Mathematical Analysis of Spacetime Geometry Around Rotating Black Holes

How does the spin of a black hole shape its horizon, its ergosphere and the surrounding spacetime geometry? Working with the Kerr metric and elementary differential geometry, we derive explicit relations between the spin \( χ=a/M \) and specific geometric characteristics. Writing \( χ=sinψ \) turns the horizon radii, area, irreducible mass \( Mcos(ψ /2) \), surface gravity, angular velocity \( tan(ψ /2)/2M \) and photon orbits into half-angle formulas, and critical spins become simple angles. The horizon keeps its equatorial circumference \( 4πM \) for every spin, while its polar circumference is \( 8Mcos(ψ /2)E(sin²ψ /2) \). Its Gaussian curvature becomes negative on polar caps for \( χ > √3/2 \), with area fraction \( 1-tan{30}∘/tan(ψ /2) \). The Kretschmann scalar on the horizon becomes negative on polar caps if and only if \( χ>1/2 \), with area fraction \( 1-tan{15}∘/tan(ψ /2) \), which reaches \( √3-1 \) at extremality. The proper volume of the ergoregion grows linearly, \( 8{π }²M³χ \), at small spin, although the extractable energy grows only quadratically. It diverges as \( 2π (√2+arsinh1)M³ln[1/(1-{χ }²)] \) at extremality. All results are verified symbolically and numerically.