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] independently.
, the Reissner–Nordström metric (i.e. the line element) is where The total mass of the central body and its irreducible mass are related by[6][7] The difference between
is due to the equivalence of mass and energy, which makes the electric field energy also contribute to the total mass.
The classical Newtonian theory of gravity may then be recovered in the limit as the ratio
For example, the Schwarzschild radius of the Earth is roughly 9 mm (3/8 inch), whereas a satellite in a geosynchronous orbit has an orbital radius
that is roughly four billion times larger, at 42164 km (26200 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.
These concentric event horizons become degenerate for 2rQ = rs, which corresponds to an extremal black hole.
Black holes with 2rQ > rs cannot exist in nature because if the charge is greater than the mass there can be no physical event horizon (the term under the square root becomes negative).
[9] 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.
[10] Theories with supersymmetry usually guarantee that such "superextremal" black holes cannot exist.
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 P cos θ dφ in the electromagnetic potential.
[clarification needed] 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
Given the Christoffel symbols, one can compute the geodesics of a test-particle.
[11][12] Instead of working in the holonomic basis, one can perform efficient calculations with a tetrad.
be a set of one-forms with internal Minkowski index
The parallel transport of the tetrad is captured by the connection one-forms
The connections can be solved for by inspection from Cartan's equation
can be constructed as a collection of two-forms by the second Cartan equation
which again makes use of the exterior derivative and wedge product.
This approach is significantly faster than the traditional computation with
[14] 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 use θ instead of φ.
All total derivatives are with respect to proper time
The metric itself is a solution when written as a differential equation
immediately yields the constant relativistic specific angular momentum
The total time dilation between the test-particle and an observer at infinity is
are the radial and transverse components of the local velocity-vector.