Kerrmetrik

The Kerr metric, also known as Kerr spacetime, is a solution to Einstein's field equations in vacuum that describes the spacetime surrounding a rotating, uncharged mass. It was discovered by Roy Kerr in 1963 and generalizes the Schwarzschild solution by incorporating angular momentum. The metric is stationary and axisymmetric, characterized by two parameters: mass M and angular momentum per unit mass a = J/M. In Boyer–Lindquist coordinates (t, r, θ, φ) the line element can be written (in geometric units where G = c = 1) as

ds^2 = - (Δ − a^2 sin^2 θ)/Σ dt^2 − (2a sin^2 θ (r^2 + a^2 − Δ))/Σ dt dφ + (Σ/Δ) dr^2 + Σ dθ^2 + sin^2 θ/Σ [ (r^2 + a^2)^2 − a^2 Δ sin^2 θ ] dφ^2,

where Δ = r^2 − 2Mr + a^2 and Σ = r^2 + a^2 cos^2 θ. The metric reduces to the Schwarzschild form when a = 0.

Key features include two Killing vector fields corresponding to time translations and rotations about the axis of symmetry, reflecting its stationary and axisymmetric nature. Horizons arise from Δ = 0, giving r± = M ± sqrt(M^2 − a^2). A black hole horizon exists for |a| ≤ M; if |a| > M the solution describes a naked singularity. The Kerr geometry also contains a ring singularity at r = 0, θ = π/2, and an ergosphere boundary where g_tt = 0, outside of which energy and angular momentum can be extracted via processes such as the Penrose mechanism.

In astrophysics, the Kerr metric models rotating black holes formed from stellar collapse and is used to study accretion disks, relativistic jets, frame dragging, and gravitational wave signatures, making it central to tests of general relativity in strong gravity.