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

[JesseM]

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".

 

[cfrogue]

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".

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.

That is not an issue LT deals with.

 

[JesseM]

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?

 

[cfrogue]

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.

 

[JesseM]

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.

 

[Dale]

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.

 

[JesseM]

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).

 

[Dale]

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.

 

[Jorrie]

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.

 

[A.T.]

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.

The center diverges by vt.

No, that's 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.

 

[cfrogue]

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.

 

[cfrogue]

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.

 

[cfrogue]

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.

No, that's 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.

Wrong I will make you a diagram.

|-------|------------------------|
O      vt                        x
       O'
        |--------- x'/λ ---------|

 

[JesseM]

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.

You're simply incorrect here, the light sphere in both frames is spherical and centered at the origin at any given value of the time coordinate. Suppose the coordinates of the light cone in O' are given by any x',y',z',t' that satisfy the following equation:

x'^2 + y'^2 + z'^2 = (ct')^2

You can see that for any given value of t', the values of x',y',t' that satisfy this equation will form a sphere of radius ct' centered at the origin (see the equation of a sphere). Then applying the Lorentz transformation to this gives:

gamma^2*(x - vt)^2 + y^2 + z^2 = c^2*gamma^2*(t - vx/c^2)^2

squaring the terms in parentheses gives:

gamma^2*(x^2 - 2xvt + v^2*t^2) + y^2 + z^2 = c^2*gamma^2*(t^2 - 2xvt/c^2 + v^2*x^2/c^4)

multiplying this out and then adding gamma^2*2xvt to both sides gives:

gamma^2*x^2 + gamma^2*v^2*t^2 + y^2 + z^2 = c^2*gamma^2*t^2 + gamma^2*v^2*x^2/c^2

Putting all the x terms on the left side and the t terms on the right gives:

gamma^2*x^2*(1 - v^2/c^2) + y^2 + z^2 = gamma^2*t^2*(c^2 - v^2)

And gamma^2 = 1/(1 - v^2/c^2), so plugging this in gives:

x^2 + y^2 + z^2 = t^2*(c^2 - v^2)/(1 - v^2/c^2)

Multiplying both numerator and denominator of the fraction on the right by c^2 gives:

x^2 + y^2 + z^2 = t^2*c^2*(c^2 - v^2)/(c^2 - v^2)

Which simplifies to:

x^2 + y^2 + z^2 = (ct)^2

So, this is the equation for the same light cone in the O frame. You can see that for any given value of t, the set of x,y,z that satisfy this equation will form a sphere of radius ct centered on the origin.

 

[cfrogue]

You're simply incorrect here, the light sphere in both frames is spherical and centered at the origin at any given value of the time coordinate. Suppose the coordinates of the light cone in O' are given by any x',y',z',t' that satisfy the following equation:

x'^2 + y'^2 + z'^2 = (ct')^2

You can see that for any given value of t', the values of x',y',t' that satisfy this equation will form a sphere of radius ct' centered at the origin (see the equation of a sphere). Then applying the Lorentz transformation to this gives:

gamma^2*(x - vt)^2 + y^2 + z^2 = c^2*gamma^2*(t - vx/c^2)^2

squaring the terms in parentheses gives:

gamma^2*(x^2 - 2xvt + v^2*t^2) + y^2 + z^2 = c^2*gamma^2*(t^2 - 2xvt/c^2 + v^2*x^2/c^4)

multiplying this out and then adding gamma^2*2xvt to both sides gives:

gamma^2*x^2 + gamma^2*v^2*t^2 + y^2 + z^2 = c^2*gamma^2*t^2 + gamma^2*v^2*x^2/c^2

Putting all the x terms on the left side and the t terms on the right gives:

gamma^2*x^2*(1 - v^2/c^2) + y^2 + z^2 = gamma^2*t^2*(c^2 - v^2)

And gamma^2 = 1/(1 - v^2/c^2), so plugging this in gives:

x^2 + y^2 + z^2 = t^2*(c^2 - v^2)/(1 - v^2/c^2)

Multiplying both numerator and denominator of the fraction on the right by c^2 gives:

x^2 + y^2 + z^2 = t^2*c^2*(c^2 - v^2)/(c^2 - v^2)

Which simplifies to:

x^2 + y^2 + z^2 = (ct)^2

So, this is the equation for the same light cone in the O frame. You can see that for any given value of t, the set of x,y,z that satisfy this equation will form a sphere of radius ct centered on the origin.

You are wrong.

I am viewing this from the coords of O only both light spheres.

In addition, you failed to note the origin of O' moves and so the origin of the light sphere in O' moves.

You see, if the light sphere is expanding in front of me at my origin and the light sphere is expanding in front of you at your origin and you are moving at vt, then the light sphere has two origins, 0 in mine and vt in yours.

So yes, you have two light spheres, the one is not spherical in O' BTW, but you failed to note the origin of the light sphere in O' is at vt.

 

[Dale]

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..

Elongated, not spherical, and not simultaneous, yes.

Since the diagram is just a single spatial dimension you can't see the not-spherical aspect, but you can see the elongated and non-simultaneous aspect as well as the different center. Look at the line t=1 vs the line t'=1 and note where each intersects the light cone. That is the light sphere in the unprimed frame at t=1 and in the primed frame at t'=1. You can see that the primed sphere is elongated as you said, and non-simultaneous as I said. You can also see how the center of the unprimed one is at x=0 and the center of the primed one is at x'=0.

 

[cfrogue]

Elongated, not spherical, and not simultaneous, yes.

Since the diagram is just a single spatial dimension you can't see the not-spherical aspect, but you can see the elongated and non-simultaneous aspect as well as the different center. Look at the line t=1 vs the line t'=1 and note where each intersects the light cone. That is the light sphere in the unprimed frame at t=1 and in the primed frame at t'=1. You can see that the primed sphere is elongated as you said, and non-simultaneous as I said. You can also see how the center of the unprimed one is at x=0 and the center of the primed one is at x'=0.

We agree.

I am exploring this "elongated sphere".

Naturally, it is just a "calculated" sphere from the perspective of O and not the "real" sphere.

And, yes, the elongation indicates the lack of simultaneity in O for O' from the POV of O.

Obviously, simultaneity will be shorter in the direction of the positive x-axis vs the negative x-axis in the coords of O calculating O'.

So, I asked the question what are the points in O such that O' sees simultaneity. That is the light sphere I constructed for O'.

But, I am disturbed that one light sphere has two different behaviors in the calculations of O with two different origins.

Being disturbed however, is not scientific.

 

[JesseM]

You are wrong.

I am viewing this from the coords of O only both light spheres.

At any given time coordinate in frame O, there is only one light sphere (assuming we are talking about light emitted in all directions from a single event in the past), and it is always centered at the origin of O and spherical in shape. The different light spheres seen by O and O' are just different ways of slicing up a single light cone, based on their different definitions of simultaneity. Do you disagree?

Also, did you look at my diagram in post 182? If so did you understand it?

 

[Dale]

So, I asked the question what are the points in O such that O' sees simultaneity.

That is directly from the Lorentz transform. Simply set e.g. t'=1 and simplify to get the equation of a line (t=mx+b) and then plot the line.

But, I am disturbed that one light sphere has two different behaviors in the calculations of O with two different origins..

Well, this is largely personal preference, but that is why I prefer the term "light cone" to "light sphere". There is only one light cone, but an infinite number of ways to "slice" that cone and get many different light spheres. Anyway, it is less disturbing to me that way and closer to how I think about relativity.

 

[cfrogue]

That is directly from the Lorentz transform. Simply set e.g. t'=1 and simplify to get the equation of a line (t=mx+b) and then plot the line.

Well, this is largely personal preference, but that is why I prefer the term "light cone" to "light sphere". There is only one light cone, but an infinite number of ways to "slice" that cone and get many different light spheres. Anyway, it is less disturbing to me that way and closer to how I think about relativity.

We agree on these elements.

You however have not yet come to grips with a light sphere evolving in one frame and another evolving in another and they are separated by vt. These are two distinct light spheres.

To be honest, I thought I cracked the inconsistency of this but I know now I have not.


It is probably sufficient that there exists two light spheres at two origins which is impossible but I have not come up with the good argument from my POV.

 

[Dale]

These are two distinct light spheres.
...
It is probably sufficient that there exists two light spheres at two origins which is impossible but I have not come up with the good argument from my POV.

You are thinking too small here, there are an infinite number of distinct light spheres, not just two.

 

[cfrogue]

You are thinking too small here, there are an infinite number of distinct light spheres, not just two.

LOL, I know that, one for each v.

 

[atyy]

It is probably sufficient that there exists two light spheres at two origins which is impossible but I have not come up with the good argument from my POV.

The centre of the light sphere in each frame is just an "assigned" centre - no event actually happens there, except when the light is emitted, and the sphere has zero radius - at this point the origins coincide and both frames agree on the centre of the light sphere. So it doesn't matter that the frames disagree about it, just as they don't agree about what is simultaneous.

 

[Jorrie]

You do not understand the light sphere.

The origin moves in I'.

LOL, I wonder who is doing the "not understanding" (or perhaps the "not writing clearly") here!

I wrote to you:

... And probably also have noticed in what respect they do show the divergence of the centers (not origins) of the light sphere.

The common light cone has a static spacetime origin at the vertex in all frames. As viewed from reference frame O, the light spheres have apparent spatial centers (or as atyy has written: "assigned" centers), one for each v AND one for each t. They do not have spacetime origins.

Look at https://www.physicsforums.com/showpost.php?p=2467732&postcount=182" again... ;)

 

[A.T.]

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.

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

Wrong I will make you a diagram.

|-------|------------------------|
O      vt                        x
       O'
        |--------- x'/λ ---------|

No arguments what exactly is wrong with my statements, just an ASCII art that has nothing to do with SR? Well here is my diagram:

+----------+
|  PLEASE  |
|  DO NOT  |
| FEED THE |
|  TROLLS  |
+----------+
    |  |    
    |  |    
  .\|.||/..

 

[cfrogue]

LOL, I wonder who is doing the "not understanding" (or perhaps the "not writing clearly") here!

I wrote to you:

The common light cone has a static spacetime origin at the vertex in all frames. As viewed from reference frame O, the light spheres have apparent spatial centers (or as atyy has written: "assigned" centers), one for each v AND one for each t. They do not have spacetime origins.

Look at https://www.physicsforums.com/showpost.php?p=2467732&postcount=182" again... ;)

The common light cone has a static spacetime origin at the vertex in all frames

This is artificial, the center moves vt in the moving frame.

OK, I have another way to look at it, tell me what you think.

I have a light timer on the left end of a rod of length r and a light source in the center of the rod. A light timer records the time when light strikes it. This will be the moving frame at v.

Now, when the center of the rod is at the origin of O, all clocks are synchronized to 0 in each frame.

Now, light will strike the left point of the rod at t = r/(λ(c+v)) in O.

What will the light timer in O' read, t' = t*λ, for time dilation?

 

[cfrogue]

The centre of the light sphere in each frame is just an "assigned" centre - no event actually happens there, except when the light is emitted, and the sphere has zero radius - at this point the origins coincide and both frames agree on the centre of the light sphere. So it doesn't matter that the frames disagree about it, just as they don't agree about what is simultaneous.

Maybe so, but the fact is if you are in a frame and light emits from a light source, light moves spherically from the light source with the light source as the center.

So, this is what to expect in the moving O' or we are ignoring a fact.

 

[cfrogue]

No arguments what exactly is wrong with my statements, just an ASCII art that has nothing to do with SR? Well here is my diagram:

+----------+
|  PLEASE  |
|  DO NOT  |
| FEED THE |
|  TROLLS  |
+----------+
    |  |    
    |  |    
  .\|.||/..

Cute art.

Sorry, I misread your statement to mean O and O' disagree with the center of O'.

After rereading it, you are saying the same as me.

I agree with you if you are saying the center of the light sphere is moving with the moving frame.

Thus, O will see it at 0 and O' will see it at vt and they disagree.

 

[JesseM]

cfrogue, can you answer my questions from post 193?

At any given time coordinate in frame O, there is only one light sphere (assuming we are talking about light emitted in all directions from a single event in the past), and it is always centered at the origin of O and spherical in shape. The different light spheres seen by O and O' are just different ways of slicing up a single light cone, based on their different definitions of simultaneity. Do you disagree?

Also, did you look at my diagram in post 182? If so did you understand it?

 

[A.T.]

I agree with you if you are saying the center of the light sphere is moving with the moving frame.

No, there is no frame in which the center of the light sphere is moving.

Thus, O will see it at 0 and O' will see it at vt and they disagree.

This sounds right, assuming "0" and "vt" refer to space points given in O-coorindantes: Each frame observes a different physical center of the light sphere. But nobody observes a moving center, two centers, or two spheres.

 

[cfrogue]

cfrogue, can you answer my questions from post 193?

What I am finding is the following.

One light sphere has two different origins.

I am not ready to say what it really is yet though.

So, no it is not the same light sphere.

 

[cfrogue]

No, there is no frame in which the center of the light sphere is moving.

This sounds right, assuming "0" and "vt" refer to space points given in O-coorindantes: Each frame observes a different physical center of the light sphere. But nobody observes a moving center, two centers, or two spheres.

We are in complete agreement

up to this point

"But nobody observes a moving center, two centers, or two spheres"

O knows the center of the light sphere for O' is located at vt. That therefore, is a moving light sphere.

 

[A.T.]

O knows the center of the light sphere for O' is located at vt. That thereforen is a moving light sphere.

Yes O knows / can calculate that the center of the light sphere observed by O' is located at vt, but O doesn't observe the center of the sphere there.

O uses his simultaneity to determine the center, and doesn't have to care about the simultaneity in O' or any other of the infinite number of frames.

 

[Dale]

Maybe so, but the fact is if you are in a frame and light emits from a light source, light moves spherically from the light source with the light source as the center.

No, it does not. The light flash moves spherically from the origin of the flash regardless of the subsequent motion of the source. That is the http://en.wikipedia.org/wiki/Postulates_of_special_relativity" : "As measured in an inertial frame of reference, light is always propagated in empty space with a definite velocity c that is independent of the state of motion of the emitting body."

 

[cfrogue]

No, it does not. The light flash moves spherically from the origin of the flash regardless of the subsequent motion of the source. That is the http://en.wikipedia.org/wiki/Postulates_of_special_relativity" : "As measured in an inertial frame of reference, light is always propagated in empty space with a definite velocity c that is independent of the state of motion of the emitting body."

You have failed to realize in the moving frame, it thinks it is at rest.

Light proceeds spherically in that frame from the light emission point in the frame.

Now, from the rest frame, that center point is at vt at any time t.

We must be scientific.