Difference between revisions of "Decoding the Graph-Wall-Tome Connection"

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<div style="font-weight:bold;line-height:1.6;">Further thoughts on the meaning of R</div>
<div style="font-weight:bold;line-height:1.6;">Further thoughts on the meaning of R</div>
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===== Computing length in non-orthogonal bases =====
Computing length in non-orthogonal bases


First, just describing the length of a vector on a curved space is hard. It is given by:
First, just describing the length of a vector on a curved space is hard. It is given by:
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* See: [https://www.youtube.com/watch?v=UfThVvBWZxM&t=14m27s the video @ 14m27s]
* See: [https://www.youtube.com/watch?v=UfThVvBWZxM&t=14m27s the video @ 14m27s]


===== Computing vector rotation due to parallel transport =====
Computing vector rotation due to parallel transport


Then, they show parallel transport when following a parallelogram, but over a curved 3D manifold. To compute the vector rotation by components, they show:
Then, they show parallel transport when following a parallelogram, but over a curved 3D manifold. To compute the vector rotation by components, they show:
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===== Putting it all together =====
Putting it all together


Now, moving to 4D, we can compute $$R_{\mu v}$$ as:
Now, moving to 4D, we can compute $$R_{\mu v}$$ as: