If correct: a catastrophe in the Lorentz transformation

[JesseM]

It is obvious by now that the subject is getting broader and broader so let me collect my mind and ask the following questions. Is it physically, that is according to relativity, possible that event A happened before event B in the x-frame while event B happened before A in the x'-frame? Under which conditions is that possible?

Yes, it is definitely possible for two frames to disagree on the order of two events, the condition where this can occur is when the spacetime interval between the events is "space-like", which among other things means that neither event is in the past or future light cone of the other event (so there can be no causal relation between the two events). See here for more discussion of the meaning of time-like, light-like, and space-like intervals.

 

[aawahab76]

Yes, it is definitely possible for two frames to disagree on the order of two events, the condition where this can occur is when the spacetime interval between the events is "space-like", which among other things means that neither event is in the past or future light cone of the other event (so there can be no causal relation between the two events). See here for more discussion of the meaning of time-like, light-like, and space-like intervals.

I completely agree with this, so any two events A and B that are causally unrelated (that is cannot be joined by a light ray, we may return to the causality concept in the future) can have A before B or B before A or A simultaneous with B depending on the two events and on the states of the two frames, right? In addition if A and have time-like intervals between them, A is before B or B before A for both frames, right? Now apply this to our initial problem with the event (10^100 m, 1 sec) in the x=frame, what are the two events here? Are they space-like separated? Are the state of the two frames capable to make A before B in the x-frame and the other way round for the x'-frame?

 

[JesseM]

I completely agree with this, so any two events A and B that are causally unrelated (that is cannot be joined by a light ray, we may return to the causality concept in the future) can have A before B or B before A or A simultaneous with B depending on the two events and on the states of the two frames, right?

That's right, in fact for any pair of events that are space-like separated it is always possible to find three frames where each of these is true.

In addition if A and have time-like intervals between them, A is before B or B before A for both frames, right?

Yes, for either time-like or light-like intervals, all frames will agree on the order.

Now apply this to our initial problem with the event (10^100 m, 1 sec) in the x=frame, what are the two events here? Are they space-like separated? Are the state of the two frames capable to make A before B in the x-frame and the other way round for the x'-frame?

Well, you were before expressing puzzlement that in one frame that event happened after the event of the origins meeting, and in another it happened before, so presumably the two events would be the event you mentioned above and the event of the origins meeting at (0 m, 0 sec). The interval between these events is space-like, because the definition of space-like is that \Delta x^2 > c^2 \Delta t^2, and here we have \Delta x^2 = (10^100 meters)^2 = 10^200 m^2, and c^2 \Delta t^2 = (299792458 m/s)^2 * (1 second)^2 = 8.98755179 * 10^16 m^2. And we've already shown, just by using the Lorentz transformation, that the relative velocity of the two frames is indeed large enough so that the order of the events is different in each frame.

 

[aawahab76]

That's right, in fact for any pair of events that are space-like separated it is always possible to find three frames where each of these is true.

Yes, for either time-like or light-like intervals, all frames will agree on the order.

Well, you were before expressing puzzlement that in one frame that event happened after the event of the origins meeting, and in another it happened before, so presumably the two events would be the event you mentioned above and the event of the origins meeting at (0 m, 0 sec). The interval between these events is space-like, because the definition of space-like is that \Delta x^2 > c^2 \Delta t^2, and here we have \Delta x^2 = (10^100 meters)^2 = 10^200 m^2, and c^2 \Delta t^2 = (299792458 m/s)^2 * (1 second)^2 = 8.98755179 * 10^16 m^2. And we've already shown, just by using the Lorentz transformation, that the relative velocity of the two frames is indeed large enough so that the order of the events is different in each frame.

ok, accepting your calculation, still how do you decide the order of the two events in each of the two given frames?

 

[JesseM]

ok, accepting your calculation, still how do you decide the order of the two events in each of the two given frames?

Just figure out the time coordinate of both events in each frame (if you know the position and time coordinates of each event in one frame, you can use the Lorentz transformation to find their time coordinates in the other). Whichever has the larger time-coordinate in a particular frame is by definition the later one in that frame.

 

[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.

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.
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.