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

[JesseM]

I cannot see that.

They are struck at the same time in O', no?

Sorry I got my t' and t mixed up, I should have written that in frame O, the strike at the left end happens at t = +gamma*v*d/(2*c^2) and the strike at the right end occurs at t = -gamma*v*d/(2*c^2), so the time between them in frame O is gamma*v*d/c^2.

 

[cfrogue]

Sorry I got my t' and t mixed up, I should have written that in frame O, the strike at the left end happens at t = +gamma*v*d/(2*c^2) and the strike at the right end occurs at t = -gamma*v*d/(2*c^2), so the time between them in frame O is gamma*v*d/c^2.

OK, this difference can be written as

(d/(2c)) ( 2*gamma*v/c )

Thus, this occurs before the stike in O at slow speeds since a rod in O is struck at t = d/(2c)

Something is wrong with this

 

[JesseM]

OK, this difference can be written as

(d/(2c)) ( 2*gamma*v/c )

Thus, this occurs before the stike in O at slow speeds since a rod in O is struck at t = d/(2c)

Something is wrong with this

Huh? What "strike in O" are you talking about? I thought there were only two strikes at either end of the rod at rest in O', are you imagining a third strike at one end of a rod at rest in O?

 

[cfrogue]

Sorry I got my t' and t mixed up, I should have written that in frame O, the strike at the left end happens at t = +gamma*v*d/(2*c^2) and the strike at the right end occurs at t = -gamma*v*d/(2*c^2), so the time between them in frame O is gamma*v*d/c^2.

Let me tell you what I have been thinking about.

The right end point of the rod in O' crosses the right end point of the stationary frame at time

vt + r/λ = r
vt = r ( 1 - 1/λ )
t = (r/v) ( 1 - 1/λ )

I also know, the center point of the light sphere is located at vt at any time t in O for the light sphere in O'

So, what I am missing is the time in O when the simultaneous strikes occur in O'.

This will give me a detailed analysis of the light sphere in O' from the coords of O.

I am having a mental block on the time in O when the simultaneous strikes occur in O'.


[Edited to add: There are two rods. One in O and one in O'. They have the same rest length in the respective frames of d. When they are centered, the light source centered on the O' rod, emits a light.]

 

[cfrogue]

Huh? What "strike in O" are you talking about? I thought there were only two strikes at either end of the rod at rest in O', are you imagining a third strike at one end of a rod at rest in O?

I gave you a detailed objective in the next post.

 

[JesseM]

Let me tell you what I have been thinking about.

The right end point of the rod in O' crosses the right end point of the stationary frame at time

vt + r/λ = r
vt = r ( 1 - 1/λ )
t = (r/v) ( 1 - 1/λ )

OK

I also know, the center point of the light sphere is located at vt at any time t in O.

The point that marks the center of the light sphere according O' is located at vt in O, but the center of the light sphere according to O is always at x=0 at any time t in O. Do you disagree?

So, what I am missing is the time in O when the simultaneous strikes occur in O'.

I already gave you that, assuming the strikes happened at t'=0 in O'. Or do you want the strikes to happen at the moment the light sphere reaches the ends of the rod at rest in O', assuming the original flash that created the sphere happened at t'=0 and x'=0 in O'? Or do you want the strikes to happen at the moment the light sphere reaches the ends of the rod in O', but you want to time the original flash so that the light sphere reaches the right end of the rod in O' at the same moment it reaches the right end of the rod in O? Or something else? You really need to explain what you're thinking in clearer terms.

 

[cfrogue]

The point that marks the center of the light sphere according O' is located at vt in O, but the center of the light sphere according to O is always at x=0 at any time t in O. Do you disagree?

Yes, I agree what O sees is a light sphere origined at x = 0.

I am trying to completely map how it proceeds in O' in terms of the time and measurements of O

I already gave you that, assuming the strikes happened at t'=0 in O'. Or do you want the strikes to happen at the moment the light sphere reaches the ends of the rod at rest in O', assuming the original flash that created the sphere happened at t'=0 and x'=0 in O'? Or do you want the strikes to happen at the moment the light sphere reaches the ends of the rod in O', but you want to time the original flash so that the light sphere reaches the right end of the rod in O' at the same moment it reaches the right end of the rod in O? Or something else? You really need to explain what you're thinking in clearer terms.

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?

 

[Dale]

What do you get for the time dilation for this?

x = λvt/ (1 + λ)

Note, y > 0 for this case and z = 0.

I'm not sure what you mean. Is x = λvt/ (1 + λ) the equation of the worldline of some object? If so, what time coordinates do you wish to evaluate it at?

 

[cfrogue]

I'm not sure what you mean. Is x = λvt/ (1 + λ) the description of the worldline of some object? If so, what time coordinates do you wish to evaluate it at?

It is an event, but y >> 0.

What is t'?

I come up with t' = t with v > 0.

 

[JesseM]

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.

You want the light to strike the right endpoint at t'=0? That would imply that the original flash was set off at t'=-d/2c, correct? This will make the problem more complicated, since the center of the light sphere will not be at x=0 in frame O...it would be easier if you had the light flash set off at the spacetime origin of both frames, so the light would reach the right end at t'=d/(2c) in frame O'.

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?

(v/c)*gamma = sqrt(v^2/c^2)/sqrt(1 - v^2/c^2) = 1/[sqrt(c^2/v^2)*sqrt(1 - v^2/c^2)] = 1/sqrt(c^2/v^2 - 1). That's not the same as the factor after (d/(2c)) in your last equation.

 

[Dale]

It is an event, but y >> 0.

What is t'?

I come up with t' = t with v > 0.

x = λvt/ (1 + λ) is the equation of a worldline in spacetime (has the form t = mx + b). It can't be an event. An event is the equivalent of a point in spacetime, e.g. the intersection of two worldlines.

 

[cfrogue]

You want the light to strike the right endpoint at t'=0? That would imply that the original flash was set off at t'=-d/2c, correct?

(v/c)*gamma = sqrt(v^2/c^2)/sqrt(1 - v^2/c^2) = 1/[sqrt(c^2/v^2)*sqrt(1 - v^2/c^2)] = 1/sqrt(c^2/v^2 - 1). That's not the same as the factor after (d/(2c)) in your last equation.

Well, that is what I meant, I gess I left something off.

To make sure, this is what I intended and I want us to agree.

t = ( d/(2c) ) ( √(1 / ((c/v)² - 1) ) )

 

[JesseM]

Well, that is what I meant, I gess I left something off.

To make sure, this is what I intended and I want us to agree.

t = ( d/(2c) ) ( √(1 / ((c/v)² - 1) ) )

Yes, that's the same as what I got.

 

[cfrogue]

Yes, that's the same as what I got.

$$t = (\frac{d}{2c}) \frac{1}{\sqrt{c^2/v^2 - 1}}$$

OK, I did it in this format.

Is this correct?

How do you enlarge these things?

 

[cfrogue]

Yes, that's the same as what I got.

Set v = c/√ 2

 

[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?