Benefits of time dilation / length contraction pairing?

[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

 

[neopolitan]

Some diagrams posted http://www.geocities.com/neopolitonian/gen.htm". Some more to come, when I get to the machine the originals were created on.

If there are comments at this stage, please keep them to where I have stuffed up (there's bound to be something).

cheers,

neopolitonian

 

[JesseM]

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.

But how do you actually derive x'_B = gamma*(x_A - vt_A)? Are you still defining x'_A and x_B like this?

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

Then if you want to write the G equations as in #174 (which will make G = 1/gamma) that'd be:

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

(by the way, I should add that although I never got into it, I wasn't really happy with your answer in post 227 to why you assumed the G factor would be the same in both equations--you cited 'Galilean invariance', but how can you use Galilean invariance as a starting assumption in a derivation that's supposed to give SR equations, when Galilean invariance explicitly contradicts SR? Maybe you just meant 'invariance of the laws of physics in different inertial frames', i.e. the 'principle of relativity' that works in both Galilean relativity and SR? Even then I think more work would be needed to justify this step, because the physical situations aren't totally symmetric, in A's frame B is moving towards the photon while in B's frame A is moving away from the photon...)

Then I guess the next step would be the substitution t'_A = x'_A/c and t_B = x_B/c...I guess the justification here is that these terms can be defined using events on the path of a photon, for example t'_A is (time in the A frame between E_A and photon reaching B) while x'_A basically means (distance in A frame between E_A and photon reaching B).

Anyway, with that substitution you'd have:

x'_A=x_A - vx'_A/c --> x_A = x'_A*(1 + v/c)

and

x_B=x'_B + vx_B/c --> x'_B = x_B*(1 - v/c)

Then you can take x_A = x'_A*(1 + v/c) and x'_B = x_B*(1 - v/c) and combine with the G equations x_A = x_B*G and x'_B = x'_A*G giving:

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

which combine to give [x'_A*(1 + v/c)/G] * (1 - v/c) = x'[/sub]A[/sub]*G which implies G^2 = (1 + v/c)*(1 - v/c), so G = 1/gamma.

So, x'_B = x'_A*G becomes x'_A = gamma*x'_B, and x_A = x_B*G becomes x_B = gamma*x_A.

But then what's next? Is there a way to combine those equations with the original definitions of x'_A and x_B, namely x'_A=x_A - vt'_A and x_B=x'_B + vt_B, to yield the desired equation x'_B = gamma*(x_A - vt_A)? If there is I'm not seeing how.

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?

Just in terms of the numbers, sure, if x'_B = 4 and x_A = 8 and t_A = 8, then x'_B = gamma*(x_A - vt_A) works out. But I don't see how to actually derive that equation from your starting definitions. Also, as I said, the physical meaning of the terms, in particular why x_A and t_A are both positive, is ambiguous to me:

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).

So, which of these definitions is the reason that x_A and t_A are both positive?

1) they are the coordinates of two different individual events--x_A is the position coordinate of E_A in the A frame while t_A is the time coordinate of the event of light passing A in the A frame

2) they represent the coordinate distance and time between a single pair of events, but taken in different order--x_A = (position of event E_A) - (position of event of light passing A), while t_A = (time of event of light passing A) - (time of event E_A)

3) Similar to 3, but they are defined as the absolute value of (coordinate of one event) - (coordinate of other event), so the order doesn't matter

4) Something else?

I assume whichever definition you adopt, the definitions for x'_B and t'_B would be analogous. Also, if you want to change the definitions a little so x_A involves taking the events in the same order as t_A, meaning that x_A will actually be negative (my 'tweak'), that would make things a lot simpler so feel free to take that option (in this case the rest of the proof should work if you do some other tweaks as well, like defining x'_A as x_A + vt'_A so that it refers to [position of light passing B] - [position of E_A]).

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.

Sounds good, all of that would be helpful. If you want to go back and redo derivations I do recommend using the tweak above where coordinate distances and times between pairs of events are always consistent about the order so that they can sometimes be negative...it's up to you though.

 

[neopolitan]

The diagrams are http://www.geocities.com/neopolitonian/g2ev2.htm".

cheers,

neopolitan

PS Just read your previous post. I am defining Galilean invariance as saying that "fundamental physical laws are invariant across all inertial frames". I didn't go into that in the diagram, but since there are physical laws which involve the speed of light as a constant (permittivity of free space comes to mind) , then I take invariant speed of light to be a fundamental physical law.

 

[JesseM]

The diagrams are http://www.geocities.com/neopolitonian/g2ev2.htm".

cheers,

neopolitan

PS Just read your previous post. I am defining Galilean invariance as saying that "fundamental physical laws are invariant across all inertial frames". I didn't go into that in the diagram, but since there are physical laws which involve the speed of light as a constant (permittivity of free space comes to mind) , then I take invariant speed of light to be a fundamental physical law.

The idea that the laws of physics are invariant across inertial frames is known as the "principle of relativity" (which can mean either Galilean relativity or SR relativity depending on the context), while Galilean invariance is defined as the principle that the laws of physics are invariant under the Galilei transform (x' = x - vt and t' = t). Using these definitions, while it makes sense to assume the principle of relativity in a derivation of the Lorentz transformation, it wouldn't make sense to assume Galilean invariance since this assumption is logically incompatible with the idea that light moves at c in all frames, so I'd suggest changing your terminology here to bring it in line with the way physicists would understand these terms. In any case, even if you're talking about the principle of relativity I don't think that's sufficient to justify the claim that the same G factor should appear in x_A = x_B*G and x'_B = x'_A*G because these symbols represent coordinate distances and times for particular events in a particular physical scenario, these equations don't represent general laws of physics.

Starting on your diagrams, I'm a little confused about something on the page here where you say "According to the Galilean boost, t'=t so the time that photon hits B is the same for A and B". But you can't assume the coordinates of the two observers are related by the Galilean boost if you're deriving the Lorentz transform (and in fact we know in our numerical example that the time coordinate the photon hits B is different in the A frame than it is in the B frame)--is this just another pedagogical remark about what would be true in Galilean physics that is not actually part of the derivation? Hopefully when you write x' = x - vt at the top of that page you'd agree that this equation cannot actually be interpreted as relating A's frame to B's frame if we're assuming they both use frames where the speed of light is c as in SR...x' and x are both just different distances in A's frame, right? (If we're talking the distance between B and E_A as a function of time, x would be the position coordinate of E_A in A's frame while x'(t) would be the distance between B and E_A in A's frame, for example.) So in this context x' = x - vt is not a Galilean boost because it's not relating the coordinates of two different frames.

Also, on that page you appear to have changed the meaning of t_A and t'_B from what they were previously. For example, in prior posts t_A represented the time in the A frame that the photon passed A (giving t_A = 8, as you wrote in post 243), but based on equation (1) from that page it seems you're now using t_A to represent the time in the A frame that the photon passes B (giving t_A = 5...in previous posts you had t'_A = 5). Do you want to change the definitions from here on out, or do you want to just treat that as an error and keep things consistent with the notation used earlier on this thread? In any case, with t_A as the time the photon passes B (t_A = 5), and x_A as the position of E_A (x_A = 8), it would no longer make sense to do the substitution t_A = x_A/c as you did in the line between (1) and (3).

 

[neopolitan]

The idea that the laws of physics are invariant across inertial frames is known as the "principle of relativity" (which can mean either Galilean relativity or SR relativity depending on the context), while Galilean invariance is defined as the principle that the laws of physics are invariant under the Galilei transform (x' = x - vt and t' = t). Using these definitions, while it makes sense to assume the principle of relativity in a derivation of the Lorentz transformation, it wouldn't make sense to assume Galilean invariance since this assumption is logically incompatible with the idea that light moves at c in all frames, so I'd suggest changing your terminology here to bring it in line with the way physicists would understand these terms. In any case, even if you're talking about the principle of relativity I don't think that's sufficient to justify the claim that the same G factor should appear in x_A = x_B*G and x'_B = x'_A*G because these symbols represent coordinate distances and times for particular events in a particular physical scenario, these equations don't represent general laws of physics.

I was using "Galilean invariance or Galilean relativity is a principle of relativity which states that the fundamental laws of physics are the same in all inertial frames." - http://en.wikipedia.org/wiki/Galilean_invariance"

I do note that subsequently in that article it gives the axioms of Newtonian relativity which are the absoluteness of space and the universality of time.

Remember I originally said something along the lines of "according to each,the other measures space oddly". I've tired to avoid that terminology. Instead I showed that there is some difference between how each measures space and how the other measures space, and kept an element of invariance (or relativity) - that neither frame is privileged.

That would mean that according to A, a spatial interval measured in A's frame (between two events) would be measured differently in B's frame and the relationship between those spatial measurements would be identical to when, according to B, a spatial interval measured in B's frame (between the same two events) would be measured differently in A' frame.

In otherwords, according to A, when A and B are colocated, the distance between A and B and the YDE (yellow dot event) is x_A. According to B, that distance (which is not when A and B are colocated), is x_B. The interval is a pure distance in the A frame, so:

x_B = (a factor times or fuction of).x_A

According to B, when A and B are colocated, the distance between A and B and the YDE (yellow dot event) is x'_B. According to A, that distance (which is not when A and B are colocated), is x'_A. The interval is a pure distance in the B frame, so:

x'_A = (a factor times or fuction of).x'_B

I've taken the step of saying it is not a function, but a factor. I'm using prior knowledge here, but if I were to be very very particular, I could say it might be a function, but let's try a factor first then once I've found that a factor works, I can say we don't need a function.

Therefore:

x_B = (a factor times).x_A
x'_A = (a factor times).x'_B

Now I need to give (a factor times) a useful notation.

I originally chose a Thai letter, but to make it easier, instead I decided to use a Roman letter. Initially, I made (a factor times) = G. But you can quickly work out that that makes G=1/gamma. So, to make it easier - I thought - I expressed everything so that (a factor times) = 1/gamma = 1/G.

Do I need to spell that out in the derivation?

Starting on your diagrams, I'm a little confused about something on the page here where you say "According to the Galilean boost, t'=t so the time that photon hits B is the same for A and B". But you can't assume the coordinates of the two observers are related by the Galilean boost if you're deriving the Lorentz transform (and in fact we know in our numerical example that the time coordinate the photon hits B is different in the A frame than it is in the B frame)--is this just another pedagogical remark about what would be true in Galilean physics that is not actually part of the derivation?

Pedagogical remark

Hopefully when you write x' = x - vt at the top of that page you'd agree that this equation cannot actually be interpreted as relating A's frame to B's frame if we're assuming they both use frames where the speed of light is c as in SR...x' and x are both just different distances in A's frame, right? (If we're talking the distance between B and E_A as a function of time, x would be the position coordinate of E_A in A's frame while x'(t) would be the distance between B and E_A in A's frame, for example.) So in this context x' = x - vt is not a Galilean boost because it's not relating the coordinates of two different frames.

Look at drawings 4 and 5 in the sequence. In those you can see x', x and vt mapped according to A (so according to A, you would know that these can be subscripted with A).

We are starting with the Galilean boost (perhaps just the equation) to go through a process to obtain the spatial Lorentz transformation (perhaps just the equation) and during that process there will points where what we have is not quite either.

x'_A=x_A - vt_A works in both GalRel and SR, correct?

Perhaps we need to define what we mean by "boost", I just mean the equation, I am not using it to compare two frames. I am using it to tell me the answer to: "with an initial (t=0)separation of x between a body and a distant location, if that body moves towards that location with a speed of v, then what is the separation between that body and the distance location at a time t?" I am effectively comparing two frames, because I can continue doing that, for all values of t and build up a description of the frame according the body and implied frame in which the body is in motion. But that just means I can use the equation as a tool later on, if I feel like it.

Also, on that page you appear to have changed the meaning of t_A and t'_B from what they were previously. For example, in prior posts t_A represented the time in the A frame that the photon passed A (giving t_A = 8, as you wrote in post 243), but based on equation (1) from that page it seems you're now using t_A to represent the time in the A frame that the photon passes B (giving t_A = 5...in previous posts you had t'_A = 5). Do you want to change the definitions from here on out, or do you want to just treat that as an error and keep things consistent with the notation used earlier on this thread?

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

I do notice that on that page, I have errors.

Firstly, time for a photon to get from YDE to A (according to A) is x_A/c and the time at which a photon from YDE gets to B (according to A) is x'_A/c. (Note the wording.)

The time for a photon to get from YDE to A (according to B) is x_B/c and the time at which a photon from YDE gets to B (according to B) is x'_B/c. (Note the wording.)

I will have to fix this because it flows on further through the document.

The end result will be the same, but it will be using better defined values. (edit - I know this, because I just jotted it down on paper and it works. It works the way I thought it did right back before I started this thread, it's just that I have a better grasp on what each of the values are.) I thought about posting the document (on geocities) and looking at it again in the cold light of day, but it's often more difficult to see mistakes in your own work.

I'll have to go back to the original document I did on this derivation (not posted in this thread) and see whether I have similar errors. That particular document was put together with much less pressure

cheers,

neopolitan

 

[JesseM]

I was using "Galilean invariance or Galilean relativity is a principle of relativity which states that the fundamental laws of physics are the same in all inertial frames." - http://en.wikipedia.org/wiki/Galilean_invariance"

I do note that subsequently in that article it gives the axioms of Newtonian relativity which are the absoluteness of space and the universality of time.

Yes, the wikipedia page was implicitly referring to Newtonian inertial frames.

Remember I originally said something along the lines of "according to each,the other measures space oddly". I've tired to avoid that terminology. Instead I showed that there is some difference between how each measures space and how the other measures space, and kept an element of invariance (or relativity) - that neither frame is privileged.

That would mean that according to A, a spatial interval measured in A's frame (between two events) would be measured differently in B's frame and the relationship between those spatial measurements would be identical to when, according to B, a spatial interval measured in B's frame (between the same two events) would be measured differently in A' frame.

But in your equations with G you weren't talking about measuring the distance between a single pair of events in two frames.

In otherwords, according to A, when A and B are colocated, the distance between A and B and the YDE (yellow dot event) is x_A. According to B, that distance (which is not when A and B are colocated), is x_B.

Previously x_B referred to the distance between E_B and the event of the photon passing A. Are you changing the definition completely now? When you say "that distance", it's unclear whether you mean the distance between B and the YDE at the moment it occurs, or the distance between A and the YDE at the moment it occurs (either way it'd be the distance in the B frame, and the moment it occurs in the B frame, presumably). Whichever way you choose, the equation x_B = G*x_A is not talking about the distance between a single pair of events in two different frames.

x_B = (a factor times or fuction of).x_A

According to B, when A and B are colocated, the distance between A and B and the YDE (yellow dot event) is x'_B.

But in B's frame the YDE doesn't occur when A and B are colocated. Do you mean the distance between B and the position the yellow dot occurred in the past? But in B's frame B isn't moving, so the distance between B and the position where the YDE was in the past will be the same as the distance between B and the YDE at the moment it occurred, which might already be the definition of x_B, unless x_B referred to the distance between A and the YDE at the moment it occurred in the B frame (see my question above).

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

Distance between what two events? Is one of them the YDE? But the YDE did occur when they were colocated in A's frame, did it not? Your way of defining these terms is extremely confusing, you really need to be much more specific. Illustrating the definitions in terms of a numerical example would be helpful, then you could say things like "the YDE occurred at x=8 and t=0 in A's frame" and "I want the distance between the YDE event, which occurred at t=-6 in B's frame, and the event on B's worldline which also occurred at t=-6 in B's frame", stuff like that.

The interval is a pure distance in the B frame, so:

x'_A = (a factor times or fuction of).x'_B

I've taken the step of saying it is not a function, but a factor. I'm using prior knowledge here, but if I were to be very very particular, I could say it might be a function, but let's try a factor first then once I've found that a factor works, I can say we don't need a function.

Therefore:

x_B = (a factor times).x_A
x'_A = (a factor times).x'_B

I don't understand what you mean by "a factor works" ('works' in what sense? Do you just mean it gives results consistent with your prior knowledge of the Lorentz transform?), and I also don't see where you justified the idea that it would be the same numerical factor in both equations.

Look at drawings 4 and 5 in the sequence. In those you can see x', x and vt mapped according to A (so according to A, you would know that these can be subscripted with A).

We are starting with the Galilean boost (perhaps just the equation) to go through a process to obtain the spatial Lorentz transformation (perhaps just the equation) and during that process there will points where what we have is not quite either.

x'_A=x_A - vt_A works in both GalRel and SR, correct?

That depends on what the terms mean. It would work if x_A referred to the distance between B and the YDE at t=0 in the A frame (assuming the YDE occurs at that time in the A frame, so it's just a renamed E_A), and x'_A referred to the distance between B and the position of the YDE at some later time t_A (what event defines the term t_A?)

Perhaps we need to define what we mean by "boost", I just mean the equation, I am not using it to compare two frames. I am using it to tell me the answer to: "with an initial (t=0)separation of x between a body and a distant location, if that body moves towards that location with a speed of v, then what is the separation between that body and the distance location at a time t?" I am effectively comparing two frames, because I can continue doing that, for all values of t and build up a description of the frame according the body and implied frame in which the body is in motion.

I don't really see how you're comparing two frames--aren't all distances and times here defined in terms of frame A?

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

If you're going to redefine all kinds of terms though, you really need to provide detailed definitions of what they mean.

Firstly, time for a photon to get from YDE to A (according to A) is x_A/c and the time at which a photon from YDE gets to B (according to A) is x'_A/c. (Note the wording.)

So x'_A is indeed the distance between B and the position of the YDE at the moment the photon passes B, all as measured in the A frame? If the YDE occurs at x=8, t=0 in the A frame then x'_A = 5?

The time for a photon to get from YDE to A (according to B) is x_B/c and the time at which a photon from YDE gets to B (according to B) is x'_B/c. (Note the wording.)

OK, so using the above numbers, the YDE occurs at t=-6, x=10 in the B frame, therefore it will reach B at time t=4. So that means x'_B = 4 in this example? But earlier you said "According to B, when A and B are colocated, the distance between A and B and the YDE (yellow dot event) is x'_B." According to that definition, if A and B were colocated at x=0 in the B frame and the YDE's position was x=10, shouldn't x'_B = 10?

 

[neopolitan]

Previously x_B referred to the distance between E_B and the event of the photon passing A. Are you changing the definition completely now?

I don't recall ever meaning that, or writing it. I can understand how you might think it is the distance between E_A and the event of the photon passing A, since this is what x_A is, in the A frame. I'm not going to trawl back through old posts to look for what I said in order to defend what might well have been a typo.

But in B's frame the YDE doesn't occur when A and B are colocated.

I know that.

Do you mean the distance between B and the position the yellow dot occurred in the past? But in B's frame B isn't moving, so the distance between B and the position where the YDE was in the past will be the same as the distance between B and the YDE at the moment it occurred, which might already be the definition of x_B, unless x_B referred to the distance between A and the YDE at the moment it occurred in the B frame (see my question above).

If you can resist redefining my terms we might avoid the issue where you think the definition of x_B has been something quite different to anything I have ever thought it has been (as in the question above).

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

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

We are getting buried in words. The addition of extra words may help, but I am wondering if it actually would.

Distance between what two events? Is one of them the YDE? But the YDE did occur when they were colocated in A's frame, did it not? Your way of defining these terms is extremely confusing, you really need to be much more specific. Illustrating the definitions in terms of a numerical example would be helpful, then you could say things like "the YDE occurred at x=8 and t=0 in A's frame" and "I want the distance between the YDE event, which occurred at t=-6 in B's frame, and the event on B's worldline which also occurred at t=-6 in B's frame", stuff like that.

Are the spacetime diagrams of no help at all? They have numbers all over them.

I don't understand what you mean by "a factor works" ('works' in what sense? Do you just mean it gives results consistent with your prior knowledge of the Lorentz transform?), and I also don't see where you justified the idea that it would be the same numerical factor in both equations.

A factor gives you a relationship which is symmetric. It could have (in another universe) demanded a function to have a symmetric relationship. But in our universe "a factor works".

Symmetry would also demand that the same numerical factor be in both equations. Remove symmetry and you have a privileged frame (not necessarily either of the frames in question, but a privileged frame somewhere that is more closely aligned to one of these two frames than to the other).

That depends on what the terms mean. It would work if x_A referred to the distance between B and the YDE at t=0 in the A frame (assuming the YDE occurs at that time in the A frame, so it's just a renamed E_A), and x'_A referred to the distance between B and the position of the YDE at some later time t_A (what event defines the term t_A?)

Try looking at all the diagrams. Synthesise, then respond. If I was sitting down next to you, you could do what you are doing, and I could point to things for you, but since we have this sort of correspondence, you are going to have to look at everything first and make an effort to synthesise.

Anyway, to try to help, I will do a separate diagram for the gen.htm series, which explicitly shows where the values manifest.

I don't really see how you're comparing two frames--aren't all distances and times here defined in terms of frame A?

x' = x - vt

In the A frame, the unprimed frame, B is moving so all the distances to locations change with time. x' is the distance between B after a period of t and a location given by x. According to B, B is stationary, so the location given by x in the A frame is not fixed. Therefore any fixed location in the A frame varies with the rate at which A moves relative to B. You can use this to convert between frames (at least in Galilean relativity, otherwise you need a Lorentz transformation). Note that "boost" is "Galilean boost", and "Lorentz transformation" is "Lorentz transformation".

If you're going to redefine all kinds of terms though, you really need to provide detailed definitions of what they mean.

Do you promise not to redefine my terms?

So x'_A is indeed the distance between B and the position of the YDE at the moment the photon passes B, all as measured in the A frame? If the YDE occurs at x=8, t=0 in the A frame then x'_A = 5?

Yes

OK, so using the above numbers, the YDE occurs at t=-6, x=10 in the B frame, therefore it will reach B at time t=4. So that means x'_B = 4 in this example? But earlier you said "According to B, when A and B are colocated, the distance between A and B and the YDE (yellow dot event) is x'_B." According to that definition, if A and B were colocated at x=0 in the B frame and the YDE's position was x=10, shouldn't x'_B = 10?

Nope. You are using a redefined value of x'_B.

cheers,

neopolitan

I'll post the updated diagram for you later.

 

[neopolitan]

http://www.geocities.com/neopolitonian/generality6_all_values.jpg".

All the values are there. Do I have to put them into words, or does the diagram suffice?

cheers,

neopolitan

 

[neopolitan]

The http://www.geocities.com/neopolitonian/g2ev2.htm" page has been revised.

cheers,

neopolitan

 

[JesseM]

Previously x_B referred to the distance between E_B and the event of the photon passing A. Are you changing the definition completely now?

I don't recall ever meaning that, or writing it. I can understand how you might think it is the distance between E_A and the event of the photon passing A, since this is what x_A is, in the A frame. I'm not going to trawl back through old posts to look for what I said in order to defend what might well have been a typo.

From post 243, here were your old definitions:

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.

Also see post 224 when you wrote x_B=x'_B + vt_B where "t_B is when the photon from E_B passes A according to B (eg t=10)"; since the distance between A and E_B is increasing over time, if x'_B was the position of E_B in the B frame (which is also what you said in the quote above), then that equation also fits perfectly with the notion that x_B is the distance between A and E_B at the time t_B when the photon passes A. Clearly you were using these definitions at one point, it wasn't a "typo".

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

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

OK, thanks. I realized you were talking about some distance at the time of YDE in B's frame, but the reason this was ambiguous was because your previous definition was "according to A, when A and B are colocated, the distance between A and B and the YDE (yellow dot event) is x_A", so when you referred to "that distance" in the next sentence it was unclear if you meant the distance between A and YDE or the distance between B and YDE at the moment YDE occurred in the B frame (when A and B weren't colocated).

Are the spacetime diagrams of no help at all? They have numbers all over them.

That was forgetful on my part, I was trying to go back through the derivation on that page in order, so when I got confused about the meaning of the terms I didn't think to skip to the end to check the diagram. So OK, I think based on the diagram I see what the definitions are (you don't show x_B in the diagram, but you explained that above), but please check to see if these are right:

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

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

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

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

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

Incidentally, if these definitions and numbers are correct then x'_A = x_A - vt_A would imply t_A is the time in the A frame that the light passes B (so t_A = 5), is that right? And in x_B = x'_B + vt'_B implies that t'_B = 10...what is the physical definition of t'_B, or of the equation x(t) = x'_B + vt' in general? The equation x'(t) = x_A - vt in the A frame had an obvious physical interpretation, x'(t) referred to the distance between B and the YDE as a function of time in the A frame, since at t=0 B was at a distance of x_A from the YDE (just as A was at that moment, since they were colocated), and B was moving towards the position of the YDE with velocity v. I suppose in this case x(t) = x'_B + vt' can be taken to give the distance between A and the event E_B as a function of time, since x'_B is the distance between E_B and A&B at t'=0 in the B frame, and A is moving away from that position at velocity v.

The only problem with this definition is that when we write x_B = x'_B + vt'_B, x_B was not originally defined to mean the distance between A and E_B at some time t'_B. But if we choose the time t'_B when the light passes A, we find that in the B frame this event occurs at position -6 and time t'_B = 10, so the distance between A and E_B at this moment is 10, just like the distance between B and the YDE, so I guess we can say that x_B can be defined as either of these. But here I was relying on my prior knowledge of the Lorentz transform to show that the distance in the B frame (YDE to B's position) is identical to the distance in the B frame (E_B to light passing A), so I think that means if you want to use the equation x_B = x'_B + vt'_B in your derivation without assuming what you're trying to prove, you really need to define x_B as the distance between E_B and the light passing A...exactly the same definition you denied when I quoted it at the beginning of this post! If instead you define x_B as the distance between B and the YDE at the moment it occurs, how can you justify the equation x_B = x'_B + vt'_B ? Why should we expect that relationship to hold if we don't already know the Lorentz transformation?

Note that if we do define x_B in terms of the distance between E_B and the event of the light passing A, and we also return to the term E_A for the YDE, then the symmetry in the definitions is much more readily apparent:

x_A is the distance between E_A and A (in the A frame). It would be 8.

x_B is the distance between E_B and the light passing A (in the B frame). It would be 10.

x'_A is the distance between E_A and the light passing B (in the A frame). It would be 5.

x'_B is the distance between E_B and B (in the B frame). It would be 4.

A factor gives you a relationship which is symmetric. It could have (in another universe) demanded a function to have a symmetric relationship. But in our universe "a factor works".
Symmetry would also demand that the same numerical factor be in both equations. Remove symmetry and you have a privileged frame (not necessarily either of the frames in question, but a privileged frame somewhere that is more closely aligned to one of these two frames than to the other).

Why do you think "symmetry" demands that the same factor/function appear in both equations though? You haven't really justified this. The meaning of the terms in the two equations doesn't appear very symmetrical if we use your definitions--in the equation x_B = (a factor times).x_A, x_A refers to the position of the YDE in the A frame while x_B refers to the position of the YDE in the B frame, but then in the equation x'_A = (a factor times).x'_B, x'_A is the distance in the A frame between YDE and the light passing B, while x'_B is the distance in the B frame between B and E_B. If we use my equivalent-but-stated-differently definitions above, then there is more of an apparent symmetry, x_B = (a factor times).x_A becomes:

the distance between E_B and the light passing A (in the B frame) = (a factor times) the distance between E_A and A (in the A frame)

And x'_A = (a factor times).x'_B becomes:

the distance between E_A and the light passing B (in the A frame) = (a factor times) the distance between E_B and B (in the B frame)

When stated this way, you can see that the second is just the first with all the A's and B's reversed. So here it is at least intuitive that it would turn out the same factor appears in both equations, although I still don't think it's really justified since the actual physical situation does not look the same in both frames (in A's frame, B is moving towards the position of E_A, while in B's frame, A is moving away from the position of E_B at the same speed).