Kerr metric

In general relativity, the Kerr metric (or Kerr vacuum) describes the geometry of spacetime around a rotating massive body, such as a rotating black hole. This famous exact solution was discovered in 1963 by the New Zealand born mathematician Roy Kerr.

The Boyer/Lindquist coordinate chart

The Boyer-Lindquist chart for the Kerr vacuum solution is given by

$ds^{2}=\rho ^{2}({\frac {dr^{2}}{\Delta }}+d\theta ^{2})+(r^{2}+a^{2})\sin ^{2}\theta d\phi ^{2}-dt^{2}+{\frac {2mr}{\rho ^{2}}}(a\sin ^{2}\theta d\phi -dt)^{2}$

where

ρ²=r² + a²cos²θ

and

Δ=r² - 2mr + a².

Here m is the mass of the black hole, and a is is a parameter describing the rotation of the black hole, related to the angular momentum J by $a=J/m$ (this quantity is often called the specific angular momentum). However, in general, the Boyer/Lindquist radial coordinate r does not have a straightforward interpretation as a radial coordinate.

Features of the Kerr Vacuum

The Kerr vacuum exhibits many noteworthy features: the maximal analytic extension includes a sequence of asympotically flat exterior regions, each associated with an ergosphere, stationary limit surfaces, event horizons, Cauchy horizons, closed timelike curves, and a ring-shaped curvature singularity. The geodesic equation can be solved exactly in closed form. In addition to two Killing vector fields (corresponding to time translation and axisymmetry), the Kerr vacuum admits a remarkable Killing tensor. There is a pair of principal null congruences (one ingoing and one outgoing). The Weyl tensor is algebraically special, in fact it has Petrov type D. The global structure is known. Topologically, the homotopy type of the Kerr spacetime can be simply characterized as a line with circles attached at each integer point.

Relation to Other Exact Solutions

The Kerr vacuum is a particular example of a stationary axially symmetric vacuum solution to the Einstein field equation. The family of all stationary axially symmetric vacuum solutions to the Einstein field equation are the Ernst vacuums. The asympotically flat Ernst vacuums can be characterized by an infinite sequence of Geroch multipole moments, the first two of which can be interpreted as the mass and angular momentum of the source of the field. Then the Kerr vacuum is characterized by saying that all the higher moments vanish. In particular, the quadrupole moment of the Kerr vacuum vanishes; in this sense, it is the simplest rotating and stationary asympotically flat vacuum.

The Kerr solution is also related to various non-vacuum solutions which model black holes. For example, the Kerr/Newman electrovacuum models a (rotating) black hole endowed with an electric charge, while the Kerr/Vaidya null dust models a (rotating) hole with infalling electromagnetic radiation.

The special case $a=0$ of the Kerr metric yields the Schwarzschild metric, which models a nonrotating black hole which is static and spherically symmetric, in the Schwarzschild coordinates. (In this case, every Geroch moment but the mass vanishes.)

The interior of the Kerr vacuum, or rather a portion of it, is locally isometric to the Chandrasekhar/Ferrari CPW vacuum, an example of a colliding plane wave model. This is particularly interesting, because the global structure of this CPW solution is quite different from that of the Kerr vacuum, and in principle, an experimenter could hope to study the geometry of (the outer portion of) the Kerr interior by arranging the collision of two suitable gravitational plane waves.

Open Problems

The Kerr vacuum is often used as a model of a black hole, but if we hold the solution to be valid only outside some compact region (subject to certain restrictions), in principle we should be able to use it as an exterior solution to model the gravitational field around a rotating massive object other than a black hole, such as a neutron star--- or the Earth. This works out very nicely for the non-rotating case, where we can match the Schwarschild vacuum exterior to a Schwarzschild fluid interior, and indeed to more general static spherically symmetric perfect fluid solutions. However, the problem of finding a rotating perfect-fluid interior which can be matched to a Kerr exterior, or indeed to any asymptotically flat vacuum exterior solution, has proven very difficult. In particular, the Walhquist fluid, which was once thought to be a candidate for matching to a Kerr exterior, is now known not to admit any such matching. At present it seems that only approximate solutions modeling slowly rotating fluid balls (the relativistic analog of oblate spheroidal balls with nonzero mass and angular momentum but vanishing higher multipole moments) are known. However, the exterior of the Neugebauer/Meinel disk, an exact dust solution which models a rotating thin disk, represents a limiting case of the Kerr vacuum.

References

• R. P. Kerr, "Gravitational field of a spinning mass as an example of algebraically special metrics", Phys. Rev. Lett. 11, 237 (1963).

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