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

[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...

 

[cfrogue]

Quite clearly on x,t (0,0) and x',t' (0,0) - the same spot on the diagram...

Yes, I see that.

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

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

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

 

[Dale]

Yes, but what is the math?

I am not sure what you mean. Do you mean that you think that the diagram correctly represents the situation in O but you don't understand how I got from the math (which you understand) to the spacetime diagram (which you don't yet trust completely) for the white lines representing the O' coordinates? Is that correct?

 

[matheinste]

OK, maybe you are right.

Where is the light sphere centered?

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

Matheinste.

 

[Jorrie]

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

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

1. Correct.

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

 

[cfrogue]

I am not sure what you mean. Do you mean that you think that the diagram correctly represents the situation in O but you don't understand how I got from the math (which you understand) to the spacetime diagram (which you don't yet trust completely) for the white lines representing the O' coordinates? Is that correct?

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

I can see the equidistant expansion of light.

I just just wondering if you agree with Jorrie that the light expands spherically from x',t' (0,0) in O'?