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.

Contents

The metric

Discovered by Hans Reissner and Gunnar Nordström, their metric can be written as

c^2 {d \tau}^{2} = \left( 1 - {\color{OliveGreen}\frac{r_{s}}{r}} %2B {\color{Red}\frac{r_{Q}^{2}}{r^{2}}} \right) c^{2} dt^{2} - \frac{dr^{2}}{1 - {\color{OliveGreen}\frac{r_{s}}{r}} %2B {\color{Red}\frac{r_{Q}^{2}}{r^{2}}}} - r^{2} d\Omega^{2}

where

τ is the proper time (time measured by a clock moving with the particle) in seconds,
c is the speed of light in meters per second,
t is the time coordinate (measured by a stationary clock at infinity) in seconds,
r is the radial coordinate (circumference of a circle centered on the star divided by 2π) in meters,
Ω is the solid angle,
\,d \Omega^2 = d \theta^2 %2B \sin^2 \theta d \phi^2
rs is the Schwarzschild radius (in meters) of the massive body, which is related to its mass M by
r_{s} = \frac{2GM}{c^{2}}
where G is the gravitational constant, and
rQ is a length-scale corresponding to the electric charge Q of the mass
r_{Q}^{2} = \frac{Q^{2}G}{4\pi\epsilon_{0} c^{4}}
where 1/4πε0 is Coulomb's force constant.[1]

The colors have been added in order to highlight the extensions to Minkowski spacetime. In the limit that the charge Q (or equivalently, the length-scale rQ) goes to zero, one recovers the Schwarzschild metric. To do this, you just remove term in \,\color{Red}\text{red}. The classical Newtonian theory of gravity may then be recovered in the limit as the ratio rs/r goes to zero. To do this, you just remove term in \,\color{OliveGreen}\text{green}. In that limit, the metric returns to the Minkowski metric for special relativity

c^{2} d\tau^{2} = c^{2} dt^{2} - dr^{2} - r^{2} d\Omega^{2}.\,

In practice, the ratio rs/r is almost always extremely small. For example, the Schwarzschild radius rs 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

Although charged black holes with r_{Q} \ll r_{s} are similar to the Schwarzschild black hole, they have two horizons: the event horizon and an internal Cauchy horizon. As usual, the event horizons for the spacetime are located where g^{rr} diverges:

(g^{rr})^{-1}= 1 - \frac{r_{s}}{r} %2B \frac{r_{Q}^{2}}{r^{2}} = \frac{1}{r^2}(r^2 - r_sr %2B r_Q^2) = 0.

The second factor r 2 − rs r + rQ2 is a quadratic in r and we find its zeros by using the quadratic formula:

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

These concentric event horizons become degenerate for 2r_{Q}=r_{s} which corresponds to an extremal black hole. Black holes with 2r_{Q}>r_{s} 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. Theories with supersymmetry usually guarantee that such "superextremal" black holes can't exist.

The electromagnetic potential is

A^{\alpha} = \left(\frac{Q}{r}, 0, 0, 0\right).

If magnetic monopoles are included into the theory, then a generalization to include magnetic charge P is obtained by replacing Q^2 by Q^2 %2B P^2 in the metric and including the term P \cos \theta d \phi in the electromagnetic potential.

See also

Notes

  1. ^ Landau 1975.

References

External links

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