Reissner-Nordström metric

In physics and astronomy, the Reissner-Nordström metric is a solution to the Einstein field equations in empty space, which corresponds to the gravitational field of a charged, non-rotating, spherically symmetric body of mass "M". Discovered by Gunnar Nordström and Hans Reissner, their metric can be written as

:$c^2\; \{d\; au\}^\{2\}\; =\; left(\; 1\; -\; frac\{r\_\{s\{r\}\; +\; frac\{r\_\{Q\}^\{2\{r^\{2\; ight)\; c^\{2\}\; dt^\{2\}\; -\; frac\{dr^\{2\{1\; -\; frac\{r\_\{s\{r\}\; +\; frac\{r\_\{Q\}^\{2\{r^\{2\}\; -\; r^\{2\}\; d\; heta^\{2\}\; -\; r^\{2\}\; sin^\{2\}\; heta\; ,\; dvarphi^\{2\}$

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, :"θ" is the colatitude (angle from North) in radians, :"φ" is the longitude in radians, and :"r_s" 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:"r"_"Q" is a length-scale corresponding to the electric charge "Q" of the mass

::$r\_\{Q\}^\{2\}\; =\; frac\{Q^\{2\}G\}\{4piepsilon\_\{0\}\; c^\{4$:where 1/4π"ε"₀ is Coulomb's force constant.Landau 1975.]

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

:$c^\{2\}\; d\; au^\{2\}\; =\; c^\{2\}\; dt^\{2\}\; -\; dr^\{2\}\; -\; r^\{2\}\; d\; heta^\{2\}\; -\; r^\{2\}\; sin^\{2\}\; heta\; dphi^\{2\},$

In practice, the ratio "r"_"s"/"r" is almost always extremely small. For example, the Schwarzschild radius "r"_"s" 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 may be reliably located by analyzing the equation

:$g^\{00\}=\; 1\; -\; frac\{r\_\{s\{r\}\; +\; frac\{r\_\{Q\}^\{2\{r^\{2\; =\; 0$

This quadratic equation for "r" has the solutions

:$r\_pm\; =\; frac\{r\_\{s\}\; pm\; sqrt\{r\_\{s\}^2\; -\; 4r\_\{Q\}^2\{2\}.$

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

:$A\_\{alpha\}\; =\; left(frac\{Q\}\{r\},\; 0,\; 0,\; 0\; ight)$.

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

References

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External links

* [http://casa.colorado.edu/~ajsh/rn.html spacetime diagrams] including Finkelstein diagram and Penrose diagram, by Andrew J. S. Hamilton
* " [http://demonstrations.wolfram.com/ParticleMovingAroundTwoExtremeBlackHoles/ Particle Moving Around Two Extreme Black Holes] " by Enrique Zeleny, The Wolfram Demonstrations Project.