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.
The method is based on the following fact (see for example [1])

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: $$\begin{equation*} R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = 8 \pi T_{\mu\nu} \end{equation*}$$

A spinor $\psi$ satisfying equation $\eqref{eq:20161115b}$ is called a Killing spinor. Definitions and conventions about the formulae above can be found at the bottom of this post.
I take the following ansatz for the metric and electromagnetic field of an extreme black hole $$\begin{align*} ds^2 &= - f(x)^2 dt^2 + g(x)^2 ( dx_1^2 + dx_2^2 + dx_3^2)\\ A_{\mu} dx^{\mu} &= \frac{1}{\sqrt{4 \pi}} h(x) dt, \quad\text{with}\quad F = dA \end{align*}$$ The functions $f(x)$, $g(x)$ and $h(x)$ do not depend on $t$. Although all calculations can be easily performed on a computer after choosing explicit gamma matrices, it gives more insight to work by hand. I take the following dual tetrad: $\theta^t = f(x) dt$ and $\theta^i = g(x) dx^i$. The components of the connection are then $$\begin{align*} \Gamma_{tti}&= \frac{1}{f g}\partial_i f\\ \Gamma_{ijk}&= \frac{1}{g^2} \left( - \delta_{ij} \partial_k g + \delta_{ik} \partial_j g\right) \end{align*}$$ thus $$\begin{equation*} \not\Gamma_t = -\frac{1}{2} \frac{1}{f g}\partial_i f\ \gamma^{ti} \\ \end{equation*}$$ and $$\begin{equation*} \not\Gamma_i = \frac{1}{2} \frac{1}{g^2} \partial_k g\ \gamma^{ik} \end{equation*}$$ also $$\begin{equation*} F = \frac{1}{\sqrt{4 \pi}} \partial_i h \ dx^i dt = \frac{1}{\sqrt{4 \pi}} \partial_i h \ g^{-1} f^{-1}\ \theta^i \theta^t \end{equation*}$$ thus $$\begin{equation*} \not F = \frac{2}{4} \frac{1}{\sqrt{4 \pi}} \partial_i h \ g^{-1} f^{-1} \gamma^{it} \end{equation*}$$ The $\mu = t$ component of equation $\eqref{eq:20161115b}$ then gives $$\begin{equation*} \left[ -\frac{1}{2} \frac{1}{fg} \partial_i f \ \gamma^{ti} + \frac{i}{2} \partial_i h \ g^{-1} f^{-1} \gamma^{it} \gamma_t \right]\psi = 0 \end{equation*}$$ A solution of this equation is given by $h=f$ and $\psi$ satisfying $i \gamma_t \psi = - \psi$. The $\mu = i$ component of equation $\eqref{eq:20161115b}$ gives $$\begin{equation*} \left[e_i +\frac{1}{2} \frac{1}{g^2} \partial_k g\ \gamma^{ik}+ \frac{i}{2} \partial_k h \ g^{-1} f^{-1} \gamma^{kt} \gamma_i \right]\psi = 0 \end{equation*}$$ Using $i \gamma_t \psi = - \psi$, this equation is equivalent with $$\begin{equation*} \left[e_i +\frac{1}{2} \frac{1}{g^2} \partial_k g\ \gamma^{ik} - \frac{1}{2} \partial_i h \ g^{-1} f^{-1} - \frac{1}{2} \partial_k h \ g^{-1} f^{-1} \gamma^{ki} \right]\psi = 0 \end{equation*}$$ This is solved by $e_i(\psi) -\dfrac{1}{2} \partial_i h \ g^{-1} f^{-1}\psi =0$ and $\dfrac{1}{2} \dfrac{1}{g^2} \partial_k\ g + \dfrac{1}{2} \partial_k h \ g^{-1} f^{-1}=0$. Thus $g = f^{-1}$. Writing $\psi = k(x) \epsilon$, with $\epsilon$ a constant spinor, we find $k = f ^{1/2}$. Finally, the Maxwell equation $\dfrac{1}{\sqrt{-g}} \partial_{\mu} (\sqrt{-g} F^{\mu\nu} ) =0$ gives $\partial_i ( f^{-2} \partial_i f) =0$, thus $f^{-1}$ is a harmonic function.
All in all, if $U(x_1,x_2,x_3)$ is a harmonic function, then $f = U^{-1}$, $g = U$, $h = U^{-1}$ is a solution of the Einstein-Maxwell equations with Killing spinor $\psi = U^{-1/2} \epsilon$ where $i \gamma_0 \epsilon = - \epsilon$. The special case $U(x_1,x_2,x_3) = 1+ \frac{m}{r}$ with $r^2 = x_1^2 + x_2^2 + x_3^3$ is the Reissner–Nordström solution with mass equal to the charge. The case with general $U$ is called the Majumdar-Papapetrou solution. Notice that the equations we solved above were all linear. This is much easier than solving the non-linear Einstein equations.

Further reading
This method is related to supersymmetry. The equation $\eqref{eq:20161115b}$ expresses that the supersymmetry variation of the gravitino is zero, and therefore that the solution is supersymmetric. The method can also be generalized to higher dimensions. For example, in [2], the authors classified all supersymmetric solution of (some version of) the five-dimensional Einstein-Maxwell equations. They based their classification on an equation analogous to $\eqref{eq:20161115b}$.

References

1. Black hole solutions in string theory, Maeda and Nozawa, 2011. hep-th/1104.1849
2. All supersymmetric solutions of minimal supergravity in five dimensions, Gauntlett, Gutowski, Hull, Pakis and Reall, 2002. hep-th/0209114

Definitions and conventions

• $\gamma^a$ are the gamma matrices that satisfy $\{ \gamma^a, \gamma^b\} = 2 \eta^{ab}$ with $\eta^{ab}=\text{diagonal}(-1,1,1,1)$
• $\gamma^{ab} = \frac{1}{2} (\gamma^a \gamma^b - \gamma^b \gamma^a)$
• The covariant derivative of a spinor is $$\begin{equation*} (\nabla_{\mu} \psi)^{\alpha} = \partial_{\mu} \psi^{\alpha} + \Gamma^{\alpha}_{\mu\beta} \psi^{\beta} \end{equation*}$$ with $\Gamma^{\alpha}_{\mu\beta} = - \frac{1}{4} \Gamma_{\mu ab} (\gamma^{ab})^{\alpha}_{\ \ \beta}$.
• $\Gamma_{\mu ab}$ are the components of the spin connection where $\omega^a_{\ \ b} = \Gamma^a_{\mu b} dx^{\mu}$ and $\Gamma_{\mu ab} = \eta{bc}\Gamma^c_{\mu a}$.

$\not F = \frac{1}{4} F_{ij} \gamma^{ij}$.

I also write $\not\Gamma_\mu = - \frac{1}{4} \Gamma_{\mu ij} \gamma^{ij}$, thus $\nabla_{\mu}\psi = \partial_{\mu}\psi + \not\Gamma_\mu \psi$

Indices like $\mu$, $\nu$, ... are curved indices; indices like $i,j, \ldots$ are flat indices, and indices like $\alpha, \beta, \ldots$ are spinor indices