Monday, December 5, 2016

On Killing spinors in general dimensions

The following property is true in four spacetime dimensions [1] [2]

If the electromagnetic field $F_{ab}$ satisfies Maxwell's equations \begin{equation*} \nabla_{[a}F_{bc]}= 0 \quad\text{and}\quad \nabla^a F_{ab} =0 \end{equation*} and there is a spinor $\psi$ such that \begin{equation}\label{eq:20161115b} (\nabla_{\mu} + i \sqrt{4 \pi} \not F \gamma_{\mu} ) \psi = 0 \end{equation} and $i \bar \psi \gamma^{\mu} \psi$ is time-like

then the Einstein equations are satisfied as well: \begin{equation*} R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = 8 \pi T_{\mu\nu} \end{equation*}

This property (property A from now on) can for example be used to obtain the metric and electromagnetic field of the Israel-Wilson-Perjés (IWP) black holes. Because I wanted to generalize the IWP black holes to higher dimensions, I wanted to find the generalization of property A in higher dimensions.

Monday, November 21, 2016

Extremal black holes via Killing spinors

The usual way to obtain the metric and electromagnetic field of a charged black hole in general relativity is to make a spherically symmetric ansatz, insert this ansatz into the Einstein-Maxwell equations and then solve the resulting set of non-linear ordinary differential equations. In this post I explain an alternative method that uses Killing spinors. This method can be used for extremal black holes. These are black holes with charge equal to the mass.

Tuesday, October 11, 2016

The isotropic harmonic oscillator

While studying Lie-algebras I read that the three-dimensional harmonic oscillator has an $SU(3)$ symmetry. I found this very unexpected; I thought it was ''obvious'' that the symmetry is only $SO(3)$.

Thursday, October 6, 2016

$F_4$ tensor products

The product of two irreducible representations of a simple Lie algebra can be decomposed into irreducible components. There are various techniques to calculate this decomposition, see for example chapter XIV in [1]. However, the decomposition can also be calculated by brute force.

Tuesday, September 27, 2016

Weight diagrams of $G_2$

This post contains pictures of weight diagrams of irreducible representations of the Lie algebra $G_2$.

Monday, September 19, 2016

The 7-dimensional representation of $G_2$

The exceptional Lie algebra $G_2$ has an irreducible representation of dimension 7. In this post I calculate the matrices of this irrep.

Saturday, September 10, 2016

Commutation relations in $G_2$

In this post I calculate the structure constants of the exceptional Lie algebra $G_2$. I assume the reader is familiar with Lie algebras, for example at the level of chapter 9 in [1].