Reissner-Nordström 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 Reissner-Nordström metric (after Hans Reissner and Gunnar Nordström ) describes electrically charged , non-rotating black holes . Mathematically speaking, it is an exact solution of the Einstein equations , which is clearly determined by the following properties:

  • asymptotically flat
  • static
  • spherical-symmetrical

Line element

The line element of the Reissner-Nordström metric has the form:

where is the total mass equivalent and the electrical charge of the object. is Newton's constant of gravity and the Coulomb's constant. The so-called natural units are used and the coordinate system rotates due to the spherical symmetry without loss of generality in such a way that both angular coordinates are reduced to a single angle, so that the metric also in the form

can be written (also in the following section). For the sake of simplicity, an electrical point charge is assumed in the coordinate origin. Magnetic fields and circulating currents are neglected. The electromagnetic four potential is thus a Coulomb potential :

what about

the Maxwell tensor gives.

Since and flow into the line element with opposite signs (the electric field exerts a negative pressure radially, which leads to gravitational repulsion), from a certain distance the attraction (decreases with it ) and from a certain proximity the repulsion (this decreases with it ab) predominate in what is known as the "Reissner Nordström Repulsion".

The total mass equivalent of the central body and its irreducible mass are in relation

.

The difference between and is due to that by the equivalence of mass and energy and the electric field energy in flows.

Metric tensor

The co- and contravariant metric is thus

Horizons and singularities

As with the Schwarzschild metric , the event horizon lies at the radius where the metric becomes singular. That means

Due to the quadratic dependence on the radius r, however, there are two solutions to this equation. Therefore there is an outer event horizon at and the inner, also called Cauchy horizon , at .

In the case

the root disappears in and the two horizons collapse into a single one. Is however

,

so the root is imaginary, with which there is no horizon. In this case one speaks of a bare singularity , which according to today's view, however, cannot exist ("Cosmic Censorship" hypothesis). Modern supersymmetric theories usually prohibit them for black holes. Elementary particles such as protons and electrons, on the other hand, have a charge that is much larger than their mass, but they are not black holes either.

For the Reissner-Nordström metric goes over to the Schwarzschild metric . Their singularities are then at and .

Since the charge of black holes is very quickly neutralized in practice by electrical currents, namely the accretion flows, electrically charged black holes play a subordinate role in astrophysics.

Christmas symbols

The non-disappearing Christoffel symbols that deal with the clues

over

are derived from the metric tensor

Gravitational time dilation

The gravitational component of the time dilation results from

whereby not only the mass of the central body but also its charge flows into it. The radial escape speed of an electrically neutral particle is related to this

.

Equations of motion

In dimensionless natural units of are the equations of motion aligned on the -plane

the equations of motion of a test particle charged with the specific charge :

and the total time dilation

The first derivatives of the coordinates associated with the contravariant components of the local 3-speed in the ratio

.

it follows

The total specific energy obtained for the test particle is included

The specific angular momentum

is also a conservation quantity of movement. and denote the radial and transverse components of the local velocity vector. The total local speed is thus

.

Web links

Individual evidence

  1. Gerald Marsh: Charge, geometry, and effective mass , pp. 2–5
  2. ^ Joint Institute for Laboratory Astrophysics, Colorado: Journey into and through a Reissner-Nordström black hole
  3. Orlando Luongo, Hernando Quevedo: Characterizing repulsive gravity with curvature eigenvalues
  4. a b Ashgar Quadir: The Reissner Nordström Repulsion
  5. Øyvind Grøn, Sigbjørn Hervik: Einstein's General Theory of Relativity , p. 274
  6. Øyvind Grøn: Poincaré Stress and the Reissner-Nordström Repulsion
  7. ^ Andrew Hamilton: The Reissner Nordström Geometry
  8. Célérier, Santos & Satheeshkumar: Hilbert repulsion in the Reissner-Nordström and Schwarzschild spacetimes , pp. 3–7
  9. Thibault Damour : Black Holes: Energetics and Thermodynamics , p. 11 ff.
  10. Brandon Carter: Global structure of the Kerr family of gravitational fields (1968)