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

[A.T.]

My problem with this method is that this causes the x locations in O, x and -x to be simultaneous in both O and O'.

Events can be simultaneous, not locations.

Thus, ct = +-x and ct' = +- x'.

These equations do not describe two events in each frame, they describe all events on the light cone in each frame. The derivation by atyy only shows that all events on the light cone in one frame will also be on the light cone in the other frame, after LT. But It says nothing about whether two simultaneous events on the light cone will still be simultaneous after LT. Just because they both still are on the light cone, doesn't mean that they still have the same t-coordinate.

They are in relative motion and therefore, the two points cannot appear synchronous to both frames.

Correct. I show this in post #30

The goal is to use x' and -x' which will be struck by the light sphere at the same time in O' and then prove the corresponding x values in O are not synchronous by using the standard LT translations.

That is what I did in post #30, but I just swaped O and O', and instead of x and -x I express the positions in the syncronised frame as cT and -cT, where T can be any value.

Then, I would like to figure out how a light sphere will have a fixed origin in O while at the same time also have a moving origin in O located at vt based on the necessary conditions of the light postulate for O'.

A light sphere has a fixed center in every frame.

 

[Dale]

My problem with this method is that this causes the x locations in O, x and -x to be simultaneous in both O and O'.

Thus, ct = +-x and ct' = +- x'.

They are in relative motion and therefore, the two points cannot appear synchronous to both frames.

You are avoiding the question, and it is a very important question. Equation 1d) is obtained directly by substitution of valid equations as you can easily verify yourself. So the only possible way for 1d) to be wrong is if algebraic substitution itself is wrong.

So, for the third time: Why do you think that algebraic substitution is not a valid operation in relativity?

I assert that substitution is valid in relativity. If you have any expression in the primed coordinates you can substitute the Lorentz transform equations and obtain the corresponding expression in the unprimed coordinates. That is the whole point of the Lorentz transform. This is just basic algebra.

 

[cfrogue]

You are avoiding the question, and it is a very important question. Equation 1d) is obtained directly by substitution of valid equations as you can easily verify yourself. So the only possible way for 1d) to be wrong is if algebraic substitution itself is wrong.

So, for the third time: Why do you think that algebraic substitution is not a valid operation in relativity?

I assert that substitution is valid in relativity. If you have any expression in the primed coordinates you can substitute the Lorentz transform equations and obtain the corresponding expression in the unprimed coordinates. That is the whole point of the Lorentz transform. This is just basic algebra.

I am sorry. Yes, it is valid. It is not useful for this application.

 

[Dale]

I am sorry. Yes, it is valid. It is not useful for this application.

If substitution is valid then equation 1d) must be correct since it was validly derived from correct equations. For you to say otherwise is illogical.

If you think substitution is OK, then on what grounds are you rejecting the derivation? Note: "I don't understand how the result fits in with other things like the relativity of simultaneity" is not a valid criticism of a derivation.

 

[cfrogue]

Events can be simultaneous, not locations.

My context was light strike points in the stationary frame.
Therefore, my context holds.

These equations do not describe two events in each frame, they describe all events on the light cone in each frame. The derivation by atyy only shows that all events on the light cone in one frame will also be on the light cone in the other frame, after LT. But It says nothing about whether two simultaneous events on the light cone will still be simultaneous after LT. Just because they both still are on the light cone, doesn't mean that they still have the same t-coordinate.

Yea, also, I have more information about this puzzle but it is still useless.

A light sphere has a fixed center in every frame.

The light postulate requires that the light sphere expands spherically in O' since the light was emitted from O' and after any time, the light sphere must be centered at vt to achieve this. Also, at the origin of O, by the light postulate, the light sphere must be expanding spherically.

I will look at your post #30 again.

 

[A.T.]

If you think substitution is OK, then on what grounds are you rejecting the derivation? Note: "I don't understand how the result fits in with other things like the relativity of simultaneity" is not a valid criticism of a derivation.

I think you miss the point. This derivation has nothing to do with simultaneity, it just shows that all events on the light cone in one frame will also be on the light cone in the other frame, after LT.

 

[Dale]

This derivation has nothing to do with simultaneity

I know that, which is why cfrogue's objections on that basis are not relevant. I am not addressing the simultaneity issue because it is being well handled by others, including yourself. I am trying to get him to understand how to use the Lorentz transform algebraically.

 

[cfrogue]

If substitution is valid then equation 1d) must be correct since it was validly derived from correct equations. For you to say otherwise is illogical.

If you think substitution is OK, then on what grounds are you rejecting the derivation? Note: "I don't understand how the result fits in with other things like the relativity of simultaneity" is not a valid criticism of a derivation.

No, I did not say your substitution was invalid. I said it is not useful.

As to your other part, I think we need to bound the problem to correctly bring in R of S.

 

[cfrogue]

OK, this arbritrary x and x' cannot be solved as far as I can tell.

So, I say the problem should be bounded.

Let O' have a rod of rest length d and a light source centered. When O and the light source are coincident, the light flashes.

So, the following are true from R of S.

t_L = d/(2*λ*(c+v))
and
t_R = d/(2*λ*(c-v))


I wonder if this will help with the solution.

 

[A.T.]

The light postulate requires that the light sphere expands spherically in O' since the light was emitted from O' and after any time, the light sphere must be centered at vt to achieve this.

No, the light sphere is centered at the origin in both frames, because the light was emitted at the origin in both frames. I doesn't matter who emitted the light. When the origins met, light was emitted there, and it expands spherically in each frame around each frames origin.

 

[Dale]

No, I did not say your substitution was invalid. I said it is not useful.

Why not? It is an essential algebraic tool for correctly deriving the equation that answers your OP. That certainly makes it useful for this thread.

Do you now accept that equation 1e) x = ct is correctly derived?

 

[cfrogue]

I think you miss the point. This derivation has nothing to do with simultaneity, it just shows that all events on the light cone in one frame will also be on the light cone in the other frame, after LT.

Can you be more specific on the ordinality of events and the light cone regarding observers?

 

[cfrogue]

No, the light sphere is centered at the origin in both frames, because the light was emitted at the origin in both frames. I doesn't matter who emitted the light. When the origins met, light was emitted there, and it expands spherically in each frame around each frames origin.

OK, so where is the origin of O' after time t in O?

 

[cfrogue]

Why not? It is an essential algebraic tool for correctly deriving the equation that answers your OP. That certainly makes it useful for this thread.

Do you now accept that equation 1e) x = ct is correctly derived?

Yes, I agree.

 

[A.T.]

OK, this arbritrary x and x' cannot be solved as far as I can tell.

So, I say the problem should be bounded.

Yes, that is what I did in #30. I picked a t-coordinate and called it T. Together with the light cone condition (x = ct or x = -ct) this gives you two simultaneous events on the light cone. You then apply LT to both and find that they are not simultaneous in the other frame.

OK, so where is the origin of O' after time t in O?

The origin of O' in O is at x=vt, but the center of the light sphere in O stays at x=0.

and vice versa:

The origin of O in O' is at x=-vt, but the center of the light sphere in O' stays at x=0.

 

[cfrogue]

Yes, that is what I did in #30. I picked a t-coordinate and called it T. Together with the light cone condition (x = ct or x = -ct) this gives you two simultaneous events on the light cone. You then apply LT to both and find that they are not simultaneous in the other frame.



The origin of O' in O is at x=vt, but the center of the light sphere in O stays at x=0.

and vice versa:

The origin of O in O' is at x=-vt, but the center of the light sphere in O' stays at x=0.

Let me see now.

The light sphere expands spherically in O' and origined in O' where the origin of O' is located at vt, but the light sphere stays origined in O.

You have not thought this through.

The origin of the light sphere must be at 0 for O and yet at the same time it must be origined at O' which is located at vt in the coords of O.

O will therefore see two light spheres.

 

[A.T.]

The origin of the light sphere must be at 0 for O

Yes

and yet at the same time it must be origined at O' which is located at vt in the coords of O.

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.

O will therefore see two light spheres.

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]

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.

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]

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.

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?

 

[Dale]

Yes, I agree.

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, let's 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]

Excellent, so let's see how this works with simultaneity.

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, let's 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 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.

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]

You have not thought this through.

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 $x_B$ axis at $x_B = +10$ and $x_B = -10$ 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 $t_B = 10$ 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 $x_b$ 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 $x_A = -8.66$ and $x_A = +8.66$ 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 $x_A = -5.77$ light-seconds and $t_A = 5.77$ seconds, whereupon that firecracker explodes. The light sphere continues to expand, and then meets the right-hand firecracker at $x_A = 17.32$ light-seconds and $t_A = 17.32$ seconds, whereupon that firecracker explodes.

To check these calculations, we plug $x_A = -5.77$, $t_A = 5.77$ for the explosion of the first firecracker, and v = 0.5 and c = 1, into the Lorentz transformation equations. We get $x_B = -10$ and $t_B = 10$ 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 $x_A$ axis at $x_A = -10$ and $x_A = +10$ 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]

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 $x_B$ axis at $x_B = +10$ and $x_B = -10$ 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 $t_B = 10$ 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 $x_b$ 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 $x_A = -8.66$ and $x_B = +8.66$ 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 $x_A = -5.77$ light-seconds and $t_A = 5.77$ seconds, whereupon that firecracker explodes. The light sphere continues to expand, and then meets the right-hand firecracker at $x_A = 17.32$ light-seconds and $t_A = 17.32$ seconds, whereupon that firecracker explodes.

To check these calculations, we plug $x_A = -5.77$, $t_A = 5.77$ for the explosion of the first firecracker, and v = 0.5 and c = 1, into the Lorentz transformation equations. We get $x_B = -10$ and $t_B = 10$ 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 $x_A$ axis at $x_A = -10$ and $x_A = +10$ 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.

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]

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.

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]

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?

You need to realize 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.

 

[Dale]

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

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]

You need to realize 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.

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?

http://www.youtube.com/watch?v=V3Kd7IGPyeg&feature=related"

 

[matheinste]

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?

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]

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.

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]

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.

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.

 

[matheinste]

cfrogue,

You must at some stage realize that you are dealing with four dimensional spacetime and not three dimensional space. The expanding sphere in space does not fully represent what is going on in spacetme where the real world's events are played out.

The coincidence of the origins and point of emission do remain the same event in spacetime and do obey all the relevant equations and the light postulate. The origins may appear to move apart in the geometric spatial representations, but in spacetime this is not the case.

Matheinste.

 

[cfrogue]

cfrogue,

You must at some stage realize that you are dealing with four dimensional spacetime and not three dimensional space. The expanding sphere in space does not fully represent what is going on in spacetme where the real world's events are played out.

The coincidence of the origins and point of emission do remain the same event in spacetime and do obey all the relevant equations and the light postulate. The origins may appear to move apart in the geometric spatial representations, but in spacetime this is not the case.

Matheinste.

Yea, that is how I am able to realize that the origin of the light sphere is at 0 and ct in O.

Let me know when you understand this.