KerrNewman

The Kerr–Newman solution is an exact, axially symmetric solution of Einstein’s field equations of general relativity coupled with Maxwell’s equations for electromagnetism. It describes the space–time geometry outside a rotating, charged, axially symmetric mass. The metric generalises the Kerr metric, which represents an uncharged rotating mass, and the Reissner–Nordström metric, which represents a non‑rotating charged mass. The Kerr–Newman metric is characterized by three parameters: the mass M, the specific angular momentum a = J/M, and the electric charge Q. It reduces to the Kerr metric when Q = 0, and to the Reissner–Nordström metric when a = 0.

The line element in Boyer–Lindquist coordinates (t, r, θ, φ) takes the form

ds² = −(Δ − a² sin²θ)/Σ dt² − 2a sin²θ (r² + a² − Δ)/Σ dt dφ + Σ/Δ dr² + Σ dθ² + sin²θ [(r² + a²)² − a² Δ sin²θ]/Σ dφ²,

where Δ = r² − 2Mr + a² + Q² and Σ = r² + a² cos²θ. The electromagnetic vector potential Aµ has the non‑zero components A_t = −Qr/(r² + a² cos²θ) and A_φ = Qar sin²θ/(r² + a² cos²θ). The horizons are located at radii r± = M ± √(M² − a² − Q²), provided the discriminant is non‑negative; otherwise the solution describes a naked singularity. Extremal black holes arise when M² = a² + Q².

The Kerr–Newman metric exhibits an ergosphere bounded by the outer static limit r = M + √(M² − a² cos²θ), within which all observers must co‑rotate with the black hole. Rotational energy can be extracted via the Penrose process or superradiant scattering. The charge introduces electromagnetic interactions, affecting charged particle orbits and the black hole’s thermodynamic quantities. The surface gravity κ, angular velocity Ω_H, and electrostatic potential Φ_H at the horizon are given by κ = (r+ − r−)/(2(r+² + a²)), Ω_H = a/(r+² + a²), and Φ_H = Q r+/(r+² + a²). These quantities enter the first law of black hole mechanics: dM = (κ/8π) dA + Ω_H dJ + Φ_H dQ. The Kerr–Newman solution therefore provides a foundational model for studying the interplay of gravity, rotation, and electromagnetism in strong‑field astrophysical contexts.