Kerr-Newman metric

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Black Hole Metrics
static rotating
uncharged Schwarzschild metric Kerr metric
loaded Reissner-Nordström metric Kerr-Newman metric
: Electric charge ; : Angular momentum

The Kerr-Newman metric (after Roy Kerr and Ezra Ted Newman ) is an exact, asymptotically flat, stationary and axially symmetric solution of the Einstein equations for electrically charged , rotating black holes . If the complex transformation that leads from the Schwarzschild metric to the Kerr solution is applied to the Reissner-Nordström metric , this leads to the Kerr-Newman solution.

Line element

The line element has the form in Boyer-Lindquist coordinates :

The space-time signature and the following abbreviations were used here:

denote the mass equivalent (including charge and rotation energy ) of the central body, the electrical charge and the angular momentum of the black hole. By selection in the theory of relativity conventional natural units with (gravitational constant, the speed of light and Coulomb's constant ) have mass , electric charge and angular momentum parameter the same dimension as a length . is the Schwarzschild radius .

The irreducible mass is related to the total, also referred to as the gravitating mass mass equivalent ratio

Since energy has to be added to a static and neutral object that is to be set in rotation or electrically charged, and this energy is itself equivalent to a mass due to the equivalence of mass and energy , the mass equivalent of a rotating and / or charged body is accordingly higher than when it is neutrally at rest. With the help of the Penrose process , energy and thus also its mass equivalent can be extracted from a black hole , but not so much that less than the irreducible mass (that of a corresponding Schwarzschild hole) would be left in the end.

The co- and contravariant metric coefficients are thus

In the case of an electrically neutral black hole , the Kerr-Newman metric is simplified to the Kerr metric . In the case of a non-rotating black hole , the Reissner-Nordström metric results and for a neutral and non-rotating object, the Schwarzschild metric .

Ergosphere and event horizon

Event horizons and ergospheres. a² + Q² runs in pseudospherical r, θ, φ-coordinates from 0 to 1 and in Cartesian x, y, z-coordinates from 1 to 0.

For the external event horizon at and inner, also Cauchy horizon mentioned wherein concerns by set and after dissolving a Boyer-Lindquist radius of

and for the inner and outer ergosphere

At , the horizon would dissolve and the metric would no longer describe a black hole. Bodies with a higher spin cannot collapse into a black hole without giving up angular momentum and / or neutralizing part of their charge by accretion of oppositely charged matter.

Equations of motion

Test particles in the strong gravitational field of a rapidly rotating and strongly charged central mass (a / M = 0.9, Q / M = 0.4)

With the electromagnetic potential

and the resulting Maxwell tensor

arise over

the equations of motion of a free-falling test particle; these are in the dimensionless natural units , which are reduced to and to , and lengths are measured in and times in :

with for the specific total energy (potential, kinetic and rest energy), for the specific axial angular momentum and for the electrical charge per mass of the test particle. is the Carter constant:

with the canonical specific impulse components

,

where , is the poloidal component of the orbital angular momentum and the orbital angle of inclination . The axial angular momentum

and the total energy of the test particle

are also constants of the movement.

is the angular velocity of a locally angular momentum-free observer induced by frame dragging .

The proper time derivatives of the coordinates are related to the local speed of 3 , which is measured relative to a local observer with no angular momentum

.

This results for the individual components

for the radial,

for the poloid,

for the axial and

for the total local speed, where

is the axial radius of gyration (local circumference by 2π), and

the gravitational component of time dilation. The radial escape velocity of an electrically neutral particle is thus

.

Individual evidence

  1. ^ Ezra (Ted) Newman and Tim Adamo: Kerr-Newman metric . Scholarpedia, 9 (10): 31791
  2. ^ Newman & Janis: Note on the Kerr Spinning-Particle Metric
  3. ^ A b Charles Misner , Kip S. Thorne , John. A. Wheeler : Gravitation , pp. 877, pp. 908. WH Freeman, San Francisco 1973, ISBN 0-7167-0344-0
  4. a b Sarani Chakraborty: Light deflection due to a charged, rotating body , page 4
  5. ^ Alan Myers: Natural System of Units in General Relativity , p. 4
  6. Thibault Damour : Black Holes: Energetics and Thermodynamics , p. 11 ff.
  7. Bhat, Dhurandhar & Dadhich: Energetics of the Kerr-Newman Black Hole by the Penrose process , p 94 ff.
  8. Joakim Bolin, Ingemar Bengtsson: The Angular Momentum of Kerr Black Holes , p. 2, p. 10, p. 11.
  9. ^ William Wheaton: Rotation Speed ​​of a Black Hole
  10. ^ Roy Kerr (Crafoord Prize Symposium in Astronomy): Spinning Black Holes . (Youtube, timestamp 36:47 )
  11. Brandon Carter: Global structure of the Kerr family of gravitational fields (1968)
  12. Orlando Luongo, Hernando Quevedo: Characterizing repulsive gravity with curvature eigenvalues
  13. a b Hakan Cebeci et al: Motion of the charged test particles in Kerr-Newman-Taub-NUT spacetime and analytical solutions
  14. Eva Hackmann, Hongxiao Xu: Charged particle motion in Kerr-Newmann space-times , p. 4