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.