Reissner metric

Reissner metric
Name	Reissner metric
Coordinates	Schwarzschild coordinates
Metric	Static spherically symmetric solution of Einstein–Maxwell equations
Parameters	Mass M, charge Q
Solution	Reissner–Nordström solution

Reissner metric The Reissner metric is a static, spherically symmetric exact solution of the Einstein–Maxwell field equations describing the exterior spacetime of a nonrotating charged mass. Originating from independent work by Hermann Weyl and Gunnar Nordström earlier and completed in form by Hans Reissner and Gunnar Nordström, it generalizes the Schwarzschild metric by including electromagnetic charge and reduces to Minkowski space in the absence of mass and charge. The solution plays a central role in studies of black holes, singularities, and classical tests of general relativity.

Introduction

The Reissner metric arises within the framework of Einstein field equations coupled to Maxwell's equations for the electromagnetic field. It models the spacetime outside a spherically symmetric, charged, nonrotating body and is characterized by two parameters associated with the ADM mass and electric charge. Historically linked to the development of relativistic electrodynamics and the era of early exact solutions such as the Schwarzschild solution and the Kerr metric, the Reissner metric informed later work on cosmic censorship conjectures and singularity theorems by Roger Penrose and Stephen Hawking.

Derivation and form

Starting from a static, spherically symmetric ansatz in Schwarzschild coordinates, one solves the coupled Einstein–Maxwell equations using a diagonal metric with functions of the radial coordinate r. The resulting line element is expressed in terms of mass M and charge Q and contains the function f(r)=1 - 2GM/(c^2 r) + GQ^2/(4πε0 c^4 r^2) in SI units, analogous to the potential term in the Reissner–Nordström solution derivation. The electromagnetic field is described by the Coulomb potential consistent with Maxwell, and the stress–energy tensor feeds back into the curvature through the Ricci tensor and Einstein tensor. The derivation parallels methods used for the Schwarzschild solution and contrasts with the rotating Kerr–Newman metric where angular momentum appears.

Properties and horizons

The metric function f(r) determines causal structure and horizon locations, producing up to two coordinate horizons when parameters satisfy Q^2 < M^2, a degenerate extreme horizon at Q^2 = M^2, and naked singularities when Q^2 > M^2, relevant to the cosmic censorship conjecture discussed by Roger Penrose. The timelike Killing vector associated with stationarity and the spherical Killing vectors associated with SO(3) symmetry define conserved quantities; the event horizon and Cauchy horizon arise in the maximal extension, whose structure was analyzed in analogy with Reissner–Nordström Carter–Penrose diagrams and informed by studies of inner horizon stability by Brandon Carter and later perturbation work by Subrahmanyan Chandrasekhar. Surface gravity, Hawking temperature, and horizon area link the solution to laws of black hole thermodynamics developed by Jacob Bekenstein and Stephen Hawking.

Geodesics and motion

Geodesic motion in the Reissner spacetime yields trajectories for massive particles and null geodesics for photons, with conserved energy and angular momentum associated with Killing vectors. Radial free-fall and circular orbits are governed by effective potential methods similar to those used for Schwarzschild geodesics studied by Karl Schwarzschild and later analytic treatments by Chandrasekhar. Photon spheres and innermost stable circular orbits influence gravitational lensing and accretion disk phenomenology relevant to observations by facilities such as Event Horizon Telescope and satellite missions like Chandra X-ray Observatory. Charged particle motion couples to the electromagnetic potential and exhibits phenomena akin to radiative energy loss considered in classical work by J. J. Thomson and relativistic treatments tied to Lorentz force law formulations.

Physical interpretation and sources

Physically, the Reissner metric represents the exterior field of a charged, nonrotating compact object; however, astrophysical relevance is limited because large net charges are expected to be neutralized by surrounding plasma and processes described in studies by Enrico Fermi and plasma physics literature. As an idealized model it serves to probe electromagnetic contributions to curvature and to compare with charged solutions in alternative theories such as Brans–Dicke theory or Kaluza–Klein theory. Interior solutions matching to the exterior Reissner metric have been constructed for charged fluids and stars, extending classical stellar models initiated by Subrahmanyan Chandrasekhar and J. R. Oppenheimer and George Volkoff.

Extensions and limits

The Reissner metric embeds into more general families, reducing to Schwarzschild metric when Q→0 and relating to the Kerr–Newman metric when angular momentum is included. Extremal limits at Q^2 = M^2 produce zero-temperature horizons with enhanced symmetry studied in string theory contexts by groups around Edward Witten and Andrew Strominger. Quantum and semiclassical extensions invoke Hawking radiation and backreaction analyses pioneered by Stephen Hawking and Roger Penrose, while numerical relativity investigations of charged collapse connect to work by M. W. Choptuik and others studying critical phenomena.

Applications and significance

Beyond serving as a testbed for classical and quantum gravity, the Reissner solution informs theoretical questions about singularity structure, censorship, and horizon thermodynamics addressed by Hawking, Penrose, and Bekenstein. It provides analytic examples for gravitational lensing, wave propagation, and black hole perturbation theory employed in research by Vishveshwara and Teukolsky. In high-energy theory and string theory, extremal charged black holes derived from Reissner-like solutions underpin microstate counting and dualities explored by Strominger, Vafa, and Polchinski.

Category:Exact solutions in general relativity