Black hole of Kerr

Description

Contrary to the case of the black hole without rotation and electric charge (called Black hole of Schwarzschild), the black hole of Kerr has a Ergosphère in addition to sound horizon of the events. Whereas this last is described by a sphere of ray $r_h$ , the ergosphère is a Ellipsoïde of revolution whose small axis is aligned with the axis of rotation of the black hole and of the same size than $r_h$ , and the main roads of size $r_ \ mathrm {stat}$ is located in the equatorial plan. Moreover, $r_ \ mathrm {stat} \ Ge r_h$ . (see the fig. 1).

Horizon of the events

The presence of the horizon of the events does not depend on the rotation of the black hole, it is a characteristic common to all the types of black holes who represents finally the gasoline even the EC what is a Black hole. The particles which cross the horizon of the events fall definitively into the black hole without possibility of escaping from it.

In the case of a black hole of Kerr, the ray of the horizon of the events is written:

$r_ {H} = \ frac {r_ \ mathrm {HS}} {2} \ left 1 + \ sqrt {1 \ left (\ frac {Jc} {GM^2} \ right) ^2} \ right$ ,

where $G$ is the gravitational Constante, $c$ is the Speed of light, $r_ \ mathrm {HS}$ is the Rayon of Schwarzschild. The value of the ray of the horizon of the black hole of Kerr thus lies between half of the ray of Schwarzschild (when the angular momentum is maximum, $J = Mc$ ) and the aforementioned ray (angular momentum no one, $J = 0$ , case of the Black hole of Schwarzschild).

Ergosphère

The ergosphère is known as limiting statics in the sense that the particles which cross it are obligatorily involved in the direction of rotation of the black hole, in other words, they have one angular momentum of the same sign there than $J$ . This drive confers kinetic moment and mechanical energy with a particle which penetrates in the ergosphère then escapes from it, so that the black hole sees his kinetic moment decreasing. It is the effect Penrose which makes it possible to pump energy with a black hole in rotation.

The ergosphère is described by the polar equation:

$r = \ frac {r_ \ mathrm {S}} {2} \ left 1 + \ sqrt {1 \ left (\ frac {Jc} {GM^2} \ cos \ theta \ right) ^2} \ right$

where, all equal notations in addition, $\ theta$ indicates the angle compared to the axis of rotation. It is about a Ellipsoïde of revolution of small axis $r_ {H}$ and of main roads $r_ \ mathrm {stat} = r_ \ mathrm {HS}$ .

Metric of Kerr

The metric one of Kerr is written in the Coordonnées of Boyer-Lindquist. It is given by (by posing $G = C = 1$ ):

$\ mathrm {D} s^2 = - \ left (1 \ frac {2Mr} {\ Sigma} \ right) \ mathrm {D} t^2 - \ frac {4aMr \ sin^2 \ theta} {\ Sigma} \ mathrm {D} T \ mathrm {D} \ phi + \ frac {\ Sigma} {\ Delta} \ mathrm {D} r^2$ $+ \ Sigma \ mathrm {D} \ theta^2 + \ left (r^2+a^2+ \ frac {2a^2Mr \ sin^2 \ theta} {\ Sigma} \ right) \ sin^2 \ theta \ mathrm {D} \ phi^2$ ,

where $\! \ a=J/M$ , $\! \ \ Sigma=r^2+a^2 \ cos^2 \ theta$ , and $\! \ \ Delta=r^2-2Mr+a^2$ .

Let us note that the horizon of the events is given by surface $\ Delta=0$ , where the coefficient of $\ mathrm {D} r^2$ diverges. The ergosphère is given by surface where $1-2Mr/\ Sigma=0$ , where the coefficient of $\ mathrm {D} t^2$ is cancelled.

In addition, by posing $J = 0$ one obtains the Métrique of Schwarzschild. In the extreme case where $J = \ pm M$ , the metric one describes an object in rotation which ceases being a black hole, but is not at the speed of rupture. Finally, let us mention that one does not know the metric one inside an object with spherical symmetry in rotation which prolongs the metric one of Kerr. On the other hand, such a solution is known in the particular case of metric of Schwarzschild.

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