Kerr metric

The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially-symmetric black hole with a spherical event horizon. The Kerr metric is an exact solution of the Einstein field equations of general relativity; these equations are highly non-linear, which makes exact solutions very difficult to find. The Kerr metric is a generalization of the Schwarzschild metric, which was discovered by Karl Schwarzschild in 1915 and which describes the geometry of spacetime around an uncharged, spherically-symmetric, and non-rotating body. The corresponding solution for a charged, spherical, non-rotating body, the Reissner–Nordström metric, was discovered soon afterwards (1916–1918). However, the exact solution for an uncharged, rotating black-hole, the Kerr metric, remained unsolved until 1963, when it was discovered by Roy Kerr. The natural extension to a charged, rotating black-hole, the Kerr–Newman metric, was discovered shortly thereafter in 1965. These four related solutions may be summarized by the following table:

Non-rotating (J = 0) Rotating (J ≠ 0)
Uncharged (Q = 0) Schwarzschild Kerr
Charged (Q ≠ 0) Reissner–Nordström Kerr–Newman

where Q represents the body's electric charge and J represents its spin angular momentum.

According to the Kerr metric, such rotating black-holes should exhibit frame dragging, an unusual prediction of general relativity. Measurement of this frame dragging effect was a major goal of the Gravity Probe B experiment. Roughly speaking, this effect predicts that objects coming close to a rotating mass will be entrained to participate in its rotation, not because of any applied force or torque that can be felt, but rather because of the curvature of spacetime associated with rotating bodies. At close enough distances, all objects – even light itself – must rotate with the black-hole; the region where this holds is called the ergosphere.

Rotating black holes have surfaces where the metric appears to have a singularity; the size and shape of these surfaces depends on the black hole's mass and angular momentum. The outer surface encloses the ergosphere and has a shape similar to a flattened sphere. The inner surface marks the "radius of no return" also called the "event horizon"; objects passing through this radius can never again communicate with the world outside that radius. However, neither surface is a true singularity, since their apparent singularity can be eliminated in a different coordinate system. Objects between these two horizons must co-rotate with the rotating body, as noted above; this feature can be used to extract energy from a rotating black hole, up to its invariant mass energy, Mc2.

Mathematical form

The Kerr metric[1][2] describes the geometry of spacetime in the vicinity of a mass M rotating with angular momentum J

\begin{align} c^{2} d\tau^{2} = & \left( 1 - \frac{r_{s} r}{\rho^{2}} \right) c^{2} dt^{2} - \frac{\rho^{2}}{\Delta} dr^{2} - \rho^{2} d\theta^{2} \\ & - \left( r^{2} + \alpha^{2} + \frac{r_{s} r \alpha^{2}}{\rho^{2}} \sin^{2} \theta \right) \sin^{2} \theta \ d\phi^{2} + \frac{2r_{s} r\alpha \sin^{2} \theta }{\rho^{2}} \, c \, dt \, d\phi \end{align}

 

 

 

 

(1)

where the coordinates r, \theta, \phi are standard spherical coordinate system, and rs is the Schwarzschild radius

r_{s} = \frac{2GM}{c^{2}}

 

 

 

 

(2)

and where the length-scales α, ρ and Δ have been introduced for brevity

\alpha = \frac{J}{Mc}

 

 

 

 

(3)

\rho^{2} = r^{2} + \alpha^{2} \cos^{2} \theta

 

 

 

 

(4)

\Delta = r^{2} - r_{s} r + \alpha^{2}

 

 

 

 

(5)

In the non-relativistic limit where M (or, equivalently, rs) goes to zero, the Kerr metric becomes the orthogonal metric for the oblate spheroidal coordinates

c^{2} d\tau^{2} = c^{2} dt^{2} - \frac{\rho^{2}}{r^{2} + \alpha^{2}} dr^{2} - \rho^{2} d\theta^{2}- \left( r^{2} + \alpha^{2} \right) \sin^{2}\theta d\phi^{2}

 

 

 

 

(6)

which are equivalent to the Boyer–Lindquist coordinates[3]

{x} = \sqrt {r^2 + \alpha^2} \sin\theta\cos\phi

 

 

 

 

(7)

{y} = \sqrt {r^2 + \alpha^2} \sin\theta\sin\phi

 

 

 

 

(8)

{z} = r \cos\theta

 

 

 

 

(9)

Wave operator

Since even a direct check on the Kerr metric involves cumbersome calculations, the contravariant components g^{ik} of the metric tensor are shown below in the expression for the square of the four-gradient operator:

\begin{align} g^{\mu\nu}\frac{\partial}{\partial{x^{\mu}}}\frac{\partial}{\partial{x^{\nu}}} = & \frac{1}{c^{2}\Delta}\left(r^{2} + \alpha^{2} + \frac{r_{s}r\alpha^{2}}{\rho^{2}}\sin^{2}\theta\right)\left(\frac{\partial}{\partial{t}}\right)^{2} + \frac{2r_{s}r\alpha}{c\rho^{2}\Delta}\frac{\partial}{\partial{\phi}}\frac{\partial}{\partial{t}} \\ & - \frac{1}{\Delta\sin^{2}\theta}\left(1 - \frac{r_{s}r}{\rho^{2}}\right)\left(\frac{\partial}{\partial{\phi}}\right)^{2} - \frac{\Delta}{\rho^{2}}\left(\frac{\partial}{\partial{r}}\right)^{2} - \frac{1}{\rho^{2}}\left(\frac{\partial}{\partial{\theta}}\right)^{2} \end{align}

 

 

 

 

(10)

Frame dragging

We may rewrite the Kerr metric (1) in the following form:

c^{2} d\tau^{2} = \left( g_{tt} - \frac{g_{t\phi}^{2}}{g_{\phi\phi}} \right) dt^{2} + g_{rr} dr^{2} + g_{\theta\theta} d\theta^{2} + g_{\phi\phi} \left( d\phi + \frac{g_{t\phi}}{g_{\phi\phi}} dt \right)^{2}.

 

 

 

 

(11)

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, where Ω is called the Killing horizon.

\Omega = -\frac{g_{t\phi}}{g_{\phi\phi}} = \frac{r_{s} r \alpha c}{\rho^{2} \left( r^{2} + \alpha^{2} \right) + r_{s} r \alpha^{2} \sin^{2}\theta}.

 

 

 

 

(12)

Thus, an inertial reference frame is entrained by the rotating central mass to participate in the latter's rotation; this is called frame-dragging, and has been tested experimentally.[4] Qualitatively, frame-dragging can be viewed as the gravitational analog of electromagnetic induction. An "ice skater", in orbit over the equator and rotationally at rest with respect to the stars, extends her arms. The arm extended toward the black hole will be torqued spinward. The arm extended away from the black hole will be torqued anti-spinward. She will therefore be rotationally sped up, in a counter-rotating sense to the black hole. This is the opposite of what happens in everyday experience. If she is already rotating at a certain speed when she extends her arms, inertial effects and frame-dragging effects will balance and her spin will not change. Due to the Principle of Equivalence gravitational effects are locally indistinguishable from inertial effects, so this rotation rate, at which when she extends her arms nothing happens, is her local reference for non-rotation. This frame is rotating with respect to the fixed stars and counter-rotating with respect to the black hole. A useful metaphor is a planetary gear system with the black hole being the sun gear, the ice skater being a planetary gear and the outside universe being the ring gear. This can be also be interpreted through Mach's principle.

The two surfaces on which the Kerr metric appears to have singularities; the inner surface is the spherical event horizon, whereas the outer surface is an oblate spheroid. The ergosphere lies between these two surfaces; within this volume, the purely temporal component gtt is negative, i.e., acts like a purely spatial metric component. Consequently, particles within this ergosphere must co-rotate with the inner mass, if they are to retain their time-like character.

Important surfaces

The Kerr metric has two physical relevant surfaces on which it appears to be singular. The inner surface corresponds to an event horizon similar to that observed in the Schwarzschild metric; this occurs where the purely radial component grr of the metric goes to infinity. Solving the quadratic equation 1/grr = 0 yields the solution:

r_\mathit{inner} = \frac{r_{s} + \sqrt{r_{s}^{2} - 4\alpha^{2}}}{2}

Another singularity occurs where the purely temporal component gtt of the metric changes sign from positive to negative. Again solving a quadratic equation gtt=0 yields the solution:

r_\mathit{outer} = \frac{r_{s} + \sqrt{r_{s}^{2} - 4\alpha^{2} \cos^{2}\theta}}{2}

Due to the cos2θ term in the square root, this outer surface resembles a flattened sphere 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 gtt 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.

As with the event horizon in the Schwarzschild metric the apparent singularities at rinner and router are an illusion created by the choice of coordinates (i.e., they are coordinate singularities). In fact, the space-time can be smoothly continued through them by an appropriate choice of coordinates.

Ergosphere and the Penrose process

Main article: Penrose process

A black hole in general is surrounded by a surface, called the event horizon and situated at the Schwarzschild radius for a nonrotating black hole, where the escape velocity is equal to the velocity of light. Within this surface, no observer/particle can maintain itself at a constant radius. It is forced to fall inwards, and so this is sometimes called the static limit.

A rotating black hole has the same static limit at its event horizon but there is an additional surface outside the event horizon named the "ergosurface" given by (r-GM)^{2} = G^{2}M^{2}-J^{2}\cos^{2}\theta in Boyer–Lindquist coordinates, which can be intuitively characterized as the sphere where "the rotational velocity of the surrounding space" is dragged along with the velocity of light. Within this sphere the dragging is greater than the speed of light, and any observer/particle is forced to co-rotate.

The region outside the event horizon but inside the surface where the rotational velocity is the speed of light, is called the ergosphere (from Greek ergon meaning work). Particles falling within the ergosphere are forced to rotate faster and thereby gain energy. Because they are still outside the event horizon, they may escape the black hole. The net process is that the rotating black hole emits energetic particles at the cost of its own total energy. The possibility of extracting spin energy from a rotating black hole was first proposed by the mathematician Roger Penrose in 1969 and is thus called the Penrose process. Rotating black holes in astrophysics are a potential source of large amounts of energy and are used to explain energetic phenomena, such as gamma ray bursts.

Features of the Kerr geometry

The Kerr geometry 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 geometry 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.

Note that the Kerr geometry is unstable with regards to perturbations in the interior region. This instability means that although the Kerr metric is axis-symmetric, a black hole created through gravitational collapse may not be so. This instability also implies that many of the features of the Kerr geometry described above would also probably not be present in such a black hole.

A surface on which light can orbit a black hole is called a photon sphere. The Kerr solution has infinitely many photon spheres, lying between an inner one and an outer one. In the nonrotating, Schwarzschild solution, with α=0, the inner and outer photon spheres degenerate, so that all the photons sphere occur at the same radius. The greater the spin of the black hole is, the farther from each other the inner and outer photon spheres move. A beam of light traveling in a direction opposite to the spin of the black hole will circularly orbit the hole at the outer photon sphere. A beam of light traveling in the same direction as the black hole's spin will circularly orbit at the inner photon sphere. Orbiting geodesics with some angular momentum perpendicular to the axis of rotation of the black hole will orbit on photon spheres between these two extremes. Because the space-time is rotating, such orbits exhibit a precession, since there is a shift in the \phi \, variable after completing one period in the \theta \, variable.

Overextreme Kerr solutions

The location of the event horizon is determined by the larger root of \Delta=0. When {r_s / 2} < \alpha (i.e. G M^2 < J c ), there are no (real valued) solutions to this equation, and there is no event horizon. With no event horizons to hide it from the rest of the universe, the black hole ceases to be a black hole and will instead be a naked singularity.[5]

Kerr black holes as wormholes

Although the Kerr solution appears to be singular at the roots of Δ = 0, these are actually coordinate singularities, and, with an appropriate choice of new coordinates, the Kerr solution can be smoothly extended through the values of r corresponding to these roots. The larger of these roots determines the location of the event horizon, and the smaller determines the location of a Cauchy horizon. A (future-directed, time-like) curve can start in the exterior and pass through the event horizon. Once having passed through the event horizon, the r coordinate now behaves like a time coordinate, so it must decrease until the curve passes through the Cauchy horizon.

The region beyond the Cauchy horizon has several surprising features. The r coordinate again behaves like a spatial coordinate and can vary freely. The interior region has a reflection symmetry, so that a (future-directed time-like) curve may continue along a symmetric path, which continues through a second Cauchy horizon, through a second event horizon, and out into a new exterior region which is isometric to the original exterior region of the Kerr solution. The curve could then escape to infinity in the new region or enter the future event horizon of the new exterior region and repeat the process. This second exterior is sometimes thought of as another universe. On the other hand, in the Kerr solution, the singularity is a ring, and the curve may pass through the center of this ring. The region beyond permits closed time-like curves. Since the trajectory of observers and particles in general relativity are described by time-like curves, it is possible for observers in this region to return to their past.

While it is expected that the exterior region of the Kerr solution is stable, and that all rotating black holes will eventually approach a Kerr metric, the interior region of the solution appears to be unstable, much like a pencil balanced on its point.[6] This is related to the idea of cosmic censorship.

Relation to other exact solutions

The Kerr geometry 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 \alpha = 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 geometry, 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 geometry, 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 geometry were computed by Hansen; they turn out to be

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

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 vacuum 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} - \alpha^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:

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

Open problems

The Kerr geometry 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 Schwarzschild 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 are known. (Slowly rotating fluid balls are the relativistic analog of oblate spheroidal balls with nonzero mass and angular momentum but vanishing higher multipole moments.) However, the exterior of the Neugebauer–Meinel disk, an exact dust solution which models a rotating thin disk, approaches in a limiting case the \alpha = M Kerr geometry. Physical thin-disk solutions obtained by identifying parts of the Kerr space-time are also known.[7]

Trajectory equations

The trajectory of a test mass around a spinning (Kerr) black hole. Unlike orbits around a Schwarzschild black hole, the orbit is not confined to a single plane, but will ergodically fill a toruslike region around the equator.

The equations of the trajectory and the time dependence for a particle in the Kerr field are as follows.

In the Hamilton-Jacobi equation we write the action S in the form:

\ S = -E_{0}t + L\phi + S_{r}(r) + S_{\theta}(\theta)

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:

\left(\frac{dS_{\theta}}{d\theta}\right)^{2} + \left(\alpha E_{0}\sin\theta - \frac{L}{\sin\theta}\right)^{2} + \alpha^{2}m^{2}\cos^{2}\theta = K
\Delta\left(\frac{dS_{r}}{dr}\right)^{2} - \frac{1}{\Delta}\left[\left(r^{2} + \alpha^{2}\right)E_{0} - \alpha L\right]^{2} + m^{2}r^{2} = -K

where K is a fourth arbitrary constant (usually called Carter's 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:

{\frac{\partial{S}}{\partial{E_{0}}}} = const
{\frac{\partial{S}}{\partial{L}}} = const
{\frac{\partial{S}}{\partial{K}}} = const

Symmetries

The group of isometries of the Kerr metric is the subgroup of the ten-dimensional Poincaré group which takes the two-dimensional locus of the singularity to itself. It retains the time translations (one dimension) and rotations around its axis of rotation (one dimension). Thus it has two dimensions. Like the Poincaré group, it has four connected components: the component of the identity; the component which reverses time and longitude; the component which reflects through the equatorial plane; and the component that does both.

In physics, symmetries are typically associated with conserved constants of motion, in accordance with Noether's theorem. As shown above, the geodesic equations have four conserved quantities: one of which comes from the definition of a geodesic, and two of which arise from the time translation and rotation symmetry of the Kerr geometry. The fourth conserved quantity does not arise from a symmetry in the standard sense and is commonly referred to as a hidden symmetry.

See also

References

Notes

  1. ↑ Kerr, Roy P. (1963). "Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics". Physical Review Letters 11 (5): 237–238. Bibcode:1963PhRvL..11..237K. doi:10.1103/PhysRevLett.11.237.
  2. ↑ Landau, L. D.; Lifshitz, E. M. (1975). The Classical Theory of Fields (Course of Theoretical Physics, Vol. 2) (revised 4th English ed.). New York: Pergamon Press. pp. 321–330. ISBN 978-0-08-018176-9.
  3. ↑ Boyer, Robert H.; Lindquist, Richard W. (1967). "Maximal Analytic Extension of the Kerr Metric". J. Math. Phys. 8 (2): 265–281. Bibcode:1967JMP.....8..265B. doi:10.1063/1.1705193.
  4. ↑ Will, Clifford M. (May 2011). "Finally, results from Gravity Probe B". Physics (American Physical Society) 4: 43. arXiv:1106.1198. Bibcode:2011PhyOJ...4...43W. doi:10.1103/Physics.4.43.
  5. ↑ Chandrasekhar, S. (1983). The Mathematical Theory of Black Holes. International Series of Monographs on Physics 69. p. 375.
  6. ↑ Penrose 1968
  7. ↑ Bičák, Jří, and Tomáš Ledvinka. "Relativistic disks as sources of the Kerr metric." Physical review letters 71.11 (1993): 1669.

Further reading

External links

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