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

[atyy]

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))

I don't understand this step. Can you explain your reasoning in more detail?

 

[cfrogue]

I don't understand this step. Can you explain your reasoning in more detail?

Yes,

given a rod in O', the right end point moves away from the expanding light sphere in O.

So, for example, let's do the right end point,

the light must travel the distance d/(2*λ) plus the distance O' moved in time t which is vt.
So, light travels ct in time t therefore,

ct = d/(2*λ) + vt.

ct - vt = d/(2*λ)

t( c - v ) = d/(2*λ)

t = d/(2*λ*(c - v))

Since this is the time in O, then it is t'(R').

You can also find this logic here.
http://www.fourmilab.ch/etexts/einstein/specrel/www/

At the end of section 2.

Einstein wrote it as tA - tB = rAB/(c-v)

 

[atyy]

Yes,

given a rod in O', the right end point moves away from the expanding light sphere in O.

So, for example, let's do the right end point,

the light must travel the distance d/(2*λ) plus the distance O' moved in time t which is vt.
So, light travels ct in time t therefore,

ct = d/(2*λ) + vt.

ct - vt = d/(2*λ)

t( c - v ) = d/(2*λ)

t = d/(2*λ*(c - v))

Since this is the time in O, then it is t'(R').

You can also find this logic here.
http://www.fourmilab.ch/etexts/einstein/specrel/www/

At the end of section 2.

Einstein wrote it as tA - tB = rAB/(c-v)

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))

So this is incorrect (I think). t'(L') is the time that O' assigns to the photon crossing L'.

The time that O assigns to the photon crossing L' is something else, let's call that t(t'(L')).

 

[cfrogue]

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

So this is incorrect (I think). t'(L') is the time that O' assigns to the photon crossing L'.

The time that O assigns to the photon crossing L' is something else, let's call that t(t'(L')).

No, these are the standard equations for O.

O' concludes the light strikes t'(L') and t'(R') at the same time.

 

[atyy]

No, these are the standard equations for O.

I think the LHS is wrong, but the RHS is correct.

O' concludes the light strikes t'(L') and t'(R') at the same time.

Yes. But O does not conclude that the light strikes L' and R' at the same time. O concludes that the light strikes L' at t(t'(L')) and the light strikes R' at t(t'(R')).

I would try

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

OK, I'm not sure about that, let me do some algebra ...

 

[cfrogue]

I think the LHS is wrong, but the RHS is correct.



Yes. But O does not conclude that the light strikes L' and R' at the same time. O concludes that the light strikes L' at t(t'(L')) and the light strikes R' at t(t'(R')).

I would try

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

OK, I'm not sure about that, let me do some algebra ...

Yes. But O does not conclude that the light strikes L' and R' at the same time.

In one sense, R of S it does not, but the light postulate in O' demands they are the same.

This would imply O' concludes the R' and L' are struck at the same time, but O cannot calculate this correctly with LT, so LT fails.

 

[cfrogue]

I think the LHS is wrong, but the RHS is correct.



Yes. But O does not conclude that the light strikes L' and R' at the same time. O concludes that the light strikes L' at t(t'(L')) and the light strikes R' at t(t'(R')).

I would try

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

OK, I'm not sure about that, let me do some algebra ...

No...

 

[atyy]

No...

Yup, that's wrong.

Editing ... it's almost right ... ok, it's right.

Calculation A

t(t'(R')) = d/(2*λ(c-v)
=d.sqrt(1-v^2/c^2)/[2(c-v)]
=d.sqrt((1-v/c).(1+v/c))/[2(c-v)]
=d.sqrt(((c-v)/c).((c+v)/c))/[2(c-v)]
=d.sqrt((c-v).(c+v))/[2c(c-v)]
=(d/2c).sqrt((c+v)/(c-v))

Calculation B

For O', t'(R')=d/2c and L'=d/2

The Lorentz transformation is "t' = ( t - vx/c^2 )λ"
In the above formula we put
" t' " as t(t'(R'))
" t " as t'(R')=d/2c
" x " as L'=d/2
" v " as -v
So
t(t'(R')) = [d/2c + vd/(2c^2)]λ
=(d/2c).(1+v/c).λ
=d/2c.(1+v/c)/sqrt(1-v^2/c^2)
=d/2c.(1+v/c)/sqrt((1-v/c)(1+v/c))
=d/2c.sqrt((1+v/c)/(1-v/c))
=d/2c.sqrt(((c+v)/c))/((c-v)/c)
=d/2c.sqrt((c+v)/(c-v)))

Calculation A from the viewpoint of O, and Calculation B as the Lorentz transformation of the viewpoint of O' into the viewpoint of O match up.

 

[A.T.]

In each frame the sphere is centered around that 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.

In frame O it is centered around O origin
In frame O' it is centered around O' origin

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.

No it is a failed logic, because you confuse things from two different frames. In each frame there is only one origin that is the center of the light sphere. What you have to realize is this:

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

A physical location is defined by a real object like the light source or the end of a rod. The center of the light sphere is not a physical location, but just a calculated coordiante. It is calculated from the coordiantes of those physical locations which are hit by the light simultaneously. But simultaneity is frame dependent and so is the center of the light sphere in respect to physical locations.

 

[Dale]

In the following reply I am going to make a strong distinction between the light cone and a light sphere; I will not use the two terms interchangeably. I am also going to make a strong distinction between the apex of the light cone and a center of a light sphere. The light cone is a single 4D geometric object in Minkowski spacetime which is the set of all events that are null separated (c²Δt²-Δx²-Δy²-Δz²=0) from the apex event. A light sphere is a 3D conic section of the light cone formed by the intersection of the light cone with any plane of simultaneity in any inertial reference frame, the center of a light sphere is the event which is equidistant from all of the events in the light sphere in the 3D plane of simultaneity used to form the light sphere.

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?

Yes, that is correct the light sphere in O at any given t>0 is centered at x=0. Also, the light sphere in O' at any given t'>0 is centered at x'=0. Furthermore, the line formed by the centers of all of the light spheres in O' is given by the equation x'=0 or x=vt, and the line formed by the centers of all of the light spheres in O is given by the equation x=0 or x'=-vt'.

I am certain that you will think this is a contradiction, but it is not. The key thing to remember is that there is not just a single light sphere, there are in fact an infinite number of light spheres one corresponding to every coordinate time in every inertial reference frame. Different frames disagree on the location of the center of the various light spheres because each light sphere refers to a completely different set of events.

In contrast, there is a single light cone, and the apex of the light cone does not change. All frames agree which event is the apex (although they may disagree on the coordinates assigned to the apex if it is not located at the mutual origin) and that event does not change in any reference frame.

 

[Dale]

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

Yes, you can see this in the spacetime diagram. Note that L follows the line x=-1 and R follows the line x=1. You can easily see where each of those lines intersect the light cone and verify that both events occur at the line t=1.

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

Yes, you can also see this in the spacetime diagram. L' follows the line x'=-1 and R' follows the line x'=1. Again, you can easily see where each of those lines intersect the light cone and verify that both events occur at the line t'=1.

 

[Jackadsa]

Yes. But O does not conclude that the light strikes L' and R' at the same time.

In one sense, R of S it does not, but the light postulate in O' demands they are the same.

This would imply O' concludes the R' and L' are struck at the same time, but O cannot calculate this correctly with LT, so LT fails.

atyy has it correct. Let me call t'(R') and t'(L') the times that an observer in O' records the light pulse hitting R' and L', and t(R') and t(L') the times that an observer in O records these same events (note that my t(R') is atyy's t(t'(R'))). Then as atyy stated:

(I will use & for gamma, since I havn't yet figured out how to use equations)

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

which you get from considering the rear and leading ends of the rod moving towards and away from the light respectively.

also, if x(R') and x(L') are the spatial coordinates of the events as recorded by an observer in O, then:

x(R') = ct(R')
x(L') = -ct(L')

Now, the observer in O can use the lorentz transformation to calculate the coordinates in O'. In particular, we are interested in what he calculates t'(R') and t'(L') to be, and whether this is consistent with t'(R') = t'(L') = d/2c.

t'(R') = &[t(R') - vx(R')/c^2]
t'(R') = &[t(R') - t(R')v/c] (I have substituted x(R') = ct(R'))
t'(R') = &t(R')(1 - v/c)
t'(R') = &(1 - v/c)d/(2&(c-v)) (I have substituted t(R') = d/(2&(c-v)))
t'(R') = d/2c

A similar calculation shows t'(L') = d/2c

So O records both events as non simultaneous. But then when he uses the lorentz transformation to figure out what O' sees, he finds that the events are indeed simultaneous in O' as expected.

This would imply O' concludes the R' and L' are struck at the same time, but O cannot calculate this correctly with LT, so LT fails.

 

[Dale]

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

I think you mean:
t(L') = d/(2*?(c+v))
and
t(R') = d/(2*?(c-v))

Again, you can see each of these in the diagram. For t(R') look at R' (x'=1) and see where it intersects the light cone, and note that event occurs at t=2 which agrees with the equation above. For t(L') look at L' (x'=-1) and see where it intersects the light cone, and note that event is halfway between the lines for t=0 and t=1, so it occurs at t=0.5 which also agrees with the equation above.

Hopefully you are beginning to see the value of spacetime diagrams.

 

[cfrogue]

I think the LHS is wrong, but the RHS is correct.



Yes. But O does not conclude that the light strikes L' and R' at the same time. O concludes that the light strikes L' at t(t'(L')) and the light strikes R' at t(t'(R')).

I would try

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

OK, I'm not sure about that, let me do some algebra ...

I have looked it over and you are correct.

 

[cfrogue]

In frame O it is centered around O origin
In frame O' it is centered around O' origin


No it is a failed logic, because you confuse things from two different frames. In each frame there is only one origin that is the center of the light sphere. What you have to realize is this:

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

A physical location is defined by a real object like the light source or the end of a rod. The center of the light sphere is not a physical location, but just a calculated coordiante. It is calculated from the coordiantes of those physical locations which are hit by the light simultaneously. But simultaneity is frame dependent and so is the center of the light sphere in respect to physical locations.

No, the center of the light sphere in each is a well defined concept.

The center diverges by vt.

 

[cfrogue]

Yup, that's wrong.

Editing ... it's almost right ... ok, it's right.

Calculation A

t(t'(R')) = d/(2*λ(c-v)
=d.sqrt(1-v^2/c^2)/[2(c-v)]
=d.sqrt((1-v/c).(1+v/c))/[2(c-v)]
=d.sqrt(((c-v)/c).((c+v)/c))/[2(c-v)]
=d.sqrt((c-v).(c+v))/[2c(c-v)]
=(d/2c).sqrt((c+v)/(c-v))

Calculation B

For O', t'(R')=d/2c and L'=d/2

The Lorentz transformation is "t' = ( t - vx/c^2 )λ"
In the above formula we put
" t' " as t(t'(R'))
" t " as t'(R')=d/2c
" x " as L'=d/2
" v " as -v
So
t(t'(R')) = [d/2c + vd/(2c^2)]λ
=(d/2c).(1+v/c).λ
=d/2c.(1+v/c)/sqrt(1-v^2/c^2)
=d/2c.(1+v/c)/sqrt((1-v/c)(1+v/c))
=d/2c.sqrt((1+v/c)/(1-v/c))
=d/2c.sqrt(((c+v)/c))/((c-v)/c)
=d/2c.sqrt((c+v)/(c-v)))

Calculation A from the viewpoint of O, and Calculation B as the Lorentz transformation of the viewpoint of O' into the viewpoint of O match up.

I will look these over and thanks.

I agree, the equations I presented were frame mixed.

 

[cfrogue]

In the following reply I am going to make a strong distinction between the light cone and a light sphere; I will not use the two terms interchangeably. I am also going to make a strong distinction between the apex of the light cone and a center of a light sphere. The light cone is a single 4D geometric object in Minkowski spacetime which is the set of all events that are null separated (c²Δt²-Δx²-Δy²-Δz²=0) from the apex event. A light sphere is a 3D conic section of the light cone formed by the intersection of the light cone with any plane of simultaneity in any inertial reference frame, the center of a light sphere is the event which is equidistant from all of the events in the light sphere in the 3D plane of simultaneity used to form the light sphere.Yes, that is correct the light sphere in O at any given t>0 is centered at x=0. Also, the light sphere in O' at any given t'>0 is centered at x'=0. Furthermore, the line formed by the centers of all of the light spheres in O' is given by the equation x'=0 or x=vt, and the line formed by the centers of all of the light spheres in O is given by the equation x=0 or x'=-vt'.

I am certain that you will think this is a contradiction, but it is not. The key thing to remember is that there is not just a single light sphere, there are in fact an infinite number of light spheres one corresponding to every coordinate time in every inertial reference frame. Different frames disagree on the location of the center of the various light spheres because each light sphere refers to a completely different set of events.

In contrast, there is a single light cone, and the apex of the light cone does not change. All frames agree which event is the apex (although they may disagree on the coordinates assigned to the apex if it is not located at the mutual origin) and that event does not change in any reference frame.

The fact is that the light sphere has two different centers based on any stationary observer.

That part is a contradiction.

However, I ran some simulations today and cannot get the light sphere in O' to exceed the origin of O if v = .5c on the negative x-axis side.

Do you also come up with this?

 

[cfrogue]

Yes, you can see this in the spacetime diagram. Note that L follows the line x=-1 and R follows the line x=1. You can easily see where each of those lines intersect the light cone and verify that both events occur at the line t=1. Yes, you can also see this in the spacetime diagram. L' follows the line x'=-1 and R' follows the line x'=1. Again, you can easily see where each of those lines intersect the light cone and verify that both events occur at the line t'=1.

My equations were wrong and I admit that for the particular situation given.

However, I am still looking at the origin of the light sphere located at two different places give the stationary observer.

 

[cfrogue]

atyy has it correct. Let me call t'(R') and t'(L') the times that an observer in O' records the light pulse hitting R' and L', and t(R') and t(L') the times that an observer in O records these same events (note that my t(R') is atyy's t(t'(R'))). Then as atyy stated:

(I will use & for gamma, since I havn't yet figured out how to use equations)

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

which you get from considering the rear and leading ends of the rod moving towards and away from the light respectively.

also, if x(R') and x(L') are the spatial coordinates of the events as recorded by an observer in O, then:

x(R') = ct(R')
x(L') = -ct(L')

Now, the observer in O can use the lorentz transformation to calculate the coordinates in O'. In particular, we are interested in what he calculates t'(R') and t'(L') to be, and whether this is consistent with t'(R') = t'(L') = d/2c.

t'(R') = &[t(R') - vx(R')/c^2]
t'(R') = &[t(R') - t(R')v/c] (I have substituted x(R') = ct(R'))
t'(R') = &t(R')(1 - v/c)
t'(R') = &(1 - v/c)d/(2&(c-v)) (I have substituted t(R') = d/(2&(c-v)))
t'(R') = d/2c

A similar calculation shows t'(L') = d/2c

So O records both events as non simultaneous. But then when he uses the lorentz transformation to figure out what O' sees, he finds that the events are indeed simultaneous in O' as expected.

Agreed

 

[cfrogue]

I think you mean:
t(L') = d/(2*?(c+v))
and
t(R') = d/(2*?(c-v))

Again, you can see each of these in the diagram. For t(R') look at R' (x'=1) and see where it intersects the light cone, and note that event occurs at t=2 which agrees with the equation above. For t(L') look at L' (x'=-1) and see where it intersects the light cone, and note that event is halfway between the lines for t=0 and t=1, so it occurs at t=0.5 which also agrees with the equation above.

Hopefully you are beginning to see the value of spacetime diagrams.

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.

 

[JesseM]

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?

 

[atyy]

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.

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

 

[cfrogue]

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?

 

[cfrogue]

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.

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

Wow, your taste in music is unique.

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

 

[cfrogue]

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.

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

Yea, I came up with Jackadsa's equations today.

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

 

[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  |
+----------+
    |  |    
    |  |    
  .\|.||/..