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.

Contents

Mathematical form

Frame dragging

Singular surfaces

Gradient operator

Features of the Kerr vacuum

Relation to other exact solutions

Multipole moments

Open problems

The equations of the trajectory and the time dependence for a particle in the Kerr field

See also

References

Mathematical form

The Kerr metric^[1][2] describes the geometry of spacetime in the vicinity of a mass ''M'' rotating with angular momentum ''J''
:$$
c^{2} d au^{2} =
left( 1 - rac{r_{s} r}{
ho^{2}}
ight) c^{2} dt^{2}
- rac{
ho^{2}}{Lambda^{2}} dr^{2}
-
ho^{2} d heta^{2} -

::::$$
left( r^{2} + lpha^{2} + rac{r_{s} r lpha^{2}}{
ho^{2}} sin^{2} heta
ight) sin^{2} heta dphi^{2} + rac{2r_{s} rlpha}{
ho^{2}} , c , dt , dphi

where ''r''_''s'' is the Schwarzschild radius
:$$
r_{s} = rac{2GM}{c^{2}}

and where the length-scales α, ρ and Λ have been introduced for brevity
:$$
lpha = rac{J}{Mc}

:$$
ho^{2} = r^{2} + lpha^{2} cos^{2} heta

:$$
Lambda^{2} = r^{2} - r_{s} r + lpha^{2}

In the non-relativistic limit where ''M'' (or, equivalently, ''r''_''s'') goes to zero, the Kerr metric becomes the orthogonal metric for the oblate spheroidal coordinates
:$$
c^{2} d au^{2} =
c^{2} dt^{2}
- rac{
ho^{2}}{r^{2} + lpha^{2}} dr^{2}
-
ho^{2} d heta^{2}
- left( r^{2} + lpha^{2}
ight) sin^{2} heta dphi^{2}

which are equivalent to the Boyer-Lindquist coordinates^[3]
:
:
:$\{z\}\; =\; r\; cos\; heta\; quad$

Frame dragging

We may re-write the Kerr metric in the following form
:$$
c^{2} d au^{2} =
left( g_{tt} - rac{g_{tphi}^{2}}{g_{phiphi}}
ight) dt^{2}
+ g_{rr} dr^{2} + g_{ heta heta} d heta^{2} +
g_{phiphi} left( dphi + rac{g_{tphi}}{g_{phiphi}} dt
ight)^{2}.

This metric is equivalent to a co-rotating reference frame that is rotating with angular speed Ω that depends on both the radius ''r'' and the colatitude θ
:$$
Omega = -rac{g_{tphi}}{g_{phiphi}} = rac{r_{s} lpha r}{
ho^{2} left( r^{2} + lpha^{2}
ight) + r_{s} lpha^{2} r sin^{2} heta}.

Thus, an inertial reference frame is entrained by the rotating central mass to participate in the latter's rotation; this is frame-dragging, which has been observed experimentally.

Singular surfaces

The Kerr metric has two surfaces on which it appears to be singular. The inner surface corresponds to a spherical event horizon similar to that observed in the Schwarzschild metric; this occurs where the purely radial component ''g_rr'' of the metric goes to infinity. Solving the quadratic equation 1/''g''_''rr'' = 0 yields the solution
:$$
r_mathit{inner} = rac{r_{s} + sqrt{r_{s}^{2} - 4lpha^{2}}}{2}

Another singularity occurs where the purely temporal component ''g_tt'' of the metric changes sign from positive to negative. Again solving a quadratic equation ''g_tt''=0 yields the solution
:$$
r_mathit{outer} = rac{r_{s} + sqrt{r_{s}^{2} - 4lpha^{2} cos^{2} heta}}{2}

This outer surface is not a sphere, but an oblate spheroid that touches the inner surface at the poles of the rotation axis, where the colatitude θ equals 0 or π; the space between these two surfaces is called the ergosphere. There are two other solutions to these quadratic equations, but they lie within the event horizon, where the Kerr metric is not used, since it has unphysical properties (see below).
A moving particle experiences a positive proper time along its worldline, its path through spacetime. However, this is impossible within the ergosphere, where ''g_tt'' is negative, unless the particle is co-rotating with the interior mass ''M'' with an angular speed at least of Ω. Thus, no particle can rotate opposite to the central mass within the ergosphere.

Gradient operator

Since even a direct check on Kerr metric involves cumbersome calculations, there would be perhaps a very good idea to introduce in here also the contravariant components $g^\{ik\}$ of the metric tensor. These are shown below in the expression for the square of the four-gradient operator:
:$$
g^{mu
u}rac{partial}{partial{x^{mu}}}rac{partial}{partial{x^{
u}}} = rac{Lambda^{2}}{
ho^{2}}left(rac{partial}{partial{r}}
ight)^{2} + rac{1}{
ho^{2}}left(rac{partial}{partial{ heta}}
ight)^{2} -
rac{2r_{s}rlpha}{c
ho^{2}Lambda^{2}}rac{partial}{partial{phi}}rac{partial}{partial{t}} -

:::::$$
rac{1}{c^{2}Lambda^{2}}left(r^{2} + lpha^{2} + rac{r_{s}rlpha^{2}}{
ho^{2}}sin^{2} heta
ight)left(rac{partial}{partial{t}}
ight)^{2} +
rac{1}{Lambda^{2}sin^{2} heta}left(1 - rac{r_{s}r}{
ho^{2}}
ight)left(rac{partial}{partial{phi}}
ight)^{2}

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.

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.

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
:
Thus, the special case of the Schwarzschild vacuum (α=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
:
ight)
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.

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 Wahlquist 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, approaches in a limiting case the ''a''=''M'' Kerr vacuum.

The equations of the trajectory and the time dependence for a particle in the Kerr field

In the Hamilton-Jacobi equation we write the action S in the form:
::::$S\; =\; -E\_\{0\}t\; +\; Lphi\; +\; S\_\{r\}(r)\; +\; S\_\{\; heta\}(\; heta)$
where $E\_\{0\}$, m, and L are the conserved energy, the rest mass and the component of the angular momentum (along the axis of symmetry of the field) of the particle consecutively, and carry out the separation of variables in the Hamilton Jacobi equation as follows:
::
ight)^{2} + left(aE_{0}sin heta - rac{L}{sin heta}
ight)^{2} + a^{2}m^{2}cos^{2} heta = K
::
ight)^{2} - rac{1}{Delta}left[left(r^{2} + a^{2}
ight)E_{0} - aL
ight]^{2} + m^{2}r^{2} = -K
where K is a new arbitrary constant. The equation of the trajectory and the time dependence of the coordinates along the trajectory (motion equation) can be found then easily and directly from these equations:
::
::
::

See also

★ Schwarzschild metric

★ Kerr-Newman metric

★ Reissner-Nordström metric

References

1. Gravitational field of a spinning mass as an example of algebraically special metrics, , RP, Kerr, Physical Review Letters, 1963
2. The Classical Theory of Fields (Course of Theoretical Physics, Vol. 2), , LD, Landau, Pergamon Press, 1975,
3. Maximal Analytic Extension of the Kerr Metric, , RH, Boyer, J. Math. Phys., 1967


★ Exact Solutions of Einstein's Field Equations, Stephani, Hans; Kramer, Dietrich; MacCallum, Malcolm; Hoenselaers, Cornelius & Herlt, Eduard, , , Cambridge University Press, 2003, ISBN 0-521-46136-7

★ The Geometry of Kerr Black Holes, O'Neill, Barrett, , , A. K. Peters, 1995, ISBN 1-56881-019-9

★ Introducing Einstein's Relativity, D'Inverno, Ray, , , Clarendon Press, 1992, ISBN 0-19-859686-3 ''See chapter 19'' for a readable introduction at the advanced undergraduate level.

★ The Mathematical Theory of Black Holes, Chandrasekhar, S., , , Clarendon Press, 1992, ISBN 0-19-850370-9 ''See chapters 6--10'' for a very thorough study at the advanced graduate level.

★ Colliding Plane Waves in General Relativity, Griffiths, J. B., , , Oxford University Press, 1991, ISBN 0-19-853209-1 ''See chapter 13'' for the Chandrasekhar/Ferrari CPW model.

★ Introduction to General Relativity, Adler, Ronald; Bazin, Maurice & Schiffer, Menahem, , , McGraw-Hill, 1975, ISBN 0-07-000423-4 ''See chapter 7''.

★ Characterization of three standard families of vacuum solutions as noted above.

★ Corrections and Comments on the Multipole Moments of Axisymmetric Electrovacuum Spacetimes, Sotiriou, Thomas P.; and Apostolatos, Theocharis A., , , Class. Quant. Grav., 2004 arXiv eprint Gives the relativistic multipole moments for the Ernst vacuums (plus the electromagnetic and gravitational relativistic multipole moments for the charged generalization).

★

★ Unknown, , B, Carter, Physical Review Letters, 1971

★ General Relativity, , RM, Wald, The University of Chicago Press, 1984,