Reissner–Nordström metric

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In physics and astronomy, the Reissner–Nordström metric is a static solution to the Einstein-Maxwell field equations, which corresponds to the gravitational field of a charged, non-rotating, spherically symmetric body of mass M.

The metric was discovered by Hans Reissner,[1] Hermann Weyl,[2] Gunnar Nordström[3] and G. B. Jeffery.[4]

The metric[edit]

In spherical coordinates (t, r, θ, φ), the line element for the Reissner–Nordström metric is

where c is the speed of light, t is the time coordinate (measured by a stationary clock at infinity), r is the radial coordinate, dΩ² is a 2-sphere defined by

rs is the Schwarzschild radius of the body given by

and rQ is a characteristic length scale given by

Here 1/4πε0 is Coulomb force constant K.

The total mass of the central body and its irreducible mass are related by[5][6]

.

The difference between M and Mirr is due to the equivalence of mass and energy, which makes the electric field energy also contribute to the total mass.

In the limit that the charge Q (or equivalently, the length-scale rQ) goes to zero, one recovers the Schwarzschild metric. The classical Newtonian theory of gravity may then be recovered in the limit as the ratio rs/r goes to zero. In that limit that both rQ/r and rs/r go to zero, the metric becomes the Minkowski metric for special relativity.

In practice, the ratio rs/r is often extremely small. For example, the Schwarzschild radius of the Earth is roughly 9 mm (3/8 inch), whereas a satellite in a geosynchronous orbit has a radius r that is roughly four billion times larger, at 42,164 km (26,200 miles). Even at the surface of the Earth, the corrections to Newtonian gravity are only one part in a billion. The ratio only becomes large close to black holes and other ultra-dense objects such as neutron stars.

Charged black holes[edit]

Although charged black holes with rQ ≪ rs are similar to the Schwarzschild black hole, they have two horizons: the event horizon and an internal Cauchy horizon.[7] As with the Schwarzschild metric, the event horizons for the spacetime are located where the metric component grr diverges; that is, where

This equation has two solutions:

These concentric event horizons become degenerate for 2rQ = rs, which corresponds to an extremal black hole. Black holes with 2rQ > rs can not exist in nature because if the charge is greater than the mass there can be no physical event horizon[8] (the term under the square root becomes negative). Objects with a charge greater than their mass can exist in nature, but they can not collapse down to a black hole, and if they could, they would display a naked singularity.[9] Theories with supersymmetry usually guarantee that such "superextremal" black holes cannot exist.

The electromagnetic potential is

If magnetic monopoles are included in the theory, then a generalization to include magnetic charge P is obtained by replacing Q² by Q² + P² in the metric and including the term Pcos θ dφ in the electromagnetic potential.[clarification needed]

Gravitational time dilation[edit]

The gravitational time dilation in the vicinity of the central body is given by

which relates to the local radial escape-velocity of a neutral particle

.

Christoffel symbols[edit]

The Christoffel symbols

with the indicies

give the nonvanishing expressions

Given the Christoffel symbols, one can compute the geodesics of a test-particle.[10][11]

Equations of motion[edit]

Because of the spherical symmetry of the metric, the coordinate system can always be aligned in a way that the motion of a test-particle is confined to a plane, so for brevity and without restriction of generality we further use Ω instead of θ and φ. In dimensionless natural units of G=M=c=K=1 the motion of an electrically charged particle with the charge q is given by

which gives

The total time dilation between the test-particle and an observer at infinity is

The first derivatives and the contravariant components of the local 3-velocity are related by

.

which gives the initial conditions

The specific orbital energy

and the specific relative angular momentum

of the test-particle are conserved quantities of motion. and are the radial and transverse components of the local velocity-vector. The local velocity is therefore

.

Alternative formulation of metric[edit]

The metric can alternatively be expressed like this:

Notice that k is a unit vector. Here M is the constant mass of the object, Q is the constant charge of the object, and η is the Minkowski tensor.

See also[edit]

Notes[edit]

  1. ^ Reissner, H. (1916). "Über die Eigengravitation des elektrischen Feldes nach der Einsteinschen Theorie". Annalen der Physik (in German). 50: 106–120. Bibcode:1916AnP...355..106R. doi:10.1002/andp.19163550905. 
  2. ^ Weyl, H. (1917). "Zur Gravitationstheorie". Annalen der Physik (in German). 54: 117–145. Bibcode:1917AnP...359..117W. doi:10.1002/andp.19173591804. 
  3. ^ Nordström, G. (1918). "On the Energy of the Gravitational Field in Einstein's Theory". Verhandl. Koninkl. Ned. Akad. Wetenschap., Afdel. Natuurk., Amsterdam. 26: 1201–1208. 
  4. ^ Jeffery, G. B. (1921). "The field of an electron on Einstein's theory of gravitation". Proc. Roy. Soc. Lond. A. 99: 123–134. Bibcode:1921RSPSA..99..123J. doi:10.1098/rspa.1921.0028. 
  5. ^ Thibault Damour: Black Holes: Energetics and Thermodynamics, S. 11 ff.
  6. ^ Ashgar Quadir: The Reissner Nordström Repulsion
  7. ^ Chandrasekhar, S. (1998). The Mathematical Theory of Black Holes (Reprinted ed.). Oxford University Press. p. 205. ISBN 0-19850370-9. Archived from the original on 29 April 2013. Retrieved 13 May 2013. And finally, the fact that the Reissner-Nordström solution has two horizons, an external event horizon and an internal 'Cauchy horizon,' provides a convenient bridge to the study of the Kerr solution in the subsequent chapters. 
  8. ^ Andrew Hamilton: The Reissner Nordström Geometry Archived 2007-07-07 at the Wayback Machine. (Casa Colorado)
  9. ^ Carter, Brandon. Global Structure of the Kerr Family of Gravitational Fields, Physical Review, page 174
  10. ^ Leonard Susskind: The Theoretical Minimum: Geodesics and Gravity, (General Relativity Lecture 4, timestamp: 34m18s)
  11. ^ Eva Hackmann, Hongxiao Xu: Charged particle motion in Kerr-Newmann space-times

References[edit]

External links[edit]