Light sphere question

A.T.

Yes
No. The position of O'-origin in O is not relevant to light propagation in O. Why should it be? O' is just one of an infinite number of frames moving relative to O. It seems you are thinking it terms of ballistic light theory to justify this claim. Stick to SR.
You have not thought this through.

matheinste

cfrogue. Let me attempt a different, but equivalent explanation.

Considering a purely spatial sphere does not tell the whole story. The following gives a rough idea of what is going on, although this explanation is only an addition as the previous posters have said it all in a different way already.

It may be easier to consider the light cone associated with the emission of a light pulse when both relatively moving observers are present at the event of emission. The light cone represents the expanding sphere with one spatial dimension supressed but has the advantage of involving the temporal dimension going upwards. The apex of the future directed light cone is a the event of emission or origin for the emission and both observers. Who or what is responsible for the emission is of no consequence as long as both obserevrs are present at the event.

A cross section of the cone in the form of an expanding circle represents the expanding sphere centered on one observer who is considered stationary. A cross section of the same light cone, tilted at an angle to the first cross section, represents the expanding sphere centered on the observer who is considered to be moving. So the two observers see different sections of the SAME light cone.

Both sections are centered about a line pointing directly upwards from the cone's apex, the time axis.

Matheinste.

cfrogue

Thank goodness you all are getting me to understand.

Sorry, I am so thick.

The light sphere must expand at the origin of O and of O' at vt.

Can you confirm or deny this?

cfrogue

I am sticking to SR.

SR says by the light postulate that the light must expand spherically in the frame of O' at its origin since that was the emission point in O'.

At any time t, that emission point is located at vt in the coords of O.

Yet, the light postulate also says the light must expand spherically in O from the emission point which is 0, whether it was emitted from a stationary or moving light source.

Can you confirm or deny this?

DaleSpam

Excellent, so let's see how this works with the relativity of simultaneity by working out a concrete example.

Starting in the unprimed frame we have ct = ±x. So, let's choose t=5 in units where c=1 and we find two events which we can label A and B that satisfy the unprimed light cone equation. The coordinates for A are x=5 and t=5, the coordinates for B are x=-5 and t=5. Now, lets say that the primed frame is moving at 0.6 c (γ=1.25), let's do the Lorentz transform and find A' and B'.

For A':
t' = ( t - vx/c² )γ = (5 - 0.6 5/1²) 1.25 = 2.5
x' = ( x - vt )γ = (5 - 0.6 5) 1.25 = 2.5

For B':
t' = ( t - vx/c² )γ = (5 - 0.6 (-5)/1²) 1.25 = 10
x' = ( x - vt )γ = ((-5) - 0.6 5) 1.25 = -10

Note that A' and B' are NOT simultaneous as you would expect due to the relativity of simultaneity. Note also that A' and B' each satisfy the light cone equation in the primed frame: ct' = ±x'. So, the fact that the equation of the light cone is the same in both reference frames does not contradict the relativity of simultaneity. This is, in fact, required by the second postulate.

cfrogue

The t' is required to be simultaneous in O' according to the light postulate. The light was emitted from O'.

I note you have t'=10 and t'=2.5.

jtbell

No, you have not thought this through completely. You have not yet assimilated the significance of relativity of simultaneity in this situation. Let's expand on this with a specific numeric example, using the notation in my previous post.

Suppose we fasten firecrackers to the [itex]x_B[/itex] axis at [itex]x_B = +10[/itex] and [itex]x_B = -10[/itex] light-seconds, equipped with light-sensitive triggers. Both firecrackers are stationary in frame B. In frame B the light expanding from the origin takes 10 seconds to reach both firecrackers, and they explode simultaneously at [itex]t_B = 10[/itex] seconds, on opposite sides of the expanding light-sphere.

To see what this looks like in frame A, suppose frame B and its attached firecrackers are moving in the +x direction at v = 0.5c. Distances along the [itex]x_b[/itex] axis are length-contracted by a factor of 0.866 as observed in frame A. When the light flash occurs at the origin, the two firecrackers are located at [itex]x_A = -8.66[/itex] and [itex]x_A = +8.66[/itex] light-seconds. Knowing the starting points, speeds, and directions of motion for the light and the firecrackers (in frame A), we can calculate that the expanding sphere of light first meets the left-hand firecracker at [itex]x_A = -5.77[/itex] light-seconds and [itex]t_A = 5.77[/itex] seconds, whereupon that firecracker explodes. The light sphere continues to expand, and then meets the right-hand firecracker at [itex]x_A = 17.32[/itex] light-seconds and [itex]t_A = 17.32[/itex] seconds, whereupon that firecracker explodes.

To check these calculations, we plug [itex]x_A = -5.77[/itex], [itex]t_A = 5.77[/itex] for the explosion of the first firecracker, and v = 0.5 and c = 1, into the Lorentz transformation equations. We get [itex]x_B = -10[/itex] and [itex]t_B = 10[/itex] which agrees with what we started with in frame B. Similarly for the explosion of the second firecracker.

To summarize: in both frames, there is a single expanding sphere of light. In frame A, the sphere meets the two firecrackers at different times, whereas in frame B, they meet simultaneously.

We can turn this around and start with two firecrackers fastened to the [itex]x_A[/itex] axis at [itex]x_A = -10[/itex] and [itex]x_A = +10[/itex] light-seconds. We get similar results, but with the frames switched: in frame A, the light-sphere meets the two firecrackers simultaneously, whereas in frame B it does not.

matheinste

cfrogue

Note that events that are simultaneous in one frame cannot be simultaneous in a frame moving relative to it.

The times at which the light to reaches points on the surface of the sphere (circlular cross section of cone) in one frame are only equal when measured in that frame. The observer in that frame considers the times at which the light reaches the points on the "other" sphere (tilted, non circular, cross section of cone) to be not simultaneous.

The same reasoning applies if the observers are interchanged.

Matheinste.

cfrogue

Well, you are off task of this thread with a new thought experiment.

Have you mathematically established the fact the light sphere is at 0 in O and also at vt in O to satisfy the light postulate in O'?

I cannot find this in the above.

What am I missing?

cfrogue

I am guessing I said the above around 4 times in this thread already.

So, I have that part figured out.

But, we still have not resolved the light sphere origin problem.

Any ideas?

matheinste

You need to realise that although in a purely spatial representation the origins, are represented by different POINTS moving apart, in four dimensional spacetime the coincidence of the origins and the emission are, and remain,the same EVENT. Events have no spatial or temporal extension and so do not move.

Matheinste.

cfrogue

Let's see.

By the light postulate, we need a light sphere expanding at the origin of O and we need a light sphere expanding at the origin of O' since O' emitted the light.

Yet, at any time t in the coordinates of O, O' is located at vt.

That would mean the light sphere is origined at 0 and at vt at the same time in O.

DaleSpam

No, this is not what the second postulate requires at all. The second postulate requires that the speed of light be the same in O' as in O:

Using event A' we determine that the speed of light in O' is |x'/t'| = |2.5/2.5| = 1
Or, using event B' we determine that the speed of light in O' is |x'/t'| = |-10/10| = 1

So the speed of light in O' is 1 which is equal to the speed of light in O. The requirement of the second postulate is met.

cfrogue

So, let's see the equations you have.

I would like you to note, the origin of O' is always located at vt from the coords of O.

See the t in the equation?

Coldplay_The Scientist

matheinste

No equations needed. You are again thinking purely spatially. The emission and the coincidence of the origins are one SPACETIME EVENT. Nothing that happens after the event altrers its coordinates.

Matheinste.

cfrogue

So where have you included that O' moves to vt?

You have not resolved anything with this.

Are you claiming that the light postulate is false?

It requires that the light sphere expands in O' at the origin.

cfrogue

Sorry, I did not see this post.

The light postulate requires in any frame from the light emission point, light proceeds spherically in all directions at c regardless of the motion of the light source.

So, yes, this is what the light postulate demands.