Accelerators

Some questions Eric posed related to the assignment:

What is $$F_A$$ geometrically?

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

What are $$R_{\mu v}$$ and $$R$$ geometrically?

Einstein field equations Einstein’s original publication, Die Feldgleichungen der Gravitation, in English

$$R$$

$$R$$ 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.

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.

Computing length in non-orthogonal bases

First, just describing the length of a vector on a curved space is hard. It is given by:

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

Some notes:

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: 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: 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 $$R_{\mu v}$$ 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?

Cohomology

What does this have to do with Penrose Stairs?

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?

From theplebistocrat:

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.
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.
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.