Difference between revisions of "Annotating the Wall"

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== Questions by Eric Weinstein ==
== Questions by Eric Weinstein ==


 
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=== What is $$F_A$$ geometrically? ===
<div style="font-weight:bold;line-height:1.6;">What is $$F_A$$ geometrically?</div>
 
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$$F_A$$ is the curvature tensor associated to the connection or vector potential $$A$$.
$$F_A$$ is the curvature tensor associated to the connection or vector potential $$A$$.
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=== What are $$R_{\mu v}$$ and $$R$$ geometrically? ===
[https://en.wikipedia.org/wiki/Einstein_field_equations Einstein field equations]
Einstein’s original publication, Die Feldgleichungen der Gravitation, in English


==== $$R$$ ====
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<div style="font-weight:bold;line-height:1.6;">What are $$R_{\mu v}$$ and $$R$$ geometrically?</div>
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[https://www.youtube.com/watch?v=UfThVvBWZxM&t=12m6s Explanation of $$R$$]
[https://www.youtube.com/watch?v=UfThVvBWZxM&t=12m6s Explanation of $$R$$]
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In the video, they focus first on the curvature of space. Hopefully they incorporate back in curvature in time, because that's less obvious.
In the video, they focus first on the curvature of space. Hopefully they incorporate back in curvature in time, because that's less obvious.
==== $$R_{\mu v}$$ ====


The same video then proceeds to explain $$R_{\mu v}$$. It progresses through some concepts.
The same video then proceeds to explain $$R_{\mu v}$$. It progresses through some concepts.


===== Computing length in non-orthogonal bases =====
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First, just describing the length of a vector on a curved space is hard. It is given by:


$$Length^{squared} = g_{11}dX^{1}dX^{1} + g_{12}dX^{1}dX^{2} + g_{21}dX^{2}dX^{1} + g_{22}dX^{2}dX^{2}$$
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<div style="font-weight:bold;line-height:1.6;">How do they relate?</div>
Some notes:
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[https://en.wikipedia.org/wiki/Cohomology Cohomology]
* This is not Pythagorean theorem, because $$dX^{1}$$ and $$dX^{2}$$ are not perpendicular.  
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* Instead, looks like a formula to get the diagonal from two opposite vertices in a parallelogram.
* If $$dX^{1}$$ and $$dX^{2}$$ are perpendicular, then $$g_{12}$$ and $$g_{21}$$ would be 0, and we would get $$Length^{squared} = g_{11}(dX^{1})^{2} + g_{22}(dX^{2})^{2}$$
* 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:
 
$$dV^{1} = dX^{1}dX^{2} (V^{1}R^{1}_{112} + V^{2}R^{1}_{212} + V^{3}R^{1}_{312})$$
 
$$dV^{2} = dX^{1}dX^{2} (V^{1}R^{2}_{112} + V^{2}R^{2}_{212} + V^{3}R^{2}_{312})$$
 
$$dV^{3} = dX^{1}dX^{2} (V^{1}R^{3}_{112} + V^{2}R^{3}_{212} + V^{3}R^{3}_{312})$$
 
 
or, using $$i$$ to summarize across all 3 components (difference vectors):
 
$$dV^{i} = dX^{1}dX^{2} (V^{1}R^{i}_{112} + V^{2}R^{i}_{212} + V^{3}R^{i}_{312})$$
 
or , using $$j$$ to index over all 3 components (original vector):


$$dV^{i} = dX^{1}dX^{2} \Sigma_{j} [(V^{j}R^{i}_{j12}]$$


See: [https://www.youtube.com/watch?v=UfThVvBWZxM&t=19m33s the video @ 19m33s]
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<div style="font-weight:bold;line-height:1.6;">What does this have to do with Penrose Stairs?</div>
Open questions:
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* Why a parallelogram?
* How to properly overlay the parallelogram onto the 3d manifold, in order to do the parallel transport?
* How does this relate to the length computation above?
 
 
===== Putting it all together =====
 
Now, moving to 4D, we can compute $$R_{\mu v}$$ as:
 
$$R_{00} = R^{0}_{000} + R^{1}_{010} + R^{2}_{020} + R^{3}_{030}$$
 
$$R_{10} = R^{0}_{100} + R^{1}_{110} + R^{2}_{120} + R^{3}_{130}$$
 
$$R_{01} = R^{0}_{001} + R^{1}_{011} + R^{2}_{021} + R^{3}_{030}$$
 
etc.
 
Indexing i over all 4 component vectors / dimensions, we get:
 
$$R_{00} = \Sigma_{i} R^{i}_{0i0}$$
 
$$R_{10} = \Sigma_{i} R^{i}_{1i0}$$
 
$$R_{01} = \Sigma_{i} R^{i}_{0i1}$$
 
etc.
 
Summarizing on $$\mu$$, we get:
 
$$R_{\mu 0} = \Sigma_{i} R^{i}_{\mu i0}$$
 
$$R_{\mu 1} = \Sigma_{i} R^{i}_{\mu i1}$$
 
etc
 
Summarizing on $$v$$, we get:
 
$$R_{\mu v} = \Sigma_{i} R^{i}_{\mu iv}$$
 
 
Open questions:
 
* If we hadn't moved from 3D to 4D, what would this all have looked like?
* What does this have to do with the parallelogram?
* Why are there two indices?
 
=== How do they relate? ===
[https://en.wikipedia.org/wiki/Cohomology Cohomology]
 
=== What does this have to do with Penrose Stairs? ===
* [https://en.wikipedia.org/wiki/Penrose_stairs Penrose stairs]
* [https://en.wikipedia.org/wiki/Penrose_stairs Penrose stairs]
* [https://en.wikipedia.org/wiki/Spinor Spinor]
* [https://en.wikipedia.org/wiki/Spinor Spinor]
We’ve heard Eric talk about Penrose stairs and spinors - essentially phenomena where you cannot return to the original state through a 360 degree rotation, but require a 720 degree rotation.
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We’ve heard Eric talk about Penrose stairs and spinors - essentially phenomena where you cannot return to the original state through a 360 degree rotation, but require a 720 degree rotation.


=== What are “Horizontal Subspaces” and what do they have to do with Vector Potentials or Gauge fields? ===
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<div style="font-weight:bold;line-height:1.6;">What are “Horizontal Subspaces” and what do they have to do with Vector Potentials or Gauge fields?</div>
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* [https://en.wikipedia.org/wiki/Vertical_and_horizontal_bundles Vertical and horizontal bundles]
* [https://en.wikipedia.org/wiki/Vertical_and_horizontal_bundles Vertical and horizontal bundles]
* [https://en.wikipedia.org/wiki/Introduction_to_gauge_theory Introduction to gauge theory]
* [https://en.wikipedia.org/wiki/Introduction_to_gauge_theory Introduction to gauge theory]
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But doesn't this require the symmetry break? How is left and right rotation in a subspace transformed into verticality? This is a crazy rabbit hole, friends. Keep your chins up. Let me know if this was helpful or leading astray.
But doesn't this require the symmetry break? How is left and right rotation in a subspace transformed into verticality? This is a crazy rabbit hole, friends. Keep your chins up. Let me know if this was helpful or leading astray.
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== Further Resources ==
== Further Resources ==
Jakob
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