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

[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.

 

[atyy]

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

Wow, I never knew the name of that song.

I'll lay off replying to my your last comment on my post, since so many have addressed it - I think A.T. in particular has addressed your comments on my posts.

 

[cfrogue]

Wow, I never knew the name of that song.

I'll lay off replying to my your last comment on my post, since so many have addressed it - I think A.T. in particular has addressed your comments on my posts.

I am glad you liked the song.
Otherwise, you are wrong.

 

[cfrogue]

Have any of you torch carriers resolved the light sphere origin problem?

I have not seen this.

 

[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.

I understand that you do not understand, but I do not understand why you do not understand. If you do not undersatand that's OK, probably quite normal. If you think what people have told you is just plain wrong then how can they possibly help.

Perhaps a break from posting and some time spent studying what has been said would help. That is not a cynical statement, you may find that rapid fire on several threads across the forum gives you no time to take stock of the explanations given.

Matheinste.

 

[atyy]

I am glad you liked the song.
Otherwise, you are wrong.

Yes, I would be wrong if I did not like the song!

 

[cfrogue]

I understand that you do not understand, but I do not understand why you do not understand. If you do not undersatand that's OK, probably quite normal. If you think what people have told you is just plain wrong then how can they possibly help.

Perhaps a break from posting and some time spent studying what has been said would help. That is not a cynical statement, you may find that rapid fire on several threads across the forum gives you no time to take stock of the explanations given.

Matheinste.

Yea, maybe you are right.

This might give you time to figure out how the origin of the light sphere moves with O'.

 

[DrGreg]

cfrogue, in one dimension of space, consider this.

At time t in the O frame, the light is at two places P and Q, x_P = ct and x_Q = −ct. The centre of the sphere, in the O frame, is halfway between these points at ½(x_P + x_Q) = 0.

Transform these two events to the O' frame and you get values for x'_P and x'_Q. But these events are not simultaneous in the O' frame, so they are not of equal distance from O'. The sphere is expanding so the earlier event is nearer to O' than the later event, and the point that is halfway in between them at ½(x'_P + x'_Q) is not at distance zero from O'. In fact, if you do the calculation you should find that the midpoint is at x' = −vt'.

But these two events aren't simultaneous in the O' frame, so that's not how you find the centre of the sphere in the O' frame. If you choose two events R and S that are simultaneous in the O' frame (i.e. with the same t' value) you will get x'_R = ct' and x'_S = −ct' with a midpoint of zero.

What this shows that if an object does not maintain a constant shape (in this case, an expanding sphere), two observers can disagree over where the centre of the object is. And it is relativity of simultaneity that is responsible for this disagreement.

 

[cfrogue]

cfrogue, in one dimension of space, consider this.

At time t in the O frame, the light is at two places P and Q, x_P = ct and x_Q = −ct. The centre of the sphere, in the O frame, is halfway between these points at ½(x_P + x_Q) = 0.

Transform these two events to the O' frame and you get values for x'_P and x'_Q. But these events are not simultaneous in the O' frame, so they are not of equal distance from O'. The sphere is expanding so the earlier event is nearer to O' than the later event, and the point that is halfway in between them at ½(x'_P + x'_Q) is not at distance zero from O'. In fact, if you do the calculation you should find that the midpoint is at x' = −vt'.

But these two events aren't simultaneous in the O' frame, so that's not how you find the centre of the sphere in the O' frame. If you choose two events R and S that are simultaneous in the O' frame (i.e. with the same t' value) you will get x'_R = ct' and x'_S = −ct' with a midpoint of zero.

What this shows that if an object does not maintain a constant shape (in this case, an expanding sphere), two observers can disagree over where the centre of the object is.

Yes, but that is not what our problem is.

We have one observer O with the origin at 0 and at vt for one light sphere.

That is the problem.

 

[DrGreg]

Yes, but that is not what our problem is.

We have one observer O with the origin at 0 and at vt for one light sphere.

That is the problem.

As I understand it, your problem is that the first observer says the centre of the sphere is fixed at x=0. The second observer says the centre of the sphere is fixed at x'=0, a location which the first observer would say is moving at x=vt. That apparent contradiction is exactly the point I am addressing.

If that's not your problem, then I don't understand what is.

 

[Dale]

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.

I agree. This is why the equation must be ct = ±x in O and ct' = ±x' in O'. See my second approach back in https://www.physicsforums.com/showpost.php?p=2462629&postcount=11".

 

[Dale]

Have any of you torch carriers resolved the light sphere origin problem?

Yes.

I have not seen this.

So I have noticed.

Have you understood and resolved the simultaneity issue in your mind yet? Do you now understand how the equations x = ct in O and x' = ct' in O' are perfectly compatible with the relativity of simultaneity? If so then I can begin addressing the "moving center" issue, but I prefer to resolve one issue at a time.

 

[A.T.]

I am sticking to SR.

Your own version of it unfortunately

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'.

Yes and this applies to both frames, not only O':
The light must also 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.

Wrong. The emission point is just a coordiante (0,0) and doesn't change with time. The light source is located at vt in the coords of O. You confuse two points in the coords of O:

x=0 : emission point, the point where the light source was at emission time t=0, center of the light sphere at any time in O

x = vt : the point where the light source is after a time t : completely irrelevant to the light sphere in 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.

Yes. Here you get it right.

 

[cfrogue]

As I understand it, your problem is that the first observer says the centre of the sphere is fixed at x=0. The second observer says the centre of the sphere is fixed at x'=0, a location which the first observer would say is moving at x=vt. That apparent contradiction is exactly the point I am addressing.

If that's not your problem, then I don't understand what is.

You have nailed one of the problems.

There are two of them.

I am happy to explore how you resolve the moving origin.

 

[cfrogue]

I agree. This is why the equation must be ct = ±x in O and ct' = ±x' in O'. See my second approach back in https://www.physicsforums.com/showpost.php?p=2462629&postcount=11".

I agree it must be this.

But, you have left of R of S.

If we look at my post of using rods in each frame of length d and a light source at the center of the O' d, the problem with this logic becomes obvious.

More spercifically,

t_L' = d/(2* λ *(c+v))

t_R' = d/(2* λ *(c-v))

where t_L' is the time when the left point -x' is struck and t_R' is the time when the right point is struck x'.

This is a direct application of R of S.

Note, it is false that these time are simultaneous. In fact, just like with the train enbankment experiment, both observers O and O' agree the right endpoint of the rod is struck after the right endpoint in the moving O'. The moving observer will claim the light shot toward the front occurred after the light shot toward the back whereas O will conclude in its frame and rod d, that both points are struck at the same time.

This is a more concrete way of looking at -x', x', -x, x.

 

[cfrogue]

Yes.So I have noticed.

LOL, I like this stuff.

Have you understood and resolved the simultaneity issue in your mind yet? Do you now understand how the equations x = ct in O and x' = ct' in O' are perfectly compatible with the relativity of simultaneity? If so then I can begin addressing the "moving center" issue, but I prefer to resolve one issue at a time.

No, I posted just before this showing a concrete example regarding the simultaneity issue.

 

[cfrogue]

Your own version of it unfortunately


Yes and this applies to both frames, not only O':
The light must also expand spherically in the frame of O at its origin since that was the emission point in O.


Wrong. The emission point is just a coordiante (0,0) and doesn't change with time. The light source is located at vt in the coords of O. You confuse two points in the coords of O:


x=0 : emission point, the point where the light source was at emission time t=0, center of the light sphere at any time in O

x = vt : the point where the light source is after a time t : completely irrelevant to the light sphere in O

Oh, yes I agree with the half answer above. But, since the light source is in O' and O' is a frame, then light must expand spherically in O' from the emission point which after any time t is located at vt in the coords of O.

 

[A.T.]

the emission point which after any time t is located at vt in the coords of O.

No, the emission point is not located at vt in O. You confuse the emission point which is a constant coordinate in both frames, with the position of the light source which changes with time in O.

I have pointed your misconception as clearly as possible in my last post. The fact that you just ignore it, and repeat the same nonsense, shows that you just don't want to get it.

 

[Dale]

I agree it must be this.

But, you have left of R of S.

If we look at my post of using rods in each frame of length d and a light source at the center of the O' d, the problem with this logic becomes obvious.
...

OK, this may be new to you, but here is a spacetime diagram showing the situation under discussion. I apologize if you are familiar with such diagrams, but I am going to assume that you are not and walk you through in detail its construction and meaning.

This diagram is drawn from the perspective of O, and the coordinates of O are indicated by the black lines and black text. The vertical axis is time and the horizontal axis is distance, both as measured by a system of rods and synchronized clocks at rest in O, and the units are such that c=1.

O' is another system of rods and synchronized clocks all at rest wrt each other, but moving at v=0.6c wrt the rods and clocks in O. The O' coordinates are obtained by the Lorentz transform equations and are indicated on the diagram by the white lines and the white text.

Also indicated are two yellow lines given by the equation ct = ±x. This represents the flash of light emitted from the origin. Because we are using units where c=1 they proceed at a 45º angle.

OK, that should cover the explanation of the diagram. Do you have any questions about the diagram? Do you see how this represents the scenario we are discussing? If we use d=1 then we even have lines specifically for the endpoints of each rod as you describe (x=±1 and x'=±1). Is this diagram acceptable to you as a tool for discussing the scenario?

 

[cfrogue]

No, the emission point is not located at vt in O. You confuse the emission point which is a constant coordinate in both frames, with the position of the light source which changes with time in O.

I have pointed your misconception as clearly as possible in my last post. The fact that you just ignore it, and repeat the same nonsense, shows that you just don't want to get it.

Let me ask you this.
Since the light source is in O', does the light postulate hold for O'?

Let's only consider O' for the time being.

 

[cfrogue]

OK, this may be new to you, but here is a spacetime diagram showing the situation under discussion. I apologize if you are familiar with such diagrams, but I am going to assume that you are not and walk you through in detail its construction and meaning.

This diagram is drawn from the perspective of O, and the coordinates of O are indicated by the black lines and black text. The vertical axis is time and the horizontal axis is distance, both as measured by a system of rods and synchronized clocks at rest in O, and the units are such that c=1.

O' is another system of rods and synchronized clocks all at rest wrt each other, but moving at v=0.6c wrt the rods and clocks in O. The O' coordinates are obtained by the Lorentz transform equations and are indicated on the diagram by the white lines and the white text.

Also indicated are two yellow lines given by the equation ct = ±x. This represents the flash of light emitted from the origin. Because we are using units where c=1 they proceed at a 45º angle.

OK, that should cover the explanation of the diagram. Do you have any questions about the diagram? Do you see how this represents the scenario we are discussing? If we use d=1 then we even have lines specifically for the endpoints of each rod as you describe (x=±1 and x'=±1). Is this diagram acceptable to you as a tool for discussing the scenario?

No, the geometry of this is not able to look at the light sphere in O'. Further, it will only see R of S for O' which is fine by me.

But, the problem here is does the light postulate hold for O'. That is the question.

If the light postulate holds in O', as it should since the rules are all the same for each frame, then the light sphere must emerge from the emission point in the frame of O'.
Do you agree?

 

[A.T.]

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.

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.

 

[cfrogue]

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?

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'.

 

[matheinste]

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.

 

[cfrogue]

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.

 

[Dale]

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'.

 

[A.T.]

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.

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'.

 

[Jorrie]

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'."

 

[cfrogue]

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'.

 

[cfrogue]

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?

 

[Jorrie]

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...