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

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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.

  • #121
cfrogue said:
The only problem is that these two origins do not remain coincident because of the relative motion of O'.

What problem? In each frame the sphere is centered around this frame's origin, and NOT AROUND BOTH ORIGINS IN ONE FRAME. Therefore It doesn't matter if they remain coincident.

cfrogue said:
Does this make sense?
Your 'problems' here? No, they don't make sense to me.
 
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  • #122
A.T. said:
What problem? In each frame the sphere is centered around this frame's origin, and NOT AROUND BOTH ORIGINS IN ONE FRAME. Therefore It doesn't matter if they remain coincident.

In each frame the sphere is centered around this frame's origin, and NOT AROUND BOTH ORIGINS IN ONE FRAME. Therefore It doesn't matter if they remain coincident.

What is "this"?
 
  • #123
Jorrie said:
One last comment for now...

No, when the light is emitted, the light source is at (0,0) in all frames, not at vt.

Please reread previous posts again: origins (and events) do not move - objects move...

It is true that you can define an origin anywhere, by setting clocks to zero. However, in the scenario sketched, the origins are fixed and do not move with time - they are defined at t=0 and that's that.

Thanks.

If I have a light source in a frame and emit the light, the light travels to equidistant points in the same time t = d/c.

Is this correct?
 
  • #124
A.T. said:
What problem? In each frame the sphere is centered around this frame's origin, and NOT AROUND BOTH ORIGINS IN ONE FRAME. Therefore It doesn't matter if they remain coincident.


Your 'problems' here? No, they don't make sense to me.


In each frame the sphere is centered around this frame's origin, and NOT AROUND BOTH ORIGINS IN ONE FRAME. Therefore It doesn't matter if they remain coincident.

So, you are saying the light sphere is centered at the origin of each frame. Yet, the frames' origins separate by vt after any time t.

It is plain and simple logic that this implies there are two different light sphere origins.
 
  • #125
cfrogue said:
I just just wondering if you agree with Jorrie that the light expands spherically from x',t' (0,0) in O'?
As I have mentioned before, I think the best way to describe it is as a single cone in 4D with the apex of the cone at the flash event. I prefer that description rather than a set of 3D spheres at different times with expanding radii, but essentially yes, I agree with Jorrie.

cfrogue said:
Dale, may I ask you if you agree with the below based on your diagram? I did not hijack this language, it is mine.

Let O and O' be two objects and let there be one light sphere. Let E(O) mean object O was struck by the light sphere.

According to the logic of the light cone, one and only one of the following trichotomy holds:
1) Object O is struck by the light sphere before object O' written as E(O) < E(O')
2) Object O is struck by the light sphere after object O' written as E(O) > E(O')
3) Both O and O' were struck by the light sphere but neither condition 1 or 2 were ever true, written as E(O) = E(O')

This trichotomy is just a restatement of causality as implemented by the light cone. Also, no observers in the universe can disagree on the ordinality of events as determined by one light sphere. This would be a violation of causality. Thus, given two events E(O) and E(O'), one and only one of the three above conditions is valid. Whichever one of the three is valid, that same condition applies to all observers in the universe.
No, this is not correct. For instance, let's say that the unprimed object is at rest in the unprimed frame at x=1, and let's say that the primed object is at rest in the primed frame (moving at v=0.6c in the unprimed frame) at the position x'=-1. You can easily see the event where the yellow light cone line intersects the black x=1 line, this event occurs at t=1 in the unprimed coordinates and at t'=0.5 in the primed coordinates. You can also easily see the event where the yellow light cone intersects the white x'=-1 line, this event occurs at t=0.5 in the unprimed and at t'=1 in the primed coordinates. So the two frames disagree about the order of the events.

All reference frames agree that the event of the flash came before either object being struck by the light cone because those events are lightlike separated, but the events of two different objects being struck by the light cone is, in general, spacelike separated and therefore the order will vary in different reference frames. I believe that we had a discussion about this exact subject in a different thread, perhaps it will help to have the diagram.
 
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  • #126
DaleSpam said:
No, this is not correct. For instance, let's say that the unprimed object is at rest in the unprimed frame at x=1, and let's say that the primed object is at rest in the primed frame (moving at v=0.6c in the unprimed frame) at the position x'=-1. You can easily see the event where the yellow light cone line intersects the black x=1 line, this event occurs at t=1 in the unprimed coordinates and at t'=0.5 in the primed coordinates. You can also easily see the event where the yellow light cone intersects the white x'=-1 line, this event occurs at t=0.5 in the unprimed and at t'=1 in the primed coordinates. So the two frames disagree about the order of the events.

All reference frames agree that the event of the flash came before either object being struck by the light cone because those events are lightlike separated, but the events of two different objects being struck by the light cone is, in general, spacelike separated and therefore the order will vary in different reference frames. I believe that we had a discussion about this exact subject in a different thread, perhaps it will help to have the diagram.

No, I do not need diagrams. I am trying to determine positions of logic.

OK, now your diagram has the center of the light cone at the origin of O and also at O', is this correct.
 
  • #127
http://casa.colorado.edu/~ajsh/sr/paradox.html

Look at the two dimensional spatial planes which show what you see in a two dimensional space diagram and how it relates to spacetime via sections through the light cone.

Matheinste.
 
  • #128
cfrogue said:
OK, now your diagram has the center of the light cone at the origin of O and also at O', is this correct.
Yes, you can see that the flash (the intersection of the yellow lines) is at the intersection of the x=0 and t=0 lines, and also at the intersection of the x'=0 and t'=0 lines. The apex of the light cone is therefore at the origin of both frames.
 
  • #129
matheinste said:
http://casa.colorado.edu/~ajsh/sr/paradox.html

Look at the two dimensional spatial planes which show what you see in a two dimensional space diagram and how it relates to spacetime via sections through the light cone.

Matheinste.

Here's a clue. Cerulean's concept of space and time may not be the same as Vermilion's.

Yea, let's see the math.
I will reword this above,
Cerulean's concept of space and time may not be the same as Vermilion's
and
Vermilion'sconcept of space and time may not be the same as Cerulean's

It is reciprocal.

I have the math when anyone is ready.
 
  • #130
DaleSpam said:
Yes, you can see that the flash (the intersection of the yellow lines) is at the intersection of the x=0 and t=0 lines, and also at the intersection of the x'=0 and t'=0 lines. The apex of the light cone is therefore at the origin of both frames.

OK, now the light sphere remains centered at O and also at O' in your diagram and O and O' are also diverging at vt.

Is this all correct?
 
  • #131
cfrogue said:
Here's a clue. Cerulean's concept of space and time may not be the same as Vermilion's.

Yea, let's see the math.
I will reword this above,
Cerulean's concept of space and time may not be the same as Vermilion's
and
Vermilion'sconcept of space and time may not be the same as Cerulean's

It is reciprocal.

I have the math when anyone is ready.

If you know the answers why are you asking us.

Anyway, enlighten us. Show us your mathematics.

Matheinste.
 
  • #132
matheinste said:
If you know the answers why are you asking us.

Anyway, enlighten us. Show us your mathematics.

Matheinste.

May I use rods?
 
  • #133
cfrogue said:
May I use rods?

No, only cones. :-p
 
  • #134
atyy said:
No, only cones. :-p

LOL, you are funny!
 
  • #135
cfrogue said:
May I use rods?

Just mathematics will do.

But if you must use rods then do so. Rods can be tricky things.

Matheinste.
 
  • #136
matheinste said:
Just mathematics will do.

But if you must use rods then do so. Rods can be tricky things.

Matheinste.

Each frame agrees on a rest distance of d for two rods one for each frame.

We will label the endpoints of the rods as L, R, L' and R'.

A light source is centered on the rod of O'.

When the two rods happen to be centered and O' is moving in relative motion, the light is flashed from the light source of O'.

Now, in the frame of O, t(L) = t(R) by the light postulate, t(x) means light strikes the point.

In O', t'(L') = t'(R')


any disagreements?
 
  • #137
cfrogue said:
Each frame agrees on a rest distance of d for two rods one for each frame.

We will label the endpoints of the rods as L, R, L' and R'.

A light source is centered on the rod of O'.

When the two rods happen to be centered and O' is moving in relative motion, the light is flashed from the light source of O'.

Now, in the frame of O, t(L) = t(R) by the light postulate, t(x) means light strikes the point.

In O', t'(L') = t'(R')


any disagreements?

No. That is unfair. Can we stick to the original problem which many of us have spent a lot of time on..

The above scenario is taken from a book which I have recently read. I'm just trying to find it to see if its word for word. I think off hand that it is Petkov.

Matheinste.
 
  • #138
A near version of your new scenario appears in Relativity and the Nature of Spacetime by Petkov. Around page 40. But I have no doubt it appears elsewhere also.

Matheinste.
 
  • #139
matheinste said:
No. That is unfair. Can we stick to the original problem which many of us have spent a lot of time on..

The above scenario is taken from a book which I have recently read. I'm just trying to find it to see if its word for word. I think off hand that it is Petkov.

Matheinste.

I assure you I am operating strickly from my logic and have not read any of this. I would not lie.
This method helps to see the answer.

Do you know what I am going to do next?
 
  • #140
matheinste said:
A near version of your new scenario appears in Relativity and the Nature of Spacetime by Petkov. Around page 40. But I have no doubt it appears elsewhere also.

Matheinste.


This is not new. This is exploring the expanding light sphere and how it strikes equidistant points in each frame.
 
  • #141
cfrogue said:
I assure you I am operating strickly from my logic and have not read any of this. I would not lie.
This method helps is see the answer.

Do you know what I am going to do next?

But we already know the answer to the originally posed scenario.

Do I know what you are going to do next. Based on experience I would say you will prevaricate.

Matheinste.
 
  • #142
matheinste said:
But we already know the answer to the originally posed scenario.

Do I know what you are going to do next. Based on experience I would say you will prevaricate.

Matheinste.

OK, do you agree with the equations?

I mean, that is the light postulate.

Further, this should not be a problem because I am only bounding the expanding light sphere.
 
  • #143
cfrogue said:
Each frame agrees on a rest distance of d for two rods one for each frame.

We will label the endpoints of the rods as L, R, L' and R'.

A light source is centered on the rod of O'.

When the two rods happen to be centered and O' is moving in relative motion, the light is flashed from the light source of O'.

Now, in the frame of O, t(L) = t(R) by the light postulate, t(x) means light strikes the point.

In O', t'(L') = t'(R')


any disagreements?

No disagreements.
 
  • #144
atyy said:
No disagreements.

Well, Matheinste is walking around with protest signs.
 
  • #145
cfrogue said:
OK, do you agree with the equations?

I mean, that is the light postulate.

Further, this should not be a problem because I am only bounding the expanding light sphere.

I agree with and understand fully the standard equations of an expanding light sphere in both three dimensional space and four dimensional spacetime. I also know that they are involved fundamentally in deriving some important SR results. However, a merely verbal statement of the light postulate suffices completely to set the scenario originally discussed.

If you cannot ask questions or argue your point based upon the original light sphere scenario I refuse to take any more of my time on this.

Anyway for now I have an excuse. Its 2.25 in the morning here and I work at six in the morning. So goodnight soon.

Matheinste.
 
  • #146
cfrogue said:
Well, Matheinste is walking around with protest signs.

That's ok. You should take Matheinste seriously (he's very good), but we can have some fun anyway.
 
  • #147
atyy said:
That's ok. You should take Matheinste seriously (he's very good), but we can have some fun anyway.

You're too kind. You have obviously missed my many poor postings. Please, please please do not let me spoil your fun. For all my protests I am enjoying it too.

Matheinste.
 
  • #148
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  • #149
Let's see, where was I?

In O', t'(L') = t'(R')

Now, there is relative motion v > 0 between O and O'.

Wait, I forgot the relativity of simultaneity.

In that case, with a rod length of d, O draws the following conclusions.

t'(L') = d/(2*λ(c+v))
and
t'(R') = d/(2*λ(c-v))


But, from the above
t'(L') = t'(R')

so

t'(L') = d/(2*λ(c+v)) = t'(R') = d/(2*λ(c-v))

d/(2*λ(c+v)) = d/(2*λ(c-v))

1/(2*λ(c+v)) = 1/(2*λ(c-v))

1/(c+v) = 1/(c-v)

(c+v) = (c-v)

2v = 0

v = 0

a contradiction.
 
  • #150
cfrogue said:
If I have a light source in a frame and emit the light, the light travels to equidistant points in the same time t = d/c.

Is this correct?

Yes, if you define the instant of the flash as x, t (0,0) - the origin.

Also, if we define the origin of the O' frame to coincide with the origin of O, then the light also travels to equidistant points in the same O'-time: t' = d'/c.

Note that according to the O frame, the light does not reach those points of the O' frame simultaneously - hence Einstein's relativity of simultaneity...
 

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