Deriving Equations for Light Sphere in Collinear Motion - O and O' Observers

[JesseM]

Set v = c/√ 2

OK, so t=d/2c.

 

[cfrogue]

OK, so t=d/2c.

the frame has moved vt, where is the center of the light sphere in O' in the coords of O?

 

[JesseM]

the frame has moved vt, where is the center of the light sphere in O' in the coords of O?

As I mentioned before, you made things more complicated by requiring that the light sphere reached x'=d/2 at t'=0, this means that the original flash (the tip of the light cone) must have happened at x'=0, t'=-d/(2c). Translating this into frame O:

x = gamma*(x' + vt) = -gamma*d*v/(2c). With v = c/sqrt(2) this becomes:
-(1/sqrt(1/2))*d/(2*sqrt(2)) = -d/2

t = gamma*(t' + vx'/c^2) = -gamma*d/(2c). With v=c/sqrt(2) this becomes:
-(1/sqrt(1/2))*d/(2c) = -d/(c*sqrt(2))

So in frame O, the flash happened at x=-d/2, t=-d/(c*sqrt(2)). If you want to figure out the coordinates in O of an object which remains at the center of the light sphere in O', you'd need an object with velocity v which passes through those coordinates, so its position as a function of time would be:

x(t) = vt - v*(-d/(c*sqrt(2))) - d/2

So at time t = d/(2c), this object would be at:

x = vd/(2c) + vd/(c*sqrt(2)) - d/2 = vd/(2c) + (vd*sqrt(2))/(2c) - dc/2c
= d*[v*(1 + sqrt(2)) - c]/2c

 

[cfrogue]

As I mentioned before, you made things more complicated by requiring that the light sphere reached x'=d/2 at t'=0, this means that the original flash (the tip of the light cone) must have happened at x'=0, t'=-d/(2c). Translating this into frame O:

x = gamma*(x' + vt) = -gamma*d*v/(2c). With v = c/sqrt(2) this becomes:
-(1/sqrt(1/2))*d/(2*sqrt(2)) = -d/2

t = gamma*(t' + vx'/c^2) = -gamma*d/(2c). With v=c/sqrt(2) this becomes:
-(1/sqrt(1/2))*d/(2c) = -d/(c*sqrt(2))

So in frame O, the flash happened at x=-d/2, t=-d/(c*sqrt(2)). If you want to figure out the coordinates in O of an object which remains at the center of the light sphere in O', you'd need an object with velocity v which passes through those coordinates, so its position as a function of time would be:

x(t) = vt - v*(-d/(c*sqrt(2))) - d/2

So at time t = d/(2c), this object would be at:

x = vd/(2c) + vd/(c*sqrt(2)) - d/2 = vd/(2c) + (vd*sqrt(2))/(2c) - dc/2c
= d*[v*(1 + sqrt(2)) - c]/2c

How do you figure x=-d/2, t=-d/(c*sqrt(2)).

The flash happened at the center of O when x = 0.


Let's keep this simple.

Only look at O for the time being.

The flash occurred at the origin and proceeds spherically and stikes the points r, -r at time r/c.

That is a fact.

Now, we looked at a general equation when O calculated O' saw its points struck at the same time.

We did that. Here is the equation.
$$t = \frac{d}{2c} \frac{1}{\sqrt{c^2/v^2 - 1}}$$
OK, that is when O believes O' sees the simultaneous strike.

I said set v = c/√2.

Guess what, they occur are at the same time in O.

 

[JesseM]

How do you figure x=-d/2, t=-d/(c*sqrt(2)).

The flash happened at the center of O when x = 0.

Not if you wanted to have it so the flash happened at the center of O', and reached the right endpoint at x'=d/2 at time t'=0 in O'. In that case the flash cannot have happened at x=0 in O.

Let's keep this simple.

Only look at O for the time being.

The flash occurred at the origin and proceeds spherically and stikes the points r, -r at d/r.

That is a fact.

OK, then when you translate into O', it won't be true that the light reached x'=d/2 at time t'=0. Everything will be simpler if you assume the flash happened at the spacetime origin, in which case it reaches the right endpoint of the rod at rest in O' at time t'=d/(2c). That would mean you'd have to revise your calculation of the time t in frame O that the light reached the right endpoint of the rod at rest in O'.

Now, we looked at a general equation when O calculated O' saw its points struck at the same time.

We did that. Here is the equation.
$$t = \frac{d}{2c} \frac{1}{\sqrt{c^2/v^2 - 1}}$$
OK, that is when O believes O' sees the simultaneous strike.

That equation was based on the assumption that the light reached the right endpoint of the rod at rest in O' at time t'=0 in O', which means if you want the flash to have happened at the center of this rod at x'=0 in O', it must have happened at t'=-d/(2c), which means (by applying the Lorentz transformation to x'=0, t'=-d/(2c)) it did not happen at x=0 in O.

I suppose you could drop the assumption that the flash happened at the center of the rod at x'=0 in O'. Then if we say it hit the right endpoint at $$t = \frac{d}{2c} \frac{1}{\sqrt{c^2/v^2 - 1}}$$ in O, and we plug in v=c/sqrt(2) so t=d/(2c). And in frame O the right endpoint has x(t) = d/(2*gamma) + vt, so at t=d/(2c) it is at:
x = d/(2*gamma) + vd/(2c)

And with v=c/sqrt(2) and gamma=sqrt(2), this becomes:

x = d/(2*sqrt(2)) + d/(2*sqrt(2)) = d/sqrt(2)

So if we want the flash to have happened at x=0, it must have happened at a time d/(c*sqrt(2)) earlier than the moment the light reached the right end in frame O, meaning in frame O the flash happened at t = d/(2c) - d/(c*sqrt(2)) = d/(2c) - (d*sqrt(2))/(2c) = d*(1 - sqrt(2))/(2c). But of course if the flash happened at x=0 and t=d*(1 - sqrt(2))/(2c), that means it did not happen at x'=0 in frame O'.

I said set v = c/√2.

Guess what, they occur are at the same time in O.

What do you mean "they"? You didn't calculate the time of the light hitting the left endpoint in O.

 

[cfrogue]

Not if you wanted to have it so the flash happened at the center of O', and reached the right endpoint at x'=d/2 at time t'=0 in O'. In that case the flash cannot have happened at x=0 in O.

So, when did it happen?

We have no distance to consider.

OK, then when you translate into O', it won't be true that the light reached x'=d/2 at time t'=0. Everything will be simpler if you assume the flash happened at the spacetime origin, in which case it reaches the right endpoint of the rod at rest in O' at time t'=d/(2c). That would mean you'd have to revise your calculation of the time t in frame O that the light reached the right endpoint of the rod at rest in O'.

The light flashed in the frame of O at the origin. That is the experiment and you are saying something different.

 

[cfrogue]

That equation was based on the assumption that the light reached the right endpoint of the rod at rest in O' at time t'=0 in O', which means if you want the flash to have happened at the center of this rod at x'=0 in O', it must have happened at t'=-d/(2c), which means (by applying the Lorentz transformation to x'=0, t'=-d/(2c)) it did not happen at x=0 in O.

What??

And,
Now to the origin of one of the two systems (k) let a constant velocity v be imparted in the direction of the increasing x of the other stationary system (K), and let this velocity be communicated to the axes of the co-ordinates, the relevant measuring-rod, and the clocks. To any time of the stationary system K there then will correspond a definite position of the axes of the moving system,
http://www.fourmilab.ch/etexts/einstein/specrel/www/

Thus, the above is not correct.

 

[JesseM]

The light flashed in the frame of O at the origin. That is the experiment and you are saying something different.

Did it happen at t=0, or do you allow it to have happened at an earlier time in O? If it happened at x=0 and t=0 in O, then by the Lorentz transformation it also happened at x'=0 and t'=0 in O', which means it's impossible that the light from the flash arrived at x'=d/2 at t'=0 in O'. On the other hand, if you're willing to drop the assumption that the flash happened at t=0 in O, and also drop the assumption that it happened at x'=0 in O', then it can work...as I showed, if you start with these assumptions:

1. The flash happened at x=0 in O
2. the light from the flash reached x'=d/2 at t'=0 in O'
3. v = c/sqrt(2)

...then you get the conclusion that the flash happened at t= - d*(sqrt(2) - 1)/(2c) in O.

 

[JesseM]

What??

And,
Now to the origin of one of the two systems (k) let a constant velocity v be imparted in the direction of the increasing x of the other stationary system (K), and let this velocity be communicated to the axes of the co-ordinates, the relevant measuring-rod, and the clocks. To any time of the stationary system K there then will correspond a definite position of the axes of the moving system,
http://www.fourmilab.ch/etexts/einstein/specrel/www/

Thus, the above is not correct.

"Thus"? You're not making any sense, there's no reason the highlighted sentence contradicts my own statement that "(by applying the Lorentz transformation to x'=0, t'=-d/(2c))".

 

[cfrogue]

Did it happen at t=0, or do you allow it to have happened at an earlier time in O? If it happened at x=0 and t=0 in O, then by the Lorentz transformation it also happened at x'=0 and t'=0 in O', which means it's impossible that the light from the flash arrived at x'=d/2 at t'=0 in O'. On the other hand, if you're willing to drop the assumption that the flash happened at t=0 in O, and also drop the assumption that it happened at x'=0 in O', then it can work...as I showed, if you start with these assumptions:

1. The flash happened at x=0 in O
2. the light from the flash reached x'=d/2 at t'=0 in O'
3. v = c/sqrt(2)

...then you get the conclusion that the flash happened at t= - d*(sqrt(2) - 1)/(2c) in O.

Let's see.

What stops us from synching the clocks at zero with the light flash since we have collinear relative motion?

Sure, any angled motion would prevent this.

 

[cfrogue]

"Thus"? You're not making any sense, there's no reason the highlighted sentence contradicts my own statement that "(by applying the Lorentz transformation to x'=0, t'=-d/(2c))".

ok, we then know we can sync the positions to zero, agreed?

 

[JesseM]

Let's see.

What stops us from synching the clocks at zero with the light flash since we have collinear relative motion?

Sure, any angled motion would prevent this.

Of course you can synch the clocks so that the flash happens at t=0 in frame O and t'=0 in frame O'. But if the flash happens at t'=0 in frame O', how can the light from the flash be at position x=d/2 at t'=0 in frame O'? Light can only expand away from the flash at c in frame O', just like any other frame. Are you suggesting the flash itself happened on the endpoint of the rod in O' at t'=0?

 

[JesseM]

ok, we then know we can sync the positions to zero, agreed?

If you want to set the positions and times so the flash happens at x'=0 and t'=0 in frame O', then it's not possible for the light from the flash to have reached x'=d/2 at t'=0, the light's position as a function of time in the right direction would have to be x'(t')=ct' so it wouldn't reach x'=d/2 until t'=d/(2c).

 

[cfrogue]

Of course you can synch the clocks so that the flash happens at t=0 in frame O and t'=0 in frame O'. But if the flash happens at t'=0 in frame O', how can the light from the flash be at position x=d/2 at t'=0 in frame O'? Light can only expand away from the flash at c in frame O', just like any other frame. Are you suggesting the flash itself happened on the endpoint of the rod in O' at t'=0?

All I want to know right now is can we sync the clocks and the positions.

All evidence says we can.

 

[JesseM]

All I want to know right now is can we sync the clocks and the positions.

All evidence says we can.

I don't know what you mean by "sync the clocks and the positions", you're speaking too vaguely. There is no way to sync the clocks and positions in frame O' so that it is both true that the flash happened at x'=0 and t'=0 and that the light from the flash reached x'=d/2 at time t'=0. On the other hand, if you're just asking whether we can construct an inertial coordinate system out of clocks and rulers of the type Einstein described for frame O', of course we can, and we can construct another one for frame O such that x'=0 and t'=0 in O' lines up with x=0 and t=0 in O.

 

[cfrogue]

I don't know what you mean by "sync the clocks and the positions", you're speaking too vaguely. There is no way to sync the clocks and positions in frame O' so that it is both true that the flash happened at x'=0 and t'=0 and that the light from the flash reached x'=d/2 at time t'=0. On the other hand, if you're just asking whether we can construct an inertial coordinate system out of clocks and rulers of the type Einstein described for frame O', of course we can, and we can construct another one for frame O such that x'=0 and t'=0 in O' lines up with x=0 and t=0 in O.

Alright, do you agree we can sync the positions?

Now if O' syncs t to zero, and so does O, what is the time difference to make these two start the event at time t = 0 in both?

 

[JesseM]

Alright, do you agree we can sync the positions?

Sync the positions of what? Again, are you just asking whether we can have x=0 and t=0 in O match up with x'=0 and t'=0 in O'? If so, that's what I just said:

if you're just asking whether we can construct an inertial coordinate system out of clocks and rulers of the type Einstein described for frame O', of course we can, and we can construct another one for frame O such that x'=0 and t'=0 in O' lines up with x=0 and t=0 in O.

But if you mean something different than "sync the positions", you have to actually explain what you're talking about rather than speaking so cryptically.

Now if O' syncs t to zero, and so does O, what is the time difference to make these two start the event at time t = 0 in both?

Time difference between what and what? And when you say "start the event", start what event? The original flash of light?

 

[cfrogue]

Sync the positions of what? Again, are you just asking whether we can have x=0 and t=0 in O match up with x'=0 and t'=0 in O'? If so, that's what I just said:

So, Yes we can?

 

[JesseM]

So, Yes we can?

Yes, we can sync things up so x=0 and t=0 in O matches up with x'=0 and t'=0 in O'. I was already assuming these events matched up in all my previous calculations (since it's a requirement for them to match up in order for the Lorentz transformation to work).

 

[cfrogue]

Yes, we can sync things up so x=0 and t=0 in O matches up with x'=0 and t'=0 in O'. I was already assuming these events matched up in all my previous calculations (since it's a requirement for them to match up in order for the Lorentz transformation to work).

OK, post 334 is based purely on LT and not opinions.

I do not see you matching all the consequences.

Post 334 uses LT to give a stunning conclusion. You and I both derived this with you being the major player.

How do you square all this with your "new" assertions?

 

[cfrogue]

Yes, we can sync things up so x=0 and t=0 in O matches up with x'=0 and t'=0 in O'. I was already assuming these events matched up in all my previous calculations (since it's a requirement for them to match up in order for the Lorentz transformation to work).

BTW, this is a song I like.

http://www.youtube.com/watch?v=okUrGG66KpQ"

This means, there is time for enjoying.

 

[JesseM]

OK, post 334 is based purely on LT and not opinions.

I do not see you matching all the consequences.

Post 334 uses LT to give a stunning conclusion. You and I both derived this with you being the major player.

In 334 you only showed that in frame O, the light would reach the right endpoint at t=d/(2c), given your previous assumptions. Nothing very stunning about this. Then you said "Guess what, they occur are at the same time in O" but you never explained what "they" was supposed to refer to.

How do you square all this with your "new" assertions?

Are you actually going to give any thought to those assertions or do you just plan to dismiss them in a knee-jerk way? Tell me whether you agree or disagree with the following:

1. If the flash happened at x'=0 and t'=0 in frame O', then it is impossible the light from the flash could reach x'=d/2 at t'=0 in frame O'.

2. If the light from the flash reached x'=d/2 at t'=0 in frame O', then the only way the flash could have happened at x'=0 in frame O' would be if at happened at time t'=-d/(2c).

3. If the light reached the right endpoint at t=d/(2c) in frame O, then since the right endpoint had a position as a function of time given by x(t)=d/(2*gamma) + vt in frame O, this means that in frame O the position of the right endpoint when the light reached it must have been x = d/(2*gamma) + vd/(2c). If we plug in v=c/sqrt(2) and gamma=sqrt(2) this becomes x=d/sqrt(2).

4. If the light reached the right endpoint at x=d/sqrt(2), t=d/(2c) in frame O, then if the flash happened at x=0 in frame O, the flash must have happened at a time d/(c*sqrt(2)) earlier than the time it reached the position x=d/sqrt(2). So, if the flash happened at x=0 in frame O, this implies it happened at time t=d/(2c) - d/(c*sqrt(2))=d*(1 - sqrt(2))/(2c)

Please think about and consider each of these and tell me whether you agree or disagree with each; if you disagree with any, please explain why.

 

[cfrogue]

In 334 you only showed that in frame O, the light would reach the right endpoint at t=d/(2c), given your previous assumptions. Nothing very stunning about this. Then you said "Guess what, they occur are at the same time in O" but you never explained what "they" was supposed to refer to.

Well, it is axiomatic that t=d/(2c) in O.

However, we are looking at O' and the simultaneity of it with regard to the left and right points hit of the rod

We did that. Here is the equation.
$$t = \frac{d}{2c} \frac{1}{\sqrt{c^2/v^2 - 1}}$$
OK, that is when O believes O' sees the simultaneous strike.

I said set v = c/√2.

Guess what, they occur are at the same time in O as does the strikes of the endpoints of O'.

 

[cfrogue]

Are you actually going to give any thought to those assertions or do you just plan to dismiss them in a knee-jerk way? Tell me whether you agree or disagree with the following:

1. If the flash happened at x'=0 and t'=0 in frame O', then it is impossible the light from the flash could reach x'=d/2 at t'=0 in frame O'.

2. If the light from the flash reached x'=d/2 at t'=0 in frame O', then the only way the flash could have happened at x'=0 in frame O' would be if at happened at time t'=-d/(2c).

3. If the light reached the right endpoint at t=d/(2c) in frame O, then since the right endpoint had a position as a function of time given by x(t)=d/(2*gamma) + vt in frame O, this means that in frame O the position of the right endpoint when the light reached it must have been x = d/(2*gamma) + vd/(2c). If we plug in v=c/sqrt(2) and gamma=sqrt(2) this becomes x=d/sqrt(2).

4. If the light reached the right endpoint at x=d/sqrt(2), t=d/(2c) in frame O, then if the flash happened at x=0 in frame O, the flash must have happened at a time d/(c*sqrt(2)) earlier than the time it reached the position x=d/sqrt(2). So, if the flash happened at x=0 in frame O, this implies it happened at time t=d/(2c) - d/(c*sqrt(2))=d*(1 - sqrt(2))/(2c)

Please think about and consider each of these and tell me whether you agree or disagree with each; if you disagree with any, please explain why.

I am really not concerned with all this.


There is a condition where all of this is false according to LT.

What I think you should do is assert that the conclusions of LT we went through are false.

 

[JesseM]

Well, it is axiomatic that t=d/(2c) in O.

Um, no it isn't. The rod is at rest in O', not O, remember?

However, we are looking at O' and the simultaneity of it with regard to the left and right points hit of the rod

In O' the strikes occurred simultaneously (as you said in post 319) and at t'=0 (as you said in post 322), that was part of the original conditions of the problem! It was when and where they occurred in O that you were wondering about.

We did that. Here is the equation.
$$t = \frac{d}{2c} \frac{1}{\sqrt{c^2/v^2 - 1}}$$

No, that was the equation for what time t in frame O the light would strike the right endpoint of the rod at rest in O', given that it reached the right endpoint at t'=0 and x'=d/2 in O'. You never calculated anything about when the light would reach the left endpoint in O.

I said set v = c/√2.

Guess what, they occur are at the same time in O as does the strikes of the endpoints of O'.

No, the time of the strikes in O' was t'=0. Here are your words from post #322:

Please look at what you gave me.

I have changed it at the end.

It will strike the right endpoint at t = ( t' + vx'/(c^2))λ in O'.

t'=0 and x'=d/2 because the rod length is d and the light is centered.

So, t = vdλ/(2*c^2)

t = (d/(2c)) (v/c)λ
t = (d/(2c)) sqrt(1/((c/v)^2 - 1))

How does the last equation look?

Perhaps you should go back and reread the discussion from post #322 on to refresh your memory.

 

[atyy]

Well, it is axiomatic that t=d/(2c) in O.

However, we are looking at O' and the simultaneity of it with regard to the left and right points hit of the rod

We did that. Here is the equation.
$$t = \frac{d}{2c} \frac{1}{\sqrt{c^2/v^2 - 1}}$$
OK, that is when O believes O' sees the simultaneous strike.

I said set v = c/√2.

Guess what, they occur are at the same time in O as does the strikes of the endpoints of O'.

No. The formula for t(R') is only correct if (x'(R')=d/2, t'(R')=0), which cannot be the case if the light is emitted at (x=x'=0, t=t'=0). You made the same error back in #334, and JesseM corrected your error in #335:

Now, we looked at a general equation when O calculated O' saw its points struck at the same time.

We did that. Here is the equation.
$$t = \frac{d}{2c} \frac{1}{\sqrt{c^2/v^2 - 1}}$$
OK, that is when O believes O' sees the simultaneous strike.

I said set v = c/√2.

Guess what, they occur are at the same time in O.

That equation was based on the assumption that the light reached the right endpoint of the rod at rest in O' at time t'=0 in O', which means if you want the flash to have happened at the center of this rod at x'=0 in O', it must have happened at t'=-d/(2c), which means (by applying the Lorentz transformation to x'=0, t'=-d/(2c)) it did not happen at x=0 in O.

I suppose you could drop the assumption that the flash happened at the center of the rod at x'=0 in O'. Then if we say it hit the right endpoint at $$t = \frac{d}{2c} \frac{1}{\sqrt{c^2/v^2 - 1}}$$ in O, and we plug in v=c/sqrt(2) so t=d/(2c). And in frame O the right endpoint has x(t) = d/(2*gamma) + vt, so at t=d/(2c) it is at:
x = d/(2*gamma) + vd/(2c)

And with v=c/sqrt(2) and gamma=sqrt(2), this becomes:

x = d/(2*sqrt(2)) + d/(2*sqrt(2)) = d/sqrt(2)

So if we want the flash to have happened at x=0, it must have happened at a time d/(c*sqrt(2)) earlier than the moment the light reached the right end in frame O, meaning in frame O the flash happened at t = d/(2c) - d/(c*sqrt(2)) = d/(2c) - (d*sqrt(2))/(2c) = d*(1 - sqrt(2))/(2c). But of course if the flash happened at x=0 and t=d*(1 - sqrt(2))/(2c), that means it did not happen at x'=0 in frame O'.

http://www.youtube.com/watch?v=7pi2_tgCSJQ"

 

[atyy]

No, the time of the strikes in O' was t'=0. Here are your words from post #322:

t'=0 and x'=d/2 because the rod length is d and the light is centered.

I didn't understand what cfrogue meant by "because the rod length is d and the light is centered", do you?

Edit: I understand it explains x'=d/2, but t'=0?

 

[JesseM]

I didn't understand what cfrogue meant by "because the rod length is d and the light is centered", do you?

Edit: I understand it explains x'=d/2, but t'=0?

Yes, I figured that just referred to the x'=d/2 part, if not I don't understand the reasoning either. But either way, I just took it as part of the statement of the problem that the light reached the right end of the rod at rest in O' at x'=d/2, t'=0.

 

[atyy]

It will strike the right endpoint at t = ( t' + vx'/(c^2))λ in O'.

t'=0 and x'=d/2 because the rod length is d and the light is centered.

What do you mean by "because the rod length is d and the light is centered". Is that supposed to explain why t'=0?

 

[Dale]

There are 5 events in question:
1) flash: t = t' = 0, x = x' = 0
2) light hits right end of unprimed rod: t = d/(2c), x = d/2
3) light hits left end of unprimed rod: t = d/(2c), x = -d/2
4) light hits right end of primed rod: t' = d/(2c), x' = d/2
5) light hits left end of primed rod: t' = d/(2c), x' = -d/2