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
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, r_{S} is the Schwarzschild radius of the body given by
and r_{Q} is a characteristic length scale given by
Here 1/4πε_{0} is Coulomb force constant.^{[1]}
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 that both r_{Q}/r and r_{S}/r go to zero, the metric becomes the Minkowski metric for special relativity.
In practice, the ratio r_{S}/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
Although charged black holes with r_{Q} ≪ r_{S} 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 g^{rr} diverges; that is, where
This equation has two solutions:
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.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}^{[citation needed]} }} 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 Q^{2} by Q^{2} + P^{2} in the metric and including the term Pcos θ dφ in the electromagnetic potential.Template:Clarify
See also
Notes
References
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External links
- spacetime diagrams including Finkelstein diagram and Penrose diagram, by Andrew J. S. Hamilton
- "Particle Moving Around Two Extreme Black Holes" by Enrique Zeleny, The Wolfram Demonstrations Project.