Reissner–Nordström metric
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, Hermann Weyl, Gunnar Nordström and George Barker Jeffery independently. The metric. In spherical coordinates formula_1, the Reissner–Nordström metric (i.e. the line element) is formula_2 where formula_3 is the speed of light, formula_4 is the proper  time, formula_5 is the time coordinate (measured by a stationary clock at infinity), formula_6 is the radial coordinate, formula_7 are the spherical angles, and formula_8 is the Schwarzschild radius of the body given by formula_9 and formula_10 is a characteristic length scale given by formula_11 Here, formula_12 is the electric constant. The total mass of the central body and its irreducible mass are related by formula_13 The difference between formula_14 and formula_15 is due to the equivalence of mass and energy, which makes the electric field energy also contribute to the total mass. In the limit that the charge formula_16  (or equivalently, the length scale formula_10) goes to zero, one recovers the Schwarzschild metric. The classical Newtonian theory of gravity may then be recovered in the limit as the ratio formula_18 goes to zero. In the limit that both formula_19 and formula_18 go to zero, the metric becomes the Minkowski metric for special relativity. In practice, the ratio formula_18 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 an orbital radius formula_6 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" ≪ "r" are similar to the Schwarzschild black hole, they have two horizons: the event horizon and an internal Cauchy horizon. As with the Schwarzschild metric, the event horizons for the spacetime are located where the metric component formula_23 diverges; that is, where formula_24 This equation has two solutions: formula_25 These concentric event  horizons become degenerate for 2"r" = "r", which corresponds to an extremal black hole. Black holes with 2"r" > "r" can not 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). 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. Theories with supersymmetry usually guarantee that such "superextremal" black holes cannot exist. The electromagnetic potential is formula_26 If  magnetic monopoles are included in the theory, then a generalization to include magnetic charge "P" is obtained by replacing "Q" by "Q" + "P" in the metric and including the term "P" cos "θ" "dφ" in the electromagnetic potential. Gravitational time dilation. The gravitational time dilation in the vicinity of the central body is given by formula_27 which relates to the local radial escape velocity of a neutral particle formula_28 Christoffel symbols. The Christoffel symbols formula_29 with the indices formula_30 give the nonvanishing expressions formula_31 Given the Christoffel symbols, one can compute the geodesics of a test-particle. Equations of  motion. 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". In dimensionless natural units of "G" = "M" = "c" = "K" = 1 the motion of an electrically charged particle with the charge "q" is given by formula_32 which yields formula_33 formula_34 formula_35 All total derivatives are with respect to proper time formula_36. Constants of the motion are provided by solutions formula_37 to the  partial differential equation formula_38 after substitution of the second derivatives given above. The metric itself is a solution when written as a differential equation formula_39 The separable equation formula_40 immediately yields the constant relativistic specific angular momentum formula_41 a third constant obtained from formula_42 is the specific energy (energy per unit rest mass) formula_43 Substituting formula_44 and formula_45 into formula_46 yields the radial equation formula_47 Multiplying under the integral sign by formula_44 yields the orbital equation formula_49 The total time dilation between the test-particle and an observer at infinity is formula_50 The first derivatives formula_51 and the contravariant components  of the local 3-velocity formula_52 are related by formula_53 which gives the initial conditions formula_54 formula_55 The specific orbital energy formula_56 and the specific relative angular momentum formula_57 of the test-particle are conserved quantities of motion. formula_58 and formula_59 are the radial and transverse components of the local velocity-vector. The local velocity is therefore formula_60 Alternative formulation of metric. The metric can be expressed in Kerr–Schild form like this: formula_61 Notice that k is a unit vector. Here "M" is the constant mass of the object, "Q" is the constant charge of the object, and "η  is the Minkowski tensor. Quantum corrections to the metric. In certain approaches to quantum gravity, the classical Reissner–Nordström metric receives quantum corrections. An example of this is given by the effective field theory approach pioneered by Barvinsky and Vilkovisky. At second order in curvature, the classical Einstein-Hilbert action is supplemented by local and non-local terms: where formula_63 is an energy scale. The exact values of the coefficients formula_64 are unknown, as they depend on the nature of the ultra-violet theory of quantum gravity. On the other hand, the coefficients formula_65 are calculable. The operator formula_66 has 	Reissner–Nordström metric
7731850	the integral representation The new additional terms in the action imply a modification of the classical solution. The quantum corrected Reissner–Nordström metric, up to order formula_68, was found by Campos Delgado: where