Kerr metric

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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.

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[edit] The Boyer/Lindquist coordinate chart

Kerr vacuum solution written in Boyer-Lindquist coordinates is given by:ds^2 = -\left(1-\frac{2Mr}{\Sigma}\right)dt^2 -\frac{4aMr\sin^2\theta}{\Sigma}dtd\phi +\frac{\Sigma}{\Delta}dr^2 + \Sigma d\theta^2 + \left(r^2+a^2+\frac{2a^2Mr\sin^2\theta} {\Sigma}\right) \sin^2\theta d\phi^2 where

Σ = r2 + a2cos2θ,
Δ = r2 − 2Mr + a2,
M is the mass of the rotating object,
a describes the rotation of the black hole, being related to the angular momentum J by a = J/M, and
all quantities are in geometrized units where c=G=1.

When the spin parameter a (which is often called the specific angular momentum) has a value of zero, there is no rotation and you have Schwarzschild solution. The case of a=M corresponds to a maximally rotating massive object. Note that

  • in general, the Boyer/Lindquist radial coordinate r does not have a straightforward interpretation as a radial coordinate, and
  • "maximal" refers to the highest value of a for which a black hole can exist, not the highest a that a rotating massive object can have.

Some important details of the Kerr metric can be discovered when it is written in the above form. The location of the event horizon is given by the surface where Δ = 0, i.e. where the coefficient of the dr2 differential diverges. The location of the ergosphere is given by the surface where 1 − 2Mr / Σ = 0, i.e. the location where the coefficient of the dt2 differential vanishes.

[edit] Features of the Kerr vacuum

The Kerr vacuum exhibits many noteworthy features: the maximal analytic extension includes a sequence of asymptotically 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.

While the Kerr vacuum is an exact axis-symmetric solution to Einstein's field equations, the solution is probably not stable in the interior region of the black hole (Penrose, 1968). The stable interior solution is probably not axis-symmetric. The instability of the Kerr metric in the interior region implies that many of the features of the Kerr vacuum described above would probably not be present in a black hole that came into being through gravitational collapse.

[edit] 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 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.

[edit] Multipole moments

Each asymptotically flat Ernst vacuum can be characterized by giving the infinite sequence of relativistic multipole moments, the first two of which can be interpreted as the mass and angular momentum of the source of the field. There are alternative formulations of relativistic multipole moments due to Hansen, Thorne, and Geroch, which turn out to agree with each other. The relativistic multipole moments of the Kerr vacuum were computed by Hansen; they turn out to be

M_n = M \, (i \, a)^n

Thus, the special case of the Schwarzschild vacuum (a=0) gives the "monopole point source" of general relativity.

Warning: do not confuse these relativistic multipole moments with the Weyl multipole moments, which arise from treating a certain metric function (formally corresponding to Newtonian gravitational potential) which appears the Weyl-Papapetrou chart for the Ernst family of all stationary axisymmetric vacuums solutions using the standard euclidean scalar multipole moments. In a sense, the Weyl moments only (indirectly) characterize the "mass distribution" of an isolated source, and they turn out to depend only on the even order relativistic moments. In the case of solutions symmetric across the equatorial plane the odd order Weyl moments vanish. For the Kerr vacuum solutions, the first few Weyl moments are given by

a_0 = M, \; \; a_1 = 0, \; \; a_2 = M \, \left( \frac{M^2}{3} - a^2 \right)

In particular, we see that the Schwarzschild vacuum has nonzero second order Weyl moment, corresponding to the fact that the "Weyl monopole" is the Chazy-Curzon vacuum solution, not the Schwarzschild vacuum solution, which arises from the Newtonian potential of a certain finite length uniform density thin rod.

In weak field general relativity, it is convenient to treat isolated sources using another type of multipole, which generalize the Weyl moments to mass multipole moments and momentum multipole moments, characterizing respectively the distribution of mass and of momentum of the source. These are multi-indexed quantities whose suitably symmetrized (anti-symmetrized) parts can be related to the real and imaginary parts of the relativistic moments for the full nonlinear theory in a rather complicated manner.

Perez and Moreschi have given an alternative notion of "monopole solutions" by expanding the standard NP tetrad of the Ernst vacuums in powers of r (the radial coordinate in the Weyl-Papapetrou chart). According to this formulation:

  • the isolated mass monopole source with zero angular momentum is the Schwarzschild vacuum family (one parameter),
  • the isolated mass monopole source with radial angular momentum is the Taub-NUT vacuum family (two parameters; not quite asymptotically flat),
  • the isolated mass monopole source with axial angular momentum is the Kerr vacuum family (two parameters).

In this sense, the Kerr vacuums are the simplest stationary axisymmetric asymptotically flat vacuum solutions in general relativity.

[edit] 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.

[edit] References

  • Stephani, Hans; Kramer, Dietrich; MacCallum, Malcolm; Hoenselaers, Cornelius & Herlt, Eduard (2003). Exact Solutions of Einstein's Field Equations. Cambridge: Cambridge University Press. ISBN 0-521-46136-7.
  • O'Neill, Barrett (1995). The Geometry of Kerr Black Holes. Wellesley, MA: A. K. Peters. ISBN 1-56881-019-9.
  • D'Inverno, Ray (1992). Introducing Einstein's Relativity. Oxford: Clarendon Press. ISBN 0-19-859686-3. See chapter 19 for a readable introduction at the advanced undergraduate level.
  • Chandrasekhar, S. (1992). The Mathematical Theory of Black Holes. Oxford: Clarendon Press. ISBN 0-19-850370-9. See chapters 6--10 for a very thorough study at the advanced graduate level.
  • Griffiths, J. B. (1991). Colliding Plane Waves in General Relativity. Oxford: Oxford University Press. ISBN 0-19-853209-1. See chapter 13 for the Chandrasekhar/Ferrari CPW model.
  • Adler, Ronald; Bazin, Maurice & Schiffer, Menahem (1975). Introduction to General Relativity, Second Edition, New York: McGraw-Hill. ISBN 0-07-000423-4. See chapter 7.
  • Sotiriou, Thomas P.; and Apostolatos, Theocharis A. (2004). "Corrections and Comments on the Multipole Moments of Axisymmetric Electrovacuum Spacetimes". Class. Quant. Grav. 21: 5727-5733. arXiv eprint Gives the relativistic multipole moments for the Ernst vacuums (plus the electrogmagnetic and gravitational relativistic multipole moments for the charged generalization).
  • Penrose R (1968). ed C. de Witt and J. Wheeler: Battelle Rencontres. W. A. Benjamin, New York.
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