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

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    Light Sphere
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Discussion Overview

The discussion revolves around deriving equations related to a spherical light pulse emitted by a moving observer O' in collinear motion relative to a stationary observer O. Participants explore the implications of the Lorentz transformation on the coordinates and proper time of both observers, focusing on the relationship between the light sphere's equations in their respective frames.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that the equations for the light sphere in O' can be expressed as ct' = ± x', while seeking to derive corresponding equations in the frame of O.
  • Others challenge the validity of the equations, suggesting that ct = ± x violates the principles of relativity.
  • One participant presents transformation equations, t' = (t - vx/c^2)λ and x' = (x - vt)λ, arguing that these must hold for the light sphere of O'.
  • There is a contention regarding the simultaneity of events in both frames, with some arguing that events simultaneous in O cannot be simultaneous in O' due to relative motion.
  • Participants discuss the implications of simultaneity and how it affects the interpretation of events described in both frames, noting that two events viewed as simultaneous in one frame may not be so in another.
  • Some participants express frustration over the mathematical derivations, indicating that they have struggled to reconcile the equations with the principles of relativity.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the validity of the equations derived for the light sphere or the implications of simultaneity in different frames. Multiple competing views remain regarding the interpretation of the equations and the conditions under which they hold.

Contextual Notes

There are unresolved mathematical steps and assumptions regarding the definitions of simultaneity and the conditions of relative motion that participants have not fully clarified. The discussion reflects ongoing uncertainty about the correct application of the Lorentz transformation to this scenario.

  • #451
atyy said:
Note that light striking the left endpoint of the primed rod is a different event from light striking the right endpoint of the primed rod.

For O, light hits the left endpoint of the unprimed rod at the same t as light hits the right endpoint of the unprimed rod.

For O, light hits the left endpoint of the primed rod at an earlier t than light hits the right endpoint of the primed rod.

For O', light hits the left endpoint of the primed rod at the same t' as light hits the right endpoint of the primed rod.

For O', light hits the left endpoint of the unprimed rod at a later t' than light hits the right endpoint of the unprimed rod.

I completely agree.

Now, when in the coords of O are the points of O' struck at the same time.

That is the question.

If you come up with two answers for O, the O' will see the strikes at the same time twice.
 
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  • #452
cfrogue said:
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.

atyy said:
BTW, this looks correct, except for the final sentence "Thus, the calculated t is the unique time in O when the points of O' are struck at the same time." So what we are having is a problem in interpreting the mathematics.

cfrogue said:
I do not think so since I showed no other t fits the bill > t or < t.

The final sentence is wrong because "the points of O' are struck at the same time" refers to 2 events. However, by the restriction x'=r, you can only consider the event of light hitting the right endpoint of the primed rod. You cannot consider the the event of light hitting the left endpoint of the primed rod, at which x'=-r. So the reference to 2 events in the final sentence is not justified.
 
  • #453
cfrogue said:
I completely agree.

Now, when in the coords of O are the points of O' struck at the same time.

That is the question.

If you come up with two answers for O, the O' will see the strikes at the same time twice.
You mean the way you did exactly that here:
cfrogue said:
O' sees the strikes as simultaneous whereas O sees them at different times.
I don't know why you keep contradicting yourself, unless there is a major underlying issue that is a mystery to me.
 
  • #454
Al68 said:
Are you referring to a time at which an observer physically sees light reflected back from the rod's endpoints?
No, I am only applying LT.
I do not use the reflected logic.

That is when my eyes will see it. I want to know when it actually happens which LT gives.


Do you think there should be two times in O when the points of O' are struck at the same time?

Al68 said:
According to SR, there are. And you say so next:

SR does not say O' will see simultaneous strikes at two different times. That contradicts reality.

Can you prove this?



Al68 said:
Do you not realize that means that t(L) is a different value than t(R), meaning more than one t in O for the single t' in O' [t'=t'(L)=t'(R)]?

That is false and you do not understand the mapping of LT.

There is more than one t in O, but only one in O'. That is LT.

Do you understand that given x^2 = 9, that -3 and 3 fit the bill?

This is the way LT works and we are exploring this.

This is R of S combined with the light postulate in O'.

O will see the strikes at different times but O' will see them at the same time.
 
  • #455
cfrogue said:
Now, when in the coords of O are the points of O' struck at the same time.

That is the question.

According to the relativity of simultaneity, this question is meaningless.
 
  • #456
atyy said:
The final sentence is wrong because "the points of O' are struck at the same time" refers to 2 events. However, by the restriction x'=r, you can only consider the event of light hitting the right endpoint of the primed rod. You cannot consider the the event of light hitting the left endpoint of the primed rod, at which x'=-r. So the reference to 2 events in the final sentence is not justified.

Not correct.

LT provides for two different times to be mapped to one time.

That is R of S and the light postulate O'.

So, O sees two different times for L' and R' of O' but O' sees them at the same time.

Where is this wrong?
 
  • #457
atyy said:
According to the relativity of simultaneity, this question is meaningless.

This is not true.

This is a function.

given x^2 and -3 and 3, both map to the same value in the other system.

Where is the problem?
 
  • #458
atyy said:
According to the relativity of simultaneity, this question is meaningless.

I need to say this.

There is no question O' sees its points hit at the same time by the light postulate.

If LT cannot handle this, according to your logic, then SR cannot map its own logic correctly.

I have shown it does work.
 
  • #459
cfrogue said:
SR does not say O' will see simultaneous strikes at two different times. That contradicts reality.

Can you prove this?
Why would I try to prove something I never said?
Al68 said:
Do you not realize that means that t(L) is a different value than t(R), meaning more than one t in O for the single t' in O' [t'=t'(L)=t'(R)]?
That is false and you do not understand the mapping of LT.
If that is false, then why do you say exactly that next:
There is more than one t in O, but only one in O'. That is LT.
I agree. Why wouldn't I agree with a statement that I made, which you claimed to be false.
 
  • #460
cfrogue said:
I have shown it does work.

The step x'=r in your derivation is not justified if you want to be able to refer to both events in the final interpretation. Once you restrict yourself to x'=r, all successive steps can only refer to events that occur at x'=r, not at x'=-r.
 
  • #461
Al68 said:
If that is false, then why do you say exactly that next:I agree. Why wouldn't I agree with a statement that I made, which you claimed to be false.

I said:
There is more than one t in O, but only one in O'. That is LT.

I am simply not getting the problem.

I said before, -3, 3 under x^2 = 9.

What is the problem?

There are two different values in O, for example, -3 and 3, and one value in O', 9.

LT is consistent.
 
  • #462
atyy said:
The step x'=r in your derivation is not justified if you want to be able to refer to both events in the final interpretation. Once you restrict yourself to x'=r, all successive steps can only refer to events that occur at x'=r, not at x'=-r.

Nope, I was careful to operate with r in the x' system.

I never crossed over in the O system with this r without r/λ.

Check the logic.
 
  • #463
cfrogue said:
I said:


I am simply not getting the problem.

I said before, -3, 3 under x^2 = 9.

What is the problem?

There are two different values in O, for example, -3 and 3, and one value in O', 9.

LT is consistent.
That's exactly what others have been telling you for hundreds of posts while you (and only you) were saying things that contradict it like:
cfrogue said:
Thus, the calculated t is the unique time in O when the points of O' are struck at the same time.
 
  • #464
cfrogue said:
Nope, I was careful to operate with r in the x' system.

I never crossed over in the O system with this r without r/λ.

Check the logic.

The problem is not crossing over into O. The problem is that the light sphere is x'=r and x'=-r, and in your final sentence you refer to events at x'=r and x'=-r, yet in the middle you have excluded all points x'=-r.
 
  • #465
Al68 said:
That's exactly what others have been telling you for hundreds of posts while you (and only you) were saying things that contradict it like:

Yes, I am now certain I know you.

Can you prove your claim so I can learn?
 
  • #466
atyy said:
The problem is not crossing over into O. The problem is that the light sphere is x'=r and x'=-r, and in your final sentence you refer to events at x'=r and x'=-r, yet in the middle you have excluded all points x'=-r.

No,the light postulate says

ct' = ±x'

I included all this in my equations.

Do you have a math equation that is different?

Can I see it?
 
  • #467
cfrogue said:
Yes, I am now certain I know you.
Who am I then?
Can you prove your claim so I can learn?
Which claim?
 
  • #468
cfrogue said:
No,the light postulate says

ct' = ±x'

I included all this in my equations.

Do you have a math equation that is different?

Can I see it?

Hence x'=-ct' and x'=+ct'
 
  • #469
atyy said:
Hence x'=-ct' and x'=+ct'

this is what I used.

Note simultaneity occurs when x' = ct'.

All of it falls in place after that.

Did you say my equations are false?

Can you show me the two different times in O' when it sees simultaneity.

I would like to see the math.
 
  • #470
Al68 said:
Who am I then?Which claim?

Just prove my math is false and then I can learn.
 
  • #471
cfrogue said:
this is what I used.

Note simultaneity occurs when x' = ct'.

All of it falls in place after that.

Did you say my equations are false?

Can you show me the two different times in O' when it sees simultaneity.

I would like to see the math.

Yes, there is one time in O' when it sees simultaneity, but there are two locations x'=ct' and x'=-ct'. In your derivation, you restrict x'=r, so you restrict to one location.
 
  • #472
atyy said:
Yes, there is one time in O' when it sees simultaneity, but there are two locations x'=ct' and x'=-ct'. In your derivation, you restrict x'=r, so you restrict to one location.

There is nothing wrong with this.

I must restrict it to one time location in O' by the light postulate.

It says, x'=ct' and x'=-ct', so I must follow the rules. There is one time for simultaneity in O'.

Do you see this?
 
  • #473
cfrogue said:
There is nothing wrong with this.

I must restrict it to one time location in O' by the light postulate.

It says, x'=ct' and x'=-ct', so I must follow the rules. There is one time for simultaneity in O'.

Do you see this?

Yes, one t' coordinate, but two x' coordinates. You excluded one x' coordinate when you used x'=r in your derivation.
 
  • #474
cfrogue said:
Just prove my math is false and then I can learn.
Math isn't true or false, claims are. Your various claims contradict each other, therefore some of them are false.
cfrogue said:
Note simultaneity occurs when x' = ct'.
This makes no sense. Simultaneity isn't an event, it's a description of multiple events.
 
  • #475
atyy said:
Yes, one t' coordinate, but two x' coordinates. You excluded one x' coordinate when you used x'=r in your derivation.

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



inxs - elegantly wasted
 
  • #476
Al68 said:
Math isn't true or false, claims are. Your various claims contradict each other, therefore some of them are false.This makes no sense. Simultaneity isn't an event, it's a description of multiple events.

Are you saying this is true in O'?

Can you prove this?

O' sees all its points struck at the same time.
 
  • #477
Al68 said:
Math isn't true or false, claims are. Your various claims contradict each other, therefore some of them are false.

Looks like I am wrong.

Can you show me?
 
  • #478
cfrogue said:
Are you saying this is true in O'?

Can you prove this?

O' sees all its points struck at the same time.
What are you talking about? O' doesn't see all of its points struck at the same time, only any two points equally distance from the origin.
 
  • #479
cfrogue said:
Looks like I am wrong.

Can you show me?
I, and others better than I, have tried repeatedly, and 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.
 
  • #480
Al68 said:
What are you talking about? O' doesn't see all of its points struck at the same time, only any two points equally distance from the origin.

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'?
 

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