Benefits of time dilation / length contraction pairing?

[neopolitan]

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

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

cheers,

neopolitan

 

[neopolitan]

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

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

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

Happy?

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

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

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

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

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

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

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

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

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

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

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

...and this?

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

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

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

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

cheers,

neopolitan

 

[JesseM]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 

[neopolitan]

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

The least important bit first - from my perspective.

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

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

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

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

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

cheers,

neopolitan

 

[JesseM]

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

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

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

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

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

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

 

[neopolitan]

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

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

There are four dots:

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

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

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

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

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

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

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

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

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

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

neopolitan

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

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

 

[JesseM]

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

There are four dots:

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

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

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

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

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

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

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

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

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

 

[neopolitan]

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

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

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

Using a different coordinate system does not move the event.

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

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

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

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

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

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

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

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

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

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

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

Does that make sense to you?

cheers,

neopolitan

 

[JesseM]

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

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

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

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

Using a different coordinate system does not move the event.

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

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

and the coordinates of the orange dot are:

x = vt'_a, t = t'_a

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

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

And the coordinates of the orange dot as:

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

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

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

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

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

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

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

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

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

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

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

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

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

 

[neopolitan]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The events colocated with each observer are:

A: colocation with B, colocation of A and photon

B: colocation with A, colocation of B and photon

You asked:

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

And said:

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

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

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

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

No explicit relativity of simultaneity.

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

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

cheers,

neopolitan

 

[JesseM]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

No explicit relativity of simultaneity.

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

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

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

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

 

[neopolitan]

#303 strand

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

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

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

I have labelled this #303 strand.

cheers,

neopolitan

 

[neopolitan]

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

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

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

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

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

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

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

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

cheers,

neopolitan

 

[JesseM]

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

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

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

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

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

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

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

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

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

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

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

 

[JesseM]

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

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

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

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

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

 

[neopolitan]

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

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

Then I can't understand why you said this:

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

I took that to be criticism, although I couldn't understand what alternative you were presenting.

By the way, you talk about coordinates and intervals. To avoid having to do that, can we agree that coordinates are intervals (just a specific one where one of the implied pair is the origin of the axis)? I have no problem with the idea of rearranging my axes to transform any interval into a single coordinate, or the reverse.

On the image, we could call the red dot a coordinate, but it is also an interval from the orange dot to the red dot. Between the red dot and the green dot is an interval, but if I redefined my axes so that (0,0) was the red dot, then the green dot would be a coordinate from the red dot. I don't see a huge difference between coordinate and interval for this very reason.

cheers,

neopolitan

 

[JesseM]

I took that to be criticism, although I couldn't understand what alternative you were presenting.

It was less of a criticism and more of a puzzlement about your statement here:

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

You seemed to be saying that one pair would speak specifically speak about "a YDE that is simultaneous in A's frame with A and B being colocated", while the other pair would speak specifically about a YDE that was simultaneous with colocation in B's frame. But there's no specificity here, since each pair of equations is totally general, either pair could deal with either type of YDE. For example, one pair of equations for the Lorentz transform would be x'=gamma*(x-vt) and t'=gamma*(t-vx/c^2); if I knew the x,t coordinates of a YDE that was simultaneous with colocation in the unprimed A frame I could plug those in and get the same event's x',t' coordinates, and likewise if I knew the x,t coordinates of a YDE that was simultaneous with colocation in the primed B frame I could plug those in and get the same event's x',t' coordinates. Of course that's assuming I started out knowing the x,t coordinates of each type of event, I suppose in the case of a YDE that was simultaneous with colocation in the primed frame it's more plausible I would start out knowing the event's x',t' coordinates, so maybe that's what you meant (but then again events aren't 'native' to any particular coordinate system, it's possible I would focus on this particular event while working in unprimed coordinates for some other reason without knowing in advance it had the property of being simultaneous with colocation in the primed frame).

By the way, you talk about coordinates and intervals. To avoid having to do that, can we agree that coordinates are intervals (just a specific one where one of the implied pair is the origin of the axis)? I have no problem with the idea of rearranging my axes to transform any interval into a single coordinate, or the reverse.

Sure, that's a good way of thinking about it.

 

[neopolitan]

I'm going to try to address the x' = x - vt thing.

Rather than go over each of your paragraphs, I will try to just explain it. I hope this will satisfy you.

The implication with x' = x - vt is that you have an unprimed observer who is considering what things would be like for someone else who is moving with a speed of v towards a location, or event, at a distance of x away.

At a time t, x will be unchanged for our unprimed observer in the unprimed frame. However, the separation between the someone else and that location, or the location where the event took (or will take or takes) place, will have changed.

Our unprimed observer can then work out that at t this equation applies:

x' = x - vt

or to be more specific, since we are describing a function of time,

x'(t) = x - vt

This equation can be used to obtain the spatial interval between the someone else and any event that happens at any time at any location under Galilean relativity.

In Galilean relativity, that interval is also the spatial coordinate for the event in the someone else frame.

Can you see that while you are 100% right about x'(t) = x - vt applying to spatial coordinates, that that is not 100% of the story?

cheers,

neopolitan

 

[neopolitan]

You seemed to be saying that one pair would speak specifically speak about "a YDE that is simultaneous in A's frame with A and B being colocated", while the other pair would speak specifically about a YDE that was simultaneous with colocation in B's frame. But there's no specificity here, since each pair of equations is totally general, either pair could deal with either type of YDE. For example, one pair of equations for the Lorentz transform would be x'=gamma*(x-vt) and t'=gamma*(t-vx/c^2); if I knew the x,t coordinates of a YDE that was simultaneous with colocation in the unprimed A frame I could plug those in and get the same event's x',t' coordinates, and likewise if I knew the x,t coordinates of a YDE that was simultaneous with colocation in the primed B frame I could plug those in and get the same event's x',t' coordinates.

x' = \gamma. (x - vt)
t' = \gamma. (t - vx/c^2)

x = \gamma. (x' + vt')
t = \gamma. (t' + vx'/c^2)

gamma = 1.25
v = 0.6c
event (x,t) = (10,0)

x' = 12.5
t' = - 7.5

This event was not simultaneous with t' = 0

But we can plug these right back into the second pair of equations and get the original x and t back out.

Alternatively, we could start with

event (x',t') = (10,0)

which is a totally different event, just one which is simultaneous (in the primed frame) with t' = 0 - and we can do the same sorts of substitutions.

Notionally, we could start with an event which is not simultaneous with either.

I'm not that fussed. It's just much simpler to start with an event that is simultaneous with the colocation of the origins of the axes and work from there.

What I do know is that no matter what event you start from, that event can be associated with a photon which could result in two more events (such as: photon colocated with A and photon colocated with B).

Am I making any sort of headway?

cheers,

neopolitan

 

[JesseM]

I'm going to try to address the x' = x - vt thing.

Rather than go over each of your paragraphs, I will try to just explain it. I hope this will satisfy you.

The implication with x' = x - vt is that you have an unprimed observer who is considering what things would be like for someone else who is moving with a speed of v towards a location, or event, at a distance of x away.

At a time t, x will be unchanged for our unprimed observer in the unprimed frame. However, the separation between the someone else and that location, or the location where the event took (or will take or takes) place, will have changed.

Our primed observer can then work out that at t this equation applies:

x' = x - vt

or to be more specific, since we are describing a function of time,

x'(t) = x - vt

So was I right in guessing that you are using x'(t) to refer to the distance in the unprimed frame between the primed observer and the location, despite the fact that the symbol is primed? If so, why do you say "our primed observer" works out this equation, if all of the terms are defined in terms of the unprimed frame? And if this is what you mean, can you see why this notation might be extremely confusing, especially since you said that the high school student was aware of this equation because he paid attention in class, and yet in any textbook which used standard notation conventions the meaning of this equation would always be in the context of the Galilei transformation relating one frame to another? If you're going to take standard textbook equations and change the meaning of the terms this is really something you need to explain in advance to avoid confusion.

This equation can be used to obtain the spatial interval between the someone else and any event that happens at any time at any location under Galilean relativity.

But it has nothing specifically to do with Galilean relativity, since if x'(t) just means the distance in the unprimed frame between B and the location at x, then the equation x'(t)=x-vt isn't even relating multiple frames, it would be equally valid in SR.

In Galilean relativity, that interval is also the spatial coordinate for the event in the someone else frame.

Yes.

Can you see that while you are 100% right about x'(t) = x - vt applying to spatial coordinate, that that is not 100% of the story?

Not really sure what you mean by "applying to spatial coordinate". Do you just mean that x'=x-vt normally is understood to relate one Galilean frame's coordinates to another's, but that we can in principle redefine the meaning of x' so that it refers to the spatial separation between a moving object and a fixed location x in a single frame? If so I agree (and I already asked you if you were making such a redefinition in several earlier posts). But when you change the physical definition of the terms you're dealing with a different physical equation even if the symbols are the same, so it's not like there are two different ways of looking at the same physical equation. What's more, writing a new physical equation using notation that has a different preexisting established meaning is kind of perverse from a pedagogical point of view (sort of like if I wrote the equation E=mc^2 and said that E stands for force and c stands for the square root of acceleration), there's already an accepted convention about the physical interpretation of primed vs. unprimed coordinates and about the equation x'=x-vt, if you want to write down an equation giving a spatial separation in the unprimed frame then it would be much better to use some different notation which wouldn't be so likely to confuse people who were trying to understand you, like s(t) = x_a - vt.

 

[JesseM]

x' = \gamma. (x - vt)
t' = \gamma. (t - vx/c^2)

x = \gamma. (x' + vt')
t = \gamma. (t' + vx'/c^2)

gamma = 1.25
v = 0.6c
event (x,t) = (10,0)

x' = 12.5
t' = - 7.5

This event was not simultaneous with t' = 0

But we can plug these right back into the second pair of equations and get the original x and t back out.

Right, that was exactly my point. Neither pair of equations deals specifically with this event as your earlier comment sounded like it was saying, we could either start with the event's x and t coordinates and use the first pair of equations to get its x' and t' coordinates, or we could start with the event's x' and t' coordinates and use the second pair of equations to get its x and t coordinates. It may be a little more "natural" to start with the coordinate system where the time coordinate is zero, so that's probably what you meant in that earlier comment. In any case I don't really think we have any remaining disagreement here.

 

[neopolitan]

So was I right in guessing that you are using x'(t) to refer to the distance in the unprimed frame between the primed observer and the location, despite the fact that the symbol is primed? If so, why do you say "our primed observer" works out this equation, if all of the terms are defined in terms of the unprimed frame?

Thankfully I only ever mentioned one observer, so it should be obvious that this was a typo. Note I mentioned "observer" and "someone else". Wherever observer was written it should have been prefixed with unprimed. Sorry.

With that fixed (I edited the post) can you revisit your questions and see if they still apply.

thanks,

neopolitan

Oh, and by the way, I never primed a frame. That assumption on your part might be confusing you. Remember my description of x'_{a[\sub] from an earlier post?}

 

[JesseM]

Thankfully I only ever mentioned one observer, so it should be obvious that this was a typo. Note I mentioned "observer" and "someone else". Wherever observer was written it should have been prefixed with unprimed. Sorry.

With that fixed (I edited the post) can you revisit your questions and see if they still apply.

Yes, aside from that one sentence about the primed observer all my other points still apply; I still think it's extremely perverse from a pedagogical point of view to take an equation with an established meaning like x'=x-vt and redefine the meaning of the terms so it the equation's physical meaning becomes completely different (without stating explicitly that this is what you're doing), and to say that a high school student is aware of this equation from his classes without mentioning that his interpretation of the symbols has nothing to do with what he actually would have been taught in class. Do you not see how ridiculously confusing this is for readers? Again, if you want to write an equation expressing only the displacement in the unprimed frame, write something like d(t) = x_a - vt.

Oh, and by the way, I never primed a frame.

It doesn't matter if you did, what matters is that the equation x'=x-vt, whenever it appears in any physics textbook, always appears in the context of the Galilei transform where it relates coordinates in an unprimed frame to coordinates in a primed frame. And you said that the high school student was aware of this equation from his classes (presumably not taught by a teacher who was making up his own idiosyncratic notation), not to mention you even said that you were starting your derivation from a "Galilean" perspective! Like I said, this is akin to writing the equation E=mc^2 without mentioning that you've mentally redefined E to mean force and c^2 to mean acceleration.

 

[neopolitan]

So, what you are saying is that I am using an equation which looks a lot like the Galilean boost, but isn't, using a process which you may not understand, and arriving at a equations which look a lot like the Lorentz Transforms but are not?

This is a rhetorical question, and I am taking the answer to be yes.

So, I've found a new set of equations. I am so proud. I hereby name them "the Neopolitonian boost" and "the Neopolitonian Transforms".

The Neopolitonian boost

This equation is used in a scenario where there are two observers (A and B) and an Event.

According to A that Event is at (x_a,0). A photon from that event reaches A at event (0,t_a)

According to A, B has a velocity of v towards the Event and the boost is used to find the interval between B and that Event (according to A) at t_a

x'_a = x_a - v.t_a

(In our numerical example, this is 5 = 8 - 0.6 * 5 )

The Neopolitonian Transform

This pair of equations is used to transform coordinates between two inertial observers (or two inertial frames). For example, in the scenario above, all the values given are according to A. The Neopolitonian Transform gives a value of x' and t' according to B (x'_b and t'_b) - which make B's frame the primed frame. This can also be called "A looking at B" since the values of x'_b and t'_b are given in terms of x_a and t_a, ie "what do B's values look like in terms of A's values?"

x'_b = \gamma.(x_a - v.t_a)

t'_b = \gamma.(t_a - v.x_a/c²)



Ok. Are you happy to talk about the process used to derive the Neopolitonian Transform from the Neopolitonian boost?

Plus, are you absolutely certain that no-one is going to accuse me of just rebadging the Galilean boost and the Lorentz Transform, because it is sooooo obvious that they are not the same as the Neopolitonian boost and the Neopolitonian Transform?

cheers,

neopolitan

PS I want make explicit the fact that I still remember that we can convert coordinates to intervals and back again. I also know that it is applicable in the above.

 

[JesseM]

So, what you are saying is that I am using an equation which looks a lot like the Galilean boost, but isn't, using a process which you may not understand, and arriving at a equations which look a lot like the Lorentz Transforms but are not?

I was mainly expressing frustration that you don't seem to consider how your idiosyncratic way of explaining yourself will lead naturally to some obvious misunderstandings which have to be laboriously worked out over a long series of subsequent posts, which could have been easily avoided if you explained the difference between your equations and the "standard" ones to begin with, or made your notation different from that of standard equations so such misunderstandings would be less likely to occur. Now I understand what you meant by x'=x-vt in post 295, but a lot of wasted time could have been avoided if you had been more explicit about the way you were redefining things at the start. If you got confused by someone's explanation of a series of equations which included the equation E=mc^2, and only after a long discussion did he make clear that he was defining E to mean force and c to mean the square root of acceleration, wouldn't you find this a little frustrating too?

So, I've found a new set of equations. I am so proud. I hereby name them "the Neopolitonian boost" and "the Neopolitonian Transforms".

The Neopolitonian boost

This equation is used in a scenario where there are two observers (A and B) and an Event.

According to A that Event is at (x_a,0). A photon from that event reaches A at event (0,t_a)

According to A, B has a velocity of v towards the Event and the boost is used to find the interval between B and that Event (according to A) at t_a

x'_a = x_a - v.t_a

(In our numerical example, this is 5 = 8 - 0.6 * 5 )

It isn't really proper to use the term "boost" here, as the word "boost" in physics refers to a transformation between two frames (for example see the last paragraph in this section of wikipedia's Lorentz transformation article). Your equation is just a simple kinematical equation for the separation between an object moving at constant coordinate speed and another object at fixed coordinate position, it would hold in absolutely any coordinate system whatsoever (like an inertial SR frame or even a non-inertial frame) and therefore has no specific relation to anything "Galilean".

The Neopolitonian Transform

This pair of equations is used to transform coordinates between two inertial observers (or two inertial frames). For example, in the scenario above, all the values given are according to A. The Neopolitonian Transform gives a value of x' and t' according to B (x'_b and t'_b) - which make B's frame the primed frame. This can also be called "A looking at B" since the values of x'_b and t'_b are given in terms of x_a and t_a, ie "what do B's values look like in terms of A's values?"

x'_b = \gamma.(x_a - v.t_a)

t'_b = \gamma.(t_a - v.x_a/c²)

Why do you call this the "Neopolitan transform"? If it really holds for events at arbitrary coordinates in the A frame, and gives you the corresponding coordinates of the same event in the B frame, then it obviously has the same physical meaning as the Lorentz transformation, unlike your equation x'_a = x_a - v.t_a which did not have the same physical meaning as the Galilei transformation equation x' = x - vt since you were not relating the coordinates of two different frames. I'm trying to get you to communicate your ideas in a non-confusing way which avoids hours of back-and-forth posts clarifying your meaning which could have been easily avoided if you had stated things more clearly at that outset--in order to do this, you need to think carefully about the physical meaning of the equations you write in relation to the physical meaning of standard textbook equations, and if you realize your equation has a different physical meaning in spite of superficial similarities to an existing equation then don't write it in exactly the same form without pointing out the difference, but likewise if your equations do have the same physical meaning as existing equations then you should be clear on this and not talk as though the equation is original to you just because you've altered the notation slightly. This issue of your not thinking through the physical meaning of equations is one that seems to come up again and again in our discussions--it arose in our discussions of the difference between the time dilation equation and the "temporal analogue for length contraction" too--so if you don't really follow the point I'm making (as suggested by your indiscriminately renaming equations above regardless of whether they do or don't have the same physical meaning as existing equations) then I'd like to try to focus on this until you understand.

But as a sort of aside to this, although the words you use to describe the equations above indicate they have the same physical meaning as the Lorentz transformation, I am in fact skeptical that your actual derivation would in fact prove something as general as the words suggest, unless you have totally changed the proof from what you offered before. I expressed my view of the physical meaning of the equations you had derived back in post 247--as far as I could tell, your derivation only proved that x'_a would equal gamma*(x_b + vt_b) in the specific case where x'_a was defined as the interval between E_a and the light passing B, while x_b and t_b were the intervals between a different pair of events, namely the event of E_b and the event of the light passing A. As I explained in those diagrams, by exploiting the symmetry of your scenario we can see that all the intervals between the first pair of events are identical to the intervals between the second pair, explaining why your equation comes out looking just like the Lorentz transformation even though it's not dealing with a single pair of events like the Lorentz transformation, but it seemed to me we only know about this symmetry because we already know how the two frames are related by the Lorentz transformation. In any case, even if we "allow" this symmetry to be exploited in the proof, the scope of the proof would still be limited to showing that the separation between events on the path of a light beam obey an equation like x'_a = gamma*(x_b + vt_b), the proof you presented simply wouldn't tell cover the case of pairs of events with a timelike or spacelike separation (though we know from other more general proofs that the equation would be the same). Finally, there was a step midway through the proof that didn't seem justified to me, where you said that we could assume the factor in x'_A = (a factor times).x'_B was the same as the factor in x_B = (a factor times).x_A--I explained my objection to this at the very bottom of post 280, for example. Basically, unless you have made really large changes to the proof you already presented, it's unlikely that these criticisms will change.

Plus, are you absolutely certain that no-one is going to accuse me of just rebadging the Galilean boost and the Lorentz Transform, because it is sooooo obvious that they are not the same as the Neopolitonian boost and the Neopolitonian Transform?

I am certain that if you stated the physical meanings of the equations explicitly in the way you did above, anyone knowledgeable about physics could be convinced that the first equation does not have the same physical meaning as the Galilean boost (because it's an intrinsic part of the definition of the 'Galilean boost' x' = x - vt that it relates one frame to another, whereas you defined your equation to only involve the coordinates of a single frame), while the second set of equations does have the same physical meaning as the Lorentz transformation (as you defined the meaning of the second set of equations in words above, not saying anything about whether your derivation would actually prove those words). If there is any doubt in your mind about either of these points I suggest we focus on the issue of the physical meaning of the Galilean boost x'=x-vt and the Lorentz transformation equations and how it relates to the equations you wrote, and leave other issues for later.

 

[neopolitan]

I disagree with your analogy, but I understand your frustration.

Can we agree that it would be silly for me to claim that I have found a new equation which looks pretty much like the Lorentz Transform, and claim it as my own?

That leaves the method for getting there and the boost.

Did you see my PS in the previous post?

It was there specifically because I wanted you to remember that fact and know that I remember it.

In Galilean relativity, the (spatial) interval from B to an event is the coordinate of the event in B's rest frame.

Do we disagree about that?

If we can agree on that, then I think I could call x' = x - vt a (spatial) boost, and I think I would not get away with calling x'_a = x - vt_a the Neopolitonian (not neopolitan) boost. It would swiftly be recognised as a slight rewording of the Galilean (spatial) boost.

The thing that would distinguish it, possibly is the implication that if it is just a slight rewording of the Galilean boost ie

x'_a = x - vt_a

then

t'_a = t_a - NOTE, I am not saying this, this is the second part of an if-then statement

It is something that I go on to disprove. But until this second part is disproved, I would consider it to be the Galilean boost.

cheers,

neopolitan

 

[JesseM]

In Galilean relativity, the (spatial) interval from B to an event is the coordinate of the event in B's rest frame.

Do we disagree about that?

They happen to be equal, but an equation telling you about the spatial interval has a different physical meaning than an equation telling you about the coordinate in B's rest frame. In a given physics equation every symbol must have a single well-defined meaning.

If we can agree on that, then I think I could call x' = x - vt a (spatial) boost

Not if you have defined x' to mean the spatial interval in A's frame. You may have outside knowledge that the spatial interval in A's frame is equal to the coordinate in B's frame, but the equation itself, with the symbols defined in this way, doesn't tell you anything about B's frame.

and I think I would not get away with calling x'_a = x - vt_a the Neopolitonian (not neopolitan) boost. It would swiftly be recognised as a slight rewording of the Galilean (spatial) boost.

No, it wouldn't be a "boost" at all because it only deals with one frame. If we define x'_a as merely the coordinate separation between an object at position x and an object B moving towards x at coordinate speed v (which started at x=0 at t=0), all in A's coordinate system, do you agree that this is a purely kinematical equation which is independent of the laws of physics, or of what type of coordinate system you're using (inertial or non-inertial)?

Think of actually writing out the physical meaning of all the terms in any equations in words. One equation is:

(distance in A's frame between B and object at position x in A's frame at time t in A's frame) = (position x in A's frame) - (velocity of B in A's frame)*(time t in A's frame)

This equation, again, would work in absolutely any type of coordinate system whatsoever. The second equation is different in that it only works when using Galilean inertial frames, and it can be written as:

(position in B's frame of object at position x in A's frame at time t in A's frame) = (position x in A's frame) - (velocity of B in A's frame)*(time t in A's frame)

Of course, we also happen to know that in Galilean relativity the following is true:

(distance in A's frame between B and object at position x in A's frame at time t in A's frame) = (position in B's frame of object at position x in A's frame at time t in A's frame)

But this is knowledge external to the first two equations themselves. Think of physics equations as very stupid things that can only give you an answer to one specific type of question, and they have no knowledge of any larger context. The only way two equations can be considered "the same" is if they are answering exactly the same physical question, and only the notation is different.

It may also help to point out that someone could easily have "discovered" the first equation before the discovery of the Galilei transformation, since the first equation doesn't involve multiple coordinate systems. Do you think it would be fair for this person to demand that the Galilei transformation be renamed after themselves, since they had already discovered the equation even though the physical meaning of what was being calculated was different and they hadn't even been thinking about the relation between multiple frames?

 

[neopolitan]

While I applaud your dedication to rigour, I think you take it too far.

Just out of curiosity, I look http://www.fourmilab.ch/etexts/einstein/specrel/www/" and searched for the word "frame". It appears exactly once, in the introduction, in the phrase "frames of reference" in the context of describing the first postulate.

While trying not to go so far as committing an "appeal to authority", I do want to know why I am being held to such high standards of rigour (specifying that the Galilean boost addresses a single question about frames) when the genius who came up with Special Relativity in his own way didn't really mention frames at all?

I'm not saying you are wrong that the Galilean boost is about frames, and the Lorentz Transformation is about frames, I am just wondering if your demands are truly warranted.

They happen to be equal, but an equation telling you about the spatial interval has a different physical meaning than an equation telling you about the coordinate in B's rest frame. In a given physics equation every symbol must have a single well-defined meaning.

Would you be happy if there was an extra step added in which I address Galilean frames, say the equation is x' = x - vt and that there is also a kinematic equation x' = x - vt and while they talk about different things, the relationship x' = x - vt holds equally for whatever values of x and t you enter into it? (Since the conditions under which the equation holds are the same for x' = x - vt and for x' = x - vt).

Not if you have defined x' to mean the spatial interval in A's frame. You may have outside knowledge that the spatial interval in A's frame is equal to the coordinate in B's frame, but the equation itself, with the symbols defined in this way, doesn't tell you anything about B's frame.

Addressed above.

No, it wouldn't be a "boost" at all because it only deals with one frame. If we define x'_a as merely the coordinate separation between an object at position x and an object B moving towards x at coordinate speed v (which started at x=0 at t=0), all in A's coordinate system, do you agree that this is a purely kinematical equation which is independent of the laws of physics, or of what type of coordinate system you're using (inertial or non-inertial)?

Addressed above.

It may also help to point out that someone could easily have "discovered" the first equation before the discovery of the Galilei transformation, since the first equation doesn't involve multiple coordinate systems. Do you think it would be fair for this person to demand that the Galilei transformation be renamed after themselves, since they had already discovered the equation even though the physical meaning of what was being calculated was different and they hadn't even been thinking about the relation between multiple frames?

No, and my suggestion to rename equations was entirely facetious.

However, since we know that the interval between B and an event is the same in both frames, and that interval is the coordinate in the B frame, then I fail to see why I can't use the equation the way I do.

I do wonder if you have the visual ability to see that what I am doing is not really invalid.

I point you back to the idea that we can shift the origins of our axes (within the relevant frame, of course) for convenience. We do that anyway, by making the interval between an observer and an event parallel to the x-axis. While it is entirely sensible to place the origin of the x-axis where our reference point is (nominally an observer), the point is that this is an arbitrary decision - arbitrary but sensible.

In short, are you happy with:

Introduce Galilean frames (hopefully already done by the education system)
Introduce a kinematic equation in the form x' = x - vt (partially done)
Point out that both equations operate on the same conditions
Go from there into the derivation of Lorentz equations

cheers,

neopolitan

PS Am I going to have to draw another diagram? I've already been thinking of the best way to try to show you that what I am doing is not as whacky as you seem to think it is. Part of the problem might be that I am an engineer, manipulating equations is partly what I do. As someone with more of a physics bent, you don't seem to like the actual use of equations (or what you might term "abuse of equations" )

 

[sylas]

While I applaud your dedication to rigour, I think you take it too far.

With respect, I don't think so. I think you need to be more rigorous and careful in your definitions. This is not an excessive hurdle, and it's getting better; but I think you need to keep being as rigourous as you can manage. The comparison with Einstein is invalid. Being rigourous does not mean "mentioning frames". It means being unambiguous and precise in whatever terminology or diagrams you use. It's not that you need full derivations and proofs of everything; just less ambiguity.

No offense intended... but I've been watching with some interest and I think the major problem is a lack of precision and rigour, and it should not be that hard to fix.

Cheers -- sylas

 

[neopolitan]

With respect, I don't think so. I think you need to be more rigorous and careful in your definitions. This is not an excessive hurdle, and it's getting better; but I think you need to keep being as rigourous as you can manage. The comparison with Einstein is invalid. Being rigourous does not mean "mentioning frames". It means being unambiguous and precise in whatever terminology or diagrams you use. It's not that you need full derivations and proofs of everything; just less ambiguity.

No offense intended... but I've been watching with some interest and I think the major problem is a lack of precision and rigour, and it should not be that hard to fix.

Cheers -- sylas

Thanks, it helps to get another perspective.

 

[JesseM]

While I applaud your dedication to rigour, I think you take it too far.

Just out of curiosity, I look http://www.fourmilab.ch/etexts/einstein/specrel/www/" and searched for the word "frame". It appears exactly once, in the introduction, in the phrase "frames of reference" in the context of describing the first postulate.

The entire paper is about what we now call frames, Einstein just doesn't use that term. When he introduces a "a system of co-ordinates in which the equations of Newtonian mechanics hold good" at the beginning of section 1, what do you think that is if not an inertial frame? And he talks about different systems of coordinates throughout the paper, sometimes just using the word "system" (it's clear he means coordinate system and not some other type of physical system from the context)--for example, part 3, where he actually derives the Lorentz transformation, is titled "Theory of the Transformation of Co-ordinates and Times from a Stationary System to another System in Uniform Motion of Translation Relatively to the Former".

While trying not to go so far as committing an "appeal to authority", I do want to know why I am being held to such high standards of rigour (specifying that the Galilean boost addresses a single question about frames) when the genius who came up with Special Relativity in his own way didn't really mention frames at all?

Again, the whole paper is about frames. The precise word is irrelevant as long as the concept is understood; I'd be equally happy with saying the Galilean boost is about relating the coordinates of an event in one "system of coordinates" to the coordinates of the same event in another "system". Whatever wording you use, this is conceptually quite different from just telling you how the coordinate separation between two objects is changing in a single coordinate system.

I'm not saying you are wrong that the Galilean boost is about frames, and the Lorentz Transformation is about frames, I am just wondering if your demands are truly warranted.

Just the fact that I was confused for so long by the meaning of the equation x' = x - vt in post 295 shows that they are warranted; I'd rather not get into more lengthy discussions over such trivial stuff in the future. Even if you incorrectly described the equation as the Galilean boost, the problem could have been avoided if you had spelled out in words what each symbol meant physically; if you had said at the outset that x' was supposed to represent a separation in the same frame that x and t referred to, then I might have offered a quick correction about terminology but there wouldn't have been all the confusion about what you were trying to demonstrate with your equations. But the combination of not giving physical definitions of your symbols at the outset, using the term "Galilean boost", and writing your equation using exactly the same notation as is usually used for the Galilean boost naturally led me to draw the wrong conclusions about the physical meaning of the equation. Hopefully you agree that, spelled out in words, this:

(position in B's frame of object at position x in A's frame at time t in A's frame) = (position x in A's frame) - (velocity of B in A's frame)*(time t in A's frame)

...is telling us something physically from this?

(distance in A's frame between B and object at position x in A's frame at time t in A's frame) = (position x in A's frame) - (velocity of B in A's frame)*(time t in A's frame)

Would you be happy if there was an extra step added in which I address Galilean frames, say the equation is x' = x - vt and that there is also a kinematic equation x' = x - vt and while they talk about different things, the relationship x' = x - vt holds equally for whatever values of x and t you enter into it? (Since the conditions under which the equation holds are the same for x' = x - vt and for x' = x - vt).

As long as you define the physical meaning of whatever equations you use I'll be OK, although from a pedagogical point of view I don't really like the approach of using identical notation for two physically different equations. In any case, is it necessary to discuss Galilean relativity at all in your derivation? Isn't the kinematical equation the only one you actually make use of?

It may also help to point out that someone could easily have "discovered" the first equation before the discovery of the Galilei transformation, since the first equation doesn't involve multiple coordinate systems. Do you think it would be fair for this person to demand that the Galilei transformation be renamed after themselves, since they had already discovered the equation even though the physical meaning of what was being calculated was different and they hadn't even been thinking about the relation between multiple frames?

No, and my suggestion to rename equations was entirely facetious.

I understood it was meant to be facetious...but my point in the above comment was, if you agree this hypothetical pre-Galileo guy shouldn't get credit for the Galilei transformation despite writing down an equation like x' = x - vt, doesn't that mean you should also agree we shouldn't use the same terminology for his kinematical equation that we do for the spatial component of the Galilei transformation, even if they look the same symbolically?

I do wonder if you have the visual ability to see that what I am doing is not really invalid.

It has nothing to do with visual abilities, I get visually why it works out that the separation in A's frame between B and the object at position x is always going to be equal to the position coordinate assigned to that object in B's frame. The point is that the equations are telling you different things physically, and that since I naturally thought you were introducing x'=x-vt in post 295 to transform into B's frame, I was confused since under the Galilei transformation the light could not be moving at c in B's frame.

In short, are you happy with:

Introduce Galilean frames (hopefully already done by the education system)
Introduce a kinematic equation in the form x' = x - vt (partially done)

As I said I don't like using the same notation for two equations with different physical meanings, and I think from a pedagogical point of view it's more confusing than helpful.

Point out that both equations operate on the same conditions

By "operate on the same conditions", I take it you mean if we pick a given x,t in A's frame, we get the same value for the answer? That's fine as long as you point out the physical meaning of the "answer" is different.

PS Am I going to have to draw another diagram? I've already been thinking of the best way to try to show you that what I am doing is not as whacky as you seem to think it is. Part of the problem might be that I am an engineer, manipulating equations is partly what I do. As someone with more of a physics bent, you don't seem to like the actual use of equations (or what you might term "abuse of equations" )

Now that I understand the physical meaning of your symbols I don't object to "what you are doing" in the derivation so far, only to how you are explaining it. And I have no dislike of equations (a scurrilous charge for a student of physics, my good sir! ), I just need to be clear on the physical meaning of any variables/constants that appear in them.

 

[neopolitan]

I am happy to use the kinematic equation from the start. Historically, how long ago was that equation (or at least that relationship) first identified? (Part of my interest is to see how long ago we could have got to SR. If the kinematic equation is all that is required, rather than Galilean relativity, that might actually push the possible date back to before the 1100's given an early Islamic mathematician's work. "In dynamics and kinematics, Biruni was the first to realize that acceleration is connected with non-uniform motion, which is part of Newton's second law of motion." - http://en.wikipedia.org/wiki/Al-Biruni" Sadly, the coverage of this fellow's work is less visible to me than that for da Vinci, Galileo and Newton, but it seems to me that if Al-Biruni got so far as to consider non-uniform motion then uniform motion was probably understood. Al-Biruni's contribution to optics as claimed in the same article is interesting as well, apparently being among the first to consider the speed of light to be finite (but faster than sound) - it makes one wonder why someone like this has been pretty much invisible. There is what seems to be an inconsistency in that article, did people prior to Al-Biruni think that the speed of sound was infinite along with the speed of light? {Since Al-Biruni is credited with not only being among to consider the speed of light to be finite, but also the first to find that the speed of light is much faster than the speed of sound. I would have thought that infinitely fast is much faster than the speed of sound.} But this is merely an aside.)

Amusingly, I did try something like your:

(position in B's frame of object at position x in A's frame at time t in A's frame) = (position x in A's frame) - (velocity of B in A's frame)*(time t in A's frame)

but with subscripts. I found it too unwieldy and distracting. So, I do see the value in it.

Anyway, since with the riders, you don't object with my derivation so far, perhaps I should try to sum up where we are right now, before we go further. If I do that, being as rigorous as I can, are you happy to work from that point onwards?

cheers,

neopolitan

 

[JesseM]

I am happy to use the kinematic equation from the start. Historically, how long ago was that equation (or at least that relationship) first identified?

Not sure, I guess as soon as people came up with the notion of measuring how distances between things change with time (which would require at least somewhat fine-grained clocks), along with the concept of speed as distance traveled/time elapsed, they could have realized that the distance between a thing moving at constant speed v and a stationary thing would be shrinking at v times the time elapsed. Maybe this would come up in seafaring or something, even if it wasn't written as an algebraic equation. On the other hand, to write an equation like x - vt for a distance between a moving object and a stationary one, you need some notion of assigning objects position coordinates on a coordinate grid (or at least a coordinate line), and of choosing your origin so the moving object starts at position x=0 at time t=0, don't know if people would have thought in those terms until Descartes invented Cartesian geometry (incidentally, Galileo was about thirty years older than Descartes so I'd guess he never actually wrote the 'Galilei transformation' in algebraic terms, even if it's implicit in his work that he was saying the laws of physics would be invariant under this transformation).

Amusingly, I did try something like your:

(position in B's frame of object at position x in A's frame at time t in A's frame) = (position x in A's frame) - (velocity of B in A's frame)*(time t in A's frame)

but with subscripts. I found it too unwieldy and distracting. So, I do see the value in it.

Yeah, it would be unwieldy to include the full description in the symbol itself, but it would be helpful if each time a new symbol is introduced, you could say something like "define x' as the position in B's frame of the event that had coordinates x,t in A's frame", something along those lines.

Anyway, since with the riders, you don't object with my derivation so far, perhaps I should try to sum up where we are right now, before we go further. If I do that, being as rigorous as I can, are you happy to work from that point onwards?

Sure, as long as being rigorous means defining the physical meaning of any new terms you introduce.

 

[neopolitan]

Can we use this notation?

x'_a = the separation (hence the x) between B and an Event (hence the prime) according to A (hence the _a)

This perhaps should be expanded a little to specify clearly that unprimed means "between A and an Event". And capitalisation of Event is used to clarify that I am referring to a specific event, not just any event.

So you have two frames (A -> Event and B -> Event), as many perspectives as you like (_a, _b, _c, _d - but we will only use two) and we will have two dimensions (x and t).

I might also need to clarify something that I have firmly in mind about my t values.

When we draw a spacetime diagram, we can draw an interval between, say, A and an Event in A's future. But in the real world (another engineering trait coming out perhaps), A will not know about that Event until enough time has elapsed for the Event to take place (in the A frame) and for a photon to travel from the Event to A.

For that reason, I see utility in moving the origin of the t_a axis to be simultaneous with the Event in the A frame. If I do that then, as a consequence, x_a = c.t_a.

Notionally, A and B are colocated at (0,0) - now this does not have to physically take place because we calculate intervals so any initial offset will cancel out.

Are you happy with this? Clarifying what I am asking:

1. Are you happy with the notation regime suggested?
2. Are you happy with the concept that A and B are notionally colocated at (0,0), but they don't actually have to be colocated at that time or place?

cheers,

neopolitan

 

[JesseM]

Can we use this notation?

If x'_a is meant to be a variable, as opposed to the separation between B and the Event at some specific time, can we write it as x'_a(t)?

I might also need to clarify something that I have firmly in mind about my t values.

When we draw a spacetime diagram, we can draw an interval between, say, A and an Event in A's future. But in the real world (another engineering trait coming out perhaps), A will not know about that Event until enough time has elapsed for the Event to take place (in the A frame) and for a photon to travel from the Event to A.

For that reason, I see utility in moving the origin of the t_a axis to be simultaneous with the Event in the A frame. If I do that then, as a consequence, x_a = c.t_a.

In other recent posts (such as 308) you used t_a to refer to a specific time interval (between A and B being colocated and the photon passing A) in A's frame, so it's potentially confusing to refer to "the t_a axis"--I assume you just mean the t-axis in A's frame? Whereas in the equation x_a = c.t_a, does t_a still refer to that time interval I mentioned? And when you talk about moving the origin to be "simultaneous with the Event in the A frame", do you just mean the origin has the same time coordinate as the Event (i.e. the t-coordinate of the Event in A's frame is 0), not that the Event is actually at the origin? So it's still true that the event has coordinates x=x_a, t=0 in A's frame, and it's still true that the photon reaches A at x=0, t=t_a? If so I don't really understand why you talk about "moving the origin", since this is exactly how things were before.

Notionally, A and B are colocated at (0,0) - now this does not have to physically take place because we calculate intervals so any initial offset will cancel out.

Do the origins of their coordinate systems still coincide at a time coordinate of 0 in both systems? If so, of course it is not necessary for observers at rest in these coordinate systems to be located at x=0 in each system, they can be at any position coordinate we like. But in this case I'd like a redefinition of t_a--does it refer to the time in A's frame that the photon passes x=0, or the time it passes A, or something else?

1. Are you happy with the notation regime suggested?
2. Are you happy with the concept that A and B are notionally colocated at (0,0), but they don't actually have to be colocated at that time or place?

See above, I'm not sure I understand what you're saying here.

 

[neopolitan]

I use t_a as a time interval in the A frame between an Event, and the reception of the photon coincident with that Event (or spawned by that Event).

I started writing the summary and lost it all. Very annoying.

What I mean about moving the origin of the axes is that the Event can take place whenever. But despite that, we shift the origin of the axes so that the Event is simultaneous in the A frame with (0,0).

In other words, we can work backwards. We get a photon today and discover from other reliable sources that the photon was released a distance of 10 light years away (in our frame), so we shift the origin of our t axis to back when the photon was released (in our frame) making today t = 10 years. Equally, we can be told that in three years from now, a photon will be released from the same location. We can shift the origin of our t axis forward 3 years from today, making today t = minus 3 years (knowing that the photon won't reach us until t = 10 years.

So I can shift the origin of the t-axis backwards or forwards as I like, which means I can consider any event, at any time.

If either of A and B were to not be located at the origin of their frame of reference, I would make it B.

I'm a bit perplexed by the idea that x'_a wouldn't be variable with different values of t_a.

To the same extent that x' in the equation x' = x - vt is variable with t, so to is x'_a in the equation x'_a = x_a - vt_a variable with t_a.

Also, I do believe that in the standard Lorentz Transformation x' is variable as you vary t (and indeed x).

I would agree that I would have to write x'_a(t_a) = x_a - vt_a, if I routinely saw the Lorentz Transformation written as:

x'(x,t)= \gamma.(x - vt)
t'(x,t)= \gamma.(t - vx/c^2)

But I don't.

I feel as if you are demanding more than is justified. I can do as requested if I must though.

Can you confirm that I absolutely must specify that x'_a varies as t_a varies?

cheers,

neopolitan

 

[neopolitan]

Summarising where I think we are at (including corrections in an attempt to be more rigorous).

We start with the kinematic equation for an observer A observing B who has a velocity of v towards a location which has a separation of x from A:

x' = x - vt

or

x'(t) = x - vt

which is, in words:

(separation between B and position x, in A's frame, at time t in A's frame) = (position x in A's frame) - (velocity of B in A's frame)*(time t in A's frame)

We then introduce the concept that if a photon is released from a distance of x_a away from A and it takes a period of t_a to reach A, then:

x_a = ct_a

which is, in words:

(location of the Event, in A's frame) = (speed of light)*(time interval between the Event and when a photon from the Event reaches A, in A's frame)

Using these values in the kinematic equation we have:

x'_a = x_a - vt_a

or

x'_a(t_a) = x_a - vt_a

which is, in words:

(separation between B and the location of the Event, in A's frame, when a photon from the Event reaches A, in A's frame) = (location of the Event, in A's frame) - (velocity of B in A's frame)*(time interval between the Event and when a photon from the Event reaches A, in A's frame)

Then, we want to know how things look in B's rest frame.

x'_b = ct'_b and
x_b = x'_b + vt'_b {or x_b(t'_b)}

which are, in words:

(location of Event, in B's frame) = (speed of light)*(time interval between the Event and reception of photon from the Event, in B's frame) and

(separation between B and the location of the Event, in B's frame, when a photon from the Event reaches B, in B's frame) = (location of the Event, in B's frame) + (velocity of A in B's frame)*(time interval between the Event and when a photon from the Event reaches B, in B's frame)

Since

t_a = time interval between the Event and when a photon from the Event reaches A, in A's frame

and

t'_b = time interval between the Event and when a photon from the Event reaches B, in B's frame

we have no expectation that t_a = t'_b

Happy with that?

cheers,

neopolitan

 

[JesseM]

I use t_a as a time interval in the A frame between an Event, and the reception of the photon coincident with that Event (or spawned by that Event).

Again, are you assuming A is at position x=0 in A's own rest frame? If not, when you say "reception of the photon" do you mean when the photon crosses A's worldline, or when the photon crosses the x=0 axis?

What I mean about moving the origin of the axes is that the Event can take place whenever. But despite that, we shift the origin of the axes so that the Event is simultaneous in the A frame with (0,0).

"Take place whenever" relative to what coordinate system? If you're using a coordinate system where it takes place at t=0, then it doesn't take place whenever, and if that's the starting point of your proof then everything else in the proof follows from that assumption and whatever conclusions you reach cannot simply be assumed to still hold if the Event is located somewhere else besides t=0 (if that's what you're getting at, I'm not sure). If it helps, suppose you end up proving that if in A's frame the spatial and temporal intervals between the Event at t=0 and some second event are x and t, then the spatial interval between these same pair of events in B's frame is gamma*(x - vt) and the temporal interval is gamma*(t - vx/c^2). In that case, even though your proof started from the assumption that the Event occurred at t=0 in A's frame, it would be easy to prove a lemma of the type I talked about back in post 249:

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).

So with this lemma added, you could then show that the relation between the intervals in A's frame and the intervals in B's frame will be the same even if you move the origins so that the Event is at some totally arbitrary set of coordinates. This lemma could be added to the very end of the proof. However, this will still not necessarily mean your proof is fully general; if in your proof you assume that the first Event and the second event (which together define the intervals you're dealing with in each frame) both lie along the path of a light ray, then you haven't proved that the same relation would hold for a pair of events with a timelike or spacelike separation.

In other words, we can work backwards. We get a photon today and discover from other reliable sources that the photon was released a distance of 10 light years away (in our frame), so we shift the origin of our t axis to back when the photon was released (in our frame) making today t = 10 years.

Why assume the origin was somewhere else to begin with? You don't even have to pick the placement of your axes until you've already received the photon, and at that point it's easy to position them so that the Event 10 light years away occurred at t=0, if that's all you're worried about. When dealing with SR problems you don't really have to concern yourself with these sorts of practical issues, just assume either an omniscient perspective on spacetime, or assume all coordinates are assigned indefinitely far into the future when all the events in the region of spacetime you're interested in are already known.

If either of A and B were to not be located at the origin of their frame of reference, I would make it B.

If B is not at the origin of its own frame, then does that mean B is not necessarily colocated with A at t=0 in A's frame? If it's not, then don't you have to modify the equation

I'm a bit perplexed by the idea that x'_a wouldn't be variable with different values of t_a.

t_a is a constant in any given physical scenario, is it not? It's the time coordinate of when the photon passes A, right? So if x'_a represents the distance between B and the position x_a as a function of time, this distance is varying with the time coordinate t in A's frame, not varying with t_a (unless you are using t_a to represent A's time variable as well as the specific time the photon passes A, something I requested you not do in my last post because it'd be confusing). On the other hand, if you just want to define x'_a as the distance between B and x_a at the specific time t_a when the photon passes A (or alternatively, at the specific time t'_a when the photon passes B), that's fine with me, in this case x'_a would be a constant rather than a variable. But you seemed to want x'_a to represent a distance that could vary with time rather than a distance at a specific time in post 306 when you said:

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

To the same extent that x' in the equation x' = x - vt is variable with t, so to is x'_a in the equation x'_a = x_a - vt_a variable with t_a.

Again, in any specific physical setup isn't t_a a constant? Of course you can vary the physical setup itself, but that's not what I meant when I said I thought you were making x'_a a variable--I thought that even given a particular setup (a particular choice of position x_a for the Event on the photon's worldline), x'_a represented the changing distance between B and x_a as a function of time in A's coordinate system, not the distance between B and x_a at some specific time like t_a (I based this on your comment from post 306 above).

Also, I do believe that in the standard Lorentz Transformation x' is variable as you vary t (and indeed x).

If you can vary x and t, sure, but if you pick some specific physical event then x', x, and t for that choice of event are all constants, echoing my comment about t_a and x_a being constants for a particular choice of physical setup in your scenario.

Can you confirm that I absolutely must specify that x'_a varies as t_a varies?

See above for a clarification of my meaning. If you want x'_a to be the distance between B and x_a at a specific time corresponding to some event in your setup like the photon passing A at t_a or the photon passing B at t'_a (and remember that you had actually defined x'_a in the latter way in our earlier discussions, not the former), then there's no need to call it a variable. But your comment in post 306 seemed to insist that x'_a is defined in such a way that it changes with time rather than being the distance between B and x_a at some specific time.

 

[neopolitan]

Again, are you assuming A is at position x=0 in A's own rest frame?

Yes


As for the rest, I'm a bit confused why something that seems so obvious to me is confusing for you.

I'll try again.

In A's rest frame, A is at rest.

In A's rest frame, B is not at rest.

In A's rest frame, the separation between where the Event takes, took or will take place and A is constant.

In A's rest frame, the separation between where the Event takes, took or will take place and B is not constant.

x_a is the separation between where the Event takes, took or will take place and A in A's rest frame.

x'_a is the separation between where the Event takes, took or will take place and B in A's rest frame.

Therefore, x_a is constant and x'_a is not constant.

It seems so simple to me, I can't quite grasp why it warrants such a long post about it.


For a specific time in A's frame, x'_a is defined, not varying, but not quite a constant either (because to me a constant is only a constant if you can vary something and once you pick a specific time, you don't have anything to vary in A's frame anymore, so long as you continue to talk about the same Event).

cheers,

neopolitan

 

[JesseM]

We start with the kinematic equation for an observer A observing B who has a velocity of v towards a location which has a separation of x from A:

x' = x - vt

or

x'(t) = x - vt

which is, in words:

(separation between B and position x, in A's frame, at time t in A's frame) = (position x in A's frame) - (velocity of B in A's frame)*(time t in A's frame)

We then introduce the concept that if a photon is released from a distance of xa away from A and it takes a period of ta to reach A, then:

x_a = ct_a

which is, in words:

(location of the Event, in A's frame) = (speed of light)*(time interval between the Event and when a photon from the Event reaches A, in A's frame)

Using these values in the kinematic equation we have:

x'_a = x_a - vt_a

You should also add the assumption that the Event occurs at time coordinate t=0 in A's frame, and that A is located at x=0, since otherwise this substitution wouldn't work.

(separation between B and the location of the Event, in B's frame, when a photon from the Event reaches B, in B's frame) = (location of the Event, in B's frame) + (velocity of A in B's frame)*(time interval between the Event and when a photon from the Event reaches B, in B's frame)

I assume you meant to write separation between A and the location of the Event in B's frame, right? After all, the separation between B and any given location in B's frame will be constant. Assuming that's what you meant, then it seems to me the equation would only hold if we assume that A and B are colocated at the same time as the Event occurs in B's frame. But that's obviously problematic, because we already assumed the Event was simultaneous with A and B being colocated in A's frame, and as we know they can't both be true in relativity.

 

[JesseM]

I'll try again.

In A's rest frame, A is at rest.

In A's rest frame, B is not at rest.

In A's rest frame, the separation between where the Event takes, took or will take place and A is constant.

In A's rest frame, the separation between where the Event takes, took or will take place and B is not constant.

x_a is the separation between where the Event takes, took or will take place and A in A's rest frame.

x'_a is the separation between where the Event takes, took or will take place and B in A's rest frame.

Therefore, x_a is constant and x'_a is not constant.

Yes, that's exactly what I originally thought you meant, until you made the confusing comment that x'_a varies with t_a, rather than saying it varies with t, which is something I was asking about (I know you don't like my habit of responding to your posts line-by-line, but your habit of responding to my posts in a 'gestalt' manner often means you don't answer the specific questions I ask about, and instead just repeat things I already understand without answering the questions I specifically asked for the purposes of clarifying). Again, I thought t_a was a constant (given a particular physical setup) just like x_a--t_a represents the time the photon passes A, and for a particular physical setup there's only one unique time that this happens. After all, you wrote x_a = c*t_a--if you call x_a a constant, then based on this equation you must call t_a a constant too. It's possible you are using the symbol t_a to represent both the abstract time variable in A's frame and the specific time coordinate when the photon passes A, but I already speculated in two previous posts that you might be doing this and asked you to please not use that sort of ambiguous notation if that's what's going on.

Then you also made the point that you shouldn't have to spell out that x'_a varies with time when x' is not written as x'(x,t) in the Lorentz transformation. I pointed out that given a particular choice of physical event, x and t are not variables, analogous to how given a particular choice of physical setup (distance between the Event on the path of the light ray and A at time t=0 in A's frame), x_a and t_a are not variables in your equations. Do you understand this point, and if so do you drop this particular argument for why it's unreasonable for me to ask you to write x'_a(t) in your equations?

For a specific time in A's frame, x'_a is defined, not varying

Sure, for any variable with a function of t, if you pick a specific time t it doesn't vary at that time! This is totally trivial, but it hardly proves that a quantity that varies with t is not a "variable". The point is, in a given physics problem with a well-defined physical setup, I'd call a symbol a "variable" if its value can change depending on the value of some other symbol, and a "constant" its value depends only on the setup. Is it my request that you differentiate between the two in your notation really so onerous? Just to help add some context, one of the reasons I make this request is that I anticipate that later in the proof you're probably going to want to talk about the value of x'_a(t) at a different time other than t_a, perhaps at the time t'_a in A's frame when the photon passes A; at that point it really could become genuinely confusing if you use exactly the same symbol, a problem that will be avoided if you write x'_a(t_a) for the first and x'_a(t'_a) for the second. And if I'm wrong, and your proof will never make use of x'_a at any time other than t_a, in that case making a big deal out of the fact that x'_a is a variable seems pointless, it would be much simpler just to define x'_a as the distance between B and the position of the event at the specific time t_a when the photon passes A.

 

[neopolitan]

Fixing (corrections and clarifications are highlighted - I accept that wherever I need to make a clarification this is as bad as being wrong, I don't intend to defend being wrong - or being so unclear as to necessitate a clarification);

We start with the kinematic equation for an observer A observing B who has a velocity of v towards a location which has a separation of x from A:

x' = x - vt

or

x'(t) = x - vt

which is, in words:

(separation between B and position x, in A's frame, at time t in A's frame) = (position x in A's frame) - (velocity of B in A's frame)*(time t in A's frame)

We then introduce the concept that if at t=0 a photon is released from a distance of x_a away from A and it takes a period of t_a to reach A and A is located at x=0, then:

x_a = ct_a

which is, in words:

(location of the Event, in A's frame) = (speed of light)*(time interval between colocation of A and B and when a photon from the Event reaches A, in A's frame)

Using these values in the kinematic equation we have:

x'_a = x_a - vt_a

or

x'_a(t_a) = x_a - vt_a

which is, in words:

(separation between B and the location of the Event, in A's frame, when a photon from the Event reaches A, in A's frame) = (location of the Event, in A's frame) - (velocity of B in A's frame)*(time interval between colocation of A and B and when a photon from the Event reaches A, in A's frame)

Then, we want to know how things look in B's rest frame. If at t'=0 the photon released as described before is at a distance of x'_b away from B and it takes a period of t'_b to reach B and B is located at x'=0, then:

x'_b = ct'_b and
x_b = x'_b + vt'_b {or x_b(t'_b)}

which are, in words:

(location of Event, in B's frame) = (speed of light)*(time interval between colocation of A and B and reception of photon from the Event, in B's frame) and

(separation between A and the location of the Event, in B's frame, when a photon from the Event reaches B, in B's frame) = (location of the Event, in B's frame) + (velocity of A in B's frame)*(time interval between colocation of A and B and when a photon from the Event reaches B, in B's frame)

Since

t_a = time interval between colocation of A and B and when a photon from the Event reaches A, in A's frame

and

t'_b = time interval between colocation of A and B and when a photon from the Event reaches B, in B's frame

we have no expectation that t_a = t'_b

I'm aware that the event simultaneous with t=0 and the event simultaneous with t'=0 are not simultaneous with each other. However, each event used is colocated with the relevant observer (notionally, if A and B are timing events, the only events they can give accurate time values to are "colocation of self with other observer" and "colocation of self with photon").

The point I have to make clear again is that we only know what happened at an event once information (or photon from the event) reach us. Then we work backwards.

If A receives a photon at t_a from an event at t=0, then when did that same photon pass B? What is t'_b in terms of t_a and x_a?

What is x'_b in terms of t_a and x_a?

Can we work it out?

I think we can.

 

[neopolitan]

Yes, that's exactly what I originally thought you meant, until you made the confusing comment that x'_a varies with t_a, rather than saying it varies with t, which is something I was asking about (I know you don't like my habit of responding to your posts line-by-line, but your habit of responding to my posts in a 'gestalt' manner often means you don't answer the specific questions I ask about, and instead just repeat things I already understand without answering the questions I specifically asked for the purposes of clarifying). Again, I thought t_a was a constant (given a particular physical setup) just like x_a--t_a represents the time the photon passes A, and for a particular physical setup there's only one unique time that this happens. After all, you wrote x_a = c*t_a--if you call x_a a constant, then based on this equation you must call t_a a constant too. It's possible you are using the symbol t_a to represent both the abstract time variable in A's frame and the specific time coordinate when the photon passes A, but I already speculated in two previous posts that you might be doing this and asked you to please not use that sort of ambiguous notation if that's what's going on.

Then you also made the point that you shouldn't have to spell out that x'_a varies with time when x' is not written as x'(x,t) in the Lorentz transformation. I pointed out that given a particular choice of physical event, x and t are not variables, analogous to how given a particular choice of physical setup (distance between the Event on the path of the light ray and A at time t=0 in A's frame), x_a and t_a are not variables in your equations. Do you understand this point, and if so do you drop this particular argument for why it's unreasonable for me to ask you to write x'_a(t) in your equations?

Sure, for any variable with a function of t, if you pick a specific time t it doesn't vary at that time! This is totally trivial, but it hardly proves that a quantity that varies with t is not a "variable". The point is, in a given physics problem with a well-defined physical setup, I'd call a symbol a "variable" if its value can change depending on the value of some other symbol, and a "constant" its value depends only on the setup. Is it my request that you differentiate between the two in your notation really so onerous? Just to help add some context, one of the reasons I make this request is that I anticipate that later in the proof you're probably going to want to talk about the value of x'_a(t) at a different time other than t_a, perhaps at the time t'_a in A's frame when the photon passes A; at that point it really could become genuinely confusing if you use exactly the same symbol, a problem that will be avoided if you write x'_a(t_a) for the first and x'_a(t'_a) for the second. And if I'm wrong, and your proof will never make use of x'_a at any time other than t_a, in that case making a big deal out of the fact that x'_a is a variable seems pointless, it would be much simpler just to define x'_a as the distance between B and the position of the event at the specific time t_a when the photon passes A.

I don't intend to use a value of x'_a that is different from its value at t_a.

I don't intend to use a value of x_b that is different from its value at t'_b.

I don't intend use a value of x_a other than such that x_a = c.t_a

I don't intend use a value of x'_b other than such that x'_b = c.t'_b.

Does that make things easier?

If I do find myself using anything other than these, I will try to mark them accordingly (but I really don't think that I will).

cheers,

neopolitan

 

[sylas]

I don't intend to use a value of x'_a that is different from its value at t_a.

What does this even mean?

Is x'_a a constant, or a variable? If you mean a constant, why the heck are you speaking of its value at certain time? If you mean a variable, why the heck would you only use a single value?

Isn't (x'_a, t'_a) just the co-ordinates of the event "photon passes A" in the frame of reference of B?

 

[neopolitan]

Isn't (x'_a, t'_a) just the co-ordinates of the event "photon passes A" in the frame of reference of B?

The post you quoted was in response to a specific concern from JesseM that I would go changing the meaning of x'_a. I don't intend to.

He got to that because I noted at one point that (in general) while in the A frame x_a does not change, x'_a does. But my (specific scenario driven) derivation centres around events which lock in values of x_a, t_a and hence x'_a. So in the scenario I describe, x_a, t_a and x'_a all have one value. So fixed are they that in earlier posts we assigned them numbers.

In answer to the question you posed here, no.

x'_a is the separation between B and where the photon was at t = 0 in the A frame (x = separation, ' = between B and the Event, _a = in the A frame or according to A).

cheers,

neopolitan

It may be worth mentioning that I am keeping the prime from the kinematic equation x' = x - vt

While I understand that this may cause concern because while I focus on observer A, and that that would make a few people consider that primes refer to B's rest frame, those people might be happier to know that the equations I end up with are:

x'_b = gamma.(x_a - vt_a)
t'_b = gamma.(t_a - vx_a/c²)

which is a confluence of the primed is the B frame, unprimed is the A frame and _b is the B frame, _a is the A frame.

 

[sylas]

The post you quoted was in response to a specific concern from JesseM that I would go changing the meaning of x'_a. I don't intend to.

He got to that because I noted at one point that (in general) while in the A frame x_a does not change, x'_a does. But my (specific scenario driven) derivation centres around events which lock in values of x_a, t_a and hence x'_a. So in the scenario I describe, x_a, t_a and x'_a all have one value. So fixed are they that in earlier posts we assigned them numbers.

In answer to the question you posed here, no.

x'_a is the separation between B and where the photon was at t = 0 in the A frame (x = separation, ' = between B and the Event, _a = in the A frame or according to A).

cheers,

neopolitan

Thanks... but I am still finding this incredibly hard to follow.

You've said "x'_a is the separation between B and where the photon was at t = 0 in the A frame". Distance WHEN? At t=0 also? t according to whom? You say it has one value. But then you've also said that x'_a can "change" in the A frame? How can that possibly be?

I have trouble following along when you speak of a "location". What is a fixed location in one frame is not a fixed location in another. I think it would be clearer if you stick to "events", so that you can sensibly speak of one event in several different frames.

You scenario is this, isn't it? It involves three particles: A, B and photon. A and B are moving at constant velocity v relative to each other. The events of interest occur in this order.

• A passes by B (co-located).
• Photon passes by B.
• Photon passes by A.

Is that right? You've also added another event of photon being "emitted".

The distance between events A and B (photon passing by A and photon passing by B) as observed by the particles A and B are related by the Doppler shift factor, are they not? Multiply, or divide the distance by
\sqrt{\frac{c-v}{c+v}}
to get the distance for the other observer. The distance is greater for the particle that the photon passes by first.

Cheers -- sylas

 

[neopolitan]

sylas,

I do appreciate your interest, but you might notice that a lot has come before this. If I reply to you, Jesse will reply to my replies to you (it's happened before) and we will end up going over old ground again which is something I am trying to avoid.

In general, x'_a(t) is variable with t. Specifically, x'_a(t) is fixed with a fixed value of t=t_a.

The scenario is framed such that x_a(0) = c.t_a, in other words a time interval of t_a after t=0 (in the A frame), a photon passes A since it was initially the right distance away to cover that distance in that time.

At the time at which the photon passes A, B has traveled a distance towards where the event took place and in A's frame that is:

x'_a(t_a) = x_a - v.t_a

(the separation between B and where the event took place at the time at which the photon from the event passes A, in the A frame) = (the separation between A and where the event took place) - (the distance that B has moved towards where the event took place in the time it took for a photon to travel from the event to A)

Note, I am not specifically writing this to explain to you, I am writing it in a format that JesseM has said is necessary for it to be explained.

Now, I suspect that the method for explaining to JesseM just possibly won't be as suitable for explaining to you. If that is the case, can I suggest that you go back into the earlier posts, find an explanation which seems to suit you more, and I can try to address your questions in a separate thread?

cheers,

neopolitan

 

[JesseM]

I don't intend to use a value of x'_a that is different from its value at t_a.

I don't intend to use a value of x_b that is different from its value at t'_b.

I don't intend use a value of x_a other than such that x_a = c.t_a

I don't intend use a value of x'_b other than such that x'_b = c.t'_b.

Does that make things easier?

In that case, why introduce the complication of saying x'_a is a variable but x_a is a constant? You could easily make x'_a a constant just by specifically defining it as the separation at time t_a when the photon passes A, not the separation at an arbitrary time t in A's frame.

Just as a reminder though, back when we were going through the numerical example where you put numbers to these values, you did define the symbol x'_a in terms of t'_a, the time the photon passed B (both x'_a and t'_a had the value 5).

 

[sylas]

Now, I suspect that the method for explaining to JesseM just possibly won't be as suitable for explaining to you. If that is the case, can I suggest that you go back into the earlier posts, find an explanation which seems to suit you more, and I can try to address your questions in a separate thread?

S'okay. I am perfectly comfortable with relativity and don't need it explained to me. I can see I am not helping here, and withdraw. Sorry for the distraction!

My main aim was to suggest, gently, that you are not doing a very good job of giving clear and unambiguous definitions of what you mean. I'm glad you didn't take offense at that; I wanted to say it without coming across as being too negative. But I still find it really hard to follow what you mean with notation or use of language, and I don't think this is just me, or because it is non-standard. The problem is that it is almost always ambiguous. You evidently have a clear idea what you mean. I don't.

It should be possible to express whatever it is you mean with less words and repetition. All you need is to avoid any potential ambiguity for what notation refers to; and then pretty much any of the regulars who have struggled to follow these threads will get it, IMO. It's not a problem of finding the "right" explanation for different people.

When explanations refer to the distance to a "location", rather than an event, there's a potential ambiguity as to what the location means in different frames and times. A "location" without an associated particle usually means a fixed distance co-ordinate, or worldline with zero velocity; but that depends on the observer and I often don't know what observer is intended. Referring to a specific event, however, is nearly always crystal clear.

The main answer here is that observers A and B measure different distances from the event "photon passes B" to the event "photon passes A". If the photon passes by B and then A, and if v is their relative velocity (+ve for moving apart), then the distance between these two events d_A for observer A is related to distance d_B for observer B by

d_A = d_B \sqrt{\frac{c-v}{c+v}}​

You can show this with the Lorentz transformations; and there may be other ways to get the right answer.

 

[JesseM]

Fixing (corrections and clarifications are highlighted - I accept that wherever I need to make a clarification this is as bad as being wrong, I don't intend to defend being wrong - or being so unclear as to necessitate a clarification);

OK, the clarified version looks clear to me.

I'm aware that the event simultaneous with t=0 and the event simultaneous with t'=0 are not simultaneous with each other. However, each event used is colocated with the relevant observer

Each event you use to define the time intervals, yes. The "Event(s)" on the photon's worldline used to define x_a and x'_b aren't colocated with the observers, of course.

If A receives a photon at t_a from an event at t=0, then when did that same photon pass B?

You mean, what time in A's frame did the photon pass B? This is a time you haven't defined a symbol for yet, although in the earlier discussion you defined this as t'_a.

What is t'_b in terms of t_a and x_a?

Note that t'_b and t_a/x_a don't refer to intervals between the same pair of events, so if you show a certain relation between these values it won't necessarily prove anything about the intervals in different frames between a single pair of events as in the the Lorentz transformation. Also note that in the Lorentz transformation equation it's not only assumed you're talking about intervals between a single pair of events, but it's also assumed that when calculating the intervals you're being consistent about the order in which you're taking the events. For example, x_a and t_a could both be understood as space and time intervals between the same pair of events (photon passing A) and (Event on photon's worldline that's simultaneous with A&B being colocated) even if you didn't choose to define them in terms of this pair, so if t_a was to be defined as (time coordinate of photon passing A) - (time coordinate of Event on photon's worldline that's simultaneous with A&B being colocated) in order to make it a positive number, then that means in order to be consistent we would have to define x_a as (position coordinate of photon passing A) - (position coordinate of Event on photon's worldline that's simultaneous with A&B being colocated), so if you assume this Event has a positive position position coordinate that would make the interval x_a negative according to the above definition. I think that in your notation you are just defining x_a as the absolute value of the distance between A and the Event, so it would be positive rather than negative; in this case the physical meaning of the equation you derive relating these quantities will be quite different from the physical meaning of the Lorentz transformation relating intervals between a single pair of events calculated using a consistent order for the events.

Just to check where you're going with this, do you intend to derive an equation that has the same physical meaning as the Lorentz transformation, or do you just intend to derive an equation which looks superficially similar but whose physical meaning is different?