A Question for Cjlr
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17-09-2015, 02:07 AM (This post was last modified: 17-09-2015 10:00 AM by ZoraPrime.)
RE: A Question for Cjlr
(16-09-2015 02:19 AM)Heywood Jahblome Wrote:  
(14-09-2015 02:26 PM)cjlr Wrote:  Spin refers to angular motion about a single axis. That's how it's defined - it can't be anything else.

In 4D space two independent rotations can occur without them summing to a single rotation in a single plane. An Object rotating in the XY plane can also simultaneously rotate in the WZ plane. The rotation in the XY plane is around an axis which is a straight line, and the rotation around the WZ plane is around another axis which is also a straight line. A dust cloud in 4D space wouldn't collapse into a single accretion disc because there isn't a single plane of rotation. There are two planes of rotation....each having its own axis.

I think I am good here.

uggh.

In 4D Euclidean space, there's six angular momentum components; there's one for each plane (xy, xz, xw, yz, yw, zw). By convention, in 3-space, we associated each plane with the vector orthogonal to that plane so we can treat angular momentum as a vector.

HOWEVER, the reason why an accretion disk forms is because angular momentum is conserved; this follows from Noether's Theorem. If you can show that a system (or rather, the action associated with the system) lacks explicit rotational dependence, then angular momentum is conserved. Just because angular momentum has six components doesn't change that fact.

Thirdly, this has nothing to do with quantum spin. In quantum spin, the system's angular momentum is conserved (through entanglement); even though individual electrons will always be measured to be "spin up" or "spin down." Again, this doesn't change in a 4D extension.

Lastly, I want to draw your attention to a key word: Euclidean. When dealing with non-Euclidean space, axes lose symmetry (e.g. in Lorentz space, the ct-axis is distinct from the x-axis by its geometry), and as such, rotations are different. Whereas a rotation in the usual sense is sufficient for the spatial coordinates of Lorentzian space, the corresponding "rotation" when involving the time-axis is hyperbolic rather than trigonometric (i.e, you use this matrix, not this one). Yes, there are three extra components of something like angular momentum in special relativity, but because the ct axis is distinct, we can treat the 3 angular momentum vectors as something else entirely.
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17-09-2015, 02:22 AM
RE: A Question for Cjlr
Holy shit... that last post was so far above my pay grade I can't even... Shocking

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(06-02-2014 03:47 PM)Momsurroundedbyboys Wrote:  And I'm giving myself a conclusion again from all the facepalming.
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17-09-2015, 02:47 AM
RE: A Question for Cjlr
(17-09-2015 02:22 AM)morondog Wrote:  Holy shit... that last post was so far above my pay grade I can't even... Shocking

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17-09-2015, 02:55 AM
RE: A Question for Cjlr
(17-09-2015 02:22 AM)morondog Wrote:  Holy shit... that last post was so far above my pay grade I can't even... Shocking

Me too.

It made my head spin like someone had been rotating my Euclidean spot!



(17-09-2015 02:07 AM)ZoraPrime Wrote:  ...
Whereas a rotation in the usual sense is sufficient for the 3D Euclidean spot of Lorentzian space,
...

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17-09-2015, 02:56 AM
RE: A Question for Cjlr
(17-09-2015 02:55 AM)DLJ Wrote:  
(17-09-2015 02:22 AM)morondog Wrote:  Holy shit... that last post was so far above my pay grade I can't even... Shocking
Me too.

It made my head spin like someone had been rotating my Euclidean spot!

Did it find your... E-Spot? Consider

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17-09-2015, 03:23 AM (This post was last modified: 17-09-2015 03:28 AM by Heywood Jahblome.)
RE: A Question for Cjlr
Quote:.... so we can treat angular momentum as a vector.

Rotation is not a vector. It is a process that combines two orthogonal directions. We say that process rotates around the a particular axis, say Z, because it isn't the dimension being combined. In 3d space Once you define an axis, rotation is the combination of the remain two dimensions but this isn't really a vector even though we often think of it as being one. There is no quantity having a specific direction as well as magnitude.....there is no vector.

You really haven't shown that in 4d space rotation in the XY plane and rotation in the ZW plane can ever be combined mathematically into a single planer rotation. So my claim still seems to be true. In 4D space a dust cloud would not collapse into single accretion disk.
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17-09-2015, 06:24 AM
RE: A Question for Cjlr
(17-09-2015 03:23 AM)Heywood Jahblome Wrote:  
Quote:.... so we can treat angular momentum as a vector.

Rotation is not a vector. It is a process that combines two orthogonal directions. We say that process rotates around the a particular axis, say Z, because it isn't the dimension being combined. In 3d space Once you define an axis, rotation is the combination of the remain two dimensions but this isn't really a vector even though we often think of it as being one. There is no quantity having a specific direction as well as magnitude.....there is no vector.

You really haven't shown that in 4d space rotation in the XY plane and rotation in the ZW plane can ever be combined mathematically into a single planer rotation. So my claim still seems to be true. In 4D space a dust cloud would not collapse into single accretion disk.

Oh fuck, you again!

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17-09-2015, 07:04 AM
RE: A Question for Cjlr
(16-09-2015 12:28 AM)Heywood Jahblome Wrote:  
(14-09-2015 04:02 PM)The Organic Chemist Wrote:  If the magnetogyric ratio of the electron was fine tuned........then god. Facepalm

This thread isn't about God. Please be quiet and let the adults talk.....thanks.

Right, the adults. He ready answered your question. I understood him, did you?

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17-09-2015, 09:58 AM (This post was last modified: 17-09-2015 10:46 PM by ZoraPrime.)
RE: A Question for Cjlr
(17-09-2015 03:23 AM)Heywood Jahblome Wrote:  
Quote:.... so we can treat angular momentum as a vector.

Rotation is not a vector. It is a process that combines two orthogonal directions. We say that process rotates around the a particular axis, say Z, because it isn't the dimension being combined. In 3d space Once you define an axis, rotation is the combination of the remain two dimensions but this isn't really a vector even though we often think of it as being one. There is no quantity having a specific direction as well as magnitude.....there is no vector.

You really haven't shown that in 4d space rotation in the XY plane and rotation in the ZW plane can ever be combined mathematically into a single planer rotation. So my claim still seems to be true. In 4D space a dust cloud would not collapse into single accretion disk.

In general, angular momentum can be represented as a bivector, which is for all intents and purposes, an (anti-symmetric) matrix*. By Noether's Theorem, this bivector is conserved, and the addition for bivectors is the same vectors (you add component by component). As such the equation L[sub]system[/sub]=constant still holds true. If you're saying we can't find an operation that involves rotating XY and ZW... we can. You just multiply the corresponding rotation matrices for XY rotation and ZW for one operator. Lastly, the fact that angular momentum in 4D has six components doesn't change how in 4D a dust cloud would collapse into a single accretion disk (or the 4D Euclidean equivalent thereof). On the contrary, the same argument for 3D for why a accretion disk forms (angular momentum is constant by Noether's theorem) is the same argument in 4D for why a accretion disk forms (angular momentum is constant by Noether's theorem). The fact the bivector has more components doesn't change that argument.

Nevermind the whole fact that 4D Euclidean space never shows up anywhere, so this whole discussion is moot on that.



*if want to be pedantic, I have to say it's isomorphic to an antisymmetric matrix, but that's a detail since isomorphic is the mathematicians way of saying "practically equivalent."



_____________________

It's occurred to me that I think you're getting concerned over the word "disk." Yes, a 4D Euclidean "disk' (in the sense of how we're using it) spans over two planes. However, there's no word for "flat object that confines itself to particular plane(s)." We'd call them disks, even if they span two planes. It's like becoming concerned because someone uses the word "spherical symmetry" when discussing 4D Euclidean space and insisting "it's not a sphere, it's a hypersphere, so we should call it hypersherical symmetry." I'm calling it an accretion disk, even if it spans two planes, because I don't see why the fact it spans over two planes instead of one plane matters when it's orientation in space (i.e. the 4D angular momentum bivector) doesn't change.
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17-09-2015, 11:29 AM
RE: A Question for Cjlr
(11-09-2015 08:20 PM)Heywood Jahblome Wrote:  In space that has 3 spatial dimensions, a dust cloud will collapse to a disc. <...snip...>

Now suppose you had 4 spatial dimensions instead of 3. Would dust clouds still have an average spin? ...

Let me paraphrase your question:

In the real world cheese cannot fly, therefore we can make cheese sandwiches. Now suppose that cheese could fly. Would we still be able to make cheese sandwiches, or would it all fly away? And what would that do to the dairy industry?

Preposterous suppositions belong in a bar with all the participants drunk. Get thee to thy local watering hole, my friend, and invent all the universes you like.

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