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}}$ ?

3. Sources

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

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