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 and Gunnar Nordström.

These four related solutions may be summarized by the following table:

Non-rotating (J = 0) Rotating (J ≠ 0)
Uncharged (Q = 0) Schwarzschild Kerr
Charged (Q ≠ 0) Reissner–Nordström Kerr–Newman

where Q represents the body's electric charge and J represents its spin angular momentum.

The metric[edit]

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


ds^2 = 
\left( 1 - \frac{r_\mathrm{S}}{r} + \frac{r_Q^2}{r^2} \right) c^2\, dt^2 - \frac{1}{1 - r_\mathrm{S}/r + r_Q^2/r^2}\, dr^2 - r^2\, d\theta^2 - r^2 \sin^2 \theta \, d\phi^2,

where c is the speed of light, t is the time coordinate (measured by a stationary clock at infinity), r is the radial coordinate, rS is the Schwarzschild radius of the body given by


r_{s} = \frac{2GM}{c^2},

and rQ is a characteristic length scale given by


r_{Q}^{2} = \frac{Q^2 G}{4\pi\varepsilon_{0} c^4}.

Here 1/4πε0 is Coulomb force constant.[1]

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.[2] As with the Schwarzschild metric, the event horizons for the spacetime are located where the metric component grr diverges; that is, where

 0 = 1/g^{rr} = 1 - \frac{r_\mathrm{S}}{r} + \frac{r_Q^2}{r^2}.

This equation has two solutions:


r_\pm = \frac{1}{2}\left(r_{s} \pm \sqrt{r_{s}^2 - 4r_{Q}^2}\right).

These concentric event horizons become degenerate for 2rQ = rS, which corresponds to an extremal black hole. Black holes with 2rQ > rS are believed not to exist in nature because they would contain a naked singularity; their appearance would contradict Roger Penrose's cosmic censorship hypothesis which is generally believed to be true.[citation needed] Theories with supersymmetry usually guarantee that such "superextremal" black holes cannot exist.

The electromagnetic potential is

A_{\alpha} = \left(Q/r, 0, 0, 0\right).

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]

See also[edit]

Notes[edit]

  1. ^ Landau 1975.
  2. ^ Chandrasekhar, S. (1998). The Mathematical Theory of Black Holes (Reprinted ed.). Oxford University Press. p. 205. ISBN 0-19850370-9. 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." 

References[edit]

External links[edit]