kerr metric


  • Open problems The Kerr geometry is often used as a model of a rotating black hole but if the solution is held to be valid only outside some compact region (subject to certain
    restrictions), in principle, it should be able to be used 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.

  • A similar situation obtains when considering the Schwarzschild metric which also appears to result in a singularity at r = rs dividing the space above and below rs into two
    disconnected patches; using a different coordinate transformation one can then relate the extended external patch to the inner patch (see Schwarzschild metric § Singularities and black holes) – such a coordinate transformation eliminates the
    apparent singularity where the inner and outer surfaces meet.

  • 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.

  • The total mass equivalent (the gravitating mass) of the body (including its rotational energy) and its irreducible mass are related by[23][24] Wave operator Since even a direct
    check on the Kerr metric involves cumbersome calculations, the contravariant components of the metric tensor in Boyer–Lindquist coordinates are shown below in the expression for the square of the four-gradient operator:[21] Frame dragging
    We may rewrite the Kerr metric (1) in the following form: 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

  • Furthermore, there are two constants of motion given by the time translation and rotation symmetries of Kerr spacetime, the energy , and the component of the orbital angular
    momentum parallel to the spin of the black hole .

  • 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 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.

  • Objects between these two surfaces must co-rotate with the rotating black hole, as noted above; this feature can in principle be used to extract energy from a rotating black
    hole, up to its invariant mass energy.

  • These four related solutions may be summarized by the following table, where Q represents the body’s electric charge and J represents its spin angular momentum: According
    to the Kerr metric, a rotating body should exhibit frame-dragging (also known as Lense–Thirring precession), a distinctive prediction of general relativity.

  • However, this is impossible within the ergosphere, where gtt is negative, unless the particle is co-rotating around the interior mass M with an angular speed at least of Ω.

  • Again solving a quadratic equation gtt = 0 yields the solution: or in natural units: Due to the 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.

  • 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 swirling curvature of spacetime itself associated with rotating bodies.

  • 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.

  • 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).

  • 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).

  • Solving the quadratic equation yields the solution: which in natural units (that give ) simplifies to: While in the Schwarzschild metric the event horizon is also the place
    where the purely temporal component gtt of the metric changes sign from positive to negative, in Kerr metric that happens at a different distance.

  • [33] 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 corresponding to these roots.

  • [10] The metric (or equivalently its line element for proper time) in Boyer–Lindquist coordinates is[11][12] where the coordinates are standard oblate spheroidal coordinates,
    which are equivalent to the cartesian coordinates[13][14] where is the Schwarzschild radius and where for brevity, the length scales and have been introduced as A key feature to note in the above metric is the cross-term This implies that
    there is coupling between time and motion in the plane of rotation that disappears when the black hole’s angular momentum goes to zero.

  • In turn, the outer boundary of the ergosphere at rE is not singular by itself even in Kerr coordinates due to non-zero term.

  • However, neither surface is a true singularity, since their apparent singularity can be eliminated in a different coordinate system.

  • This instability means that although the Kerr metric is axis-symmetric, a black hole created through gravitational collapse may not be so.

  • 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.

  • Once having passed through the event horizon, the coordinate now behaves like a time coordinate, so it must decrease until the curve passes through the Cauchy horizon.

  • Due to the equivalence principle, 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.

  • 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.

  • A moving particle experiences a positive proper time along its worldline, its path through spacetime.

  • Ergosphere and the 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.

  • In the case of a rotating black hole, at close enough distances, all objects – even light – must rotate with the black hole; the region where this holds is called the ergosphere.

  • [3][4] Rotating black holes have surfaces where the metric seems to have apparent singularities; the size and shape of these surfaces depends on the black hole’s mass and
    angular momentum.

  • 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.

  • The first measurement of this frame dragging effect was done in 2011 by the Gravity Probe B experiment.

  • Therefore, if a black hole rotates with the spin , its total mass-equivalent is higher by a factor of in comparison with a corresponding Schwarzschild black hole where is
    equal to .

  • Consequently, particles within this ergosphere must co-rotate with the inner mass, if they are to retain their time-like character.

  • [20] Mass of rotational energy If the complete rotational energy of a black hole is extracted, for example with the Penrose process,[21][22] the remaining mass cannot shrink
    below the irreducible mass.

  • For the Kerr vacuum solutions, the first few Weyl moments are given by 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.

  • [13] This instability also implies that many of the features of the Kerr geometry described above may not be present inside such a black hole.

  • Physical thin-disk solutions obtained by identifying parts of the Kerr spacetime are also known.

  • The reason for this is that in order to get a static body to spin, energy needs to be applied to the system.

  • [35] 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.

  • 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.

  • Thus, no particle can move in the direction opposite to central mass’s rotation within the ergosphere.

  • The relativistic multipole moments of the Kerr geometry were computed by Hansen; they turn out to be Thus, the special case of the Schwarzschild vacuum (a = 0) gives the “monopole
    point source” of general relativity.

  • The gravitational time-dilation between a ZAMO at fixed and a stationary observer far away from the mass is In Cartesian Kerr–Schild coordinates, the equations for a photon
    are[32] where is analogous to Carter’s constant and is a useful quantity If we set , the Schwarzschild geodesics are restored.

  • 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.

  • 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.

  • It retains the time translations (one dimension) and rotations around its axis of rotation (one dimension).

  • However, the exterior of the Neugebauer–Meinel disk, an exact dust solution which models a rotating thin disk, approaches in a limiting case the Kerr geometry.

  • Within this sphere the dragging is greater than the speed of light, and any observer/particle is forced to co-rotate.

  • 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.

  • It is related to the total angular momentum of the particle and is given by Since there are four (independent) constants of motion for degrees of freedom, the equations of
    motion for a test particle in Kerr spacetime are integrable.

  • 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

  • [29] The first is the invariant mass of the test particle, defined by the relation where is the four-momentum of the particle.

  • The Kerr solution has infinitely many photon spheres, lying between an inner one and an outer one.

  • In addition to two Killing vector fields (corresponding to time translation and axisymmetry), the Kerr geometry admits a remarkable Killing tensor.

  • However, fast spinning black holes have less distance between multiplicity images.

  • 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.

  • These are multi-indexed quantities whose suitably symmetrized and 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.


Works Cited

[‘1. Warning: Do not confuse the relativistic multipole moments computed by Hansen with the Weyl multipole moments discussed below.
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