Surface Geometry of Some Meaningful Extreme Kerr-Newman Black Holes

Abstract

We address the properties of extremal black holes by considering the Christodoulou-Ruffini/Hawking mass-energy formula. By simple geometrical arguments, we found that the mass/energy formula is satisfied by two meaningful extremal black holes where mass (m), charge (Q), and angular momentum/spin (L) are proportional to the black holes irreducible mass (m_{ir}) expressed by irrational numbers. These black holes have been studied in the Christodoulou diagram and their topology in E^3 has been investigated by differential geometry. We show that one of the analyzed Kerr-Newman black holes corresponds to the case where the Gaussian curvature becomes zero at the poles. in the first case, the coefficients are linked to irrational numbers. In the second extremal black hole examined, the fundamental quantities m, Q, and L are linked to the irreducible mass by coefficients that depend solely on the "golden ratio number" -\phi_-. In this case, we show that if this extremal black hole satisfies the Pythagorean fundamental forms relation at the umbilic points, then both the scale parameter (corresponding to twice the irreducible mass) and the Gauss curvature of the surface at the poles are equal to the golden ratio numbers. For these two extremal black holes, we calculate the energy extractible by reversible transformations finding that, in percentage, the energy extractable from the latter black hole is higher than the former one.