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Reissner–Nordström metric

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 analogous solution for a charged, rotating body is given by the Kerr-Newman metric.

The metric was discovered between 1916 and 1921 by Hans Reissner,[1] Hermann Weyl,[2] Gunnar Nordström[3] and George Barker Jeffery.[4]

The metricEdit

In spherical coordinates  , the Reissner–Nordström metric (aka the line element) is

 

where   is the speed of light,   is the time coordinate (measured by a stationary clock at infinity),   is the radial coordinate,   are the spherical angles, and

  is the Schwarzschild radius of the body given by

 

and   is a characteristic length scale given by

 

Here   is Coulomb force constant  .

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

 .

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

In practice, the ratio   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   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 holesEdit

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 (is not g_{rr} divergent, or equivalently g^{rr}=0?); 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  in the electromagnetic potential.[clarification needed]

Gravitational time dilationEdit

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 symbolsEdit

The Christoffel symbols

 

with the indices

 

give the nonvanishing expressions

 
 
 
 
 
 
 
 
 

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

Equations of motionEdit

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 metricEdit

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 alsoEdit

NotesEdit

  1. ^ Reissner, H. (1916). "Über die Eigengravitation des elektrischen Feldes nach der Einsteinschen Theorie". Annalen der Physik (in German). 50 (9): 106–120. Bibcode:1916AnP...355..106R. doi:10.1002/andp.19163550905.
  2. ^ Weyl, H. (1917). "Zur Gravitationstheorie". Annalen der Physik (in German). 54 (18): 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 (697): 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

ReferencesEdit

External linksEdit