that your derivation assumes all four events have a light-like separation from one another. So your equation x'_A = gamma*(x_B - vt_B) does not have the same physical meaning as the Lorentz equation x' = gamma*(x - vt) despite the fact that it looks similar, because in the Lorentz equation x' and x either represent the coordinates of a single event in the primed and unprimed frame (which can be at any arbitrary position, not necessarily on one of the frame's spatial axes), or else x' and x can represent the coordinate intervals in two frames between a single pair of events (which can also be located at arbitrary positions, and which need not have a light-like separation from one another).

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.

But they're not indistinguishable, not when you keep in mind the physical meaning of the terms in your equations vs. the physical meaning of the terms in the Lorentz transformation.

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.

Not if you explained the physical meaning of the terms. I'm going to try to draw some diagrams of my own to show the difference in the meaning of the terms visually.

But that's ok, I've come up with new equations. I'm happy with that.

Yes, new equations which are only applicable to the very specific definitions of the events you've given (all lying on the path of a single light ray, all lying on either the space or the time axis of one of the two frames), as opposed to the Lorentz transformation which can apply to any arbitrary event or pair of events.

 

[JesseM]

OK, here are the four diagrams I drew up, apologies for the messiness. The first one shows my understanding of what the terms in your equation mean in a spacetime diagram. The second shows what the terms mean in the spatial part of the Lorentz transform equation delta-x = gamma*(delta-x' + v*delta-t'). The third shows a particular symmetry in the scenario that you use to define your terms, and in the fourth diagram this symmetry shows how the meaning of your terms x_B and t_B can be changed so that now the modified version of your equation does deal with only a single pair of events, making it a special case of the Lorentz transformation equation, which explains why both equations look the same. But notice that your derivation only works in the case where the pair of events have a light-like separation, and where one event is on the space axis of one frame and the other event is on the time axis of the second frame, whereas the Lorentz transformation deals works for arbitrary pairs of events that don't need to have these properties.

By the way, to make your equation more consistent with the Lorentz transformation equation I made a slight tweak to your definitions--where you defined x'_A and x_B in terms of the "distance" between a pair of events, and so made these positive, I picked the rule that they were defined in terms of (position coordinate of later event) - (position coordinate of earlier event), which means x'_A = -5 rather than 5, and x_B = -10 rather than 10. So the tweaked version of your equation relating them ends up being x'_A = gamma*(x_B + v*t_B).

 

[neopolitan]

It's certainly invariant in B's frame, but if you think it's "Lorentz invariant" you misunderstand the meaning of the term. "Lorentz invariant" means "invariant under the Lorentz transform", i.e. something which is the same in all inertial frames, like the invariant interval ds^2 = dx^2 - c^2*dt^2. The distance between B and the location of E_A is not something that's the same in every frame (in fact in most frames it's not constant with time), so it's not Lorentz invariant.

Fair cop, I was thinking originally about the separation between B and the event (either of them) when A and B are colocated, which is invariant and Lorentz invariant, and B takes that separation in the B frame to be invariant (but that's not Lorentz invariant).

I'm not sure how you think this is relevant. Yes, obviously E_A, E_B, the event of the photon passing B, and the event of the photon passing A all lie on the worldline of a single photon. This doesn't change the fact that x'_A is defined as the difference in position between the first and third event in the A frame, while x_B is defined as the difference in position between the second and fourth event in the B frame, and that your derivation assumes all four events have a light-like separation from one another. So your equation x'_A = gamma*(x_B - vt_B) does not have the same physical meaning as the Lorentz equation x' = gamma*(x - vt) despite the fact that it looks similar, because in the Lorentz equation x' and x either represent the coordinates of a single event in the primed and unprimed frame (which can be at any arbitrary position, not necessarily on one of the frame's spatial axes), or else x' and x can represent the coordinate intervals in two frames between a single pair of events (which can also be located at arbitrary positions, and which need not have a light-like separation from one another).

What is an axis to you?

The x-axis is normally a line with a constant value of t, in my diagram it happens to be 0. Does it have to be 0?

Since in this instance I only want to know what the x value is in both frames, t can be whatever. Can't it?

So, if relative to A, B is traveling at v, I can use an x_A-axis with any t_A value, and an x_B-axis with any t_B value and a t_B-axis with any x_B value. And if I chose the right values, I can use my diagram (and my derivation) to work out that if an event lies at x_A from A in the A frame, then it lies at x'_B from B in the B frame ... irrespective of what the t_A value of the event is.

Similarly, I can use a t_A-axis with any x_A value, and an x_B-axis with any t_B value and a t_B-axis with any x_B value. And if I chose the right values, I can use my diagram (and my derivation) to work out that if an event happens at t_A in the A frame (which is relative to an event common to A and B, usually colocation, but not necessarily), then happens at t'_B in the B frame (again relative to an event common to A and B) ... irrespective of what the x_A value of the event is.

But I stress yet again, these diagrams all retrospective. My derivation doesn't call for them. I'm only using the diagrams to try to explain to JesseM what the physical meaning of the values in the original derivation are (and to be honest, I was not originally as curious about that).

But they're not indistinguishable, not when you keep in mind the physical meaning of the terms in your equations vs. the physical meaning of the terms in the Lorentz transformation.

Write it on a piece of paper, compare them.

My end equations: x'=gamma.(x-vt) and t'=gamma.(t-vx/c²)

Lorentz Transformations: x'=gamma.(x-vt) and t'=gamma.(t-vx/c²)

I can't see the difference, can you see the difference? (If there is a difference then cutting and pasting isn't what it used to be.)

Not if you explained the physical meaning of the terms. I'm going to try to draw some diagrams of my own to show the difference in the meaning of the terms visually.

Ok, I look forward to that. I will look forward to you showing me that even if I moved the axes (ie if I did not feel obliged to use an (x,0) axis and a (0,t) axis) that I couldn't line up my diagram to match yours.

Yes, new equations which are only applicable to the very specific definitions of the events you've given (all lying on the path of a single light ray, all lying on either the space or the time axis of one of the two frames), as opposed to the Lorentz transformation which can apply to any arbitrary event or pair of events.

I disagree. The little red dot that doesn't lie on the world line defined by E_A and E_B disagrees too.

cheers,

neopolitan

(I may shortly have to take a break from this. Other things are demanding my attention.)

 

[JesseM]

What is an axis to you?

The x-axis is normally a line with a constant value of t, in my diagram it happens to be 0. Does it have to be 0?

Since in this instance I only want to know what the x value is in both frames, t can be whatever. Can't it?

The x-value of what? Neither x'_A nor x_B refer to the x-coordinate of any individual event in either frame. Rather, they both refer to the difference in x-coordinates between a pair of events.

Write it on a piece of paper, compare them.

My end equations: x'=gamma.(x-vt) and t'=gamma.(t-vx/c²)

Lorentz Transformations: x'=gamma.(x-vt) and t'=gamma.(t-vx/c²)

I can't see the difference, can you see the difference? (If there is a difference then cutting and pasting isn't what it used to be.)

See, this is the basic problem that comes up in a lot of our discussions, you don't seem to understand that equations in physics are not just defined by how they look algebraically, but also by the actual physical meaning of the terms involved. This is why I often disagreed whenever you would say that any equation of the form t1 = t2 / gamma was the "temporal analogue of length contraction" even though the physical meaning of t1 and t2 was different as far as I could tell.

Here's a word problem: "my age is gamma times what your age was v times as many years in the past as your little brother's current age." Well, if we define t=your brother's current age, x as your current age, and my age as x', then algebraically this would be represented as x' = gamma*(x - vt). But would it be accurate to refer to this equation, with the terms defined in this way, as "the spatial component of the Lorentz transformation equation"?

Ok, I look forward to that. I will look forward to you showing me that even if I moved the axes (ie if I did not feel obliged to use an (x,0) axis and a (0,t) axis) that I couldn't line up my diagram to match yours.

Well, see the previous post. Your equation conceptually relates two different pairs of events, and even though we can exploit a symmetry in that setup to show it's equivalent to relating a single pair, your derivation only shows that the equation holds for a pair of events with a light-like separation, whereas the second of my 4 diagrams shows the Lorentz transformation equation works for events with arbitrary separation (time-like in that diagram). Also, your derivation as presented does assume that all the events lie on axes that go through the origins of your two coordinate systems...if you wanted to avoid that condition, I suppose you could prove a lemma that shows that the distance and time intervals between a pair of events in a given coordinate system will be unchanged in a second coordinate system with the origin at a different location but which is at rest relative to the first (i.e. a simple coordinate transformation of the form x' = x + X₀ and t' = t + T₀ where X₀ and T₀ are constants).

I disagree. The little red dot that doesn't lie on the world line defined by E_A and E_B disagrees too.

But isn't the idea that you draw a new light ray through that red dot, and shift the position of the coordinate axes so the red dot now lies on A's spatial x_A axis, and redefine the meaning of E_A and E_B in terms of these changes? If not, maybe you could give a non-joking explanation of those diagrams from post 236. But if you are shifting the positions of the axes, then without a lemma of the type I mentioned above, you haven't proved anything about how the coordinates of the red dot in the original two coordinate systems were related (I should add that now that I think about such a lemma, which I hadn't prior to this post, it occurs to me that it would be very trivial to prove).

 

[neopolitan]

I'm not going to try to explain in your terms, since your diagrams show that you have gone off on some tangent.

I've tried explaining in your terms and that doesn't seem to work.

Can you try to understand in my terms?

Here is a very busy little diagram and a less busy diagram. I've gone all the way back to the beginning so anything I have said in between to try to explain in your terms is defunct, so please try to start from here.

The diagrams may well contain all you need to understand. If they don't, then we can discuss the diagrams in my terms. Once we are both happy that you understand what I am actually talking about in my terms, then we can try to convert things into your terms. Does that sound fair?

(Note that the temporal component is not there, let's get the spatial component sorted out before we complicate things.)

cheers,

neopolitan

PS I noticed that the definitions were difficult to read on the very busy little diagram, so I have attached them separately in a more readable format.

 

[JesseM]

I'm not going to try to explain in your terms, since your diagrams show that you have gone off on some tangent.

That's rather dismissive of you. Just focus on my first diagram, since my next three diagrams were merely intended to show why your equation looks just like the Lorentz transformation equation, and can actually be interpreted as a special case of it. Do you see any significant differences between my first diagram and your first diagram? They look the same to me, except that I included visual depictions of the meaning of symbols like x'_A and x_B whereas you didn't include them in your diagram. I also don't understand what you mean by "your terms", since except for an unimportant tweak about the signs of the distances (to make your equation consistent with the Lorentz transformation equation, which I thought is what you wanted), I've used the same terms that you used, following your definitions from post 243.

I've tried explaining in your terms and that doesn't seem to work.

Can you try to understand in my terms?

How have I not been? Again, please explain where you see any significant difference between my first diagram and your first diagram.

 

[neopolitan]

That's rather dismissive of you. Just focus on my first diagram, since my next three diagrams were merely intended to show why your equation looks just like the Lorentz transformation equation, and can actually be interpreted as a special case of it. Do you see any significant differences between my first diagram and your first diagram? They look the same to me, except that I included visual depictions of the meaning of symbols like x'_A and x_B whereas you didn't include them in your diagram. I also don't understand what you mean by "your terms", since except for an unimportant tweak about the signs of the distances (to make your equation consistent with the Lorentz transformation equation, which I thought is what you wanted), I've used the same terms that you used, following your definitions from post 243.

How have I not been? Again, please explain where you see any significant difference between my first diagram and your first diagram.

My first diagram posted on this thread was yours. Otherwise there was one at https://www.physicsforums.com/showpost.php?p=2160139&postcount=156". I don't think that is the one you mean though.

You could mean the first one posted http://www.geocities.com/neopolitonian/index.htm".

Or maybe you mean the first of my most recent drawings (a couple of posts ago).

In any event, I sort of see what you are getting at but your first drawing (and in fact the rest) implies that I am focussed on something that I am not focussed on. Since we don't agree about what I am talking about, it is better than I start again, rather than trying to talk to a drawing which isn't about what I am talking about. Sorry if that sounds dismissive.

I put quite a bit of time into the most recent diagrams. Did they help at all? Hopefully you can now better understand what I was getting at when I last mentioned Lorentz invariance.

cheers,

neopolitan

PS About the unimportant tweak, move your x_B so it ends in Event E_A, rather than beginning at photon hits A, and you will see that it crosses the t_B axis at t = -6. Then move your t_B so it spans t = -6 and the event which is the colocation of the photon and B. (<- this was an edit)

That's more like what I had in mind.

PPS Diagram added which shows what I mean.

 

[JesseM]

My first diagram posted on this thread was yours. Otherwise there was one at https://www.physicsforums.com/showpost.php?p=2160139&postcount=156". I don't think that is the one you mean though.

You could mean the first one posted http://www.geocities.com/neopolitonian/index.htm".

Or maybe you mean the first of my most recent drawings (a couple of posts ago).

Sorry, lot of diagrams posted on this thread, I meant to compare the first of my diagrams from the most recent post where I posted diagrams (post 247) with the first of your diagrams from the most recent post where you posted diagrams (post 250).

In any event, I sort of see what you are getting at but your first drawing (and in fact the rest) implies that I am focussed on something that I am not focussed on. Since we don't agree about what I am talking about, it is better than I start again, rather than trying to talk to a drawing which isn't about what I am talking about. Sorry if that sounds dismissive.

I was focused on the meaning of the individual terms in your equation which looked similar to the spatial Lorentz transformation equations. Correct me if I'm wrong, but I thought that what we're arguing about here is whether you've really derived the Lorentz transformation, or whether (as I claim) a close look at the meaning of the terms in the equation you derived shows you did not actually derive an equation which applies to the coordinates of arbitrary events or coordinate intervals between arbitrary pairs of events as with the Lorentz transformation, but only an equation that applies to events which have some more specific properties that were part of your original derivation (like the fact that the events have a light-like separation between them). I don't see how we can settle this without actually focusing on the physical meaning of individual terms like x'_A and x_B, which was what I was trying to depict in that first diagram.

I put quite a bit of time into the most recent diagrams. Did they help at all? Hopefully you can now better understand what I was getting at when I last mentioned Lorentz invariance.

What comment about Lorentz invariance do you mean, and which part of the diagram is supposed to relate to it specifically? I looked at the two diagrams, but as I said I don't really see how they contain any information that I didn't already understand and hadn't included in my own diagram.

PS About the unimportant tweak, move your x_B so it ends in Event E_A, rather than beginning at photon hits A, and you will see that it crosses the t_B axis at t = -10. Then move your t_B so it span t = -10 and the event which is the colocation of A and B.

What I called my "tweak" wasn't about changing the actual events spanned by the intervals (I did show how you could do that in diagram 4 using the symmetry argument from diagram 3, but in the other diagrams I kept the events the same), it was just about being consistent with the order of the events so that if t_B referred to (time in B frame of light passing A) - (time in B frame of E_B), then x_B should also take the events in that order, i.e. (position in B frame of light passing A) - (position in B frame of E_B) which would make x_B negative, as opposed to reversing the order and defining x_B as (position in B frame of E_B) - (position in B frame of light passing A). The reason for this tweak is just that this is how it's done in the Lorentz transformation equation dealing with intervals between a pair of events, so making your equation have a consistent convention makes it easier to see how your equation can be interpreted as a special case of the Lorentz transformation equation.

But OK, as something unrelated to my own tweak, if you take a spatial interval in the B frame which has length 10 (as x_B did) and you place one end at E_A, then since E_A has coordinates x=10 and t=-6 in the B frame, the other end of this interval will be at position x=0 and t=-6, so it seems to me it crosses the t axis of the B frame at -6 rather than -10. Unless I've gotten the algebra wrong, which is quite possible (if you think it's wrong, is it because you disagree about the coordinates of E_A in the B frame?)

 

[neopolitan]

Sorry, lot of diagrams posted on this thread, I meant to compare the first of my diagrams from the most recent post where I posted diagrams (post 247) with the first of your diagrams from the most recent post where you posted diagrams (post 250).

Ok, thanks.

I was focused on the meaning of the individual terms in your equation which looked similar to the spatial Lorentz transformation equations. Correct me if I'm wrong, but I thought that what we're arguing about here is whether you've really derived the Lorentz transformation, or whether (as I claim) a close look at the meaning of the terms in the equation you derived shows you did not actually derive an equation which applies to the coordinates of arbitrary events or coordinate intervals between arbitrary pairs of events as with the Lorentz transformation, but only an equation that applies to events which have some more specific properties that were part of your original derivation (like the fact that the events have a light-like separation between them). I don't see how we can settle this without actually focusing on the physical meaning of individual terms like x'_A and x_B, which was what I was trying to depict in that first diagram.

See my previous post, a new drawing!

What comment about Lorentz invariance do you mean, and which part of the diagram is supposed to relate to it specifically? I looked at the two diagrams, but as I said I don't really see how they contain any information that I didn't already understand and hadn't included in my own diagram.

https://www.physicsforums.com/showpost.php?p=2175785&postcount=245", where I mentioned Lorentz invariance but incorrectly (what was in my head did not end up in pixels).

Did the fact that there is only one Lorentz invariant interval, which is clearly identified at least in the second diagram, not make anything clearer?

What I called my "tweak" wasn't about changing the actual events spanned by the intervals (I did show how you could do that in diagram 4 using the symmetry argument from diagram 3, but in the other diagrams I kept the events the same), it was just about being consistent with the order of the events so that if t_B referred to (time in B frame of light passing A) - (time in B frame of E_B), then x_B should also take the events in that order, i.e. (position in B frame of light passing A) - (position in B frame of E_B) which would make x_B negative, as opposed to reversing the order and defining x_B as (position in B frame of E_B) - (position in B frame of light passing A). The reason for this tweak is just that this is how it's done in the Lorentz transformation equation dealing with intervals between a pair of events, so making your equation have a consistent convention makes it easier to see how your equation can be interpreted as a special case of the Lorentz transformation equation.

Hopefully the diagram in the previous post clarifies things.

But OK, as something unrelated to my own tweak, if you take a spatial interval in the B frame which has length 10 (as x_B did) and you place one end at E_A, then since E_A has coordinates x=10 and t=-6 in the B frame, the other end of this interval will be at position x=0 and t=-6, so it seems to me it crosses the t axis of the B frame at -6 rather than -10. Unless I've gotten the algebra wrong, which is quite possible (if you think it's wrong, is it because you disagree about the coordinates of E_A in the B frame?)

I made a correction, after doing the diagram and clearly also after you posted this. -6 is right.

cheers,

neopolitan

 

[JesseM]

I guess I was paying too much attention to your first diagram in post 250 and not enough to your second, because now that I look at it more carefully, I understand the top part but I'm having trouble understanding the bottom part...

I made a correction, after doing the diagram and clearly also after you posted this. -6 is right.

So I take it in bottom "In the B frame" part of the diagram, the caption "Location of A at -10 before the photon spawned" should instead by "Location of A at -6 before the photon spawned"?

The top "In the A frame" part of the diagram seems straightforward enough, on the right when you say "Location of event: photon spawned" you mean E_A, correct? So the shorter line in the A-frame diagram is the distance between E_A and the position where the photon passed B (x'=5) while the longer line A-frame diagram is the distance between E_A and the event of the position where the photon passed A (x=8).

But I'm confused by the bottom "in the B frame" part of the diagram. When you say "Location of event: photon spawned" in the bottom part, which event are you referring to, E_A or E_B? In the B frame the distance between E_B and the photon passing B is 4, so that would seem to be what the shorter line in the B-frame diagram refers to. But what should the longer line in the B-frame diagram refer to? At t=-6 A is at position x=3.6 in B's frame, so the distance between A at that moment and E_B's position is only 0.4, while the distance between A at that moment and E_A's position is 6.4. In the first case the bottom line should actually be shorter than the top line that goes from E_B to the photon passing B, not longer. But in the second case the B-frame diagram would be using a different event for "location of event: photon spawned" for the bottom line than it uses for the top line, which would be confusing.

Also, when you say the "this is the only interval which is Lorentz invariant", in the A frame diagram you seem to be pointing to the interval between the events E_A and the photon passing B (events which have a spatial separation of 5 in the A frame), while in the B frame diagram you seem to be pointing to the interval between the events E_B and the photon passing B (events which have a spatial separation of 4 in the B frame). Am I misunderstanding? Also, when you say the "interval" is Lorentz invariant, are you referring to the interval of coordinate distance between some pair of events, or to the spacetime interval dx^2 - c^2dt^2 between some pair of events?

Finally, I do understand what you're talking about here:

PS About the unimportant tweak, move your x_B so it ends in Event E_A, rather than beginning at photon hits A, and you will see that it crosses the t_B axis at t = -6. Then move your t_B so it spans t = -6 and the event which is the colocation of the photon and B. (<- this was an edit)

It's true that in the B frame, the spatial distance between the event at x=0, t=-6 and E_A is 10, and the time between this event and the photon passing B is 10 (since the photon passes B at t=4 in this frame). So, this is the same as x_B and t_B when they were defined in terms of E_A and the photon passing A, and I can see why this works based on the symmetry of the diagram, similar to my own diagram #3 in post 247 but with the second isosceles triangle flipped over. However, I don't see how this relates to what I was referring to when I talked about the "tweak", which again didn't involve changing the events that x_B and t_B were defined in terms of. And if this is supposed to be related to your second "less busy diagram" in post 250, I'll have to ask you to elaborate because I don't see that either.

 

[neopolitan]

I guess I was paying too much attention to your first diagram in post 250 and not enough to your second, because now that I look at it more carefully, I understand the top part but I'm having trouble understanding the bottom part...

So I take it in bottom "In the B frame" part of the diagram, the caption "Location of A at -10 before the photon spawned" should instead by "Location of A at -6 before the photon spawned"?

Frustratingly enough, I saw that the -10 was in the diagram after I got home, and the program I drew it in is at work.

The top "In the A frame" part of the diagram seems straightforward enough, on the right when you say "Location of event: photon spawned" you mean E_A, correct? So the shorter line in the A-frame diagram is the distance between E_A and the position where the photon passed B (x'=5) while the longer line A-frame diagram is the distance between E_A and the event of the position where the photon passed A (x=8).

But I'm confused by the bottom "in the B frame" part of the diagram. When you say "Location of event: photon spawned" in the bottom part, which event are you referring to, E_A or E_B? In the B frame the distance between E_B and the photon passing B is 4, so that would seem to be what the shorter line in the B-frame diagram refers to. But what should the longer line in the B-frame diagram refer to? At t=-6 A is at position x=3.6 in B's frame, so the distance between A at that moment and E_B's position is only 0.4, while the distance between A at that moment and E_A's position is 6.4. In the first case the bottom line should actually be shorter than the top line that goes from E_B to the photon passing B, not longer. But in the second case the B-frame diagram would be using a different event for "location of event: photon spawned" for the bottom line than it uses for the top line, which would be confusing.

Also, when you say the "this is the only interval which is Lorentz invariant", in the A frame diagram you seem to be pointing to the interval between the events E_A and the photon passing B (events which have a spatial separation of 5 in the A frame), while in the B frame diagram you seem to be pointing to the interval between the events E_B and the photon passing B (events which have a spatial separation of 4 in the B frame). Am I misunderstanding? Also, when you say the "interval" is Lorentz invariant, are you referring to the interval of coordinate distance between some pair of events, or to the spacetime interval dx^2 - c^2dt^2 between some pair of events?

There is no event E_B. There is an event which spawns a photon (the event formerly known as E_A) and there is the event when that photon passes B. Remember I said I was going back to the beginning, so I am trying another tack.

The event E_A, if you still want to call it that, and the event when the photon passes B are both unique events, and there is a unique spacetime interval between them which is Lorentz invariant. The magnitude of the spatial component of this spacetime interval in the A frame and the B frame are in both diagrams (in the two dimensional one, x'_B is displaced.

Attached are modified diagrams, highlighting something. They are messy because I don't have all the tools I need, but you should see that I have cut a bit out of mine and moved it up. If you can do it with t (your diagram) you can do it x (my diagram).

Finally, I do understand what you're talking about here:

It's true that in the B frame, the spatial distance between the event at x=0, t=-6 and E_A is 10, and the time between this event and the photon passing B is 10 (since the photon passes B at t=4 in this frame). So, this is the same as x_B and t_B when they were defined in terms of E_A and the photon passing A, and I can see why this works based on the symmetry of the diagram, similar to my own diagram #3 in post 247 but with the second isosceles triangle flipped over. However, I don't see how this relates to what I was referring to when I talked about the "tweak", which again didn't involve changing the events that x_B and t_B were defined in terms of. And if this is supposed to be related to your second "less busy diagram" in post 250, I'll have to ask you to elaborate because I don't see that either.

This will have to wait, I am currently busier than my drawing.

cheers,

neopolitan

 

[JesseM]

There is no event E_B. There is an event which spawns a photon (the event formerly known as E_A) and there is the event when that photon passes B. Remember I said I was going back to the beginning, so I am trying another tack.

OK, I didn't realize that by going back to the beginning you meant starting the proof again without referring to E_B. So in the second diagram from post 250, the shorter line in the "In the B frame" part of the diagram is supposed to go from the photon-spawning event (formerly known as E_A) to the event of the photon passing B? But if the spawning event occurs at x=8,t=0 in the A frame, then in the B frame it must occur at position x=1.25*(8 - 0.6*0) = 10, while of course B is always at position x=0...so shouldn't that line say x'=10 rather than x'=4? (or x=-10 if you want to define it as 'position of photon passing B' - 'position of spawning event' as in my 'tweak') And since A is at position x=3.6 at t=-6, should the bottom line representing the distance between A and the spawning event at that time be x = x' + vt = 10 + 0.6*-6 = 6.4? (or x = x' - vt = -10 - 0.6*-6 = -6.4 in the tweaked version, since A's position is further in the -x direction than the spawning event).

Attached are modified diagrams, highlighting something. They are messy because I don't have all the tools I need, but you should see that I have cut a bit out of mine and moved it up. If you can do it with t (your diagram) you can do it x (my diagram).

Is the circled line segment intended to represent the distance between the spawning event and the event of the photon passing B, as measured in the B frame? If so it needs to be longer, because you don't want the ends of the segment to lie on vertical lines of constant x extending from each event in the A frame as you seem to have drawn it, rather you want the two ends of the segment to lie on two slanted lines of constant x in the B frame (lines parallel to B's time axis) which extend from the two events. If this isn't clear I can draw my own sketch to illustrate.

This will have to wait, I am currently busier than my drawing.

No problem, take your time.

 

[neopolitan]

Recall in post #191, I said:

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

I'm still doing that. Which means I am still trying to work this out. I don't know if you still have this in mind.

OK, I didn't realize that by going back to the beginning you meant starting the proof again without referring to E_B. So in the second diagram from post 250, the shorter line in the "In the B frame" part of the diagram is supposed to go from the photon-spawning event (formerly known as E_A) to the event of the photon passing B? But if the spawning event occurs at x=8,t=0 in the A frame, then in the B frame it must occur at position x=1.25*(8 - 0.6*0) = 10, while of course B is always at position x=0...so shouldn't that line say x'=10 rather than x'=4? (or x=-10 if you want to define it as 'position of photon passing B' - 'position of spawning event' as in my 'tweak') And since A is at position x=3.6 at t=-6, should the bottom line representing the distance between A and the spawning event at that time be x = x' + vt = 10 + 0.6*-6 = 6.4? (or x = x' - vt = -10 - 0.6*-6 = -6.4 in the tweaked version, since A's position is further in the -x direction than the spawning event).

(1) The distance between A and the event when it happens at 0 is 8, according to A. (x_A = 8)

(2) A period of 5 later, at 5, according to A, the photon passes B. (t'_A = 5)

(3) A period of 8 later, at 8, according to A, the photon passes A. (t_A = 8)

(4) According to A, at that time, B has moved 3 towards the event's location, so the separation between B and where the photon was when A and B were colocated is 5. (x'_A = 5)

(5) According to B, at that time, B has not moved and the separation between B and where the photon was when A and B were colocated is 4. (x'_B = 4)

(6) The distance between B and the event when it happens at -6 is 10, according to B (x_B = 10)

(7) A period of 10 later, at 4, according to B, the photon passes B. (t_B = 10)

(8) According to B, at that time, A has moved 6 away from the event's location, so the separation between A where the photon was when A and B were colocated is 10. (x_B = 10)

(9) According to A, at that time, A has not moved and the separation between A and where the photon was when A and B were colocated is 8. (x_A = 8)

Events

(E1) A and B are colocated

(E2) Photon is emitted

(E3) Photon passes B

(E4) Photon passes A

After a walk and some further thought, I am beginning to wonder if the spatial intervals being measured are not:

From location where photon passes B to where photon was when A and B were colocated.

According to A, B has moved 3 closer to what was 8 away = 5.

According to B, B has not moved, but meets the photon at 4 so at 0, the photon was at 4.

I can't spend more time on this, but it may shed some light.

Is the circled line segment intended to represent the distance between the spawning event and the event of the photon passing B, as measured in the B frame? If so it needs to be longer, because you don't want the ends of the segment to lie on vertical lines of constant x extending from each event in the A frame as you seem to have drawn it, rather you want the two ends of the segment to lie on two slanted lines of constant x in the B frame (lines parallel to B's time axis) which extend from the two events. If this isn't clear I can draw my own sketch to illustrate.

I think it depends on which direction you are going (B to A) or (A to B). I've tried to show both. Now, I am just trying to show one.

I think to show what you want to see, I could take a section of the x_A axis and cross the x_B axis (reflecting events A-B colocation and photon-crosses x_B axis).

I'm going the other way.

Must go,

cheers,

neopolitan

 

[JesseM]

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

I'm still doing that. Which means I am still trying to work this out. I don't know if you still have this in mind.

By "works" do you just mean the fact that the equation you get ends up looking just like the Lorentz transformation? My diagrams from post 247 were intended to show why this was the case, showing how your equation could be interpreted as a special case of the Lorentz transform when dealing with two events that have a light-like separation.

(1) The distance between A and the event when it happens at 0 is 8, according to A. (x_A = 8)

(2) A period of 5 later, at 5, according to A, the photon passes B. (t'_A = 5)

(3) A period of 8 later, at 8, according to A, the photon passes A. (t_A = 8)

(4) According to A, at that time, B has moved 3 towards the event's location, so the separation between B and where the photon was when A and B were colocated is 5. (x'_A = 5)

When you say "at that time", you're referring to the time in (2) rather than (3) I take it. Also, when you refer to "where the photon was when A and B were colocated" in A's frame, that was the earlier definition of E_A.

(5) According to B, at that time, B has not moved and the separation between B and where the photon was when A and B were colocated is 4. (x'_B = 4)

But in B's frame, "where the photon was when A and B were colocated" is how the event E_B was defined earlier, so you're still including this event in your definition of x'_B.

(6) The distance between B and the event when it happens at -6 is 10, according to B (x_B = 10)

I take it by "the event when it happens" you still mean the event formerly known as E_A. So, was it a mistake in the second diagram from post 250 when in the B frame diagram you had the distance of 10 be the distance between the photon-spawning and "Location of A" at the time that should be -6? The distance between B and E_A in the B frame is 10 (this is time-invariant in the B frame, so the time of -6 is irrelevant here), but the distance between A and E_A at -6 is 6.4.

(7) A period of 10 later, at 4, according to B, the photon passes B. (t_B = 10)

(8) According to B, at that time, A has moved 6 away from the event's location, so the separation between A where the photon was when A and B were colocated is 10. (x_B = 10)

Now when you refer to "where the photon was when A and B were colocated" you seem to mean in the B frame, but that would be the event we defined as E_B, so you still seem to be including this event in your definition of x_B. Also, when you say "at that time", are you referring to the time of 4 in the B frame from (7)? At that time A is at position -0.6*4 = -2.4 on B's x-axis, so the distance between A and E_B at that time is not 10, it's 6.4 just like the distance between A and E_A at a time of -6 in B's frame. Your original definition of x_B was the distance from E_B and A at the time the photon passes A in B's frame (t=10 in B's frame), and in that case the distance is 10. So either your above verbal definition is mistaken, or you got the value of x_B wrong with that definition.

(9) According to A, at that time, A has not moved and the separation between A and where the photon was when A and B were colocated is 8. (x_A = 8)

Yes, the separation between A and E_A is always 8 in the A frame.

Events

(E1) A and B are colocated

(E2) Photon is emitted

(E3) Photon passes B

(E4) Photon passes A

After a walk and some further thought, I am beginning to wonder if the spatial intervals being measured are not:

From location where photon passes B to where photon was when A and B were colocated.

But "where photon was when A and B were colocated" depends on which frame's definition of simultaneity you're using, so again you seem to be talking about both E_A and E_B.

According to A, B has moved 3 closer to what was 8 away = 5.

In A's frame, the distance between B and E_A at the moment the photon passes B is 5, yes. This was your definition of x'_A, both in older posts and above.

According to B, B has not moved, but meets the photon at 4 so at 0, the photon was at 4.

The photon was at position x=4 on B's space axis at time t=0 in B's frame, yes. This was the position in B's frame of the event E_B, which is how you defined x'_B in older posts, and also above although you didn't use the term E_B any more (if you're going to keep talking about the event on the photon's worldline that happens at t=0 in B's frame when A and B were colocated, then can we bring back the notation of E_A and E_B?)

Is the circled line segment intended to represent the distance between the spawning event and the event of the photon passing B, as measured in the B frame? If so it needs to be longer, because you don't want the ends of the segment to lie on vertical lines of constant x extending from each event in the A frame as you seem to have drawn it, rather you want the two ends of the segment to lie on two slanted lines of constant x in the B frame (lines parallel to B's time axis) which extend from the two events. If this isn't clear I can draw my own sketch to illustrate.

I think it depends on which direction you are going (B to A) or (A to B). I've tried to show both. Now, I am just trying to show one.

I don't understand what you mean by "direction" and "B to A" vs. "A to B". Are you talking about converting something from one frame to another? If so what, specifically?

I think to show what you want to see, I could take a section of the x_A axis and cross the x_B axis (reflecting events A-B colocation and photon-crosses x_B axis).

The photon crosses the x_B axis at event E_B, are you just talking about a horizontal line in the A frame between E_B and A's time axis (x=0)? If so, that is definitely not what I "wanted to see" above, I was talking about "the distance between the spawning event (E_A) and the event of the photon passing B, as measured in the B frame". As I said, the way to represent this would be to draw in two lines parallel to B's time axis which go through these two events (E_A and the photon passing B), then draw a segment parallel to B's space axis with each end touching one of the parallel lines. On the other hand, your circled diagram in post 256 seemed to be based on imagining two vertical lines parallel to A's time axis, one line going through the event E_A and the other line going through the event of the photon passing B, and then drawing a line segment parallel to B's space axis with each end touching one of the parallel lines. This would not be the distance between E_A and the event of the photon passing B in either frame. Am I misunderstanding what you were trying to represent in that diagram?

 

[phyti]

Neo;
How close does this drawing match yours, and is this what you are trying to show?

View attachment neo problem0428.doc

 

[neopolitan]

By "works" do you just mean the fact that the equation you get ends up looking just like the Lorentz transformation? My diagrams from post 247 were intended to show why this was the case, showing how your equation could be interpreted as a special case of the Lorentz transform when dealing with two events that have a light-like separation.

Originally it wasn't a special case. The only way I could give you numbers (which is your preferred approach, nothing wrong with it) was to present a special case.

When you say "at that time", you're referring to the time in (2) rather than (3) I take it. Also, when you refer to "where the photon was when A and B were colocated" in A's frame, that was the earlier definition of E_A.

Cut and paste error, (3) should have been one up (the numbering came later), so you are right, the "at that time" in (4) refers to (2).

Yes, there is the event formally known as "E_A".

But in B's frame, "where the photon was when A and B were colocated" is how the event E_B was defined earlier, so you're still including this event in your definition of x'_B.

I know. I was just laying out all the intervals, noting that some intervals either appear in different places, or I have just reworded the description of the exact same interval. Note what I said further down in my post.

I take it by "the event when it happens" you still mean the event formerly known as E_A. So, was it a mistake in the second diagram from post 250 when in the B frame diagram you had the distance of 10 be the distance between the photon-spawning and "Location of A" at the time that should be -6? The distance between B and E_A in the B frame is 10 (this is time-invariant in the B frame, so the time of -6 is irrelevant here), but the distance between A and E_A at -6 is 6.4.

Yes, it is B and the event at t=-6 so that time 10 later at 4, the photon hits B after having traveled 10. This accords with (6) meaning x_B, right? (Noting that x unprimed is the location of the event formerly known as E_A, the subscript means according to B.)

I have an evening walk inspired idea for showing the relationships visually, which I will address shortly.

Now when you refer to "where the photon was when A and B were colocated" you seem to mean in the B frame, but that would be the event we defined as E_B, so you still seem to be including this event in your definition of x_B. Also, when you say "at that time", are you referring to the time of 4 in the B frame from (7)? At that time A is at position -0.6*4 = -2.4 on B's x-axis, so the distance between A and E_B at that time is not 10, it's 6.4 just like the distance between A and E_A at a time of -6 in B's frame. Your original definition of x_B was the distance from E_B and A at the time the photon passes A in B's frame (t=10 in B's frame), and in that case the distance is 10. So either your above verbal definition is mistaken, or you got the value of x_B wrong with that definition.

That value appears twice as (6) and (8). I know that. Clearly if we are tying ourselves to unique physical definitions for each term (and I am not necessarily saying that we shouldn't), then one of these is the wrong definition of x_B, if not both.

But "where photon was when A and B were colocated" depends on which frame's definition of simultaneity you're using, so again you seem to be talking about both E_A and E_B.

Yes, and no, but then again yes. But sort of no. Hopefully the diagram will make this clearer (and I know it can't make things less clear.)

The photon was at position x=4 on B's space axis at time t=0 in B's frame, yes. This was the position in B's frame of the event E_B, which is how you defined x'_B in older posts, and also above although you didn't use the term E_B any more (if you're going to keep talking about the event on the photon's worldline that happens at t=0 in B's frame when A and B were colocated, then can we bring back the notation of E_A and E_B?)

I certainly don't want a separate photon spawning event. I return to the diagram that I need to draw again, in which the event formally known as E_B sort of makes a reappearance. This should make more sense, once I finish responding, find time to actually draw the diagram and post it.

I don't understand what you mean by "direction" and "B to A" vs. "A to B". Are you talking about converting something from one frame to another? If so what, specifically?

Yes, converting the spatial component of a spacetime interval from one frame to another.

My diagram shows converting the spatial component of a spacetime interval in the A frame (example: x'_A ... a horizontal line, length 5) to the spatial component of a spacetime interval in the B frame (example: x'_B ... a line parallel to the x_B axis line, length 4).

You seemed to talking about converting the spatial component of a spacetime interval in the B frame (example: x_B ... a line parallel to the x_B axis line, length 10) to the spatial component of a spacetime interval in the A frame (example: x_A ... a horizontal line, length 8).

See what I mean?

The photon crosses the x_B axis at event E_B, are you just talking about a horizontal line in the A frame between E_B and A's time axis (x=0)? If so, that is definitely not what I "wanted to see" above, I was talking about "the distance between the spawning event (E_A) and the event of the photon passing B, as measured in the B frame". As I said, the way to represent this would be to draw in two lines parallel to B's time axis which go through these two events (E_A and the photon passing B), then draw a segment parallel to B's space axis with each end touching one of the parallel lines. On the other hand, your circled diagram in post 256 seemed to be based on imagining two vertical lines parallel to A's time axis, one line going through the event E_A and the other line going through the event of the photon passing B, and then drawing a line segment parallel to B's space axis with each end touching one of the parallel lines. This would not be the distance between E_A and the event of the photon passing B in either frame. Am I misunderstanding what you were trying to represent in that diagram?

I think I explained that just above. If not, let me know.

Diagram to follow, as other priorities permit.

cheers,

neopolitan

 

[neopolitan]

Neo;
How close does this drawing match yours, and is this what you are trying to show?

There's only one photon emission, so I would prefer any diagram to have only one location. The diagram isn't what I would have drawn, but I am not saying it is wrong. I'd like to get the diagram I do want to draw done without having to work out another depiction.

I do know that some of the figures you have noted do not appear on my diagrams (6.4 and 2.4) which seem to relate to a distinctly different event.

cheers,

neopolitan

 

[JesseM]

Originally it wasn't a special case. The only way I could give you numbers (which is your preferred approach, nothing wrong with it) was to present a special case.

When I say "special case" I'm not talking about this specific numerical example though, I'm talking about the fact that all the intervals are between events that lie on the same light ray and therefore have a light-like separation, and your derivation wouldn't be applicable to events with a time-like or space-like separation. The Lorentz transform deals with intervals between arbitrary events which may not have a light-like separation, like in the second diagram from my post 247.

(6) The distance between B and the event when it happens at -6 is 10, according to B (x_B = 10)

I take it by "the event when it happens" you still mean the event formerly known as E_A. So, was it a mistake in the second diagram from post 250 when in the B frame diagram you had the distance of 10 be the distance between the photon-spawning and "Location of A" at the time that should be -6? The distance between B and E_A in the B frame is 10 (this is time-invariant in the B frame, so the time of -6 is irrelevant here), but the distance between A and EA at -6 is 6.4.

Yes, it is B and the event at t=-6 so that time 10 later at 4, the photon hits B after having traveled 10. This accords with (6) meaning x_B, right?

Yeah, the distance between B and E_A is 10 in the B frame (regardless of time).

(7) A period of 10 later, at 4, according to B, the photon passes B. (tB = 10)

(8) According to B, at that time, A has moved 6 away from the event's location, so the separation between A where the photon was when A and B were colocated is 10. (xB = 10)

Now when you refer to "where the photon was when A and B were colocated" you seem to mean in the B frame, but that would be the event we defined as E_B, so you still seem to be including this event in your definition of x_B. Also, when you say "at that time", are you referring to the time of 4 in the B frame from (7)? At that time A is at position -0.6*4 = -2.4 on B's x-axis, so the distance between A and E_B at that time is not 10, it's 6.4 just like the distance between A and EA at a time of -6 in B's frame. Your original definition of x_B was the distance from E_B and A at the time the photon passes A in B's frame (t=10 in B's frame), and in that case the distance is 10. So either your above verbal definition is mistaken, or you got the value of xB wrong with that definition.

That value appears twice as (6) and (8). I know that. Clearly if we are tying ourselves to unique physical definitions for each term (and I am not necessarily saying that we shouldn't), then one of these is the wrong definition of x_B, if not both.

Well, I think (8) has to be wrong if my numbers above are right (the distance between A and E_B being 6.4 at time t=4 in the B frame).

The photon was at position x=4 on B's space axis at time t=0 in B's frame, yes. This was the position in B's frame of the event E_B, which is how you defined x'_B in older posts, and also above although you didn't use the term E_B any more (if you're going to keep talking about the event on the photon's worldline that happens at t=0 in B's frame when A and B were colocated, then can we bring back the notation of E_A and E_B?)

I certainly don't want a separate photon spawning event.

I wasn't suggesting a separate photon spawning event. Are we using the same definition of "event"? Normally in SR an event just refers to a particular geometric point in spacetime (such that all frames agree on the spacetime interval between it and other events), there doesn't need to be anything of interest actually happening at that point. So if we define E_B as "the point on the photon's worldline that's simultaneous with A&B in B's frame", that's enough to define a unique "event" even if nothing special is happening to the photon at that point on its worldline.

I don't understand what you mean by "direction" and "B to A" vs. "A to B". Are you talking about converting something from one frame to another? If so what, specifically?

Yes, converting the spatial component of a spacetime interval from one frame to another.

My diagram shows converting the spatial component of a spacetime interval in the A frame (example: x'_A ... a horizontal line, length 5) to the spatial component of a spacetime interval in the B frame (example: x'_B ... a line parallel to the x_B axis line, length 4).

You seemed to talking about converting the spatial component of a spacetime interval in the B frame (example: x_B ... a line parallel to the x_B axis line, length 10) to the spatial component of a spacetime interval in the A frame (example: x_A ... a horizontal line, length 8).

No, I wasn't talking about conversion at all, I was just talking about drawing a line segment to represent the spatial distance in the B frame between two specific events. If you want to draw the distance in the B frame between the event E_A and the event of the photon passing B, you draw one line parallel to B's time axis that goes through E_A (representing the set of events which have the same position coordinate as E_A in B's frame), another line parallel to B's time axis that goes through the event of the photon passing B (representing the set of events which have the same position coordinate as the photon passing B in B's frame), and then a line segment parallel to B's space axis whose endpoints lie on these parallel lines (representing the distance in B's frame between the position coordinate of the first event in B's frame and the position coordinate of the second event in B's frame). None of this involves any conversions to the A frame, although the lines will be skewed if you draw this in the context of an A frame diagram. It's a lot easier to visualize if you imagine doing this in the context of a B frame drawing, where you'd just draw two vertical lines going through the events, and then the horizontal distance between these lines would be the same as the distance between the events in the B frame.

I don't really understand what you mean by "the spatial component of the spacetime interval"--what spacetime interval, specifically? Do you agree a spacetime interval is always defined in terms of a pair of events? If so, are you talking about the same events I was, the event E_A and the event of the photon passing B? And is the line segment in your drawing from post 256, which is parallel to B's space axis, supposed to represent the spatial distance in B's frame between these two events?

 

[neopolitan]

Here is the diagram I have mentioned.

According to B, B does not move. According to A, B does move, so the photon which eventually passes B is, at the event photon passes the x_B axis (formerly known as event E_B) a spacetime interval away from the photon spawning event (formerly known as E_A) as shown by the green line.

I've been far too busy today to describe the green lines. Hopefully you'll work it out.

cheers,

neopolitan

Just quickly, the diagram should be viewed together with the second drawing from post #250. What I am focussing on is x' (both of them, ie x'_A (5) and x'_B (4)). With a little effort you can find vt'_A (3), vt_B (6), x_A (8) and x_B (10). I think these make more physical sense.

Something that remains to be seen is if this is a special case or not. I don't think so, but how to convince anyone else is a task for another day (or week even).

 

[JesseM]

Ah, I see what you mean. Yes, that relationship works. I suppose we could name the event where the bottom red and green lights meet as E_C, which could be defined as "the event that is at the same position in the A frame as the light passing B, and is simultaneous in the B frame with E_A." Then it would be true that in the B frame, the distance from B to E_B is equal to the distance from E_C to E_A.

But aren't we drawing on our prior knowledge of how spacetime diagrams in SR work (which are based on already knowing the full Lorentz transform) to conclude that this relationship holds? Are you saying you could derive this relationship from first principles without first knowing (or first deriving) the full Lorentz transform? If not, how does this relate to your attempt to derive the Lorentz transform?

 

[neopolitan]

Ah, I see what you mean. Yes, that relationship works. I suppose we could name the event where the bottom red and green lights meet as E_C, which could be defined as "the event that is at the same position in the A frame as the light passing B, and is simultaneous in the B frame with E_A." Then it would be true that in the B frame, the distance from B to E_B is equal to the distance from E_C to E_A.

But aren't we drawing on our prior knowledge of how spacetime diagrams in SR work (which are based on already knowing the full Lorentz transform) to conclude that this relationship holds? Are you saying you could derive this relationship from first principles without first knowing (or first deriving) the full Lorentz transform? If not, how does this relate to your attempt to derive the Lorentz transform?

Yes, I draw on my existing knowledge to draw this and how does it relate? By giving physical meanings to the terms in the equation.

I suspect that what I might be doing is akin to what we do when we take two events with a spacelike separation and define the line joining them as the x-axis so the spatial interval between them is x. To the extent that that is a special case, I agree that what I am doing is a special case.

I think that the two absolutely key events are 1) the event formerly known as E_A and 2) the event where we say that A and B are colocated: this is the spacetime interval of interest. This might sound like a special case, but really A and B don't really ever need to be colocated, we can rearrange axes and label A and B appropriately and get equations that work. (It's sort of like working out how far a boat is off an island when we are sitting on the shore of the mainland. We can work out how far the island is from us, how far the boat is from us and then how far the boat is from the island, and we can make the line joining the island and the boat the x-axis and make the island the origin of the x axis, even though we may never actually visit the island and after making our measurements we take off vertically in a balloon. I don't know about you, but I always have a tendency to think of where I am headed as my own personal x-axis - even though I could say that is the axis along which things seem to approach me )

cheers,

neopolitan

 

[JesseM]

Well, in your original derivation, the terms in the equation x'_A = gamma*(x_{B - vtB) definitely referred to coordinate distances and times between pairs of events with a lightlike separation...if you're saying that you think you could derive a similar-looking equation but where the terms had a different physical meaning, and derive it from first principles without relying on preexisting knowledge of how spacetime diagrams in SR look, then to convince me of that I think you'd really have to go back and go through the steps of the altered derivation from the beginning. I don't know if that's worth the effort at this point though, it's up to you. Also, are you saying you think you could derive it for events at arbitrary pairs of coordinates, or only in the case where we've oriented the x-axis so both events are simultaneous (or colocated) in one of the frames? If the latter that's still not as general as the Lorentz transformation, which can be applied to events that need not be simultaneous or colocated in either of the two frames.}

 

[neopolitan]

Well, in your original derivation, the terms in the equation x'_A = gamma*(x_{B - vtB) definitely referred to coordinate distances and times between pairs of events with a lightlike separation...if you're saying that you think you could derive a similar-looking equation but where the terms had a different physical meaning, and derive it from first principles without relying on preexisting knowledge of how spacetime diagrams in SR look, then to convince me of that I think you'd really have to go back and go through the steps of the altered derivation from the beginning. I don't know if that's worth the effort at this point though, it's up to you. Also, are you saying you think you could derive it for events at arbitrary pairs of coordinates, or only in the case where we've oriented the x-axis so both events are simultaneous (or colocated) in one of the frames? If the latter that's still not as general as the Lorentz transformation, which can be applied to events that need not be simultaneous or colocated in either of the two frames.}

<sub>My original derivation at https://www.physicsforums.com/showpost.php?p=2165684&postcount=174".

In that post I said to put (7) and (4) into (2), where:

(2) x'B = G.x'A

(4) x'A = xA - v.tA

(7) G = \gamma

giving

x'B = G.( xA - v.tA )

so that x'A actually disappears. I'm not saying that x'A has no meaning at all, but I do wonder if it should (or at least could) perhaps be used as an interim value. The value xA in this equation is the spatial separation between the origin of the xA axis and the event formerly known as EA. These two events don't have a lightlike separation. (And here I can point out that I am totally aware that events don't have to have anything happen at them. I was mistakenly under the impression that you wanted to tie "happenings" to events as part of your desire to have physical meanings to all the various values. The notional colocation of A and B event is part of this, even though in reality A and B don't ever have to be colocated - they can start separated and head off in opposite directions in a 1+1 universe, or just have a nearest point of approach in a 3+1 universe.)

There is no altered derivation from post #174. It stands. Each of the values may well have a better physical definition, but those better definitions don't change the derivation.

Everything between #174 and here has been to get those better definitions and while it has been quite a journey, I don't think that it has been in vain. At first I was confident that my derivation works, but that confidence was based on a rather nebulous mental picture which was clearly difficult to express in words. Now I am confident that my derivation works, and my confidence is based on a much clearer mental picture which I believe I can present in a spacetime diagram.

A clarification, do you realize that the image given in https://www.physicsforums.com/showpost.php?p=2178288&postcount=264" only refers to the spatial component of a spacetime interval?

cheers,

neopolitan</sub>

 

[JesseM]

My original derivation at https://www.physicsforums.com/showpost.php?p=2165684&postcount=174".

In that post I said to put (7) and (4) into (2), where:

(2) x'_B = G.x'_A

(4) x'_A = x_A - v.t_A

(7) G = \gamma

giving

x'_B = G.( x_A - v.t_A )

so that x'_A actually disappears.

OK, I was thinking of the equation you derived in post 227, x'_A = (x_B - vt_B).gamma, and that was also the Lorentz-like equation you talked about in later posts. I think your derivation in post 227 was using different definitions than the earlier one you quote above from post 174, because in later posts you had x'_B = 4 and x'_A = 5, but if G=gamma that would mean x'_A = G*x'_B (in fact you wrote this equation in post 227), which is the reverse of what you have above.

I'm not saying that x'_A has no meaning at all, but I do wonder if it should (or at least could) perhaps be used as an interim value. The value x_A in this equation is the spatial separation between the origin of the x_A axis and the event formerly known as E_A. These two events don't have a lightlike separation.

I don't think the equation from post 174 could actually be derived using your later definitions that you were using in post 227 and later posts; in post 227 you wrote:

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

Substituting, the equations you could get from this would be either x'_A = (x_B - vt_B)*gamma, which is what you derived, or x'_B + vt_B = x_A*gamma, which doesn't really look like a Lorentz transformation equation at all.

Everything between #174 and here has been to get those better definitions and while it has been quite a journey, I don't think that it has been in vain. At first I was confident that my derivation works, but that confidence was based on a rather nebulous mental picture which was clearly difficult to express in words. Now I am confident that my derivation works, and my confidence is based on a much clearer mental picture which I believe I can present in a spacetime diagram.

As I said I don't think you can actually derive x'_B = G.( x_A - v.t_A ) using the definitions from post 227 and subsequently. Even if you could, this would not really be much like the Lorentz transformation equation, because it doesn't relate the coordinates of a single event or single pair of events in two different frames--x'_B represents the position of E_B in the B frame (or equivalently the distance between E_B and the event of light passing B in the B frame), while x_A represents the position of E_A in the A frame (or equivalently the distance between E_A and the event of light passing A in the A frame) and t_A represents the time of the light passing A in the the A frame (or equivalently the time between E_A and the event of light passing A in the A frame, the same pair of events you might use to define x_A, except that if you want to define both x_A and t_A in terms of this pair of events then one of them must be negative if you're consistent about the order you take the events).

A clarification, do you realize that the image given in https://www.physicsforums.com/showpost.php?p=2178288&postcount=264" only refers to the spatial component of a spacetime interval?

Yes, I understood that.

 

[neopolitan]

OK, I was thinking of the equation you derived in post 227, x'_A = (x_B - vt_B).gamma, and that was also the Lorentz-like equation you talked about in later posts. I think your derivation in post 227 was using different definitions than the earlier one you quote above from post 174, because in later posts you had x'_B = 4 and x'_A = 5, but if G=gamma that would mean x'_A = G*x'_B (in fact you wrote this equation in post 227), which is the reverse of what you have above.

I did think about that shortly after I wrote #227. You can eliminate x'_B and x_A or x_B and x'_A depending on what you are after. When I wrote #227, I was really just showing that G could be gamma or 1/gamma, depending on where you initially put G but what you ended up with (a Lorentz transformation - or "Lorentz-like") didn't change. I should have either not gone any further than that or, if I did, I should have eliminated x_B and x'_A to remain consistent with #174.

I don't think the equation from post 174 could actually be derived using your later definitions that you were using in post 227 and later posts; in post 227 you wrote:

Substituting, the equations you could get from this would be either x'_A = (x_B - vt_B)*gamma, which is what you derived, or x'_B + vt_B = x_A*gamma, which doesn't really look like a Lorentz transformation equation at all.

As I said I don't think you can actually derive x'_B = G.( x_A - v.t_A ) using the definitions from post 227 and subsequently. Even if you could, this would not really be much like the Lorentz transformation equation, because it doesn't relate the coordinates of a single event or single pair of events in two different frames--x'_B represents the position of E_B in the B frame (or equivalently the distance between E_B and the event of light passing B in the B frame), while x_A represents the position of E_A in the A frame (or equivalently the distance between E_A and the event of light passing A in the A frame) and t_A represents the time of the light passing A in the the A frame (or equivalently the time between E_A and the event of light passing A in the A frame, the same pair of events you might use to define x_A, except that if you want to define both x_A and t_A in terms of this pair of events then one of them must be negative if you're consistent about the order you take the events).

With the understanding of what x'_B is, and how it relates to x'_A and x_A and x_B in the diagram at #264, do you agree that the equations at #174 work?

I repeat yet again that I have been trying to work out why this thing works. And I think it does.

Part of the process of trying to work out why works was a discussion which we have conducted which has involved what I can see are some false starts and some dead ends from which I have had to retreat and start off again. Add to that the problems I have had with coding up replies (misplaced primes, misplaced subscripts) and it's a mess.

How about we take a pause for a bit while I fix what is wrong in #174 (as far as I can tell), then I make a temporal version of what is explained in the diagram in #264, I can repose the question from three paragraphs up and we can go from there? When I redo #174, I will put what is in the post into a diagram, because I find it so much better to use a WYSIWYG interface than the Latex reference interface - especially when I am strapped for time.

cheers,

neopolitan