Rigorous physical/mathematical proof of relativity of certain aspects of S.R.

[ΔxΔp≥ћ/2]

Hi guys,

I have studied special relativity for a while now and am doing a project for one of my physics classes on it. The relativity of simultaneity is a concept that I easily grasped when I began reading S.R. and time dilation was the hardest (easiest now). I grasped it (simultaneity) easily because it I found it quite intuitive. The problem is, the way I see it presented, the relativity of simultaneity seems completely Newtonian and to be expected if light has a finite speed. I see it causing an asymmetry and contributing no symmetries, this seems contrary to a good physical law. If these concepts where true, one could easily establish (I think) that events ARE simultaneous when perceived as simultaneous by a system of co-ordinates at rest relative to the two events. Does this all dissolve if one of the events is moving relative to the other?

The presentation that I see the most often is the one with the train and two poles. Like seen at about 2:13 :
http://www.youtube.com/watch?v=AZ6N85lNgHY&NR=1

I have a lot of books on S.R. and I am starting to get into the complicated ones. I noticed that in the original Einstein paper, On the Electrodynamics of Moving Bodies, he makes no reference to any such thought experiment and rather opts for a mathematical approach that I cannot follow. A preliminary look into (I have not read it yet) Six NOT SO Easy Pieces by Feynman seems to show the same thing. I am guessing/hoping that all this train stuff is just a poor explanation/metaphor or this is something that has been corrected by general relativity. I would really like to be able to talk sensibly about this stuff to laypeople, to go on about causality and concepts such how S.R. changed our view of action at a distance.

Much to my dismay, I have also had much difficulty finding satisfactory explanations of E=mc^2 and changes to mass/energy in a body moving at a relativistic velocity. Some of my books make a poor attempt at explaining relativistic momentum and I believe that the above comes from that equation and some combination of the kinetic energy and work equations.

As I write this stuff, I get the impression that I am close to grasping these concepts. I would appreciate it if any physics undergrads or higher qualified individuals with formal S.R. training could help me.

Oups, I just noticed that I messed up the title, too bad you can't edit those things.

 

[Doc Al]

relativity of simultaneity

The problem is, the way I see it presented, the relativity of simultaneity seems completely Newtonian and to be expected if light has a finite speed.

Not sure what you mean. The speed of light is finite (of course); the interesting thing is that the speed is invariant--nothing Newtonian about that!

I see it causing an asymmetry and contributing no symmetries, this seems contrary to a good physical law.

Where do you see an asymmetry?

If these concepts where true, one could easily establish (I think) that events ARE simultaneous when perceived as simultaneous by a system of co-ordinates at rest relative to the two events. Does this all dissolve if one of the events is moving relative to the other?

Can you give an example of what's bothering you? (An event is something that happens at a specific place and time--I don't understand what you mean by events moving.)

 

[JesseM]

If these concepts where true, one could easily establish (I think) that events ARE simultaneous when perceived as simultaneous by a system of co-ordinates at rest relative to the two events. Does this all dissolve if one of the events is moving relative to the other?

An event just happens at a single instant in time at a single position in space, it doesn't have an extended existence over time so it doesn't have a velocity of its own, therefore it doesn't make any sense to talk about being at rest relative to an event or moving relative to it.

I noticed that in the original Einstein paper, On the Electrodynamics of Moving Bodies, he makes no reference to any such thought experiment and rather opts for a mathematical approach that I cannot follow.

But in section 1 of that paper he does discuss the notion that he constructs coordinate systems in relativity so that each frame defines clocks to be synchronized based on the assumption that light travels at c in that frame:

We have not defined a common ''time'' for A and B, for the latter cannot be defined at all unless we establish by definition that the ''time'' required by light to travel from A to B equals the ''time'' it requires to travel from B to A. Let a ray of light start at the ''A time'' $$t_A$$ from A towards B, let it at the ''B time'' $$t_B$$ be reflected at B in the direction of A, and arrive again at A at the ''A time'' $${t'}_A$$.

In accordance with definition the two clocks synchronize if

$$t_B - t_A = {t'}_A - t_B$$

...in other words, two clocks at rest with respect to one another are defined to be "synchronized" if the time for light to go from A to B is equal to the time for light to go from B to A (using the clocks' own readings as the light passes them to measure the time). It's a natural consequence of this that different observers will disagree about simultaneity...if you synchronized your clocks A and B this way, but I see A and B as being in motion, then naturally if B is in front and A is behind in my frame, if I assume light travels at the same speed in both directions in my frame that'll mean the light takes longer to go from A to B (since B is moving away from A's position when the light was emitted) than it takes to go from B to A (since A is moving towards B's position when the light was reflected). The train thought-experiment is just a variation on this, where we show that if both observers assume the light from the flashes moves at c in their own frame, they cannot agree on whether the flashes happened simultaneously or not.

I am guessing/hoping that all this train stuff is just a poor explanation/metaphor or this is something that has been corrected by general relativity.

No, it's a perfectly good way of showing that if each observer assumes light moves at c, they must disagree about simultaneity...what is your objection to this thought-experiment, exactly?

 

[jcsd]

Hi guys,

I have studied special relativity for a while now and am doing a project for one of my physics classes on it. The relativity of simultaneity is a concept that I easily grasped when I began reading S.R. and time dilation was the hardest (easiest now). I grasped it (simultaneity) easily because it I found it quite intuitive. The problem is, the way I see it presented, the relativity of simultaneity seems completely Newtonian and to be expected if light has a finite speed. I see it causing an asymmetry and contributing no symmetries, this seems contrary to a good physical law. If these concepts where true, one could easily establish (I think) that events ARE simultaneous when perceived as simultaneous by a system of co-ordinates at rest relative to the two events. Does this all dissolve if one of the events is moving relative to the other?

If two different events have spacelike separation then there will be a inertial frame in which they are simultaneous in, they will not be simultaneous in any other inertial frame.

An event is something that is at postion (x,y,z) at time t, it doesn't have a velocity.

The presentation that I see the most often is the one with the train and two poles. Like seen at about 2:13 :
http://www.youtube.com/watch?v=AZ6N85lNgHY&NR=1

I have a lot of books on S.R. and I am starting to get into the complicated ones. I noticed that in the original Einstein paper, On the Electrodynamics of Moving Bodies, he makes no reference to any such thought experiment and rather opts for a mathematical approach that I cannot follow. A preliminary look into (I have not read it yet) Six NOT SO Easy Pieces by Feynman seems to show the same thing. I am guessing/hoping that all this train stuff is just a poor explanation/metaphor or this is something that has been corrected by general relativity. I would really like to be able to talk sensibly about this stuff to laypeople, to go on about causality and concepts such how S.R. changed our view of action at a distance.

What exactly is your problem with the 'train' it is what relativity predicts. Of course in realtity the train is moving only a tiny fraction of c relative to the observer on the track that the difference in times of the lightning strikesviewed by the observer on the train would be so tiny as to be unmeasurable (i.e. well within the error bars of any practcial experiment).

Much to my dismay, I have also had much difficulty finding satisfactory explanations of E=mc^2 and changes to mass/energy in a body moving at a relativistic velocity. Some of my books make a poor attempt at explaining relativistic momentum and I believe that the above comes from that equation and some combination of the kinetic energy and work equations.

If you want to know hoe E=mc^2 was derievd read "On the Electrodynamics of Moving Bodies".

The other 'explanations' are using the fact that the time compoent of the four momentum vector in a given frame of an object is it's total energy in that frame. In the rest frame of the object the time component is mc² hencethe total energy of the object in the frame is mc².

 

[ΔxΔp≥ћ/2]

Ok, I will try to be as precise as possible. And by the way, boy you guys are prompt :)

Situation #1:

Lets imagine a guy, say Al, who is at rest relative to his system of reference A. This system is represented by Cartesian co-ordinates rigidly attached to Al, with Al at the origin. Also at rest relative to this system of reference A are two really, really bright lights, L to Al's left and R to Al's right. These lights are equidistant to Al and lying on the x-axis.

Now, moving relative to Al with a velocity v along the x-axis of the A system is Bob.

Al will state that the turning on of the two lights is simultaneous if the light from both lights reach him at the same time.

I do not think that Bob has any validity in saying that these events are not simultaneous if he has time to move towards one event and away from the other once they have occurred.

By the way, I do agree that both Al and Bob are equally valid in saying that they are at rest and the other is the one moving. I just do not see how it applies to the above situation.

Situation #2:

Take the above situation and add a third person, we will call him Charlie. Charlie is at rest to Al and is situated on the x-axis of the co-ordinate system A. However, Charlie is closer to the light L, the one to Al's left.

Say the same thing that happened in situation #1 happens here. Al, who is in between light L and light R sees the turn on simultaneously.

Sometime later, Charlie meets up with Al and they discuss the lighting of the lights. Charlie will assert that the events where not simultaneous, because he was closer to the light L.

Al calls Charlie an idiot, says he is full of baloney and smacks him on the side of the head.

 

[ΔxΔp≥ћ/2]

Ok... ...I just re-read the brief part of The Character of Physical Law that deals with the relativity of simultaneity and I think I am starting to get it.

An event is something that is at postion (x,y,z) at time t, it doesn't have a velocity.

Ah right... ...what was I thinking?!

I think I was mixing up causality and simultaneity.

Situation #1-Solved

Situation #2-What does relativity think of poor Charlie?

 

[matheinste]

Hello Original Poster.

I guess that from what you say about grasping (easily)the relativity of simultaneity from the speed of light being constant (but having difficulty with other concepts)that you mean that the time at which observers at different locations see two events is not the same. This is not what the relativity of simultaneity is about. It follows from the speed of light being the same for all observers. Observers in relative motion may see two events at the same time but will disagree as to whether the events happened at the same time.

I hope my guess is wrong,and if it is,i apologise for my underestimation of your grasp of the meaning.

Matheinste.

 

[jcsd]

Ok, I will try to be as precise as possible. And by the way, boy you guys are prompt :)

Situation #1:

Lets imagine a guy, say Al, who is at rest relative to his system of reference A. This system is represented by Cartesian co-ordinates rigidly attached to Al, with Al at the origin. Also at rest relative to this system of reference A are two really, really bright lights, L to Al's left and R to Al's right. These lights are equidistant to Al and lying on the x-axis.

Now, moving relative to Al with a velocity v along the x-axis of the A system is Bob.

Al will state that the turning on of the two lights is simultaneous if the light from both lights reach him at the same time.

I do not think that Bob has any validity in saying that these events are not simultaneous if he has time to move towards one event and away from the other once they have occurred.

By the way, I do agree that both Al and Bob are equally valid in saying that they are at rest and the other is the one moving. I just do not see how it applies to the above situation.

An event as has been noted is soemthing with a spatial postion and a temproal postion. When you are tlaking about "having time to moving towards one event and away from the other event" your talking spatially only, so your not talking about events, your talking about a spatial postions. Specifically spatial postions in Al's rest frame. In Bob's rest frame it is Al not him who is moving

Situation #2:

Take the above situation and add a third person, we will call him Charlie. Charlie is at rest to Al and is situated on the x-axis of the co-ordinate system A. However, Charlie is closer to the light L, the one to Al's left.

Say the same thing that happened in situation #1 happens here. Al, who is in between light L and light R sees the turn on simultaneously.

Sometime later, Charlie meets up with Al and they discuss the lighting of the lights. Charlie will assert that the events where not simultaneous, because he was closer to the light L.

Al calls Charlie an idiot, says he is full of baloney and smacks him on the side of the head.

This is not correct, because Charlie will factor in the fact his postion is not equidistant from the lights and agree that with Al's observation that they were turned on simultaneously.

 

[JesseM]

Ok, I will try to be as precise as possible. And by the way, boy you guys are prompt :)

Situation #1:

Lets imagine a guy, say Al, who is at rest relative to his system of reference A. This system is represented by Cartesian co-ordinates rigidly attached to Al, with Al at the origin. Also at rest relative to this system of reference A are two really, really bright lights, L to Al's left and R to Al's right. These lights are equidistant to Al and lying on the x-axis.

Now, moving relative to Al with a velocity v along the x-axis of the A system is Bob.

Al will state that the turning on of the two lights is simultaneous if the light from both lights reach him at the same time.

I do not think that Bob has any validity in saying that these events are not simultaneous if he has time to move towards one event and away from the other once they have occurred.

But the event of a light turning on is instantaneously brief, you can't move towards it because it doesn't persist over time (and it's irrelevant how the bulb moves after it turns on, because we're only interested in when each observer first sees the light from the event of the bulb turning on, not when they see the light that the bulb emitted after it had already been on for some finite time-interval). Perhaps what you mean is that Bob moves towards the position on Al's x-axis where one bulb was turned on, and away from the position on Al's x-axis where the other bulb was turned on. This is true, but now suppose Bob has his own cartesian coordinate system with his own x'-axis which is sliding along parallel to Al's x-axis. Viewed in Bob's coordinate system, Bob is of course at rest with respect to the positions on the x'-axis where each bulb was turned on. If light moves at c in Bob's frame, and the distance on the x'-axis between Bob's position and the position where a given bulb turned on is $$\Delta x'$$, then it must take a time of $$\Delta x' / c$$ in Bob's frame for the light from the bulb first turning on to reach his own position.

Take the above situation and add a third person, we will call him Charlie. Charlie is at rest to Al and is situated on the x-axis of the co-ordinate system A. However, Charlie is closer to the light L, the one to Al's left.

Say the same thing that happened in situation #1 happens here. Al, who is in between light L and light R sees the turn on simultaneously.

Sometime later, Charlie meets up with Al and they discuss the lighting of the lights. Charlie will assert that the events where not simultaneous, because he was closer to the light L.

Al calls Charlie an idiot, says he is full of baloney and smacks him on the side of the head.

Sure, Charlie's an idiot, of course you must take into account your distance from each event at the moment it happened when judging the time it happened--you're supposed to take the time on your clock that you saw the light from the event, then subtract [distance/c] from that time to find the time of the event in your frame, where distance is defined in terms of a ruler at rest relative to yourself. Charlie didn't do this. But I don't see what this example has to do with Bob's claim that the events happened non-simultaneously, because Bob did follow this procedure.

 

[ΔxΔp≥ћ/2]

I read Does the Inertia of a Body Depend Upon Its Energy-Content? and didn't understand much. It talks about radiation...

...Happy New Year!...

...electrodynamics and a few other things that I have no experience with. Is there an easier way to get to the E=mc^2 or is it hopeless to explain it to high school students?

And for those of you wondering about poor Charlie, do not fret. He dabbled in science for a while, but learned it all exept the crummy math. So he decided to do a B.A. in philosophy.

 

[JesseM]

I read Does the Inertia of a Body Depend Upon Its Energy-Content? and didn't understand much. It talks about radiation...

...Happy New Year!...

...electrodynamics and a few other things that I have no experience with. Is there an easier way to get to the E=mc^2 or is it hopeless to explain it to high school students?

The diagram on this page may make it clearer what Einstein was doing in his derivation. You might take a look at http://www.grallator.co.uk/emc2.htm too.

 

[ΔxΔp≥ћ/2]

Thanks a lot!
That does help!