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

[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'?

 

[cfrogue]

1. Correct.

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

Yes, they could not.

Also, it expands from x,t(0,0).

Oh, at any time t in O, where is the origin of the light source in O'?

 

[Jorrie]

Yes, they could not.

Also, it expands from x,t(0,0).

Oh, at any time t in O, where is the origin of the light source in O'?

The origin here represents a one-time emission event, with coordinates (0,0) in both frames depicted. An event's coordinates do not change in the inertial frame it was observed in - it stays the same forever.

So, "at any time t in O, where is the origin of the light source in O'?" is rather meaningless - it stays (0,0) in both frames.

 

[cfrogue]

The origin here represents a one-time emission event, with coordinates (0,0) in both frames depicted. An event's coordinates do not change in the inertial frame it was observed in - it stays the same forever.

So, "at any time t in O, where is the origin of the light source in O'?" is rather meaningless - it stays (0,0) in both frames.

it stays (0,0) in both frames

Yea, so where is the origin of O' in the coordinates of O? Is it not at vt given the relative motion of O'?

 

[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?

Dale, may I ask you if you agree with the below based on your diagram? I did not hijack this language, it is mine.

Let O and O' be two objects and let there be one light sphere. Let E(O) mean object O was struck by the light sphere.

According to the logic of the light cone, one and only one of the following trichotomy holds:
1) Object O is struck by the light sphere before object O' written as E(O) < E(O')
2) Object O is struck by the light sphere after object O' written as E(O) > E(O')
3) Both O and O' were struck by the light sphere but neither condition 1 or 2 were ever true, written as E(O) = E(O')

This trichotomy is just a restatement of causality as implemented by the light cone. Also, no observers in the universe can disagree on the ordinality of events as determined by one light sphere. This would be a violation of causality. Thus, given two events E(O) and E(O'), one and only one of the three above conditions is valid. Whichever one of the three is valid, that same condition applies to all observers in the universe.

 

[Jorrie]

it stays (0,0) in both frames

Yea, so where is the origin of O' in the coordinates of O? Is it not at vt given the relative motion of O'?

OK, I think I can see (partially) where your problem originates.

The coordinates of the one-time flash event remains where it is (0,0) in both frames. The physical source (the 'flashbulb') may have been moving relative to both the x,t and x',t' frames, but it is now irrelevant where it is - it is no longer emitting light!

 

[cfrogue]

OK, I think I can see (partially) where your problem originates.

The coordinates of the one-time flash event remains where it is (0,0) in both frames. The physical source (the 'flashbulb') may have been moving relative to both the x,t and x',t' frames, but it is now irrelevant where it is - it is no longer emitting light!

Agreed.

Now, if the light source is stationary to O, will the light source and light flash origin be the same?

 

[atyy]

OK, I think I can see (partially) where your problem originates.

'originates' - plural indeed!

 

[cfrogue]

'originates' - plural indeed!

LOL, maybe so.

Can you state under what condition the origin of the light sphere and the light source remain at the same point?

 

[Jorrie]

Agreed.

Now, if the light source is stationary to O, will the light source and light flash origin be the same?

Sure!

Irrespective of the inertial frame of the source, as long as it was at the common origin when that light cone was emitted, that's its origin. Remember, 'origin' here simply means: where it was at time zero, which is when the flash event occurred. Where the light source is later is irrelevant. [Edit: the source, static in O, will 'move up the t-axis', but that does not influence the origin.]

 

[atyy]

LOL, maybe so.

Can you state under what condition the origin of the light sphere and the light source remain at the same point?

I defer to Jorrie. I only hang around to learn about good music.

 

[cfrogue]

I defer to Jorrie. I only hang around to learn about good music.

Your call.

http://www.youtube.com/watch?v=EE_iAKz7I7Q"

 

[Jorrie]

OK, I think I can see (partially) where your problem originates.

The coordinates of the one-time flash event remains where it is (0,0) in both frames. The physical source (the 'flashbulb') may have been moving relative to both the x,t and x',t' frames, but it is now irrelevant where it is - it is no longer emitting light!

Agreed.

Another 'partial' of your problem that I can guess is an insufficient understanding of Minkowski spacetime diagrams. Since they enable visualization of ~99% of Special Relativity, it is a must learn. :)

I have noticed that you prefer the mathematical route, which is good only if you understand the underlying theory very well. Nothing like the diagram to help with that.

Need to be going - will be back later...

 

[cfrogue]

Another 'partial' of your problem that I can guess is an insufficient understanding of Minkowski spacetime diagrams. Since they enable visualization of ~99% of Special Relativity, it is a must learn. :)

I have noticed that you prefer the mathematical route, which is good only if you understand the underlying theory very well. Nothing like the diagram to help with that.

LOL, you are funny.

Sure!
Irrespective of the inertial frame of the source, as long as it was at the common origin when that light cone was emitted, that's its origin. Remember, 'origin' here simply means: where it was at time zero, which is when the flash event occurred. Where the light source is later is irrelevant. [Edit: the source, static in O, will 'move up the t-axis', but that does not influence the origin.]

OK, the light postulate is clear. If light emits from a stationary light source, the emission point and the light source remain coincident. The light source is in O' and thus, this rule must be followed. From the coordinates of O, the light source is located at vt.

Also, the light postulate says the light will expand spherically in the frame regardless of the motion of the light source.

Therefore, the light will expand spherically from the origin in O located at x,t(0,0).

You already agreed, the light expands spherically from x,t(0,0) and also from x't'(0,0).

The only problem is that these two origins do not remain coincident because of the relative motion of O'.

Does this make sense?

 

[atyy]

OK, the light postulate is clear. If light emits from a stationary light source, the emission point and the light source remain coincident. The light source is in O' and thus, this rule must be followed. From the coordinates of O, the light source is located at vt.

Also, the light postulate says the light will expand spherically in the frame regardless of the motion of the light source.

Therefore, the light will expand spherically from the origin in O located at x,t(0,0).

You already agreed, the light expands spherically from x,t(0,0) and also from x't'(0,0).

The only problem is that these two origins do not remain coincident because of the relative motion of O'.

Does this make sense?

Thanks for the music!

Everything you said above is correct. The two frames will not agree on the assignment of the centre of the expanding light sphere at later times, but there is no event at the assigned centre at later times, so there is no disagreement about a real event. The only event at an "assigned centre" is the emission of a light pulse when the origins O and O' coincide.

 

[Jorrie]

OK, the light postulate is clear. If light emits from a stationary light source, the emission point and the light source remain coincident. The light source is in O' and thus, this rule must be followed. From the coordinates of O, the light source is located at vt.

The only problem is that these two origins do not remain coincident because of the relative motion of O'.

Does this make sense?

One last comment for now...

No, when the light is emitted, the light source is at (0,0) in all frames, not at vt.

Please reread previous posts again: origins (and events) do not move - objects move...

It is true that you can define an origin anywhere, by setting clocks to zero. However, in the scenario sketched, the origins are fixed and do not move with time - they are defined at t=0 and that's that.

 

[A.T.]

The only problem is that these two origins do not remain coincident because of the relative motion of O'.

What problem? In each frame the sphere is centered around this frame's origin, and NOT AROUND BOTH ORIGINS IN ONE FRAME. Therefore It doesn't matter if they remain coincident.

Does this make sense?

Your 'problems' here? No, they don't make sense to me.

 

[cfrogue]

What problem? In each frame the sphere is centered around this frame's origin, and NOT AROUND BOTH ORIGINS IN ONE FRAME. Therefore It doesn't matter if they remain coincident.

In each frame the sphere is centered around this frame's origin, and NOT AROUND BOTH ORIGINS IN ONE FRAME. Therefore It doesn't matter if they remain coincident.

What is "this"?

 

[cfrogue]

One last comment for now...

No, when the light is emitted, the light source is at (0,0) in all frames, not at vt.

Please reread previous posts again: origins (and events) do not move - objects move...

It is true that you can define an origin anywhere, by setting clocks to zero. However, in the scenario sketched, the origins are fixed and do not move with time - they are defined at t=0 and that's that.

Thanks.

If I have a light source in a frame and emit the light, the light travels to equidistant points in the same time t = d/c.

Is this correct?

 

[cfrogue]

What problem? In each frame the sphere is centered around this frame's origin, and NOT AROUND BOTH ORIGINS IN ONE FRAME. Therefore It doesn't matter if they remain coincident.


Your 'problems' here? No, they don't make sense to me.

In each frame the sphere is centered around this frame's origin, and NOT AROUND BOTH ORIGINS IN ONE FRAME. Therefore It doesn't matter if they remain coincident.

So, you are saying the light sphere is centered at the origin of each frame. Yet, the frames' origins separate by vt after any time t.

It is plain and simple logic that this implies there are two different light sphere origins.

 

[Dale]

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

As I have mentioned before, I think the best way to describe it is as a single cone in 4D with the apex of the cone at the flash event. I prefer that description rather than a set of 3D spheres at different times with expanding radii, but essentially yes, I agree with Jorrie.

Dale, may I ask you if you agree with the below based on your diagram? I did not hijack this language, it is mine.

Let O and O' be two objects and let there be one light sphere. Let E(O) mean object O was struck by the light sphere.

According to the logic of the light cone, one and only one of the following trichotomy holds:
1) Object O is struck by the light sphere before object O' written as E(O) < E(O')
2) Object O is struck by the light sphere after object O' written as E(O) > E(O')
3) Both O and O' were struck by the light sphere but neither condition 1 or 2 were ever true, written as E(O) = E(O')

This trichotomy is just a restatement of causality as implemented by the light cone. Also, no observers in the universe can disagree on the ordinality of events as determined by one light sphere. This would be a violation of causality. Thus, given two events E(O) and E(O'), one and only one of the three above conditions is valid. Whichever one of the three is valid, that same condition applies to all observers in the universe.

No, this is not correct. For instance, let's say that the unprimed object is at rest in the unprimed frame at x=1, and let's say that the primed object is at rest in the primed frame (moving at v=0.6c in the unprimed frame) at the position x'=-1. You can easily see the event where the yellow light cone line intersects the black x=1 line, this event occurs at t=1 in the unprimed coordinates and at t'=0.5 in the primed coordinates. You can also easily see the event where the yellow light cone intersects the white x'=-1 line, this event occurs at t=0.5 in the unprimed and at t'=1 in the primed coordinates. So the two frames disagree about the order of the events.

All reference frames agree that the event of the flash came before either object being struck by the light cone because those events are lightlike separated, but the events of two different objects being struck by the light cone is, in general, spacelike separated and therefore the order will vary in different reference frames. I believe that we had a discussion about this exact subject in a different thread, perhaps it will help to have the diagram.

 

[cfrogue]

No, this is not correct. For instance, let's say that the unprimed object is at rest in the unprimed frame at x=1, and let's say that the primed object is at rest in the primed frame (moving at v=0.6c in the unprimed frame) at the position x'=-1. You can easily see the event where the yellow light cone line intersects the black x=1 line, this event occurs at t=1 in the unprimed coordinates and at t'=0.5 in the primed coordinates. You can also easily see the event where the yellow light cone intersects the white x'=-1 line, this event occurs at t=0.5 in the unprimed and at t'=1 in the primed coordinates. So the two frames disagree about the order of the events.

All reference frames agree that the event of the flash came before either object being struck by the light cone because those events are lightlike separated, but the events of two different objects being struck by the light cone is, in general, spacelike separated and therefore the order will vary in different reference frames. I believe that we had a discussion about this exact subject in a different thread, perhaps it will help to have the diagram.

No, I do not need diagrams. I am trying to determine positions of logic.

OK, now your diagram has the center of the light cone at the origin of O and also at O', is this correct.

 

[matheinste]

http://casa.colorado.edu/~ajsh/sr/paradox.html

Look at the two dimensional spatial planes which show what you see in a two dimensional space diagram and how it relates to spacetime via sections through the light cone.

Matheinste.

 

[Dale]

OK, now your diagram has the center of the light cone at the origin of O and also at O', is this correct.

Yes, you can see that the flash (the intersection of the yellow lines) is at the intersection of the x=0 and t=0 lines, and also at the intersection of the x'=0 and t'=0 lines. The apex of the light cone is therefore at the origin of both frames.

 

[cfrogue]

http://casa.colorado.edu/~ajsh/sr/paradox.html

Look at the two dimensional spatial planes which show what you see in a two dimensional space diagram and how it relates to spacetime via sections through the light cone.

Matheinste.

Here's a clue. Cerulean's concept of space and time may not be the same as Vermilion's.

Yea, let's see the math.
I will reword this above,
Cerulean's concept of space and time may not be the same as Vermilion's
and
Vermilion'sconcept of space and time may not be the same as Cerulean's

It is reciprocal.

I have the math when anyone is ready.

 

[cfrogue]

Yes, you can see that the flash (the intersection of the yellow lines) is at the intersection of the x=0 and t=0 lines, and also at the intersection of the x'=0 and t'=0 lines. The apex of the light cone is therefore at the origin of both frames.

OK, now the light sphere remains centered at O and also at O' in your diagram and O and O' are also diverging at vt.

Is this all correct?

 

[matheinste]

Here's a clue. Cerulean's concept of space and time may not be the same as Vermilion's.

Yea, let's see the math.
I will reword this above,
Cerulean's concept of space and time may not be the same as Vermilion's
and
Vermilion'sconcept of space and time may not be the same as Cerulean's

It is reciprocal.

I have the math when anyone is ready.

If you know the answers why are you asking us.

Anyway, enlighten us. Show us your mathematics.

Matheinste.

 

[cfrogue]

If you know the answers why are you asking us.

Anyway, enlighten us. Show us your mathematics.

Matheinste.

May I use rods?

 

[atyy]

May I use rods?

No, only cones.

 

[cfrogue]

No, only cones.

LOL, you are funny!

 

[matheinste]

May I use rods?

Just mathematics will do.

But if you must use rods then do so. Rods can be tricky things.

Matheinste.

 

[cfrogue]

Just mathematics will do.

But if you must use rods then do so. Rods can be tricky things.

Matheinste.

Each frame agrees on a rest distance of d for two rods one for each frame.

We will label the endpoints of the rods as L, R, L' and R'.

A light source is centered on the rod of O'.

When the two rods happen to be centered and O' is moving in relative motion, the light is flashed from the light source of O'.

Now, in the frame of O, t(L) = t(R) by the light postulate, t(x) means light strikes the point.

In O', t'(L') = t'(R')


any disagreements?

 

[matheinste]

Each frame agrees on a rest distance of d for two rods one for each frame.

We will label the endpoints of the rods as L, R, L' and R'.

A light source is centered on the rod of O'.

When the two rods happen to be centered and O' is moving in relative motion, the light is flashed from the light source of O'.

Now, in the frame of O, t(L) = t(R) by the light postulate, t(x) means light strikes the point.

In O', t'(L') = t'(R')


any disagreements?

No. That is unfair. Can we stick to the original problem which many of us have spent a lot of time on..

The above scenario is taken from a book which I have recently read. I'm just trying to find it to see if its word for word. I think off hand that it is Petkov.

Matheinste.

 

[matheinste]

A near version of your new scenario appears in Relativity and the Nature of Spacetime by Petkov. Around page 40. But I have no doubt it appears elsewhere also.

Matheinste.

 

[cfrogue]

No. That is unfair. Can we stick to the original problem which many of us have spent a lot of time on..

The above scenario is taken from a book which I have recently read. I'm just trying to find it to see if its word for word. I think off hand that it is Petkov.

Matheinste.

I assure you I am operating strickly from my logic and have not read any of this. I would not lie.
This method helps to see the answer.

Do you know what I am going to do next?

 

[cfrogue]

A near version of your new scenario appears in Relativity and the Nature of Spacetime by Petkov. Around page 40. But I have no doubt it appears elsewhere also.

Matheinste.

This is not new. This is exploring the expanding light sphere and how it strikes equidistant points in each frame.

 

[matheinste]

I assure you I am operating strickly from my logic and have not read any of this. I would not lie.
This method helps is see the answer.

Do you know what I am going to do next?

But we already know the answer to the originally posed scenario.

Do I know what you are going to do next. Based on experience I would say you will prevaricate.

Matheinste.

 

[cfrogue]

But we already know the answer to the originally posed scenario.

Do I know what you are going to do next. Based on experience I would say you will prevaricate.

Matheinste.

OK, do you agree with the equations?

I mean, that is the light postulate.

Further, this should not be a problem because I am only bounding the expanding light sphere.

 

[atyy]

Each frame agrees on a rest distance of d for two rods one for each frame.

We will label the endpoints of the rods as L, R, L' and R'.

A light source is centered on the rod of O'.

When the two rods happen to be centered and O' is moving in relative motion, the light is flashed from the light source of O'.

Now, in the frame of O, t(L) = t(R) by the light postulate, t(x) means light strikes the point.

In O', t'(L') = t'(R')


any disagreements?

No disagreements.

 

[cfrogue]

No disagreements.

Well, Matheinste is walking around with protest signs.

 

[matheinste]

OK, do you agree with the equations?

I mean, that is the light postulate.

Further, this should not be a problem because I am only bounding the expanding light sphere.

I agree with and understand fully the standard equations of an expanding light sphere in both three dimensional space and four dimensional spacetime. I also know that they are involved fundamentally in deriving some important SR results. However, a merely verbal statement of the light postulate suffices completely to set the scenario originally discussed.

If you cannot ask questions or argue your point based upon the original light sphere scenario I refuse to take any more of my time on this.

Anyway for now I have an excuse. Its 2.25 in the morning here and I work at six in the morning. So goodnight soon.

Matheinste.

 

[atyy]

Well, Matheinste is walking around with protest signs.

That's ok. You should take Matheinste seriously (he's very good), but we can have some fun anyway.

 

[matheinste]

That's ok. You should take Matheinste seriously (he's very good), but we can have some fun anyway.

You're too kind. You have obviously missed my many poor postings. Please, please please do not let me spoil your fun. For all my protests I am enjoying it too.

Matheinste.

 

[cfrogue]

That's ok. You should take Matheinste seriously (he's very good), but we can have some fun anyway.

OK

http://www.youtube.com/watch?v=k99h5aikc4g"

 

[cfrogue]

Let's see, where was I?

In O', t'(L') = t'(R')

Now, there is relative motion v > 0 between O and O'.

Wait, I forgot the relativity of simultaneity.

In that case, with a rod length of d, O draws the following conclusions.

t'(L') = d/(2*λ(c+v))
and
t'(R') = d/(2*λ(c-v))


But, from the above
t'(L') = t'(R')

so

t'(L') = d/(2*λ(c+v)) = t'(R') = d/(2*λ(c-v))

d/(2*λ(c+v)) = d/(2*λ(c-v))

1/(2*λ(c+v)) = 1/(2*λ(c-v))

1/(c+v) = 1/(c-v)

(c+v) = (c-v)

2v = 0

v = 0

a contradiction.

 

[Jorrie]

If I have a light source in a frame and emit the light, the light travels to equidistant points in the same time t = d/c.

Is this correct?

Yes, if you define the instant of the flash as x, t (0,0) - the origin.

Also, if we define the origin of the O' frame to coincide with the origin of O, then the light also travels to equidistant points in the same O'-time: t' = d'/c.

Note that according to the O frame, the light does not reach those points of the O' frame simultaneously - hence Einstein's relativity of simultaneity...