Reissner–Nordström

Reissner–Nordström
Name	Reissner–Nordström metric
Field	Karl Schwarzschild-related solutions
Discovered	Gunnar Nordström, Hermann Reissner
Year	1916–1918
Applications	General relativity, Black hole thermodynamics, Astrophysics

Reissner–Nordström is an exact, static solution of Einstein field equations describing the gravitational field outside a non-rotating, spherically symmetric charged mass in general relativity. It generalizes the Schwarzschild metric by incorporating electric charge via a coupling to the Maxwell equations, producing a family of spacetimes parameterized by mass and charge that exhibit richer causal structure, inner horizons, and distinct thermodynamic behavior. The solution has played a central role in studies by researchers associated with Roger Penrose, Stephen Hawking, and Roy Kerr on singularity theorems, cosmic censorship, and quantum effects near horizons.

Introduction

The solution emerged historically from separate lines of work by Hermann Reissner and Gunnar Nordström in the 1910s and was integrated into mainstream theoretical physics through analyses by David Hilbert and later commentators such as Robert Wald and Chandrasekhar. It is often taught alongside the Schwarzschild solution and the rotating Kerr metric as one of the simplest nontrivial exact solutions of Einstein–Maxwell theory. The spacetime depends on two parameters, typically denoted M and Q, which correspond to the ADM mass associated with Arnowitt–Deser–Misner methods and the electric charge measured at spatial infinity linked to Gauss's law for electromagnetism.

Metric and solution

In standard coordinates introduced by Karl Schwarzschild and deployed by later authors like Paul Dirac, the line element is expressed using the mass parameter M and charge parameter Q. The electromagnetic field is sourced by a static, spherically symmetric electric potential analogous to the Coulomb potential familiar from James Clerk Maxwell and studied by Michael Faraday. The metric reduces to the Schwarzschild metric when Q = 0 and to the Minkowski metric in the limit M → 0 and Q → 0, consistent with linearized gravity analyses by Albert Einstein. In variants one may adopt isotropic coordinates used in treatments by Eddington and Finkelstein to probe horizon-penetrating properties explored later by Brandon Carter.

Properties and singularities

The curvature invariants, such as the Kretschmann scalar often computed in texts by Misner, Thorne & Wheeler, reveal a physical singularity at the origin r = 0, analogous to the central singularity in the Schwarzschild black hole and studied in the singularity theorems of Roger Penrose. Unlike the neutral case, the presence of charge modifies tidal forces and the behavior of geodesics analyzed by George Ellis and Bernard Schutz. The electromagnetic stress–energy contributes to the energy conditions considered in proofs by Stephen Hawking and Alan Rendall, and influences global existence results examined by Yvonne Choquet-Bruhat.

Horizons and causal structure

Depending on the relative magnitudes of M and Q, the spacetime exhibits distinct horizon structures first discussed in causal studies by Kasner and formalized by Carter: when |Q| < M there are two horizons, an outer event horizon and an inner Cauchy horizon; the extremal case |Q| = M yields a degenerate horizon studied in the context of Bekenstein and Hawking; when |Q| > M no horizon shields the singularity, leading to a naked singularity that implicates the cosmic censorship conjecture advanced by Roger Penrose. The Penrose diagrammatic analysis used by Penrose and Hawking illustrates the causal regions, including trapped surfaces and regions with closed timelike curves in certain extended constructions analyzed by Israel.

Charged black hole thermodynamics

Thermodynamic properties of these charged solutions were central to the development of black hole thermodynamics by Jacob Bekenstein and Stephen Hawking, with the surface gravity and horizon area governing temperature and entropy relations analogous to the Laws of thermodynamics reformulated for gravitational systems. The extremal limit has zero surface gravity, raising questions about residual entropy examined by Gibbons and Strominger in semiclassical and string-theoretic contexts. Quantum field theory in curved spacetime pioneered by Unruh and Fulling yields Hawking radiation spectra dependent on charge, and backreaction studies by Page and York explore evaporation end states, including possible approach to extremality or remnant scenarios debated by Susskind and Preskill.

Extensions and related solutions

The Reissner–Nordström family has spawned numerous extensions and generalizations: rotating charged solutions such as the Kerr–Newman metric combine charge with angular momentum explored by Newman and Carter; higher-dimensional charged black holes appear in theories studied by Tangherlini and in supergravity contexts investigated by Strominger and Townsend; coupling to non-Abelian gauge fields yields colored black holes analyzed by Bartnik and McKinnon; string-theory inspired dilatonic charged solutions were developed by Garfinkle, Horowitz, and Strominger. Mathematical extensions include maximal analytic extensions constructed in the spirit of Kruskal coordinates and studied by Israel and Penrose, while numerical relativity investigations by groups associated with Pretorius and Lehner probe nonlinear stability and inner horizon instabilities related to mass inflation described by Poisson and Israel.

Category:Black holes