# nLab Kerr spacetime

### Context

#### Gravity

gravity, supergravity

# Contents

## Idea

The Kerr spacetime(s) is a (family of) certain Lorentzian manifolds / spacetimes. The Kerr spacetime describes the ambient vacuum spacetime of a spherically symmetric rotating mass density, it can be extended in a way that this mass density degenerates to a singularity of spatial radius zero. This mathematical idealization is often said to describe a rotating black hole.

The Kerr spacetimes are parametrized by two parameters $m$ and $a$ that have the physical interpretation of mass and angular momentum per unit mass respectively, of the rotating object they describe. In the degenerate case of a = 0 the Kerr spacetimes reduce to the Schwartzschild spacetime?s.

## Abstract

The definition states the components of the metric tensor in a specific coordinate system, the Boyer-Lindquist coordinates, and compares those to the Minkowski and Schwartzschild metrics. Some properties can be read off directly from the metric tensor, this is done in the properties paragraph.

## Definition

The simplest description of the Kerr metric is by using spherical coordinates $r,\varphi ,\theta$ on the “space” ${ℝ}^{3}$ and a time coordinate $t\in ℝ$. In the context of the Kerr metric these coordinates are called Boyer-Lindquist coordinates.

The Kerr metric has two real parameters $m,a\ge 0$.

We define two functions mainly as an abbreviation:

${\rho }^{2}={r}^{2}+{a}^{2}{\mathrm{cos}}^{2}\left(\theta \right)$\rho^2 = r^2 + a^2 \cos^2(\theta )
$▵={r}^{2}-2mr+{a}^{2}$\triangle = r^2 - 2 m r + a^2

The following table lists the components of the metric of Minkowski, Schwartzschild and Kerr spacetimes respectivly using the canonical coordinates and the Boyer-Lindquist coordinates for the Kerr spacetime:

metric Minkowski Schwartzschild Kerr
${g}_{\mathrm{tt}}$ -1 $-1+2\frac{m}{r}$ $-1+2\frac{\mathrm{mr}}{{\rho }^{2}}$
${g}_{\mathrm{rr}}$ +1 $\frac{r}{r-2m}$ $\frac{{\rho }^{2}}{▵}$
${g}_{\theta \theta }$ ${r}^{2}$ ${r}^{2}$ ${\rho }^{2}$
${g}_{\varphi \varphi }$ ${r}^{2}{\mathrm{sin}}^{2}\left(\theta \right)$ ${r}^{2}{\mathrm{sin}}^{2}\left(\theta \right)$ $\left({r}^{2}+{a}^{2}+\frac{2mr{a}^{2}{\mathrm{sin}}^{2}\left(\theta \right)}{{\rho }^{2}}\right){\mathrm{sin}}^{2}\theta$
${g}_{\mathrm{ij}}$ $i\ne j$ all zero all zero all zero except ${g}_{t\varphi }={g}_{\varphi t}=-\frac{2mra{\mathrm{sin}}^{2}\left(\theta \right)}{{\rho }^{2}}$

One gets the Kerr-Newman metric? for an electrically charged source with charge $e$ by replacing the definition of $▵$ with

$▵={r}^{2}-2mr+{a}^{2}+{e}^{2}$\triangle = r^2 - 2 m r + a^2 + e^2

The family of Kerr spacetimes is classified by the relation of the parameters $a$ and $m$:

• $0=a$ gives Schwartzschild spacetime

• $0<{a}^{2}<{m}^{2}$ gives slowly rotating Kerr spacetime (slow Kerr)

• ${a}^{2}={m}^{2}$ gives extreme Kerr spacetime and

• ${m}^{2}<{a}^{2}$ gives rapidly rotating Kerr spacetime (fast Kerr)

## Properties

### Direct Consequences from the Definition

There are several coordinate singularities, inlcuding the z-axis (where $\mathrm{sin}\left(\theta \right)=0$) and where $\rho =0$ and $▵=0$. Points where $▵=0$ define the horizons of Kerr spacetime.

Both ${\partial }_{t}$ and ${\partial }_{\varphi }$ are Killing vector fields, expressing the time invariance and the axial symmetry of the model respectively. Combining the sign changes $t\to -t,\varphi \to -\varphi$ gives an isometry: Letting time running backwards reverses the rotation.

Kerr spacetime is asymptotically flat, that is the Kerr metric approximates the Minkowski metric for large $r$.

### Boyer-Lindquist Blocks

The Boyer-Lindquist coordinates are defined on a subset of ${ℝ}^{2}×{𝒮}^{2}$ with $t,r$ defined on a copy of $ℝ$ respectively (actually $r$ is not supposed to take negative values, this definition is for convenience only). There are three subsets where the coordinates fail:

1. The horizon H where $▵=0$.

2. The ring singularity $\Sigma$ where ${\rho }^{2}=0$

3. The axis A where $\mathrm{sin}\left(\theta \right)=0$.

###### Definition

The Boyer-Lindquist blocks I, II, III are the following open subsets of ${ℝ}^{2}×{𝒮}^{2}-\Sigma$:

1. For slow Kerr, there are two horizons at ${r}_{±}$,

$I:r>{r}_{+}$I: r \gt r_+
$\mathrm{II}:{r}_{-}II: r_- \lt r \lt r_+
$\mathrm{III}:r<{r}_{-}$III: r \lt r_-
1. For extreme Kerr, there is a single horizon at $r=m$,

$I:r>m$I: r \gt m
$\mathrm{III}:rIII: r \lt m
1. For fast Kerr, there is no horizon and ${ℝ}^{2}×{𝒮}^{2}-\Sigma$ can be considered as one block I = III.

Block I is also called the Kerr exterior and can be visualized as close to the Newtonian concept of space and time with a central force field.

###### Theorem

Causality of I and II The Boyer-Lindquist blocks I and II are causal.

For a definition of causality see spacetime.

###### Theorem

Noncausality of III The Boyer-Lindquist block III is vicious, that is for any two events $p,q$ in III there is a timelike future-pointing curve in III from p to q.

### Higher symmetries

The Kerr spactime admits an extra Killing tensor and Killing-Yano tensor (…) See for instance (JL).

## References

Most textbooks about General Relativity have chapter about the Kerr spacetime, here is a monograph that specializes on the topic:

• Barrett O’Neill, The geometry of Kerr black holes. (ZMATH entry)

• Jacek Jezierski, Maciej Łukasik, Conformal Yano-Killing tensor for the Kerr metric and conserved quantities (arXiv:gr-qc/0510058)

Revised on September 17, 2011 11:44:15 by Urs Schreiber (82.113.121.151)