Reissner-Nordström metric

The Reissner-Nordström metric is the exact solution to Einstein-Maxwell equations describing a non-rotating, electrically charged black hole in general relativity. Discovered independently by Hans Reissner in 1916 and Gunnar Nordström in 1918, it generalizes the Schwarzschild solution to include electromagnetic charge, producing a black hole with two horizons — an outer event horizon and an inner Cauchy horizon — when the charge is sufficiently small relative to mass.

The metric reveals that electric charge fundamentally alters black hole geometry. Unlike the Schwarzschild case, the charged black hole interior is not a simple singularity surrounded by an event horizon. Instead, the Reissner-Nordström geometry contains a timelike singularity hidden behind two concentric horizons, with a region between them that can be traversed by infalling observers. This structure makes the Reissner-Nordström metric the simplest setting in which to study inner horizon physics, mass inflation, and the breakdown of predictability that motivates the chronology protection conjecture.

When charge equals mass in geometric units, the two horizons coincide and the black hole becomes extremal — a borderline case with zero temperature and maximal entropy density. Extremal black holes play a central role in string theory and the AdS/CFT correspondence, where they correspond to ground states of dual quantum field theories.