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 \( M\cos\left(\psi /2\right) \), surface gravity, angular velocity \( \tan\left(\psi /2\right)/2M \) 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 \( 8M\cos\left(\psi /2\right)E\left(\sin^{2}\frac{\psi }{2}\right) \). Its Gaussian curvature becomes negative on polar caps for \( \chi \gt \sqrt{3}/2 \), with area fraction \( 1-\tan{30}^{\circ }/\tan\left(\psi /2\right) \). The Kretschmann scalar on the horizon becomes negative on polar caps if and only if \( χ>1/2 \), with area fraction \( 1-\tan{15}^{\circ }/\tan\left(\psi /2\right) \), which reaches \( \sqrt{3}-1 \) at extremality. The proper volume of the ergoregion grows linearly, \( 8{\pi }^{2}M^{3}\chi \), at small spin, although the extractable energy grows only quadratically. It diverges as \( 2\pi \left(\sqrt{2}+\mathrm{arsinh}1\right)M^{3}\ln\left[1/\left(1-{\chi }^{2}\right)\right] \) at extremality. All results are verified symbolically and numerically.