Types of quantum field thories
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 and 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.
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.
The simplest description of the Kerr metric is by using spherical coordinates on the “space” and a time coordinate . In the context of the Kerr metric these coordinates are called Boyer-Lindquist coordinates.
The Kerr metric has two real parameters .
We define two functions mainly as an abbreviation:
\rho^2 = r^2 + a^2 \cos^2(\theta )
\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:
|all zero||all zero||all zero except|
One gets the Kerr-Newman metric? for an electrically charged source with charge by replacing the definition of with
\triangle = r^2 - 2 m r + a^2 + e^2
The family of Kerr spacetimes is classified by the relation of the parameters and :
gives Schwartzschild spacetime
gives slowly rotating Kerr spacetime (slow Kerr)
gives extreme Kerr spacetime and
gives rapidly rotating Kerr spacetime (fast Kerr)
There are several coordinate singularities, inlcuding the z-axis (where ) and where and . Points where define the horizons of Kerr spacetime.
Both and are Killing vector fields, expressing the time invariance and the axial symmetry of the model respectively. Combining the sign changes 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 .
The Boyer-Lindquist coordinates are defined on a subset of with defined on a copy of respectively (actually is not supposed to take negative values, this definition is for convenience only). There are three subsets where the coordinates fail:
The horizon H where .
The ring singularity where
The axis A where .
The Boyer-Lindquist blocks I, II, III are the following open subsets of :
For slow Kerr, there are two horizons at ,
I: r \gt r_+
II: r_- \lt r \lt r_+
III: r \lt r_-
For extreme Kerr, there is a single horizon at ,
I: r \gt m
III: r \lt m
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.
Causality of I and II The Boyer-Lindquist blocks I and II are causal.
For a definition of causality see spacetime.
Noncausality of III The Boyer-Lindquist block III is vicious, that is for any two events in III there is a timelike future-pointing curve in III from p to q.
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)