If correct: a catastrophe in the Lorentz transformation

[Dale]

I understand very well the meaning of simultaneity in classical and relativistic physics and all related subjects.

I don't think that you do, but please don't feel discouraged. It is a challenging concept for almost all students.

The point is that the universe simply doesn't care about simultaneity, only about causality. Causes always precced effects in all frames, but otherwise the ordering of events is purely an arbitrary human convention determined by our choice of coordinate system.

However, I cannot, and I believe many others, accept this easily that our physical intuition is so remote, or as we think, from the mathematical structure of the theory.

I think one of the big lessons of the last century is that our physical intuition has evolved in a very classical world and that when we are dealing with physical situations outside of our normal classical scales our intuition is not terribly useful.

 

[aawahab76]

I don't think that you do, but please don't feel discouraged. It is a challenging concept for almost all students.

The point is that the universe simply doesn't care about simultaneity, only about causality. Causes always precced effects in all frames, but otherwise the ordering of events is purely an arbitrary human convention determined by our choice of coordinate system.

I think one of the big lessons of the last century is that our physical intuition has evolved in a very classical world and that when we are dealing with physical situations outside of our normal classical scales our intuition is not terribly useful.

OK I am not discouraged at all but indeed I understand very well relativity and related subjects and trying to concentrate our discussion o a different direction than the usual play with equations and graphs. Of course that maight be the way the universe wrok but as you said that should not discourage us from trying different routs to understand those concepts. My point is that physics is completely different from mathematics until it is proved so and here I am trying to understand the physical process that leads both frames in their judgments and measurements.
My question now: does the statement " in the x-frame, event A happened at t=1 sec and B at t=2 sec so A happened before B" has a meaning?

 

[Dale]

My question now: does the statement " in the x-frame, event A happened at t=1 sec and B at t=2 sec so A happened before B" has a meaning?

Yes. This statement has meaning and is correct because you specified the reference frame. It is perfectly fine to make frame-dependent statements as long as you specify the frame.

 

[harrylin]

OK I am not discouraged at all but indeed I understand very well relativity and related subjects and trying to concentrate our discussion o a different direction than the usual play with equations and graphs. Of course that maight be the way the universe wrok but as you said that should not discourage us from trying different routs to understand those concepts. My point is that physics is completely different from mathematics until it is proved so and here I am trying to understand the physical process that leads both frames in their judgments and measurements.
[..]

Regretfully the history of scientific discovery is often neglected in physics education, while knowing how a theory emerged can help to better understand the implied physical process.

The point that Poincare explained[1] with elaboration (still in the 19th century), is that in order to make calculations, astronomers simply postulated that the one-way speed of light is isotropic in all directions. The "true" or "absolute" one-way speed of light could not be established, and according to relativity such a thing is even impossible to do[2].

Special relativity is based on Maxwell's theory of electrodynamics, and we may choose any inertial frame and pretend that it is the "rest frame" of light waves[2].

It uses Poincare's method, defining distant time as the local time plus the half the two-way transmission time[2b].

Therefore, the relativity of simultaneity was one century ago perhaps less a problem for students than it is nowadays. :tongue2:

1. http://en.wikisource.org/wiki/The_Measure_of_Time (sections X to XIII)

2. http://www.fourmilab.ch/etexts/einstein/specrel/www/ (introduction).
2b. (same, section 1)

 

[Dale]

Regretfully the history of scientific discovery is often neglected in physics education, while knowing how a theory emerged can help to better understand the implied physical process.

This is personal preference, but my feeling is exactly the opposite. I think that too much history is included in physics education to the detriment of learning a theory. E.g. Einstein's thought experiments only confused me and it wasn't until I found a more modern geometrical treatment that relativity finally "clicked" for me.

 

[bobc2]

... My point is that physics is completely different from mathematics until it is proved so and here I am trying to understand the physical process that leads both frames in their judgments and measurements.
My question now: does the statement " in the x-frame, event A happened at t=1 sec and B at t=2 sec so A happened before B" has a meaning?

aawahab76, I will try to give you some meaning in a little different context. Imagine that the universe is really physically a 4-dimensional space and that all objects in the universe are actually 4-dimensional objects, fixed--frozen--, the objects are just there extending along what are called world lines. The situation is depicted in the upper left sketch below. Here is a simple beam as a real 4-dimensional object. You see one 3-D cross-section of the beam below with the normal X1, X2, X3 coordinates. For the 4-dimensional sketch we had to suppress X3 in order to view the 4th dimension.

A curious mystery of this 4-dimensional world is that in some way you have to imagine observers (occupying a living 4-D body object) moving along their X4 world line at the speed of light, c. At any instant of an observer's time he can experience just a 3-dimensional cross-section of that 4-dimensional universe. And to make matters more strange his instantaneous cross-section view is slanted so that his X1 axis always rotates so as to make the light photon world line bisect the angle between the X4 and X1 axis as shown in the sketch to the right. So, if you consider this 4-dimensional world as actually working this way physically, then you might find your physical picture.

Once you accept this model of the 4-dimensional universe as having real 4-dimensional objects, then the problems relating to simultenaity become trivially simple. Notice in the lower right sketch that there is an event 1 and event 2 (red dots). Event 1 occurs first for the blue observer, but those same two events occur in opposite time sequence for the black observer. The two events are definite fixed events in the 4-dimensional universe. But, it's just that the blue and black observers experience two totally different instantaneous 3-D spaces within a 4-D universe. And to emphasize again, we are not talking about equations and graphs, here--rather we attempt to picuture a real physical 4-dimensional universe with real 4-dimensional physical objects.

However, now you are beginning to cross over into the subject of metaphysics and philosophy of special relativity, subjects that are frought with many different views. To purse special relativity in this context in this forum, you should visit the philosophy forum (It wasn't clear to me whether the questions in your mind were more of this nature).

 

[aawahab76]

aawahab76, I will try to give you some meaning in a little different context. Imagine that the universe is really physically a 4-dimensional space and that all objects in the universe are actually 4-dimensional objects, fixed--frozen--, the objects are just there extending along what are called world lines. The situation is depicted in the upper left sketch below. Here is a simple beam as a real 4-dimensional object. You see one 3-D cross-section of the beam below with the normal X1, X2, X3 coordinates. For the 4-dimensional sketch we had to suppress X3 in order to view the 4th dimension.

A curious mystery of this 4-dimensional world is that in some way you have to imagine observers (occupying a living 4-D body object) moving along their X4 world line at the speed of light, c. At any instant of an observer's time he can experience just a 3-dimensional cross-section of that 4-dimensional universe. And to make matters more strange his instantaneous cross-section view is slanted so that his X1 axis always rotates so as to make the light photon world line bisect the angle between the X4 and X1 axis as shown in the sketch to the right. So, if you consider this 4-dimensional world as actually working this way physically, then you might find your physical picture.

Once you accept this model of the 4-dimensional universe as having real 4-dimensional objects, then the problems relating to simultenaity become trivially simple. Notice in the lower right sketch that there is an event 1 and event 2 (red dots). Event 1 occurs first for the blue observer, but those same two events occur in opposite time sequence for the black observer. The two events are definite fixed events in the 4-dimensional universe. But, it's just that the blue and black observers experience two totally different instantaneous 3-D spaces within a 4-D universe. And to emphasize again, we are not talking about equations and graphs, here--rather we attempt to picuture a real physical 4-dimensional universe with real 4-dimensional physical objects.

However, now you are beginning to cross over into the subject of metaphysics and philosophy of special relativity, subjects that are frought with many different views. To purse special relativity in this context in this forum, you should visit the philosophy forum (It wasn't clear to me whether the questions in your mind were more of this nature).

thanks friend, but it seems that you are complicating our education here.

 

[aawahab76]

Yes. This statement has meaning and is correct because you specified the reference frame. It is perfectly fine to make frame-dependent statements as long as you specify the frame.

So in one frame we can say (at least as a convention) that event A happened before B even though the two events are space-like separated, right? That of course is done by comparing their time coordinates (that is t in the x-frame).

 

[Fredrik]

So in one frame we can say (at least as a convention) that event A happened before B even though the two events are space-like separated, right? That of course is done by comparing their time coordinates (that is t in the x-frame).

We can say that because they are spacelike separated. If two events aren't spacelike separated, they have the same time ordering in all inertial coordinate systems.

I don't know if it has been mentioned, but the statement "A happened before B" means nothing more than "the coordinate system that we have chosen to consider assigns a smaller time coordinate to A than to B". Edit: I see now that this is very similar to what you're saying in the text I'm quoting. I still think it doesn't get mentioned often enough in these threads. SR is much less confusing to a person who has realized that statements about someone's point of view are really statements about the coordinate system we choose to associate with his motion.

 

[bobc2]

thanks friend, but it seems that you are complicating our education here.

My mistake. I misunderstood your pursuit. My appologies.

 

[aawahab76]

We can say that because they are spacelike separated. If two events aren't spacelike separated, they have the same time ordering in all inertial coordinate systems.

I don't know if it has been mentioned, but the statement "A happened before B" means nothing more than "the coordinate system that we have chosen to consider assigns a smaller time coordinate to A than to B". Edit: I see now that this is very similar to what you're saying in the text I'm quoting. I still think it doesn't get mentioned often enough in these threads. SR is much less confusing to a person who has realized that statements about someone's point of view are really statements about the coordinate system we choose to associate with his motion.

You mean if the two events are space-like separated, right? I mean in the second paragraph.

 

[Fredrik]

No, I meant in general. It was just a minor point about how to explain SR pedagogically that doesn't have anything to do with what you've been discussing in this thread. (I haven't followed the discussion by the way). I'm saying e.g. that the statement "From Charlies's point of view, A is earlier than B" is just a slightly misleading way of saying "the coordinate system we associate with Charlie's motion assigns a smaller time coordinate to A than to B".

Here's an example that explains why I think it helps to understand this. Consider the two statements:

1. From Alice's point of view, Bob's aging rate is 60% of hers.
2. From Bob's point of view, Alice's aging rate is 60% of his.

Most people (who don't know SR) would say that these two statements are obviously contradicting each other. But once they understand that the first is a statement about numbers assigned by the coordinate system associated with Alice's motion, and that the second is a statement about numbers assigned by the coordinate system associated with Bob's motion, I think they will find it easier to start thinking about the possibility that they're not contradictory at all.

 

[aawahab76]

No, I meant in general. It was just a minor point about how to explain SR pedagogically that doesn't have anything to do with what you've been discussing in this thread. (I haven't followed the discussion by the way). I'm saying e.g. that the statement "From Charlies's point of view, A is earlier than B" is just a slightly misleading way of saying "the coordinate system we associate with Charlie's motion assigns a smaller time coordinate to A than to B".

Here's an example that should explain why it helps to understand this. Consider the two statements:

1. From Alice's point of view, the ticking rate of Bob's clock is 60% of the ticking rate of her own clock.
2. From Bob's point of view, the ticking rate of Alice's clock is 60% of the ticking rate of his own clock.

Most people (who don't know SR) would say that these two statements are obviously contradicting each other. But once they understand that the first is a statement about numbers assigned by the coordinate system associated with Alice's motion, and that the second is a statement about numbers assigned by the coordinate system associated with Bob's motion, I think they will find it easier to start thinking about the possibility that they're not contradictory at all.

Still, it seems that time ordering of two events that are time-like separated are the same irrespective of which frame is used.

 

[Dale]

So in one frame we can say (at least as a convention) that event A happened before B even though the two events are space-like separated, right? That of course is done by comparing their time coordinates (that is t in the x-frame).

Yes.

 

[Fredrik]

Still, it seems that time ordering of two events that are time-like separated are the same irrespective of which frame is used.

Yes, I said that they are. (Second sentence in #44).

 

[aawahab76]

Yes, I said that they are. (Second sentence in #44).

yes, my apology for you.

 

[harrylin]

This is personal preference, but my feeling is exactly the opposite. I think that too much history is included in physics education to the detriment of learning a theory. E.g. Einstein's thought experiments only confused me and it wasn't until I found a more modern geometrical treatment that relativity finally "clicked" for me.

I have no problems with such thought experiments, but a thought experiment isn't a physical process - and geometry is even less a physical process. :tongue:
Instead, I referred to physical insight from physical theories based on physical measurements.
However, the OP seems not to recognize the usefulness of the physical basis of relativity so I won't bother.

 

[aawahab76]

Yes.

Now back to our problem.

1- The event (1 sec, 10^100 m) (let us call from now on P) happened after the event (0,0) (let us call from now on O) in the x-frame.
2- So when the origin of the x'-frame passes through the origin of the x-frame, the observer in the origin of the x-frame is very sure that P did not happen yet.
3- However, for the x'-frame, P already happened.
4- the observer in the x'-frame can deliver a message to the observer in x-frame stating that and event P already happened at (t'≈-10^81 sec, x'≈10^100 m).
5- the observer in x-frame read that message quickly and deliver in his turn an argent message, by say light signals (so I am assuming that the two messages delivery, receiving, reading and reaction will last less than a second in the x-frame) telling the observer in the x'-frame that the event P at (1 sec, 10^100 m) did not happen. Just to have some action, P might instead of a flash of light be a deadly explosion.

Now before continuing, is there any problem in what I said above?

 

[bobc2]

Now back to our problem.

1- The event (1 sec, 10^100 m) (let us call from now on P) happened after the event (0,0) (let us call from now on O) in the x-frame.
2- So when the origin of the x'-frame passes through the origin of the x-frame, the observer in the origin of the x-frame is very sure that P did not happen yet.
3- However, for the x'-frame, P already happened.
4- the observer in the x'-frame can deliver a message to the observer in x-frame stating that and event P already happened at (t'≈-10^81 sec, x'≈10^100 m).
5- the observer in x-frame read that message quickly and deliver in his turn an argent message, by say light signals (so I am assuming that the two messages delivery, receiving, reading and reaction will last less than a second in the x-frame) telling the observer in the x'-frame that the event P at (1 sec, 10^100 m) did not happen. Just to have some action, P might instead of a flash of light be a deadly explosion.

Now before continuing, is there any problem in what I said above?

Statement 4. How did the x'-frame observer know that the the event had happened?

 

[aawahab76]

Statement 4. How did the x'-frame observer know that the the event had happened?

Because with respect to x'-frame, P is given by (t'≈-10^81 sec, x'≈10^100 m) while O (the event of meeting of the two observers) is (0',0') so t' for P is before t' for O. That what it seems to be.

 

[bobc2]

Because with respect to x'-frame, P is given by (t'≈-10^81 sec, x'≈10^100 m) while O (the event of meeting of the two observers) is (0',0') so t' for P is before t' for O. That what it seems to be.

That doesn't seem to explain how the observer in x' knew about the event when he met up with the other observer.

 

[bobc2]

Because with respect to x'-frame, P is given by (t'≈-10^81 sec, x'≈10^100 m) while O (the event of meeting of the two observers) is (0',0') so t' for P is before t' for O. That what it seems to be.

If you follow the photon world line from Event 1 to Event 3, you will see that the guy in x' (my blue guy) has not received the information about the light flash at Event 1 in time to communicate that information with the guy in the x coordinates (my black guy). To further complicate the discussion, you can see that the x' guy (blue) actually gets information about the light flash at Event 2 before he gets the information about the flash at Event 1, even though, in his coordinate system, Event 1 occurs first.

That's why it helps to do the spacetime diagrams, even if you could care less about whether the objects are 4-dimensional or not. However, if you are a true operationalist, then you may wish to ignore events out far away and be concerned only with directly observed light flashes as you experience them; then you can avoid spacetime diagrams.

 

[Dale]

4- the observer in the x'-frame can deliver a message to the observer in x-frame stating that and event P already happened at (t'≈-10^81 sec, x'≈10^100 m).
...
Now before continuing, is there any problem in what I said above?

Yes, the observer at x'=0 will not get the information about the event at (t',x') = (-10^81,10^100) until t' = 10^91, it will be far too late for him to deliver a message to the x-frame observer.

see post 22 above: https://www.physicsforums.com/showpost.php?p=3110316&postcount=22

 

[aawahab76]

Yes, the observer at x'=0 will not get the information about the event at (t',x') = (-10^81,10^100) until t' = 10^91, it will be far too late for him to deliver a message to the x-frame observer.

see post 22 above: https://www.physicsforums.com/showpost.php?p=3110316&postcount=22

Yes I agree with this and with what bobc2 said. So let us then and before any thing else answer this question: assume I am the observer in the x-frame. Now I build my frame using the usual method of rulers and synchronized clocks. When an event at P happens, how can I register its coordinates? Of course this question can be generalized to how can an observer register a particle path, too.

 

[bobc2]

Yes I agree with this and with what bobc2 said. So let us then and before any thing else answer this question: assume I am the observer in the x-frame. Now I build my frame using the usual method of rulers and synchronized clocks. When an event at P happens, how can I register its coordinates? Of course this question can be generalized to how can an observer register a particle path, too.

aawahab76, here is a spacetime diagram showing the point P and the points on the time and space axes for the x frame and the x' frame(my blue coordinates). I did not compute the values of the coordinates for the actual point P, but perhaps someone will.

 

[JesseM]

Yes I agree with this and with what bobc2 said. So let us then and before any thing else answer this question: assume I am the observer in the x-frame. Now I build my frame using the usual method of rulers and synchronized clocks. When an event at P happens, how can I register its coordinates? Of course this question can be generalized to how can an observer register a particle path, too.

Well, you can assume that next to each clock is a camera recording local events in the neighborhood of that clock, and the feeds from each camera are being sent to the central observer at the speed of light. So the observer may not learn about a given event until long after it happens, but when he does learn about it he can just look at the clock and ruler-marking that were right next to the event when it happened to see what position and time coordinates should (retroactively) be assigned to the event.

 

[bobc2]

Well, you can assume that next to each clock is a camera recording local events in the neighborhood of that clock, and the feeds from each camera are being sent to the central observer at the speed of light. So the observer may not learn about a given event until long after it happens, but when he does learn about it he can just look at the clock and ruler-marking that were right next to the event when it happened to see what position and time coordinates should (retroactively) be assigned to the event.

That is a really nice way of doing it. You have answered the question--I seem to have a knack for answering questions that have not been asked. Again, my appologies, aawahab76.

 

[aawahab76]

Well, you can assume that next to each clock is a camera recording local events in the neighborhood of that clock, and the feeds from each camera are being sent to the central observer at the speed of light. So the observer may not learn about a given event until long after it happens, but when he does learn about it he can just look at the clock and ruler-marking that were right next to the event when it happened to see what position and time coordinates should (retroactively) be assigned to the event.

Beautiful efforts from all contributors. Now I can imagine the following physical processes and conditions that characterize our original problem. I will organize them in bullets and keep in mind that we are treating one dimensional problem.

1- An observer in the x-frame has built his frame with the usual rulers and synchronized clocks (by the known method of Poincare and Einstein) with coordinates (t,x). We call this observer and its frame F. I think it would be understood when F ( and similarly for F' below) means the frame or the observer (located at the origin of spatial coordinates).
2- The same has been done by the x'-frame observer with coordinates (t',x'). This is called F'. F' moves with the speed v=10^(-10) m/s with respect to F (so F moves with the speed -10^(-10) m/s with respect to F").
3- An event is an absolute physical process and is independent of coordinates (or frames). An event can be represented in coordinates given by the reading of the clock and the reading of the ruler located at the event. This is done separately by each observer using his or her coordinates.
4- F and F' agree to set their time coordinates such that the event of their meeting is given by (0,0) in F and by (0',0') by F'. (0,0) is named O, and (0',0') is named O'.
5- An event P happen at (t,x)=(1 sec, 10^100 m) in F and at (t',x') in F' ( that is in F, P happened after O). We assume that the event is a flash of light.
6- The Lorentz transformation gives the corresponding coordinates of P at F', that is (t'≈-10^81 sec, x'≈10^100 m).
7- F receives the light from P at the event R given in F by (10^100/(3*10^8)=3.3*10^92 sec, 0).
8- Using the Lorentz transformation, R is given in F' by (3.3*10^92 sec, -10^91 m).
9- F' receives the light from P at the event R' given by (≈-10^10^81+10^100/(3*10^8)≈3.3*10^92 sec, 0').
10- R' is given in F by using Lorentz transformation by (≈3.3*10^92 sec, ≈9.9*10^90 m).
11- In summary, in F we have the following events O, P, R and R’. Order of these events is O, P, and both R and R’ at the same time.
12- Similarly, in F’ we have O’, P, R’ and R. Their order is P, O’, and both R and R’ at the same time.
13- Thus, our original problem can be cast in the following threefold points:
i- how does O before P in F but P before O' in F' (notice that O' is the coordinate representation in F' of the same event O)?
ii- why does a very small relative speed lead to such a huge difference in time for the event P in F' relative to F?
iii- why the Lorentz transformation (LT) does not in general reduce to the Galilean transformation (GT) ( but I think this may not be true if we define the non-relativistic limit to be the limit when c goes to infinity in which case LT reduces in general to GT). Any way, this third point may not be of interest at this moment.

to be continued but please feel free to comment, correct .. etc.

 

[Dale]

Yes I agree with this and with what bobc2 said. So let us then and before any thing else answer this question: assume I am the observer in the x-frame. Now I build my frame using the usual method of rulers and synchronized clocks. When an event at P happens, how can I register its coordinates? Of course this question can be generalized to how can an observer register a particle path, too.

I like JesseM's approach:

Well, you can assume that next to each clock is a camera recording local events in the neighborhood of that clock, and the feeds from each camera are being sent to the central observer at the speed of light. So the observer may not learn about a given event until long after it happens, but when he does learn about it he can just look at the clock and ruler-marking that were right next to the event when it happened to see what position and time coordinates should (retroactively) be assigned to the event.

Equivalently you can use radar ranging techniques.

 

[bobc2]

aawahab76, I probably am answering a question that hasn't been asked again. However, I don't seem to get the same ordering of events as you've listed when I do the space-time diagram. This time I used a symmetric diagram in order to be sure that the scaling will be the same for both F and F'.

By the way, I have not used such small speeds as you wished, because it would require a much larger screen to achieve the required resolution. But, the basic relationships would not change. If you imagine rotating the axes so that they asymptotically approach the perpendiculare black coordinates, you will see that the points on the t and t' axes just get closer and closer together, so that the order of events are as they appear in my diagram.

 

[aawahab76]

aawahab76, I probably am answering a question that hasn't been asked again. However, I don't seem to get the same ordering of events as you've listed when I do the space-time diagram. This time I used a symmetric diagram in order to be sure that the scaling will be the same for both F and F'.

By the way, I have not used such small speeds as you wished, because it would require a much larger screen to achieve the required resolution. But, the basic relationships would not change. If you imagine rotating the axes so that they asymptotically approach the perpendiculare black coordinates, you will see that the points on the t and t' axes just get closer and closer together, so that the order of events are as they appear in my diagram.

Thanks, I think graphs may seem more complicated at this point, but could you check using the Lorentz transformation.

 

[aawahab76]

Beautiful efforts from all contributors. Now I can imagine the following physical processes and conditions that characterize our original problem. I will organize them in bullets and keep in mind that we are treating one dimensional problem.

1- An observer in the x-frame has built his frame with the usual rulers and synchronized clocks (by the known method of Poincare and Einstein) with coordinates (t,x). We call this observer and its frame F. I think it would be understood when F ( and similarly for F' below) means the frame or the observer (located at the origin of spatial coordinates).
2- The same has been done by the x'-frame observer with coordinates (t',x'). This is called F'. F' moves with the speed v=10^(-10) m/s with respect to F (so F moves with the speed -10^(-10) m/s with respect to F").
3- An event is an absolute physical process and is independent of coordinates (or frames). An event can be represented in coordinates given by the reading of the clock and the reading of the ruler located at the event. This is done separately by each observer using his or her coordinates.
4- F and F' agree to set their time coordinates such that the event of their meeting is given by (0,0) in F and by (0',0') by F'. (0,0) is named O, and (0',0') is named O'.
5- An event P happen at (t,x)=(1 sec, 10^100 m) in F and at (t',x') in F' ( that is in F, P happened after O). We assume that the event is a flash of light.
6- The Lorentz transformation gives the corresponding coordinates of P at F', that is (t'≈-10^81 sec, x'≈10^100 m).
7- F receives the light from P at the event R given in F by (10^100/(3*10^8)=3.3*10^92 sec, 0).
8- Using the Lorentz transformation, R is given in F' by (3.3*10^92 sec, -10^91 m).
9- F' receives the light from P at the event R' given by (≈-10^10^81+10^100/(3*10^8)≈3.3*10^92 sec, 0').
10- R' is given in F by using Lorentz transformation by (≈3.3*10^92 sec, ≈9.9*10^90 m).
11- In summary, in F we have the following events O, P, R and R’. Order of these events is O, P, and both R and R’ at the same time.
12- Similarly, in F’ we have O’, P, R’ and R. Their order is P, O’, and both R and R’ at the same time.
13- Thus, our original problem can be cast in the following threefold points:
i- how does O before P in F but P before O' in F' (notice that O' is the coordinate representation in F' of the same event O)?
ii- why does a very small relative speed lead to such a huge difference in time for the event P in F' relative to F?
iii- why the Lorentz transformation (LT) does not in general reduce to the Galilean transformation (GT) ( but I think this may not be true if we define the non-relativistic limit to be the limit when c goes to infinity in which case LT reduces in general to GT). Any way, this third point may not be of interest at this moment.

to be continued but please feel free to comment, correct .. etc.

My intention of the above list, to be completed below, is to find contradictory result or to grasp the physical sitiuation which is my aim from the begining.

 

[Dale]

R' and R are two events which are lightlike separated, therefore they cannot occur at the same time. You are just experiencing some roundoff error and you probably have to use an arbitrary precision math package and look at the 10th decimal place to see the difference. I haven't checked your numbers, but I provided some numbers in a previous post which you can use to compare.

i) because O and P are spacelike separated, as we have already discussed
ii) it is a small difference in time, as I said several times already
iii) it does reduce to the Galilean transformation in the limit as c -> infinity

 

[bobc2]

My intention of the above list, to be completed below, is to find contradictory result or to grasp the physical sitiuation which is my aim from the begining.

Perhaps you are leading us into a question about quantum uncertainty. In my picture an interesting situation for the instantaneous spaces at very large distances develops as we let the speeds of the observers approach zero. Again, I probably get into the wrong question and have strayed way off point.

However, I leave it to someone else to do the transformation calculations.

 

[GrayGhost]

My intention of the above list, to be completed below, is to find contradictory result or to grasp the physical sitiuation which is my aim from the begining.

While reading thru this, refer to the attached graphic of your scenario ...

What you are lacking, is the understanding of relative simultaneity. Systems of relative v > 0 disagree on the measure of space and time for remote events. This disagreement leads to what is known as "relative simultaneity". IOWs, they disagree as to what are simultaneous events. Graphically, this manifests itself as a rotation in spacetime between 2 systems moving relatively, which is why the time axes (and likewise space axes) are not parallel. They instead become angularly rotated wrt one another. In a Galillean system, the axes would indeed be parallel, but not so in relativity theory. BobC2 has shown you some graphics which present the case. I'll do the same here, while trying to minimize what's on the illustration to keep it simple. If you used your LT transforms to calculate all the events that occur per x,t along the depicted slanted blue dashed line, you will find that the LT results for all those events all occur at time t'=0 of the moving system. IOWs, the blue dashed line is the x',t' system's sense-of-NOW across the all of space "as he experiences it", at his time t'=0.

Clearly, the further you move the distant event downrange along +x (always at time t=1s), the further said event must have occurred back-in-time per the x',t' system, because it falls further and further below the blue dashed line which is the x',t' system's sense of NOW space-wide ... ie his own sense-of-simultaneity for t'=0.

Hope that helps as to the physical meaning you seek.

GrayGhost .