Editing Graph, Wall, Tome
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== Background == | |||
'''Imagine:''' | |||
It's 1915, and you've made one of the greatest discoveries in hundreds of years. You visit your mother and show her your work: | |||
$$R_{\mu v}-\frac{1}{2}Rg_{\mu v} = 8 \pi T_{\mu v}$$ | |||
'That's nice dear', she responds, unaware of the implications of your discovery. | |||
The | The problem, is that although this equation carries with it the secrets of gravity, to a layman it is merely a bunch of letters. | ||
Now, consider this single image. Β | |||
[[File:sheetsunx.gif||Curved Space-Time]] | |||
Instantly, the meaning becomes clear. Gravity warps space(time), and matter, planets, and even light follows a path, that is curved by the warped geometry. | |||
Fundamental physics is an unknown world to most people. Equations, symbols, and incomprehensible terms abound, and unless you've studied post-grad mathematics and physics, this world is inaccessible to you. | |||
Although there are several great resources to map the way toward complete understanding. Most people will not undertake the journey to understand the source code to the world that we all inhabit. | |||
The | == The GWT Project == | ||
The | Bringing an understanding of fundamental physics is one aim of The Portal. | ||
There currently exist 3 resources that themselves contain all that you need for an almost complete understanding of the world. | |||
The Graph | # '''The Graph''' - A paragraph written by Ed Witten | ||
# '''The Wall''' - The iconic wall of Stony Brook University | |||
# '''The Tome''' - The book 'The Road to Reality' by Roger Penrose | |||
These resources are available to everyone, but will be sought by almost none. The aim of the GWT project to convert these resources into a medium that can be widely disseminated, and which can not be ignored. | |||
And the Tome | This project will require bi-directional information transfer, and the minds of people with many different aptitudes. | ||
Β | |||
* We need mathematicians, topologists, geometers, and physicists to understand these resources, and all of their implications. | |||
* We need explainers, and educators, to convey this information to a wider audience. | |||
* And we need artists, linguists, and programmers to create intuitive visualisations. | |||
Β | |||
The Portal will create a community of people, working together to that achieve these aims. | |||
Β | |||
Success will generate yet further insights, perhaps opening up a more fundamental understanding of the nature of reality. | |||
Β | |||
Β | |||
Β | |||
This was first collected in a Google Doc titled [https://docs.google.com/document/d/1Bo5ny0UyC8gEHiAaDR2Al2OSGscUWPS8NFI-hvB1z4o/edit?pli=1 Graph, Wall, Tome - Problem Solving]. | |||
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== The Graph == | |||
Β | |||
<blockquote> | |||
If one wants to summarise our knowledge of physics in the briefest possible terms, there are three really fundamental observations: | |||
Β | |||
# [https://en.wikipedia.org/wiki/Spacetime Spacetime] is a [https://en.wikipedia.org/wiki/Pseudo-Riemannian_manifold pseudo-Riemannian manifold] M, endowed with a [https://en.wikipedia.org/wiki/Metric_tensor metric tensor] and governed by [https://en.wikipedia.org/wiki/Geometry geometrical laws]. | |||
# Over M is a [https://en.wikipedia.org/wiki/Principal_bundle principal bundle] $$P_{G}$$ with a [https://en.wikipedia.org/wiki/Non-abelian_group non-abelian structure group] G. | |||
# [https://en.wikipedia.org/wiki/Fermion Fermions] are sections of $$(\hat{S}_{+} \otimes V_{R}) \oplus (\hat{S}\_ \otimes V_{\bar{R}})$$. $$R$$ and $$\bar{R}$$ are not [https://en.wikipedia.org/wiki/Isomorphism isomorphic]; their failure to be isomorphic explains why the light fermions are light. | |||
# Add something about Higgs | |||
Β | |||
All of this must be supplemented with the understanding that the geometrical laws obeyed by the metric tensor, the [https://en.wikipedia.org/wiki/Introduction_to_gauge_theory gauge fields], and the fermions are to be interpreted in [https://en.wikipedia.org/wiki/Quantum_mechanics quantum mechanical] terms. | |||
</blockquote> | </blockquote> | ||
=== | === Origin === Β | ||
This is a modified version of the paragraph by Edward Witten as [https://twitter.com/EricRWeinstein/status/928296366853328896?s=20 posted by Eric via Twitter]. | |||
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[[file:The-graph.png|300px]] | |||
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Eric Weinstein suggested several alterations, that have been included above: | |||
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* In (ii), βvector bundle Xβ should be changed to principal G-bundle. | |||
* Also in (ii), βnonabelian gauge group Gβ should be changed to nonabelian structure group G. | |||
* In (iii), <math>\ R</math> and <math>\tilde R</math> should be (complex) linear representations of G and so they are not equivalent. | |||
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== The Wall == | |||
Β | |||
[[File:The-wall.png|right|400px]] | |||
This image is carved into a wall at Stony Brook University. It contains many of the most fundamental equations of physics, providing a formulaic representation of all reality. | |||
[http://www.math.stonybrook.edu/~tony/scgp/wall-story/wall-story.html Source] | |||
( | Several of the equations have been identified as having direct connections to statements in 'The Graph' (identified by numbers) | ||
=== Equations === | |||
'''1: Einstein's General Relativity equation:''' | |||
: $$R_{\mu v}-\frac{1}{2}Rg_{\mu v} = 8 \pi T_{\mu v}$$ | |||
Β | |||
'''2: Maxwell's equations:''' | |||
: $$\nabla \times \mathbf{B} = \frac{1}{c} \frac{\partial \mathbf{E}}{\partial t}$$ | |||
: $$\nabla \times \mathbf{E} = -\frac{1}{c} \frac{\partial \mathbf{B}}{\partial t}$$ | |||
: $$\nabla \cdot \mathbf{B} = 0$$ | |||
: $$\nabla \cdot \mathbf{E} = 0$$ | |||
Β | |||
'''2: Yang-Mills equations:''' | |||
: $$d^*_A F_A \propto J$$ | |||
Β | |||
'''3: Dirac equation''': | |||
: $$(i \not{D}_A - m)\psi = 0$$ | |||
Β | |||
'''4: Klein-Gordon equation:''' (this is not included in 'The Wall', but it has been suggested that perhaps it should have been) | |||
: $$\frac{1}{c^2} \frac{\partial^2}{\partial t^2} \psi - \nabla^2 \psi + \frac{m^2 c^2}{\hbar^2} \psi = 0$$ | |||
Β | |||
Einstein's mass-energy equation: | |||
: $$E = mc^2$$ | |||
Β | |||
Kepler's 2nd law: | |||
: $$\frac{d\theta}{dt} \propto \frac{1}{r^2}$$ | |||
Β | |||
Newton's force-acceleration equation: | |||
: $$\mathbf{F} = m\mathbf{a}$$ | |||
Β | |||
Keplers 3rd law: | |||
: $$T^2 \propto a^3$$ | |||
Β | |||
Newtons gravitational law: | |||
: $$F = \frac{G m_1 m_2}{r^2}$$ | |||
Β | |||
Schrodinger's equation: | |||
: $$i \hbar \frac{\partial \psi}{\partial t} = - \frac{\hbar^2)}{2 m} \nabla^2 \psi + V \psi$$ | |||
Β | |||
Atiyah-Singer theorem: | |||
: $$dim\, ker \not{D}_E - dim \, coker \not{D}_E = \int_M \hat{A}(M) \cdot ch(E)$$ | |||
Β | |||
Defining relation of supersymmetry: | |||
: $$\{Q,Q\} = P$$ | |||
Β | |||
Stokes' theorem: | |||
: $$\int_M d\omega = \int_{\partial M}\omega$$ | |||
Β | |||
The boundary of a boundary is zero: | |||
: $$\partial\partial = 0$$ | |||
Β | |||
Heisenberg's indeterminacy relation: | |||
: $$\Delta x \Delta p \geq \frac{\hbar}{2}$$ | |||
Β | |||
Euler's formula for Zeta-function: | |||
: $$\sum\limits_{n=1}^{\infty} \frac{1}{n^{s}} =Β \prod\limits_{p} \frac{1}{1 - \frac{1}{p^s}}$$ | |||
Β | |||
Β | |||
Eric talked about some of the important equations on the wall. There are 2 different recorded versions of the conversation if you want to listen to it. | |||
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== The Tome == | |||
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[[File:The-tome.png|right|200px]] | |||
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This book by Roger Penrose constains a comprehensive account of the physical universe. | |||
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To gain an understanding and intuition for the information contained in 'The Graph', and 'The Wall', reading this book will provide a great head-start. | |||
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With 34 chapters spread over 1000 pages, including diagrams, equations, and descriptions, there is are multiple avenues for understanding all concepts. | |||
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=== Book Details === | |||
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* ISBN: 978-0679776314 | |||
* [https://www.amazon.com/Road-Reality-Complete-Guide-Universe/dp/0679776311 Road to Reality by Roger Penrose (2004)] | |||
* There appears to be a [https://www.amazon.com/Road-Reality-Complete-Guide-Universe-ebook/dp/B01BS7NTA6 Kindle Edition] that isn't available in the US. If anyone in the community has a way to get a Kindle version of the book, please add it here. | |||
* Purchase the book somehow, then get the [https://www.academia.edu/351112/The_Road_to_Reality_Sir_Roger_Penrose pdf here] | |||
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== Questions == | |||
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Some questions Eric posed related to the assignment: | |||
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=== What is $$F_A$$ geometrically? === | |||
[https://en.wikipedia.org/wiki/Atiyah%E2%80%93Singer_index_theorem Atiyah-Singer index theorem] | |||
Β | |||
=== What are $$R_{\mu v}$$ and $$R$$ geometrically? === | |||
[https://en.wikipedia.org/wiki/Einstein_field_equations Einstein field equaitions] | |||
Einsteinβs original publication, Die Feldgleichungen der Gravitation, in English | |||
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==== $$R$$ ==== | |||
Β | |||
[https://www.youtube.com/watch?v=UfThVvBWZxM&t=12m6s Explanation of $$R$$] | |||
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$$R$$ 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. | |||
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$$R$$ 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. | |||
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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. | |||
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==== $$R_{\mu v}$$ ==== | |||
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The same video then proceeds to explain $$R_{\mu v}$$. It progresses through some concepts. | |||
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===== Computing length in non-orthogonal bases ===== | |||
Β | |||
First, just describing the length of a vector on a curved space is hard. It is given by: | |||
Β | |||
$$Length^{squared} = g_{11}dX^{1}dX^{1} + g_{12}dX^{1}dX^{2} + g_{21}dX^{2}dX^{1} + g_{22}dX^{2}dX^{2}$$ | |||
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Some notes: | |||
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* This is not Pythagorean theorem, because $$dX^{1}$$ and $$dX^{2}$$ are not perpendicular. | |||
* 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] | |||
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===== Computing vector rotation due to parallel transport ===== | |||
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Then, they show parallel transport when following a parallelogram, but over a curved 3D manifold. To compute the vector rotation by components, they show: | |||
Β | |||
$$dV^{1} = dX^{1}dX^{2} (V^{1}R^{1}_{112} + V^{2}R^{1}_{212} + V^{3}R^{1}_{312})$$ | |||
Β | |||
$$dV^{2} = dX^{1}dX^{2} (V^{1}R^{2}_{112} + V^{2}R^{2}_{212} + V^{3}R^{2}_{312})$$ | |||
Β | |||
$$dV^{3} = dX^{1}dX^{2} (V^{1}R^{3}_{112} + V^{2}R^{3}_{212} + V^{3}R^{3}_{312})$$ | |||
Β | |||
Β | |||
or, using $$i$$ to summarize across all 3 components (difference vectors): | |||
Β | |||
$$dV^{i} = dX^{1}dX^{2} (V^{1}R^{i}_{112} + V^{2}R^{i}_{212} + V^{3}R^{i}_{312})$$ | |||
Β | |||
or , using $$j$$ to index over all 3 components (original vector): | |||
Β | |||
$$dV^{i} = dX^{1}dX^{2} \Sigma_{j} [(V^{j}R^{i}_{j12}]$$ | |||
Β | |||
See: [https://www.youtube.com/watch?v=UfThVvBWZxM&t=19m33s the video @ 19m33s] | |||
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Open questions: | |||
Β | |||
* 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? | |||
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Β | |||
===== Putting it all together ===== | |||
Β | |||
Now, moving to 4D, we can compute $$R_{\mu v}$$ as: | |||
Β | |||
$$R_{00} = R^{0}_{000} + R^{1}_{010} + R^{2}_{020} + R^{3}_{030}$$ | |||
Β | |||
$$R_{10} = R^{0}_{100} + R^{1}_{110} + R^{2}_{120} + R^{3}_{130}$$ | |||
Β | |||
$$R_{01} = R^{0}_{001} + R^{1}_{011} + R^{2}_{021} + R^{3}_{030}$$ | |||
Β | |||
etc. | |||
Β | |||
Indexing i over all 4 component vectors / dimensions, we get: | |||
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$$R_{00} = \Sigma_{i} R^{i}_{0i0}$$ | |||
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$$R_{10} = \Sigma_{i} R^{i}_{1i0}$$ | |||
Β | |||
$$R_{01} = \Sigma_{i} R^{i}_{0i1}$$ | |||
Β | |||
etc. | |||
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Summarizing on $$\mu$$, we get: | |||
Β | |||
$$R_{\mu 0} = \Sigma_{i} R^{i}_{\mu i0}$$ | |||
$$R_{\mu 1} = \Sigma_{i} R^{i}_{\mu i1}$$ | |||
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 Cohomologhy] | |||
=== What does this have to do with Penrose Stairs? === | |||
* [https:// | * [https://en.wikipedia.org/wiki/Penrose_stairs Penrose stairs] | ||
* [ | * [https://en.wikipedia.org/wiki/Spinor Spinor] | ||
Weβve 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 βHorizontal Subspacesβ and what do they have to do with Vector Potentials or Gauge fields? === | |||
* [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/Symmetry_(physics) Symmetry] | |||
From '''theplebistocrat''': | |||
<blockquote>Generally, we're wanting to understand how fermions arise from - or are embedded within / upon - topological "spaces" that have distinct rules which govern operations within those topological spaces, and then how those rules produce higher dimensional operations in corresponding spaces. Β | |||
Just intuitively, and geometrically speaking, the image that I'm getting when describing all of this and trying to hold it in my head is the image of a sort of Penrose Tower of Babel, where the fundamental underlying structures reach upwards (but also downwards and inwards?) before reaching a critical rotation that corresponds to a collapse of structure into a higher dimensional fiber bundle. Β | |||
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. | |||
</blockquote> | |||
== | == Resources & References == | ||
* [https://drive.google.com/drive/u/1/folders/1yPZHTNy47jUpmD-RMRCVtitfD-gQBRhP Folder for Project - Graph Wall Tome] | |||
* [https://www.dropbox.com/s/xdickldblj574mf/eric%20wall%20-%20tome%20-%20graph.m4a?dl=0 Recording of original call w/ Eric] | |||
* <span class="highlight">[[Ericβs Most Important Set of Books]]<span> |