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Decoding the Graph-Wall-Tome Connection: Difference between revisions
Decoding the Graph-Wall-Tome Connection (view source)
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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> | ||
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
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 | |||
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 | |||
Now, moving to 4D, we can compute $$R_{\mu v}$$ as: | Now, moving to 4D, we can compute $$R_{\mu v}$$ as: | ||