Relativity of Simultaneity

[vuquta]

Assume the typical co-location of origins and a light pulse at the co-location with two frames in relative motion.

 

The Relativity of Simultaneity contends if two points are simultaneous in the rest frame, then these points will not be simultaneous in the moving frame.

 

Both postulates of SR contend light is always spherical from the light emission point in the moving frame and any point with the associated time value can be translated into the coordinates of the rest frame.

 

So, here is the question. Just consider the x-axis.

 

Say two points R1 and R2 are simultaneous in the rest frame. What two points in the moving frame when translated to the rest frame coordinates are simultaneous to the moving frame when R1 and R2 are simultaneous in the rest frame?

[J.C.MacSwell]

Assume the typical co-location of origins and a light pulse at the co-location with two frames in relative motion.

 

The Relativity of Simultaneity contends if two points are simultaneous in the rest frame, then these points will not be simultaneous in the moving frame.

 

Both postulates of SR contend light is always spherical from the light emission point in the moving frame and any point with the associated time value can be translated into the coordinates of the rest frame.

 

So, here is the question. Just consider the x-axis.

 

Say two points R1 and R2 are simultaneous in the rest frame. What two points in the moving frame when translated to the rest frame coordinates are simultaneous to the moving frame when R1 and R2 are simultaneous in the rest frame?

 

If any two separated points, R1 and R2, are on the x axis and are simultaneous in the rest frame, they will be simultaneous in the moving frame if and only if the relative movement is orthogonal to the x axis.

(ill worded, see Swantsont below)

 

Events on the x axis would not be simultaneous in both frames unless the relative movement is orthogonal to it.

Edited August 3, 2010 by J.C.MacSwell

[vuquta]

If any two separated points, R1 and R2, are on the x axis and are simultaneous in the rest frame, they will be simultaneous in the moving frame if and only if the relative movement is orthogonal to the x axis.

 

Yes, we are considering only the x-axis. So, it is assumed they are different x coordinates.

 

But, what I am asking, given R1 and R2 are simultaneous to the rest frame, they are not simultaneous to the moving frame.

 

So, when R1 and R2 are simultaneous to the rest frame, what two points are simultaneous to the moving frame when translated to the rest frame coordinates.

[swansont]

Yes, we are considering only the x-axis. So, it is assumed they are different x coordinates.

 

But, what I am asking, given R1 and R2 are simultaneous to the rest frame, they are not simultaneous to the moving frame.

 

So, when R1 and R2 are simultaneous to the rest frame, what two points are simultaneous to the moving frame when translated to the rest frame coordinates.

 

The question is ill-formed. Points are not simultaneous, events are. Events have locations and times, which are relative to observers. The events will not be simultaneous in the moving frame. You have to have different events, which will then not be simultaneous in the original frame.

[vuquta]

The question is ill-formed. Points are not simultaneous, events are. Events have locations and times, which are relative to observers. The events will not be simultaneous in the moving frame. You have to have different events, which will then not be simultaneous in the original frame.

 

You have to have different events, which will then not be simultaneous in the original frame

Exactly. This is exactly what I am asking.

 

What are these two "events" when translated into the coordinates of the rest frame. Yes, they will not be simultaneous in the view of the rest frame, but what are they?

 

What are the coordinates in terms of the rest frame that are viewed simultaneous in the moving frame while R1 and R2 are simultaneous in the rest frame.

 

Let's just call the rest coordinates (r/c,-r,0,0)=R1 and (r/c,r,0,0)=R2.

[Sisyphus]

You're looking for an equation? You can translate between the coordinates of two different reference frames with the Lorentz transformations:

 

[math] t' = \frac{t - {v\,x/c^2}}{\sqrt{1-v^2/c^2}}\ [/math]

 

and

 

[math]x' = \frac{x - v\,t }{\sqrt{1-v^2/c^2}}\ [/math]

 

where they have a relative velocity of v along the x axis.

[vuquta]

You're looking for an equation? You can translate between the coordinates of two different reference frames with the Lorentz transformations:

 

[math] t' = \frac{t - {v\,x/c^2}}{\sqrt{1-v^2/c^2}}\ [/math]

 

and

 

[math]x' = \frac{x - v\,t }{\sqrt{1-v^2/c^2}}\ [/math]

 

where they have a relative velocity of v along the x axis.

 

Yes, I agree.

 

But, that is not what I am asking.

 

It is all agreed (r/c,-r,0,0) and (r/c,r,0,0) will not be simultaneous in the moving frame.

 

But, what I am asking since (r/c,-r,0,0) and (r/c,r,0,0) will not be simultaneous in the moving frame, then what points are?

[Sisyphus]

Any two sets of values for {t, x} with equal t1.

[swansont]

Any two sets of values for {t, x} with equal t1.

 

Right. Just do a reverse transform.

[Janus]

Assume the typical co-location of origins and a light pulse at the co-location with two frames in relative motion.

 

The Relativity of Simultaneity contends if two points are simultaneous in the rest frame, then these points will not be simultaneous in the moving frame.

 

Both postulates of SR contend light is always spherical from the light emission point in the moving frame and any point with the associated time value can be translated into the coordinates of the rest frame.

 

So, here is the question. Just consider the x-axis.

 

Say two points R1 and R2 are simultaneous in the rest frame. What two points in the moving frame when translated to the rest frame coordinates are simultaneous to the moving frame when R1 and R2 are simultaneous in the rest frame?

Your question is meaningless.

 

Here's why:

 

Look at this animation from the embankment frame of the standard train example for the Relativity of Simultaneity:

 

 

Call the event of the the front of the train being even with the right red dot "Event A"

Call the event of the the rear of the train being even with the left red dot "Event B"

 

As shown in the animation, these events are simultaneous in the embankment frame.

 

Now look at the animation for the same events from the train's frame.

 

 

Note that events A and B are not simultaneous and happen at different instants.

 

Now you question is:

"What two points in the moving frame when translated to the rest frame coordinates are simultaneous to the moving frame when R1 and R2 are simultaneous in the rest frame?"

 

But that assumes that there is some moment in the Train frame that corresponds to the moment A and B are simultaneous in the embankment frame. But there isn't.

There are single events that correspond to single events in the embankment frame. Event A and Event B both happen. and even though they both happen at the the "moment of simultaneity" in the embankment frame, you cannot transform them into a single "moment" in the train frame.

 

You could pick one or the other, but which one is more representative of the "moment of the simultaneity"? And whichever one you pick will have different events that are simultaneous to it in the train frame.

 

To make things worse, there aren't just two events that are simultaneous in the embankment frame when A an B are simultaneous, there are an infinite number of them. Every point along the train is even with some point of the embankment when A and B occur in the embankment frame, and each of those co-locations are an "event". Events that will not be simultaneous to each other in the train frame. And each one of those events could make an equal claim to being representative of the "moment of simultaneity" of the embankment. And each of them have different events that are simultaneous to them in the train frame.

 

So to put it simply, your question has no answer because what you are asking for is disallowed by the very nature of the scenario.

[vuquta]

Your question is meaningless.

 

Here's why:

 

Look at this animation from the embankment frame of the standard train example for the Relativity of Simultaneity:

 

 

Call the event of the the front of the train being even with the right red dot "Event A"

Call the event of the the rear of the train being even with the left red dot "Event B"

 

As shown in the animation, these events are simultaneous in the embankment frame.

 

Now look at the animation for the same events from the train's frame.

 

 

Note that events A and B are not simultaneous and happen at different instants.

 

Now you question is:

"What two points in the moving frame when translated to the rest frame coordinates are simultaneous to the moving frame when R1 and R2 are simultaneous in the rest frame?"

 

But that assumes that there is some moment in the Train frame that corresponds to the moment A and B are simultaneous in the embankment frame. But there isn't.

There are single events that correspond to single events in the embankment frame. Event A and Event B both happen. and even though they both happen at the the "moment of simultaneity" in the embankment frame, you cannot transform them into a single "moment" in the train frame.

 

You could pick one or the other, but which one is more representative of the "moment of the simultaneity"? And whichever one you pick will have different events that are simultaneous to it in the train frame.

 

To make things worse, there aren't just two events that are simultaneous in the embankment frame when A an B are simultaneous, there are an infinite number of them. Every point along the train is even with some point of the embankment when A and B occur in the embankment frame, and each of those co-locations are an "event". Events that will not be simultaneous to each other in the train frame. And each one of those events could make an equal claim to being representative of the "moment of simultaneity" of the embankment. And each of them have different events that are simultaneous to them in the train frame.

 

So to put it simply, your question has no answer because what you are asking for is disallowed by the very nature of the scenario.

 

Well it has an answer.

 

If we apply the light postulate in the stationary frame, meaning,

 

 

[math] \sqrt{x^2+y^2+z^2} = ct [/math]

 

And use the following set of points,

 

[math] \frac{(x - vr\gamma/c)^2}{(r\gamma)^2} + \frac{y^2}{r^2} + \frac{z^2}{r^2} = 1 [/math]

 

We will find for each intersection of the two,

 

t' = r/c and

 

 

[math] x'^2+y^2+z^2 = (ct')^2=r^2 [/math]

 

Whence, based on the light postulate in the stationary frame, we have the light sphere of radius r in the moving frame using the equation for the ellipsoid above in the coordinates of the stationary frame while at the same time, the light sphere in the stationary frame emerges spherically from (0,0,0).

Edited August 4, 2010 by vuquta