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

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SUMMARY

The discussion centers on deriving equations for a light sphere emitted by a moving observer O' in collinear motion relative to a stationary observer O. The equations governing the light sphere are established as ct' = ± x' for O' and x^2 + y^2 + z^2 = (ct)^2 for O. The Lorentz transformations are utilized to relate the coordinates and proper time between the two observers, specifically t' = (t - vx/c^2)λ and x' = (x - vt)λ. The conversation emphasizes the non-simultaneity of events in different frames, asserting that simultaneity in one frame does not translate to the other when relative motion is present.

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  • #91
cfrogue said:
Let me ask you this.
Before I go on answering more nonsensical questions based on wrong premises and misconceptions, I want to know if you finally understand the difference between the emission point and the light source position in O. If not, I see no basis for communication with you.
cfrogue said:
Since the light source is in O',...
Another misconception. The light source is not only " in O' ", it exist independently of any frame. The frames just assign coordinates to it. Sorry, you talk gibberish again.
 
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  • #92
A.T. said:
Before I go on answering more nonsensical questions based on wrong premises and misconceptions, I want to know if you finally understand the difference between the emission point and the light source position in O. If not, I see no basis for communication with you.

Well, the light source and emission point can be different.

For example, in O, with O' emitting light, the light source moves and the emission point is at the origin in O.

Now, if the light source had been at the origin in O at 0, then the light source and emission point would have been the same.

This is a simple application of the light postulate.

Oh wait, we forgot to think about O'. The light source is stationary to O'. Now, what does the light sphere do in O'? In O', the light souce and light emission points are the same in O'. Thus, the light expand spherically in O' from the emission point.

Is this not correct?



A.T. said:
Another misconception. The light source is not only " in O' ", it exist independently of any frame. The frames just assign coordinates to it. Sorry, you talk gibberish again.

Let's look at the light postulate.
Any ray of light moves in the ``stationary'' system of co-ordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body.
http://www.fourmilab.ch/etexts/einstein/specrel/www/

Now, does stationary mean at absolute rest or stationary to the frame?

If stationary means relative to the frame, then the light sphere expand spherically in O' from the light source in O'.

If I am writing gibberish, then tell me it is false that the light expands spherically from the light source in O'.
 
  • #93
cfrogue said:
I am happy to explore how you resolve the moving origin.

What you are considering is a two or three spatial dimensional representation of the scenario. What you are seeing is the projection of four dimensional spacetime onto two or three spatial dimensions. In four dimensional spacetime, which we cannot visualize, the origins remain coincident. The coincidence of the emission and the origins is a spacetime event and cannot move in space or time as it has no spatial or temporal extension.

The apparent movement in these projections is because the moving observer assigns to the event chageing coordinate values. Same event, differing assigned coordinates. This reprentation makes no claims about the centrality of the moving observer with respect to the light circle (sphere), in fact in this representation the moving observer does not remain central to the expanding CIRCLE of light represented in the same diagram. It is not expected to. However, interchange the observers and the situation is reversed. The other one now is represented as central. Each observer remains central from his own viewpoint. There is nothing to resolve, this representation is exactly as expected for the given scenario.

The best representation, though not perfect, is the projection of the cross sections of the light cone onto the x/y plane. In this representation the event is represented as the origin of a light cone, the same light cone for both observers and emitter, it does not matter if one of the observers is the emittrer or whether the emitter is considered to be moving or not. But although they all share the same light cone, the cross sections of the expanding light cone, which represent the planes of simultaneity for the two observers, are not the same shape when projected on to the x/y axes. One of cross sections is circular and one is not, as it is tilted at an angle in the cone representation. The tilted one represents the plane of simultaneity of the moving observer in moving observer's frame. The tilted one shows, in the three dimensional light cone representation, one extreme of the cross section as being lower down the time axis of the stationary observer than the other extreme. This means that the times at which the light front reaches points on the perimeter of the projection of that cross section are not simultaneous in the stationary observer's frame and so the moving observer is not considered to be central according to the stationary observer. But for the circular cross section they are simultaneous and so the stationary observer considers himself central. The difference reflects the relative motion of the observers. We are at liberty to take either as being at rest and changeing the drawing to suit. The effects are reciprocal.

Matheinste.
 
  • #94
matheinste said:
What you are considering is a two or three spatial dimensional representation of the scenario. What you are seeing is the projection of four dimensional spacetime onto two or three spatial dimensions. In four dimensional spacetime, which we cannot visualize, the origins remain coincident. The coincidence of the emission and the origins is a spacetime event and cannot move in space or time as it has no spatial or temporal extension.

The apparent movement in these projections is because the moving observer assigns to the event chageing coordinate values. Same event, differing assigned coordinates. This reprentation makes no claims about the centrality of the moving observer with respect to the light circle (sphere), in fact in this representation the moving observer does not remain central to the expanding CIRCLE of light represented in the same diagram. It is not expected to. However, interchange the observers and the situation is reversed. The other one now is represented as central. Each observer remains central from his own viewpoint. There is nothing to resolve, this representation is exactly as expected for the given scenario.

The best representation, though not perfect, is the projection of the cross sections of the light cone onto the x/y plane. In this representation the event is represented as the origin of a light cone, the same light cone for both observers and emitter, it does not matter if one of the observers is the emittrer or whether the emitter is considered to be moving or not. But although they all share the same light cone, the cross sections of the expanding light cone, which represent the planes of simultaneity for the two observers, are not the same shape when projected on to the x/y axes. One of cross sections is circular and one is not, as it is tilted at an angle in the cone representation. The tilted one represents the plane of simultaneity of the moving observer in moving observer's frame. The tilted one shows, in the three dimensional light cone representation, one extreme of the cross section as being lower down the time axis of the stationary observer than the other extreme. This means that the times at which the light front reaches points on the perimeter of the projection of that cross section are not simultaneous in the stationary observer's frame and so the moving observer is not considered to be central according to the stationary observer. But for the circular cross section they are simultaneous and so the stationary observer considers himself central. The difference reflects the relative motion of the observers. We are at liberty to take either as being at rest and changeing the drawing to suit. The effects are reciprocal.

Matheinste.

The rules are clear.

The light must expand spherically in the stationary frame at the emission point in O at 0 while at the same time, it must expand spherically in O' from the light source.

None of the above addresses this physical situation.

All relativity is built on the light sphere only expanding spherically in O and ignores its operation in O'.

However, I am now asking why the light postulate has not been applied to O' and at the same time applied to O.

This necessarily creates two light spheres with two different origins.
 
  • #95
cfrogue said:
No, the geometry of this is not able to look at the light sphere in O'.
Why not? Look at the yellow lines in the O' coordinates. The equation of the yellow lines in the primed coordinates is ct' = ±x', which we have already agreed is the correct equation for the light cone in O'.
 
  • #96
cfrogue said:
All relativity is built on the light sphere only expanding spherically in O and ignores its operation in O'.
Wrong. Relativity is built on the light sphere expanding spherically in all inertial frames.
cfrogue said:
However, I am now asking why the light postulate has not been applied to O' and at the same time applied to O.
There is no such thing as 'the same time' for O and O'.
 
  • #97
cfrogue said:
The light must expand spherically in the stationary frame at the emission point in O at 0 while at the same time, it must expand spherically in O' from the light source.

None of the above addresses this physical situation.

Referring to the Minkowski diagram DaleSpam attached to https://www.physicsforums.com/showpost.php?p=2464800&postcount=88". Did you notice the symmetry of both O and O', i.e., the light cone passing through x,t (1, 1) and (-1,1); and also through x',t' (1,1) and (-1,1)?

If you did notice and do not understand it fully, my advice is to rather ask clarity on that and avoid making statements like: "All relativity is built on the light sphere only expanding spherically in O and ignores its operation in O'."
 
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  • #98
DaleSpam said:
Why not? Look at the yellow lines in the O' coordinates. The equation of the yellow lines in the primed coordinates is ct' = ±x', which we have already agreed is the correct equation for the light cone in O'.

Yes, but what is the math?

ct' = ±x' is agreed.

What are the coordinates of ±x' in O?

I think you had is as

1) Lorentz transform approach:

We have the standard form of the Lorentz transform
1a) t' = ( t - vx/c^2 )γ
1b) x' = ( x - vt )γ

And we have any arbitrary equation in the primed frame
1c) x' = ct'

To obtain the corresponding equation in the unprimed frame we simply substitute 1a) and 1b) into 1c)

1d) ( x - vt )γ = c(( t - vx/c^2 )γ)

Which simplifies to
1e) x = ct

For the record, I agree with the math as you have it here.


Thus, ct' = ±x' and ct = ±x .

All this is OK.
But, we still have not produced the x1 and x2, x1 ≠ x2 to correspond to ±x'.

But, if x1 ≠ x2, then the light sphere is not functioing correctly in O.

And, if ct' = ±x' and ct = ±x, then one light sphere is at origin 0 in O and the other is at vt or 0 in O'.
 
  • #99
Jorrie said:
Referring to the Minkowski diagram DaleSpam attached to https://www.physicsforums.com/showpost.php?p=2464800&postcount=88". Did you notice the symmetry of both O and O', i.e., the light cone passing through x,t (1, 1) and (-1,1); and also through x',t' (1,1) and (-1,1)?

If you did notice and do not understand it fully, my advice is to rather ask clarity on that and avoid making statements like: "All relativity is built on the light sphere only expanding spherically in O and ignores its operation in O'."

OK, maybe you are right.

Where is the light sphere centered?
 
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  • #100
cfrogue said:
OK, maybe you are right.

Where is the light sphere centered?

Quite clearly at x,t (0,0) and x',t' (0,0) - the same spot on the diagram...
 
  • #101
Jorrie said:
Quite clearly on x,t (0,0) and x',t' (0,0) - the same spot on the diagram...

Yes, I see that.

From the coordinates of O, the light sphere is centered at x,t (0,0) and for O' they are at x',t' (0,0).

This is the common origin when the light is emitted, is this correct?

Now, does the light expand spherically from x',t' (0,0)?
 
  • #102
cfrogue said:
Yes, but what is the math?
I am not sure what you mean. Do you mean that you think that the diagram correctly represents the situation in O but you don't understand how I got from the math (which you understand) to the spacetime diagram (which you don't yet trust completely) for the white lines representing the O' coordinates? Is that correct?
 
  • #103
cfrogue said:
OK, maybe you are right.

Where is the light sphere centered?

The light sphere is centred on the emission event. If the emission event consists of light emission and the coincidence in spacetime of one or more observers with this light emission, then the light sphere is centred on this event and therefore all obsevers present.

Matheinste.
 
  • #104
cfrogue said:
This is the common origin when the light is emitted, is this correct?

Now, does the light expand spherically from x',t' (0,0)?

1. Correct.

2. Yes. How else if it's two edges respectively pass through x' = -1 and x' = 1 at t' = 1?
 
  • #105
DaleSpam said:
I am not sure what you mean. Do you mean that you think that the diagram correctly represents the situation in O but you don't understand how I got from the math (which you understand) to the spacetime diagram (which you don't yet trust completely) for the white lines representing the O' coordinates? Is that correct?

Well I can see the common emission point of the light in the diagram.

I can see the equidistant expansion of light.

I just just wondering if you agree with Jorrie that the light expands spherically from x',t' (0,0) in O'?
 
  • #106
Jorrie said:
1. Correct.

2. Yes. How else if it's two edges respectively pass through x' = -1 and x' = 1 at t' = 1?


Yes, they could not.

Also, it expands from x,t(0,0).

Oh, at any time t in O, where is the origin of the light source in O'?
 
  • #107
cfrogue said:
Yes, they could not.

Also, it expands from x,t(0,0).

Oh, at any time t in O, where is the origin of the light source in O'?

The origin here represents a one-time emission event, with coordinates (0,0) in both frames depicted. An event's coordinates do not change in the inertial frame it was observed in - it stays the same forever.

So, "at any time t in O, where is the origin of the light source in O'?" is rather meaningless - it stays (0,0) in both frames.
 
  • #108
Jorrie said:
The origin here represents a one-time emission event, with coordinates (0,0) in both frames depicted. An event's coordinates do not change in the inertial frame it was observed in - it stays the same forever.

So, "at any time t in O, where is the origin of the light source in O'?" is rather meaningless - it stays (0,0) in both frames.

it stays (0,0) in both frames

Yea, so where is the origin of O' in the coordinates of O? Is it not at vt given the relative motion of O'?
 
  • #109
DaleSpam said:
I am not sure what you mean. Do you mean that you think that the diagram correctly represents the situation in O but you don't understand how I got from the math (which you understand) to the spacetime diagram (which you don't yet trust completely) for the white lines representing the O' coordinates? Is that correct?

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.
 
  • #110
cfrogue said:
it stays (0,0) in both frames

Yea, so where is the origin of O' in the coordinates of O? Is it not at vt given the relative motion of O'?

OK, I think I can see (partially) where your problem originates.

The coordinates of the one-time flash event remains where it is (0,0) in both frames. The physical source (the 'flashbulb') may have been moving relative to both the x,t and x',t' frames, but it is now irrelevant where it is - it is no longer emitting light!
 
  • #111
Jorrie said:
OK, I think I can see (partially) where your problem originates.

The coordinates of the one-time flash event remains where it is (0,0) in both frames. The physical source (the 'flashbulb') may have been moving relative to both the x,t and x',t' frames, but it is now irrelevant where it is - it is no longer emitting light!

Agreed.

Now, if the light source is stationary to O, will the light source and light flash origin be the same?
 
  • #112
Jorrie said:
OK, I think I can see (partially) where your problem originates.

'originates' - plural indeed! :smile:
 
  • #113
atyy said:
'originates' - plural indeed! :smile:

LOL, maybe so.

Can you state under what condition the origin of the light sphere and the light source remain at the same point?
 
  • #114
cfrogue said:
Agreed.

Now, if the light source is stationary to O, will the light source and light flash origin be the same?

Sure!

Irrespective of the inertial frame of the source, as long as it was at the common origin when that light cone was emitted, that's its origin. Remember, 'origin' here simply means: where it was at time zero, which is when the flash event occurred. Where the light source is later is irrelevant. [Edit: the source, static in O, will 'move up the t-axis', but that does not influence the origin.]
 
  • #115
cfrogue said:
LOL, maybe so.

Can you state under what condition the origin of the light sphere and the light source remain at the same point?

I defer to Jorrie. I only hang around to learn about good music. :smile:
 
  • #116
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  • #117
Jorrie said:
OK, I think I can see (partially) where your problem originates.

The coordinates of the one-time flash event remains where it is (0,0) in both frames. The physical source (the 'flashbulb') may have been moving relative to both the x,t and x',t' frames, but it is now irrelevant where it is - it is no longer emitting light!

cfrogue said:
Agreed.

Another 'partial' of your problem that I can guess is an insufficient understanding of Minkowski spacetime diagrams. Since they enable visualization of ~99% of Special Relativity, it is a must learn. :)

I have noticed that you prefer the mathematical route, which is good only if you understand the underlying theory very well. Nothing like the diagram to help with that.

Need to be going - will be back later...
 
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  • #118
Jorrie said:
Another 'partial' of your problem that I can guess is an insufficient understanding of Minkowski spacetime diagrams. Since they enable visualization of ~99% of Special Relativity, it is a must learn. :)

I have noticed that you prefer the mathematical route, which is good only if you understand the underlying theory very well. Nothing like the diagram to help with that.


LOL, you are funny.


Sure!
Irrespective of the inertial frame of the source, as long as it was at the common origin when that light cone was emitted, that's its origin. Remember, 'origin' here simply means: where it was at time zero, which is when the flash event occurred. Where the light source is later is irrelevant. [Edit: the source, static in O, will 'move up the t-axis', but that does not influence the origin.]

OK, the light postulate is clear. If light emits from a stationary light source, the emission point and the light source remain coincident. The light source is in O' and thus, this rule must be followed. From the coordinates of O, the light source is located at vt.

Also, the light postulate says the light will expand spherically in the frame regardless of the motion of the light source.

Therefore, the light will expand spherically from the origin in O located at x,t(0,0).

You already agreed, the light expands spherically from x,t(0,0) and also from x't'(0,0).

The only problem is that these two origins do not remain coincident because of the relative motion of O'.

Does this make sense?
 
  • #119
cfrogue said:
OK, the light postulate is clear. If light emits from a stationary light source, the emission point and the light source remain coincident. The light source is in O' and thus, this rule must be followed. From the coordinates of O, the light source is located at vt.

Also, the light postulate says the light will expand spherically in the frame regardless of the motion of the light source.

Therefore, the light will expand spherically from the origin in O located at x,t(0,0).

You already agreed, the light expands spherically from x,t(0,0) and also from x't'(0,0).

The only problem is that these two origins do not remain coincident because of the relative motion of O'.

Does this make sense?

Thanks for the music!

Everything you said above is correct. The two frames will not agree on the assignment of the centre of the expanding light sphere at later times, but there is no event at the assigned centre at later times, so there is no disagreement about a real event. The only event at an "assigned centre" is the emission of a light pulse when the origins O and O' coincide.
 
  • #120
cfrogue said:
OK, the light postulate is clear. If light emits from a stationary light source, the emission point and the light source remain coincident. The light source is in O' and thus, this rule must be followed. From the coordinates of O, the light source is located at vt.

The only problem is that these two origins do not remain coincident because of the relative motion of O'.

Does this make sense?

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.
 

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