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

<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.

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

* 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 $$Length^{squared} = g_{11}(dX^{1})^{2} + g_{22}(dX^{2})^{2}$$

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

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

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

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):

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

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:

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

* [[Theory_of_Geometric_Unity|Eric Weinstein's Geometric Unity project]].