Decoding the Graph-Wall-Tome Connection: Difference between revisions

An important aspect of the [[Graph,_Wall,_Tome#Eric_Weinstein.27s_Prompt|prompt]] is that neither the [[Graph,_Wall,_Tome#The_Graph|Graph]], nor the [[Graph,_Wall,_Tome#The_Wall|Wall]] or the [[Graph,_Wall,_Tome#The_Tome|Tome]] are that important.

 

''What really matters are the common threads that run through all of them. ''

 

* Identify the common threads (the "unifying idea") in the Graph, Wall, and Tome.

* Create and collect resources that make it easy to understand them.  

 

<div style="font-weight:bold;line-height:1.6;">What is $$F_A$$ geometrically?</div>

$$F_A$$ is the curvature tensor associated to the connection or vector potential $$A$$.

<div style="font-weight:bold;line-height:1.6;">What are $$R_{\mu v}$$ and $$R$$ geometrically?</div>

[https://www.youtube.com/watch?v=UfThVvBWZxM&t=12m6s Explanation of $$R$$]

$$R$$ is a scalar value, describing the "curvature of the spacetime manifold" at each point along the manifold. It's based on a concept of 'parallel transport', where you move a vector around some path on the manifold.  

$$R$$ can be computed at each point on the manifold, and describes the difference in the vector's angle after following an infinitesimally small path around the neighborhood of that point, vs. what it was originally. The video does a great job of visualizing when and why that vector angle change would happen, with flat vs. curved manifolds.

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

 

 

===== Computing length in non-orthogonal bases =====

$$Length^{squared} = g_{11}dX^{1}dX^{1} + g_{12}dX^{1}dX^{2} + g_{21}dX^{2}dX^{1} + g_{22}dX^{2}dX^{2}$$

* This is not Pythagorean theorem, because $$dX^{1}$$ and $$dX^{2}$$ are not perpendicular.  

* If $$dX^{1}$$ and $$dX^{2}$$ are perpendicular, then $$g_{12}$$ and $$g_{21}$$ would be 0, and we would get $$Length^{squared} = g_{11}(dX^{1})^{2} + g_{22}(dX^{2})^{2}$$

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

$$dV^{1} = dX^{1}dX^{2} (V^{1}R^{1}_{112} + V^{2}R^{1}_{212} + V^{3}R^{1}_{312})$$

$$dV^{2} = dX^{1}dX^{2} (V^{1}R^{2}_{112} + V^{2}R^{2}_{212} + V^{3}R^{2}_{312})$$

$$dV^{3} = dX^{1}dX^{2} (V^{1}R^{3}_{112} + V^{2}R^{3}_{212} + V^{3}R^{3}_{312})$$

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):

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

===== Putting it all together =====

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

$$R_{00} = R^{0}_{000} + R^{1}_{010} + R^{2}_{020} + R^{3}_{030}$$

$$R_{10} = R^{0}_{100} + R^{1}_{110} + R^{2}_{120} + R^{3}_{130}$$

$$R_{01} = R^{0}_{001} + R^{1}_{011} + R^{2}_{021} + R^{3}_{030}$$

$$R_{00} = \Sigma_{i} R^{i}_{0i0}$$

$$R_{10} = \Sigma_{i} R^{i}_{1i0}$$

$$R_{01} = \Sigma_{i} R^{i}_{0i1}$$

Summarizing on $$\mu$$, we get:

$$R_{\mu 0} = \Sigma_{i} R^{i}_{\mu i0}$$

$$R_{\mu 1} = \Sigma_{i} R^{i}_{\mu i1}$$

Summarizing on $$v$$, we get:

$$R_{\mu v} = \Sigma_{i} R^{i}_{\mu iv}$$