## Monday, October 31, 2011

### 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 ${\Omega_1 \otimes \mathcal{E}}$ ?