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

[cfrogue]

I, and others better than I, have tried repeatedly, but still haven't given up yet.

But I would just read Einstein's 1905 paper for this. It's not the only source out there, and maybe not the best, but it's certainly more than good enough for this topic.

Yes, may I see the math please?

 

[Al68]

Well, actually if O' were a rigid body sphere, all points are struck at the same time, the same as O.

So, yes, all the points of O' are struck at the same time.

Say, do you have the math to refute the light postulate in O'?

There's a big difference in saying that all points in O' are struck at the same time and saying only that all points on a sphere in O' are struck at the same time.

The light postulate says all points on the sphere in O', not all points in O', would be hit at the same time in O'.

I have to assume that's what you really meant.

 

[cfrogue]

There's a big difference in saying that all points in O' are struck at the same time and saying only that all points on a sphere in O' are struck at the same time.

The light postulate says all points on the sphere in O', not all points in O', would be hit at the same time in O'.

I have to assume that's what you really meant.

Yes, that is what I meant. Just the surface of the sphere would O' see see simultaneity.

Thanks

 

[Al68]

Yes, may I see the math please?

Sure. What specifically do you want to see the math for?

 

[cfrogue]

Sure. What specifically do you want to see the math for?

Well, you said there exists two points in O that are simultaneous in O'

Can I see this?

 

[atyy]

I used the light postulate to conclude in O' x +- ct' are simultaneous.
inxs - elegantly wasted

Thanks for the music! Yes, x'= +-ct' means for one t' there are two x's. You omitted the +- in your derivation below. In the final sentence there you refer to "points of O' are struck at the same time". However, by omitting the +- in your derivation, you are excluding one point. Since there are only two points on the x' axis for fixed t', only one point remains. So the plural "points" is not justified.

I proceed by reductio ad absurdum there exists only time t in O for this t'.

Assume there exists a tx < t such that the points in O' are struck at the same time.
Then tx < r/(λ(c-v))
tx = (t' + vx'/c^2)λ < r/(λ(c-v))
We have by the SR spherical light sphere,
ct' = x', t' = x'/c
Also, by selection x' = r.
( x'/c + vx'/c^2)λ < r/(λ(c-v))
(r/c + rv/c^2)λ < r/(λ(c-v))
(1/c + 1v/c^2)λ < 1/(λ(c-v))
((c + v)/c^2)λ < 1/(λ(c-v))
(c + v) < c^2/(λ^2(c-v))
(c + v) < (c^2/(c-v))((c^2 - v^2)/c^2)
c + v < (c^2 - v^2)/(c - v)
c + v < c + v
0 < 0

This is a contradiction. The same argument hold for tx > t.
Thus, the calculated t is the unique time in O when the points of O' are struck at the same time.

 

[Al68]

Well, you said there exists two point in O that are simultaneous in O'

Can I see this?

I never said that. "Points" can't be simultaneous, events can be. I said that two events simultaneous in O' (light reaching either rod end) occur at two different times in O.

For that single t' in O', t = gamma(t' - vx'/c^2). Since there are two different values for x', there will be two different values for t.

 

[cfrogue]

Thanks for the music! Yes, x'= +-ct' means for one t' there are two x's. You omitted the +- in your derivation below. In the final sentence there you refer to "points of O' are struck at the same time". However, by omitting the +- in your derivation, you are excluding one point. Since there are only two points on the x' axis for fixed t', only one point remains. So the plural "points" is not justified.

I do not need to include "You omitted the +- in your derivation below" +-, because the light postulate says the points are simultaneous.

This is axiomatic and I do not need to prove it.

So, given the simultaneity by the light postulate, I need only consider one point.

 

[cfrogue]

I never said that. "Points" can't be simultaneous, events can be. I said that two events simultaneous in O' (light reaching either rod end) occur at two different times in O.

For that single t' in O', t = gamma(t' - vx'/c^2). Since there are two different values for x', there will be two different values for t.

It is OK to have two different values for t.



It is not OK to have two different values for the simultaneity of O' ie, two different t'.

You said it was.

Do you have the proof?

 

[Al68]

I do not need to include "You omitted the +- in your derivation below" +-, because the light postulate says the points are simultaneous.

This is axiomatic and I do not need to prove it.

So, given simultaneity by the light postulate, I need only consider one point.

That's not true. The light postulate only says the events are simultaneous in O'. They are still two separate events that correspond to two different locations in O' and therefore two different times in O.

 

[Al68]

It is not OK to have two different values for the simultaneity of O' ie, two different t'.

You said it was.

Do you have the proof?

When did I say that?

 

[atyy]

I do not need to include "You omitted the +- in your derivation below" +-, because the light postulate says the points are simultaneous.

This is axiomatic and I do not need to prove it.

So, given the simultaneity by the light postulate, I need only consider one point.

If what you say is true, then you should be able to obtain the same result with x'=-r.

 

[Dale]

Looks like I missed a lot last night. I don't know if this was already resolved, but cfrogue's proof is incorrect. Specifically:

We have by the SR spherical light sphere,
ct' = x', t' = x'/c
Also, by selection x' = r.

These two lines are wrong. It should be:
"We have by the SR spherical light sphere,
ct' = ±x', t' = ±x'/c
Also, by selection ±x' = r."

The fact that there are two times in the unprimed frame follows from that.

 

[cfrogue]

If what you say is true, then you should be able to obtain the same result with x'=-r.

This is curious.

When x'=-r in O', that is at the time r/(λ(c+v)) in O.
When x'=+r in O', that is at the time r/(λ(c-v)) in O.

Thus, two different times in O are producing simultaneity in O'.

I'm thinking about all this.

What is your view at this point?

 

[cfrogue]

Looks like I missed a lot last night. I don't know if this was already resolved, but cfrogue's proof is incorrect. Specifically:These two lines are wrong. It should be:
"We have by the SR spherical light sphere,
ct' = ±x', t' = ±x'/c
Also, by selection ±x' = r."

The fact that there are two times in the unprimed frame follows from that.

Yes, I agree and just caught that also at the same time you posted.

 

[cfrogue]

Looks like I missed a lot last night. I don't know if this was already resolved, but cfrogue's proof is incorrect. Specifically:These two lines are wrong. It should be:
"We have by the SR spherical light sphere,
ct' = ±x', t' = ±x'/c
Also, by selection ±x' = r."

The fact that there are two times in the unprimed frame follows from that.

This seems strange.

As t starts at zero in O and advances to t1 = r/(λ(c+v)), ct' < |x'|.
At that point in the time of O, ct' = -x'.

Then the negative x' continues to go more negative as time increases in O.

Then when the time in O reaches, t2 = r/(λ(c-v)), ct' = +x'.

Is this correct?

 

[Dale]

Yes, I agree and just caught that also at the same time you posted.

No problem. I have updated my spacetime diagram to highlight the events where the light cone strikes the ends of the rods. The green dots are the events where the light cone strikes the ends of the unprimed rod. The red dots are the events where the light cone strikes the ends of the primed rod.

 

[cfrogue]

No problem. I have updated my spacetime diagram to highlight the events where the light cone strikes the ends of the rods. The green dots are the events where the light cone strikes the ends of the unprimed rod. The red dots are the events where the light cone strikes the ends of the primed rod.

You do produce some excellent graphics.

 

[cfrogue]

No problem. I have updated my spacetime diagram to highlight the events where the light cone strikes the ends of the rods. The green dots are the events where the light cone strikes the ends of the unprimed rod. The red dots are the events where the light cone strikes the ends of the primed rod.

This implies there are two times in O t1, t2, such that the light stikes of the endpoints of the rod of O' are simultaneous in O' and t1 < t2.

Is this correct?

 

[atyy]

This is curious.

When x'=-r in O', that is at the time r/(λ(c+v)) in O.
When x'=+r in O', that is at the time r/(λ(c-v)) in O.

Thus, two different times in O are producing simultaneity in O'.

I'm thinking about all this.

What is your view at this point?

My view is that you are calculating the same thing as you did here, which was correct, and you had no problem interpreting:

Let's see if I understand R of S.

O sees the strikes of O' at
t_L = d/(2cλ(c+v))
t_R = d/(2cλ(c-v))

t_L < t_R
Is this R of S?

[Edited for correction]

[t_L = d/(2λ(c+v))
t_R = d/(2λ(c-v))]

 

[cfrogue]

My view is that you are calculating the same thing as you did here, which was correct, and you had no problem interpreting:

Perhaps.

Using the expanding light sphere in O, and allowing time to increase in O based on the expanding light sphere, I have found ct' = |r| at two different times in O', ie t1', t2'. I am not sure what this means though.

 

[atyy]

Perhaps.

Using the expanding light sphere in O, and allowing time to increase in O based on the expanding light sphere, I have found ct' = |r| at two different times in O', ie t1', t2'. I am not sure what this means though.

I would suggest you study DaleSpam's diagram in #497 and JesseM's diagram in #182 very carefully. In DaleSpam's diagram, there are two light spheres, marked by a pair of green points and a pair of red points respectively. There is only one light cone, which are the yellow lines, showing the left going photon and the right going photon.

 

[cfrogue]

I would suggest you study DaleSpam's diagram in #497 and JesseM's diagram in #182 very carefully. In DaleSpam's diagram, there are two light spheres, marked by a pair of green points and a pair of red points respectively. There is only one light cone, which are the yellow lines, showing the left going photon and the right going photon.

I did look at them.

There is a light sphere in O and one in O'.

LT says, for two times in O, the condition ct' = ±r is true.

Also, LT says there are two different times in O' in which ct' = ±r is true based on the emerging light sphere in O.

That seems to mean there are two in O'.

I am not sure though.

Do the diagrams show this?

 

[Dale]

This implies there are two times in O t1, t2, such that the light stikes of the endpoints of the rod of O' are simultaneous in O' and t1 < t2.

Is this correct?

Yes, where t1 is the t coordinate of the event indicated by the red dot on the left and t2 is the t coordinate of the event indicated by the red dot on the right. In this drawing (c=1, v=0.6, d=2) we have t1=0.5 and t2=2.0 so t1<t2 is correct.

 

[Dale]

I have found ct' = |r| at two different times in O', ie t1', t2'. I am not sure what this means though.

It means simultaneity is relative.

 

[Dale]

There is a light sphere in O and one in O'.

Yes, the light sphere at t=1 in O is indicated by the green dots and the light sphere at t'=1 in O' is indicated by the red dots.

LT says, for two times in O, the condition ct' = ±r is true.

Yes. Note that the left and right red dots are at t=0.5 and t=2.0 respectively.

Also, LT says there are two different times in O' in which ct' = ±r is true

No. Note that the left and right red dots are both at t'=1.

 

[cfrogue]

Yes, the light sphere at t=1 in O is indicated by the green dots and the light sphere at t'=1 in O' is indicated by the red dots.Yes. Note that the left and right red dots are at t=0.5 and t=2.0 respectively.No. Note that the left and right red dots are both at t'=1.

Are you using a particular relative v?

 

[Dale]

Yes, as I mentioned in post 504 I used v=0.6, c=1, and d=2 (or r=1) for this drawing. The exact times will be different for different v, c, or d, but it is always given by the Lorentz transform.

 

[cfrogue]

Yes, as I mentioned in post 504 I used v=0.6, c=1, and d=2 (or r=1) for this drawing. The exact times will be different for different v, c, or d, but it is always given by the Lorentz transform.

OK, thanks .6c is a good one to use.

Please evaluate the below and tell me what you think. Perhaps, I made a math error or
something.


Assume v = 3/5(c). Let r be one half to rest distance of the rod in O'
λ = 5/4, (c-v) = 2/5, (c + v) = 8/5

When t reaches r/(2c), the radius of the light sphere is ct = r/2 in O.
For the left light beam, its x location is -r/2 or x = -ct = -r/2.
t' = ( t - xv/c^2)λ
t' = ( t + tv/c )λ = t ( 1 + v/c )λ = t ( 1 + 3/5 )5/4 = 2t = r/c.

Thus, (ct')^2 = r^2 and therefore, the light sphere in O' strikes both points at +r and -r.


Then when t reaches 2r/c, x = 2r, so
t' = ( 2r/c - 2rv/c^2)λ = 2r/c(1 - v/c)λ = 2r/c(2/5)5/4 = r/c
Thus, (ct')^2 = r^2 and therefore, the light sphere in O' strikes both points at +r and -r.

Hence, there are two different times in O when O' sees the simultaneous strikes.
This is different from R of S.

Thus, time proceeds from 0 to t = r/(2c) then the light sphere in O' strikes the endpoints at the same time. Then time proceeds forward as the light sphere expands in O and then again, another simultaneous strikes occurs at t = 2r/c.

 

[Dale]

For the left light beam, its x location is -r/2 or x = -ct = -r/2.
t' = ( t - xv/c^2)?
t' = ( t + tv/c )? = t ( 1 + v/c )? = t ( 1 + 3/5 )5/4 = 2t = r/c

Don't forget to do the same for the right:

For the right light beam, its x location is r/2 or x = ct = r/2.
t' = ( t - xv/c^2)?
t' = ( t - tv/c )? = t ( 1 - v/c )? = t ( 1 - 3/5 )5/4 = t/2 = r/(4c)