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

[cfrogue]

This is most certainly not true. You have been given multiple correct math derivations, all leading to the same conclusion all of which you have summarily rejected with no substantiation.

I reiterate my question: what specifically is wrong with my eq 1d)? I have gone back and verified that the Lorentz transform equations were copied down correctly in 1a) and 1b). So the only possible objection is that you think substitution is no longer a valid algebraic operation in relativity. Why do you think that?

cfrogue, atyy already presented some pretty convincing math, but let me try two other ways:

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


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.

 

[matheinste]

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

What does it mean for two points to appear synchronous?

Matheinste.

 

[cfrogue]

To keep things simple, suppose that the origins of frames A and B coincide at x_A = x_B = 0 and t_A = t_B = 0, and that in frame A, frame B moves in the +x direction with speed v. Then coordinates in the two frames are related by the Lorentz transformation

x_B = \gamma (x_A - vt_A)

y_B = y_A

z_B = z_A

t_B = \gamma \left( t_A - \frac{v x_A}{c^2} \right)

where as usual

\gamma = \frac {1} {\sqrt {1 - v^2 / c^2}}

Suppose that when the origins of the two frames coincide, a light flash goes off at the momentarily mutual origin. In frame A, the light expands as a sphere centered at the point x_A = y_A = z_A = 0, described by the equation

x_A^2 + y_A^2 + z_A^2 = c^2 t_A^2

In frame B, the the light expands as a sphere centered at the point x_B = y_B = z_B = 0, described by the equation

x_B^2 + y_B^2 + z_B^2 = c^2 t_B^2

(Recall that the equation of a sphere centered at the origin is x^2 + y^2 + z^2 = R^2. In frame A, R_A = ct_A and in frame B, R_B = ct_B.)

You can verify that these two equations are consistent by substituting the Lorentz transformation equations into the second one and showing that it reduces to the first one. This is the same thing that DaleSpam did in post #11, except that I'm using three spatial dimensions and he used only one.

The points x_A = y_A = z_A = 0 and x_B = y_B = z_B = 0 are the same point only when t_A = t_B = 0. At all other times, in either frame, they are different points. Therefore, at those other times the light sphere is centered at different points in the two frames.

I am OK with everything above.

However, these conditions must be met.
1) When the light sphere strikes two equidistant x points in O', say x' and -x', this cannot be synchronous to O.
2) When two equidistant x points in O are struck, this cannot be synchronous in O'

Finally, one light sphere must have two different origins in space. I di not see how this is possible. Since we are able to translate O' to O, then O will conclude once light sphere will have two different origins in its own space and worse, the light sphere origin of O' moves with the frame of O'. This is the very definition of light speed anisotropy.

Anyway, what I was hoping to see is x' and -x' translated into the coordinates of O such that it is clear, these corresponding x1 and x2 and clearly not synchronous in O but are synchronous in O'.

 

[cfrogue]

OK, I think I understand your scenario. So just do it the other way round.

Event A is (xa'=ct', ta'=t')
Event B is (xb'=-ct', tb'=t')

Use your formulas to calculate ta and tb.

But, ta' = tb' since x' and -x' are synchronous in O'.

So, this does not work.

 

[cfrogue]

What do you mean by events in one frame being simultaneous with the events in a different frame? Two events can be simultaneous (or not simultaneous) within one frame, not across frames.

Two events are simultaneous in a frame if their t coordinates in that frame are equal. For example the two light sphere events

A : (xa = cT, ta = T)
B : (xb = -cT, tb = T)

are simultaneous in the frame O where they have the above coordinates, because

ta = tb

But in the frame O' where they have the coordinates

A : (xa' = ( cT-vT)*gamma, ta' = (T-vT/c)*gamma)
B : (xb' = (-cT-vT)*gamma, tb' = (T+vT/c)*gamma)

they are not simultaneous, because

ta' <> tb'


You can then swap the apostrophe annotation my formulas, so the events are simultaneous in O' but not O.

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.

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.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?