Extreme Kerr Newman Black Holes: Differential Geometry, Symmetry, and the Golden Ratio

Abstract

We investigate the geometry of extreme Kerr-Newman black holes and its role in selecting distinct black hole states. Motivated by Smarr's question of what physical information is encoded in event-horizon symmetries, we identify, besides the Smarr extreme family, a special family in which the mass, charge, angular momentum, and irreducible mass are constrained by a single irrational number. In the Christodoulou diagram, this family is selected by a discrete symmetry of appropriately scaled charge and angular-momentum variables. The local differential geometry of the event horizon provides an additional selection principle, which, up to the discrete symmetries (Q \rightarrow\pm Q) and (J \rightarrow\pm J), identifies a unique extreme Kerr-Newman black hole exhibiting the highest degree of local spherical symmetry compatible with the rotating Kerr-Newman geometry. Remarkably, for this distinct extreme configuration, all relevant physical and geometrical quantities, including the energy, electric charge, angular momentum, and irreducible mass, are solely related through the golden ratio. We also examine reversible and irreversible transformations and the associated extractable energy. Intrinsic horizon geometry can therefore constrain the macroscopic parameters and single out distinct extreme Kerr-Newman configurations.