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<div style="float:right;padding:20px;">__TOC__</div>
{{InfoboxProject
|project=Graph-Wall-Tome Connection
|image=[[File:Ascending and Descending.jpg]]
|topic=[[Graph, Wall, Tome]]
|leader=Aardvark
|startdate=21 April 2020
|customlabel1=New Graph-Wall Mind Map
|customdata1=[https://drive.google.com/file/d/1ZMT8nDyF6qT5nh6hiCjrHyX4giKKmGSA/view?usp=sharing Link]
|customlabel2=Graph Mind Map
|customdata2=[https://drive.google.com/file/d/16r60-56mhYCx4KKBvZv_6HDLOLw43X7y/view?usp=sharing Link]
|customlabel3=Google Drive
|customdata3=[https://drive.google.com/drive/folders/1706CBEJQEMppV60OU8OtcXxicluk2T3Y?usp=sharing Drive]
|customlabel4=Master Planning
|customdata4=[https://docs.google.com/document/d/1t9AvvFZzODw1WiGRZwRsFFZdPdBzYVJGLHiqWNrMtIA/edit?usp=sharing Doc]
|link1title=Website
|link1=[https://graphwalltome.com/ Homepage]
|link2title=
|link2=
|link3title=
|link3=
|link4title=Discord
|link4=[https://discord.gg/Z3u3pPm Invite]
}}


An important aspect of the [[Graph,_Wall,_Tome#Eric_Weinstein.27s_Prompt|prompt]] is that neither the Graph, nor the Wall or the Tome are that important.  
An important aspect of the [[Graph, Wall, Tome#Prime Directive|prompt]] is that neither the [[Graph, Wall, Tome#Graph|Graph]], nor the [[Graph, Wall, Tome#Wall|Wall]] or the [[Graph, Wall, Tome#Tome|Tome]] are that important on their own. Only together does their explanatory power exceed the limits of any part alone. The common threads that run through all of them are what matter: what is common among them is what should be expanded, and what is particular to one should be promoted and developed if it is a part of fundamental physics toolkit, or removed if it isn't.
 
''What really matters are the common threads that run through all of them. ''


The goals of this project are to:
The goals of this project are to:
* Identify the common threads (the "unifying idea") in the Graph, Wall, and Tome.
* Identify the common threads (the "unifying idea") in the Graph, Wall, and Tome. These will indicate how to [[Editing the Graph|Edit the Graph]] and [[Defacing the Wall|Deface the Wall]].
* Create and collect resources that make it easy to understand them.
* Create and collect resources that make it easy to understand them, thus contributing to [[Rewriting the Tome]].
 
[[Decoding the Graph-Wall-Tome Connection/Project|Database Information]]


== Guiding Questions and Comments by Eric Weinstein ==
== Guiding Questions and Comments by Eric Weinstein ==
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div style="font-weight:bold;line-height:1.6;">What is $$F_A$$ geometrically?</div>
<div style="font-weight:bold;line-height:1.6;">What is <math>F_A</math> geometrically?</div>
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
$$F_A$$ is the curvature tensor associated to the connection or vector potential $$A$$.
<math>F_A</math> is the curvature tensor associated with the connection or vector potential <math>A</math>.
</div></div>
</div></div>




<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div style="font-weight:bold;line-height:1.6;">What are $$R_{\mu v}$$ and $$R$$ geometrically?</div>
<div style="font-weight:bold;line-height:1.6;">What are <math>R_{\mu v}</math> and <math>R</math> geometrically?</div>
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
[https://www.youtube.com/watch?v=UfThVvBWZxM&t=12m6s Explanation of <math>R</math>]
<math>R</math> 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.
<math>R</math> 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.


[https://www.youtube.com/watch?v=UfThVvBWZxM&t=12m6s Explanation of $$R$$]
In the video, they focus first on the curvature of space. Hopefully they incorporate back in curvature in time, because that's less obvious.


$$R$$ 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.  
The same video then proceeds to explain <math>R_{\mu v}</math>. It progresses through some concepts.
</div></div>


$$R$$ 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.


In the video, they focus first on the curvature of space. Hopefully they incorporate back in curvature in time, because that's less obvious.
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div style="font-weight:bold;line-height:1.6;">Further thoughts on the meaning of R</div>
<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:
 
<math>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}</math>
 
Some notes:
 
* This is not Pythagorean theorem, because <math>dX^{1}</math> and <math>dX^{2}</math> are not perpendicular.
* Instead, looks like a formula to get the diagonal from two opposite vertices in a parallelogram.
* If <math>dX^{1}</math> and <math>dX^{2}</math> are perpendicular, then <math>g_{12}</math> and <math>g_{21}</math> would be 0, and we would get <math>Length^{squared} = g_{11}(dX^{1})^{2} + g_{22}(dX^{2})^{2}</math>
* 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:
 
<math>dV^{1} = dX^{1}dX^{2} (V^{1}R^{1}_{112} + V^{2}R^{1}_{212} + V^{3}R^{1}_{312})</math>
 
<math>dV^{2} = dX^{1}dX^{2} (V^{1}R^{2}_{112} + V^{2}R^{2}_{212} + V^{3}R^{2}_{312})</math>
 
<math>dV^{3} = dX^{1}dX^{2} (V^{1}R^{3}_{112} + V^{2}R^{3}_{212} + V^{3}R^{3}_{312})</math>
 
 
or, using <math>i</math> to summarize across all 3 components (difference vectors):
 
<math>dV^{i} = dX^{1}dX^{2} (V^{1}R^{i}_{112} + V^{2}R^{i}_{212} + V^{3}R^{i}_{312})</math>
 
or , using <math>j</math> to index over all 3 components (original vector):
 
<math>dV^{i} = dX^{1}dX^{2} \Sigma_{j} [(V^{j}R^{i}_{j12}]</math>
 
See: [https://www.youtube.com/watch?v=UfThVvBWZxM&t=19m33s the video @ 19m33s]
 
Open questions:
 
* 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 <math>R_{\mu v}</math> as:
 
<math>R_{00} = R^{0}_{000} + R^{1}_{010} + R^{2}_{020} + R^{3}_{030}</math>
 
<math>R_{10} = R^{0}_{100} + R^{1}_{110} + R^{2}_{120} + R^{3}_{130}</math>
 
<math>R_{01} = R^{0}_{001} + R^{1}_{011} + R^{2}_{021} + R^{3}_{030}</math>
 
etc.
 
Indexing i over all 4 component vectors / dimensions, we get:
 
<math>R_{00} = \Sigma_{i} R^{i}_{0i0}</math>
 
<math>R_{10} = \Sigma_{i} R^{i}_{1i0}</math>
 
<math>R_{01} = \Sigma_{i} R^{i}_{0i1}</math>
 
etc.
 
Summarizing on <math>\mu</math>, we get:
 
<math>R_{\mu 0} = \Sigma_{i} R^{i}_{\mu i0}</math>
 
<math>R_{\mu 1} = \Sigma_{i} R^{i}_{\mu i1}</math>
 
etc
 
Summarizing on <math>v</math>, we get:
 
<math>R_{\mu v} = \Sigma_{i} R^{i}_{\mu iv}</math>


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


[[Further thoughts on the meaning of R]]
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?
</div></div>
</div></div>


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== Direct Connections between the Graph, the Wall, and the Tome ==
== Direct Connections between the Graph, the Wall, and the Tome ==


=== Connections between the Graph and the Wall===
=== Connections between the Graph and the Wall===




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[[File:Graph To Wall.png|800px|center]]
[[File:Graph To Wall.png|800px|center]]
</div></div>
</div></div>




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[[File:Wall-Tome Connections.png|center]]
[[File:Wall-Tome Connections.png|center]]
</div></div>
</div></div>




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== Resources ==
* [https://drive.google.com/file/d/1ZMT8nDyF6qT5nh6hiCjrHyX4giKKmGSA/view?usp=sharing New Graph-Wall Mind Map]
* [https://drive.google.com/file/d/16r60-56mhYCx4KKBvZv_6HDLOLw43X7y/view?usp=sharing Map of Witten's Physics and Geometry Essay]
* [https://docs.google.com/document/d/1t9AvvFZzODw1WiGRZwRsFFZdPdBzYVJGLHiqWNrMtIA/edit?usp=sharing Master Planning Doc]
* [https://drive.google.com/drive/folders/1706CBEJQEMppV60OU8OtcXxicluk2T3Y?usp=sharing Drive Folder]
* [https://docs.google.com/document/d/18rN-zfv41xeH3WFNOrZNb7Clz-yu1dgulP4bcEPKbcY/edit?usp=sharing List of Reference Material]


== Resources ==
* [[Theory_of_Geometric_Unity|Eric Weinstein's Geometric Unity project]].
[[Category:Graph, Wall, Tome]]
[[Category:Graph, Wall, Tome]]
[[Category:Projects]]
[[Category:Projects]]
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