Confusion in Black Holes

I've been trying to make my way through Majid's paper, ``Almost commutative Riemannian geometry: wave operators'', particularly the section where he constructs the model for a Schwarzschild black hole. I've not had much success. I'm meant to reconstruct the model over a $ {\mathbb{F}_p}$, but I'm stuck on basic definitions. I'll discuss two of those things in this blog, first the vector bundle (aka projective module) and then the Grassmann connection.

1. NCG Vector Bundle for The Schwarzschild Solution

To construct the model, we start by reconsidering our notion of 3-dimensional space. Rather than thinking of coordinates $ {(x_1, x_2, x_3)}$, we're going to recast ``space'' as a ``coordinate algebra'', in particular, an algebra of polynomials $ {k[x_1, x_2, x_3]}$ over a field $ {k}$ (we'll let $ {k = \mathbb{R}}$ for now, but my task is to redo this section of the paper with $ {k=\mathbb{F}_p}$. Moreover, we're working in a sphere, so we also request that our algebra contain functions rational in $ {r}$, where $ {r^2 = x_1^2 + x_2^2 + x_3^2}$. Hence our ``space'' is the algebra $ {A = k[x_1, x_2, x_3, r, r^{-1}]}$ modded out by the aforementioned relation.

For such an NCG space (nevermind that $ {A}$ is actually commutative here), we define a vector bundle as a projective module. An easy way to get a projective module is to take a few copies of $ {A}$ under the image of an idempotent $ {E \in M_n(A)}$, e.g, let $ {E}$ be such an idempotent, then $ {\mathcal{E} = Im(E)}$ is our vector bundle. In this case we're taking the 3 by 3 matrix:

$ \displaystyle E = \begin{pmatrix} 1 - \frac{x_1^2}{r^2} & - \frac{x_1 x_2}{r^2} & - \frac{x_1 x_3}{r^2} \\ - \frac{x_2 x_1}{r^2} & 1 - \frac{x_2^2}{r^2} & - \frac{x_2 x_3}{r^2} \\ - \frac{x_3 x_1}{r^2} & - \frac{x_3 x_2}{r^2} & 1 - \frac{x_3^2}{r^2} \end{pmatrix}$

Thus our vector bundle is the subspace $ {\mathcal{E} = Im(E) \subset A^3}$. We expect elements of this vector bundle to be ``3-vectors'' with entries from $ {A}$. Yet in the paper, Majid states that $ {\omega_i = \text{d}x_i - \frac{x_i \text{d}r}{r}}$ for $ {i = 1, \, 2, \, 3}$ spans the 2-dimensional bundle $ {\mathcal{E}}$. But (judging by the $ {\text{d}x_i}$ and $ {\text{d}r}$ terms) each $ {\omega_i}$ is in our bimodule of 1-forms $ {\Omega_1}$. Where am I going wrong?

2. Grassmann Connections in NCG

Let's assume I'm not hopelessly confused about that vector bundle thing. Recall that a connection on an NCG vector bundle is a linear map $ {\nabla_{\mathcal{E}}: \mathcal{E} \rightarrow \Omega_1 \otimes \mathcal{E}}$ that obeys the following rule:

$ \displaystyle \nabla_{\mathcal{E}} (a s) = \text{d} a \otimes s + a\nabla_{\mathcal{E}}(s) \; \forall a \in A \; s \in \mathcal{E} $

According to a proposition I've read in Majid's lecture notes, if $ {\mathcal{E} = Im(E)}$ for a projector $ {E \in M_n(A)}$, then we have a ``Grassmann connection''

$ \displaystyle \nabla_{\mathcal{E}} (E v) = E \text{d}(Ev) = E\left(\text{d}(E)v + E(\text{d}v)\right) = E\text{d}(E)v + E(\text{d}v)$

Where $ {v \in A^n}$ and $ {\text{d}}$ acts on $ {v}$ and $ {E}$ component-wise. But the image of the connection is suppose to live in $ {\Omega_1 \otimes \mathcal{E}}$. $ {\text{d}v}$ lives in $ {\Omega_1^n}$, and $ {\text{d}E}$ lives in $ {M_n(\Omega_1)}$. How do we get to our tensor product space $ {\Omega_1 \otimes \mathcal{E}}$ ?

3. Sources

1. Section 4 of Almost commutative Riemannian geometry: wave operators
   

Starting my PhD - Noncommutative Geometry and Black Holes

I've had a great summer holiday. I've done a few programming projects. I've restarted my position with Universal Pictures International. It's time to start writing about my mathematics consistently. My PhD supervisor has given me a small project based on this paper: he has this noncommutative geometric model of a black hole, and I'm suppose to see what happens if I re-do each step over a finite field. It's been two weeks and, as usual, I've made very little progress on it. I don't actually understand what's going on. So let's spend a moment or two trying to clear the fog and see what a ``noncommutative geometric model'' means.

1. Noncommutative Geometry Re-visited

Regular readers (ha! as if there are any...) will know that my previous experience with NCG was with the Bost-Connes system, a topic I hope to take up again soon. My supervisor's work; however, is considerably different. We keep the general philosophy of starting from an algebra and trying to construct a geometry from it, but instead of working over Operator Algebras we're sticking to less-analytic algebraic structures and imposing additional objects on them that are meant to emmulate geometric notions. Let's go over this in detail and discuss some of these objects - keeping in mind that I don't know any geometry, canot provide any motivation for these concepts, and generally don't have a clue what I'm doing.

1.1. Differential Forms

Most readers will know that one can impose a Differentiable Structure on a topological manifold and use it to start doing some geometry. We'll dispose of the topological manifold and replace it with an algebra $ {A}$ and impose on it the notion of a 1-form. NCG 1-forms live in ``differential calculus`` $ {\Omega_1}$, which we define to be a bimodule (id est, a module on both sides) over $ {A}$. The ``differentiable structure'' comes in the form of a linear map $ {d: A \rightarrow \Omega_1}$ that obeys the product rule:

$ \displaystyle d(ab) = d(a) \cdot b + a \cdot d(b) \; \forall a, b \in A$

Additionally, we require that $ { (a, b) \mapsto a \cdot d(b) }$ spans the bimodule $ {\Omega_1}$.
Some readers may wonder what sort of algebras $ {A}$ can have a differential calculus. Actually, each algebra $ {A}$ necessarily has one, but we won't discuss that in this post.

1.2. Vector Bundles & Connections

My knowledge of modern geometry ends at this point and I am solely trusting the NCG literature. Initially, when seeing these terms, I think of the long definitions needed in Differential Geometry and start chasing down the terms in various textbooks. We don't need that here, and can describe these objects with simple algebraic ideas. That said, the next tool in our discussion is the NCG notion of a Vector Bundle. The definition here is a chain of algebraic structures: a NCG vector bundle is a finitely generated projective module over $ {A}$. A projective module is the image of a free module under a projection/idempotent. A free module is a module with basis vectors. E.g., $ {A^n}$ is a free module and if $ {E \in M_n(A)}$ is idempotent, then $ {E A^n}$ is a vector bundle.
Now if $ {\mathcal{E}}$ is an NCG vector bundle, and $ {\Omega_1}$ a differential calculus, both over $ {A}$, we can also define the NCG notion of a connection: A connection is a linear map

$ \displaystyle \nabla : \mathcal{E} \rightarrow \Omega_1 \otimes \mathcal{E}$

with the following condition:

$ \displaystyle \nabla (as) = d(a) \otimes s + a \nabla(s) \; \forall a\in A, \; s \in \mathcal{E}$

Next time we'll (Lord willing) discuss metrics, curvature, and how they all fit together to get a model for a black hole.

2. Sources

1. LTCC Lecture notes in NCG by S Majid
2. Section 4 of Almost commutative Riemannian geometry: wave operators
3. Section 2 of Noncommutative Riemannian geometry on graphs