Reissner–Nordström metric

Reissner–Nordström metric
Name	Reissner–Nordström metric
Field	General relativity
Discovered	1916–1918
Discoverer	Hans Reissner; Gunnar Nordström

Reissner–Nordström metric The Reissner–Nordström metric is an exact solution of the Einstein field equations describing the exterior spacetime of a spherically symmetric charged mass. It generalizes the Schwarzschild metric by including an electromagnetic field sourced by the Maxwell's equations solution for a point charge, and it played a foundational role in studies by Karl Schwarzschild, Albert Einstein, Hermann Weyl, David Hilbert and contemporaries in early relativistic gravitation. The solution has influenced investigations in black hole thermodynamics, quantum field theory in curved spacetime, string theory, loop quantum gravity and tests of classical no-hair theorem variants.

Introduction

The metric was derived independently by Hans Reissner and Gunnar Nordström during the 1910s after the publication of Einstein field equations; it incorporates the electromagnetic field tensor from James Clerk Maxwell's theory and the stress–energy contributions studied by Hermann Minkowski and Richard Feynman in electromagnetism. It complements other exact solutions such as Reissner metric precursors, the Kerr metric for rotating masses developed by Roy Kerr, and the charged rotating Kerr–Newman metric found by researchers influenced by Brandon Carter and Ted Newman. The solution has been central to discussions involving singularities following analyses by Roger Penrose, Stephen Hawking, and investigations into cosmic censorship proposed by Roger Penrose and debated by Shing-Tung Yau and others.

Metric and coordinate forms

In standard Schwarzschild-like coordinates (t, r, θ, φ) the line element is often written analogously to forms used by Karl Schwarzschild and later by Weyl: g_tt = 1 - 2M/r + Q^2/r^2 and g_rr = (1 - 2M/r + Q^2/r^2)^{-1}, where parameters M and Q relate to mass and charge as in treatments by ADM formalism contributors such as Richard Arnowitt, Stanley Deser, Charles Misner, and later expositions by John Wheeler. Alternative coordinate systems used in analyses include isotropic coordinates employed by Arthur Eddington and George Temple, advanced and retarded null coordinates analogous to those introduced by Paul Painlevé and Allan Gullstrand, and double-null coordinates used in studies by Demetrios Christodoulou and Piotr Chruściel. The metric couples to the electromagnetic four-potential often chosen in gauge conventions referenced by Julian Schwinger and Hans Bethe; the Maxwell field can be expressed using forms popularized by Élie Cartan and later geometric formulations by Roger Penrose.

Properties and causal structure

The solution exhibits horizons determined by roots of Δ(r) = r^2 - 2Mr + Q^2, studied in the context of horizon theorems by Stephen Hawking and James Hartle. Depending on the relation between |Q| and M, one obtains nonextremal black holes with two distinct horizons, an extremal case with a degenerate horizon investigated by Gibbons and Gary Horowitz, or a naked singularity when |Q| > M—this naked case fed arguments in the cosmic censorship conjecture debated by Roger Penrose and Demetrios Christodoulou. The causal structure, represented in Penrose diagram analyses refined by Brandon Carter and James B. Hartle, reveals regions analogous to those in the Maximal analytic extension of Schwarzschild spacetimes and includes interior timelike singularities examined by S. Chandrasekhar and Subrahmanyan Chandrasekhar. The global structure has been compared with the Reissner–Nordström–de Sitter and Reissner–Nordström–anti-de Sitter families explored in cosmological contexts by Alexander Friedmann, Georges Lemaître, and Willem de Sitter.

Geodesics and particle motion

Test particle motion in this spacetime follows geodesic equations derived using variational principles popularized by Noether and techniques employed by Lev Landau and Evgeny Lifshitz. Timelike and null geodesics show perihelion precession and light deflection analyzable with methods credited to Christian Doppler predecessors and classic treatments by Einstein and Kip Thorne. Charged particle motion requires coupling to the Lorentz force from James Clerk Maxwell and quantum corrections considered by Julian Schwinger and Gerard 't Hooft. Circular orbits, innermost stable circular orbits (ISCO), and scattering cross-sections have been computed in analogy with work on the Kerr metric by Bardeen, Press, and Teukolsky. Analyses of wave propagation and quasinormal modes reference techniques from Vladimir Belinski, Frans Pretorius, and studies motivated by gravitational-wave detections by LIGO Scientific Collaboration.

Thermodynamics and horizons

Black hole thermodynamics for charged solutions extends the four laws formulated by James Bardeen, Brandon Carter, and Stephen Hawking; surface gravity, horizon area, and electromagnetic potential enter relations analogous to those in the Bekenstein–Hawking entropy framework developed by Jacob Bekenstein and Stephen Hawking. The extremal limit has zero temperature features discussed in string-theory microstate counting by Andrew Strominger and Cumrun Vafa and studied in semiclassical approaches by Don Page and Gerard 't Hooft. Euclidean methods used by Gibbons and Hawking evaluate partition functions and contribute to path-integral derivations linked to proposals by John Preskill and Leonard Susskind.

Extensions and related solutions

Extensions include charged cosmological variants like Reissner–Nordström–de Sitter and anti-de Sitter cousins relevant to the AdS/CFT correspondence developed by Juan Maldacena and generalized charged rotating metrics such as Kerr–Newman metric whose discovery involved Ted Newman and collaborators. Higher-dimensional charged black hole solutions appear in work by Gary Horowitz and Roberto Emparan and in supergravity constructions studied by Stuart Samuel and M. J. Duff. Solutions with nonlinear electrodynamics sources relate to investigations by Mariano Cataldo and Eloy Ayón-Beato; quantum-corrected and regular black hole models draw on ideas from Bardeen and later proposals by Ayon-Beato and Marcelo Cataldo. Mathematical classifications of electromagnetic vacuum solutions follow formalisms by Élie Cartan, Petrov and the algebraic schemes of Andrzej Trautman.

Category:Exact solutions in general relativity