Recognitions:

## Light sphere question

 Quote by cfrogue No, the equations i presented are just plain wrong. No, I do not see the value of spacetime diagrams. They do not confess a diverging center of the light sphere and thus, they are incomplete. They overlay the two origins of the frames on top of each other. This does not show the behavior of the light sphere in O' moving with the origin at vt relative to the fixed origin in O at 0.
The light sphere doesn't "move" in either frame, each frame sees that the sphere expands out symmetrically from that frame's origin and remains centered around that frame's origin.

Do you understand in each frame, the "light sphere" at any given moment is really the intersection between the light cone and a surface of simultaneity in that frame? And that since the two frames have different surfaces of simultaneity, they are not referring to the same set of points in spacetime when they talk about a "light sphere" at a given moment? For example, pick an event E on the left side of the light cone. Then in frame A, the light sphere at the moment of E would contain some event E1 on the right side of the light cone which is simultaneous with E in A's frame. But in frame B, that same event E1 would not be part of the light sphere at the moment of E, instead frame B would say that the light sphere at the moment of E contains some different event E2 on the right side of the light cone which is simultaneous with E in B's frame. So they are each talking about a different set of events when they refer to the "light sphere at the moment of E".

Any disagreement/confusion here?

Recognitions:
 Quote by cfrogue I will look these over and thanks.
No problem. I missed a couple of brackets, so feel free to ask about those. Jackadsa presented cleaner notation and a sleeker argument, so glad you agree with him too.

Tool - 'Jambi'

 Quote by JesseM The light sphere doesn't "move" in either frame, each frame sees that the sphere expands out symmetrically from that frame's origin and remains centered around that frame's origin. Do you understand in each frame, the "light sphere" at any given moment is really the intersection between the light cone and a surface of simultaneity in that frame? And that since the two frames have different surfaces of simultaneity, they are not referring to the same set of points in spacetime when they talk about a "light sphere" at a given moment? For example, pick an event E on the left side of the light cone. Then in frame A, the light sphere at the moment of E would contain some event E1 on the right side of the light cone which is simultaneous with E in A's frame. But in frame B, that same event E1 would not be part of the light sphere at the moment of E, instead frame B would say that the light sphere at the moment of E contains some different event E2 on the right side of the light cone which is simultaneous with E in B's frame. So they are each talking about a different set of events when they refer to the "light sphere at the moment of E". Any disagreement/confusion here?
Do you understand the light sphere is centered in the moving frame at vt?

 Quote by atyy No problem. I missed a couple of brackets, so feel free to ask about those. Jackadsa presented cleaner notation and a sleeker argument, so glad you agree with him too. Tool - 'Jambi'
Wow, your taste in music is unique.

Anyway, the center of the light sphere is in two locations in the rest frame.

 Quote by atyy No problem. I missed a couple of brackets, so feel free to ask about those. Jackadsa presented cleaner notation and a sleeker argument, so glad you agree with him too. Tool - 'Jambi'
Yea, I came up with Jackadsa's equations today.

I hate posting equations without flushing them out. But, what the heck.

Recognitions:
 Quote by cfrogue Do you understand the light sphere is centered in the moving frame at vt?
No, that's incorrect. In the stationary frame A it's centered at x=0, and in the moving frame B it's centered at x'=0. It is true that an object which remains at x'=0 in the moving frame (and thus stays at the center of the sphere in the moving frame) is moving at x(t) = vt in the stationary frame, but the stationary frame does not define the position of this object to be the center of the light sphere at any given moment (since this object is not at equal distances from the left and right side of the light sphere in the coordinates of the stationary frame)

Now before you ask more questions, can you please do me the courtesy of answering whether you understand/agree with the points about the relativity of simultaneity I raised in my previous post, like I asked you to? Again:

Do you understand in each frame, the "light sphere" at any given moment is really the intersection between the light cone and a surface of simultaneity in that frame? And that since the two frames have different surfaces of simultaneity, they are not referring to the same set of points in spacetime when they talk about a "light sphere" at a given moment? For example, pick an event E on the left side of the light cone. Then in frame A, the light sphere at the moment of E would contain some event E1 on the right side of the light cone which is simultaneous with E in A's frame. But in frame B, that same event E1 would not be part of the light sphere at the moment of E, instead frame B would say that the light sphere at the moment of E contains some different event E2 on the right side of the light cone which is simultaneous with E in B's frame. So they are each talking about a different set of events when they refer to the "light sphere at the moment of E".

 Quote by JesseM No, that's incorrect. In the stationary frame A it's centered at x=0, and in the moving frame B it's centered at x'=0. It is true that an object which remains at x'=0 in the moving frame (and thus stays at the center of the sphere in the moving frame) is moving at x(t) = vt in the stationary frame, but the stationary frame does not define the position of this object to be the center of the light sphere at any given moment (since this object is not at equal distances from the left and right side of the light sphere in the coordinates of the stationary frame) Now before you ask more questions, can you please do me the courtesy of answering whether you understand/agree with the points about the relativity of simultaneity I raised in my previous post, like I asked you to? Again: Do you understand in each frame, the "light sphere" at any given moment is really the intersection between the light cone and a surface of simultaneity in that frame? And that since the two frames have different surfaces of simultaneity, they are not referring to the same set of points in spacetime when they talk about a "light sphere" at a given moment? For example, pick an event E on the left side of the light cone. Then in frame A, the light sphere at the moment of E would contain some event E1 on the right side of the light cone which is simultaneous with E in A's frame. But in frame B, that same event E1 would not be part of the light sphere at the moment of E, instead frame B would say that the light sphere at the moment of E contains some different event E2 on the right side of the light cone which is simultaneous with E in B's frame. So they are each talking about a different set of events when they refer to the "light sphere at the moment of E".

But, to remain consistent with the light postulate, thye light sphere is centered in the rest frame and is centered in the moving frame.

LT works all this out.

The only problem is that the center is in two different places in the rest frame, at 0 and vt.

That is not an issue LT deals with.

Recognitions:
 Quote by cfrogue I agree with your comments you wanted me to see. But, to remain consistent with the light postulate, thye light sphere is centered in the rest frame and is centered in the moving frame. LT works all this out. The only problem is that the center is in two different places in the rest frame, at 0 and vt.
If each frame defines the "center" of the sphere to be the point that's equidistant from all the points on the surface of the sphere at a given moment (according to that frame's definition of simultaneity), then the rest frame will not say that the center is at vt, because in the rest frame x=vt is not equidistant from all the points on the surface of the sphere at time t. Do you disagree with any part of that? If so, which part?

 Quote by JesseM If each frame defines the "center" of the sphere to be the point that's equidistant from all the points on the surface of the sphere at a given moment (according to that frame's definition of simultaneity), then the rest frame will not say that the center is at vt, because in the rest frame x=vt is not equidistant from all the points on the surface of the sphere at time t. Do you disagree with any part of that? If so, which part?
The center of the moving frame's sphere is at vt.

x' = (x - vt)λ.

If you look at the Cartesian diagram of this, the center of the light sphere is at vt since x'^2 = (ct')^2.

Recognitions:
 Quote by cfrogue The center of the moving frame's sphere is at vt.
Not in the moving frame it's not, it's at x'=0 in the moving frame. Again, an object which remains at the position that the moving frame defines to be "the center" (i.e. it remains at x'=0 in the moving frame) will be moving at vt in the stationary frame, but in the stationary frame this object is not at "the center" of the sphere if the stationary frame defines "center" in the way I did in my previous post. Again, please tell me if you disagree with any part of this, and if so which specific part.

Mentor
 Quote by cfrogue The fact is that the light sphere has two different centers based on any stationary observer.
No, there is no such thing as "the" light sphere. There are an infinite number of light spheres, each with a single center. In fact, every event on the interior of the light cone is the center of some light sphere.
 Recognitions: Science Advisor Here's my own attempt at a diagram, which shows what point each frame considers to be the "center" of the sphere it sees at the moment of an event E on the left side of the light cone, and illustrates how in each frame the center is indeed equidistant from E and an event on the right hand side of the light cone which that frame defines to be simultaneous with E (and thus defines the right side of the light sphere at the moment of E in that frame).

Mentor
 Quote by cfrogue No, I do not see the value of spacetime diagrams. They do not confess a diverging center of the light sphere and thus, they are incomplete. They overlay the two origins of the frames on top of each other. This does not show the behavior of the light sphere in O' moving with the origin at vt relative to the fixed origin in O at 0.
The spacetime diagram does in fact show the behavior of both the light spheres and the light cone, you just don't understand yet. Please do not give up at it. For me, the discovery of spacetime diagrams and four-vectors was pivotal in my understanding. Once I had those everything suddenly "clicked" into place.

Recognitions:
Gold Member
 Quote by cfrogue No, I do not see the value of spacetime diagrams. They do not confess a diverging center of the light sphere and thus, they are incomplete.
If you would listen to DaleSpam/JessM (and answer all their questions), you probably would have noticed the value of spacetime diagrams by now.

And probably also have noticed in what respect they do show the divergence of the centers (not origins) of the light sphere. Just place a static observer in each frame, momentarily co-located at the origin and moving at frame relative speed away from each other. Each sits at the 3D center of his/her own light sphere forever.

The Minkowski diagrams are really worth a try.

 Quote by cfrogue No, the center of the light sphere in each is a well defined concept.
Of course it is well defined, based on the simultaneity: The center of the light sphere is equidistant to all coordinates of those physical locations, which are hit by the light simultaneously. And simultaneity is frame dependent.
 Quote by cfrogue The center diverges by vt.
No, thats just position of the light source in frame O. The position of the light source is not the center of the light sphere in O, only in O'. Again:

The frames don't agree which physical location coincides with the center of the light sphere.

 Quote by DaleSpam The spacetime diagram does in fact show the behavior of both the light spheres and the light cone, you just don't understand yet. Please do not give up at it. For me, the discovery of spacetime diagrams and four-vectors was pivotal in my understanding. Once I had those everything suddenly "clicked" into place.
You are good, thanks.

I have been doing simulatios of the two different light spheres one at 0 and one at vt from the POV of O.

The light sphere in O' is elongated and not spherical at all from the POV of O.

Light is an amazing creature.

 Quote by Jorrie If you would listen to DaleSpam/JessM (and answer all their questions), you probably would have noticed the value of spacetime diagrams by now. And probably also have noticed in what respect they do show the divergence of the centers (not origins) of the light sphere. Just place a static observer in each frame, momentarily co-located at the origin and moving at frame relative speed away from each other. Each sits at the 3D center of his/her own light sphere forever. The Minkowski diagrams are really worth a try.
Yea, I can use LT to visualize everything, but thanks.

You do not understand the light sphere.

The origin moves in I'.

Here is an origin for example vt.

When you look at the equations, you will see the origin of the light sphere in O' moves.

Does this not seem natural?

I mean, the light is expanding spherically at the origin of O while at the same time it is expanding spherically at the origin of O' located at vt in the coords of O.