Discontinuities in the Evolution of Properties of Kerr Black Holes in the Extremal Limit

Ever since its discovery by Roy Kerr in 1963, the geometry around rotating, electrostatically-neutral black holes, otherwise known as Kerr black holes, has significantly contributed to theoretical developments in the fields of general relativity, thermodynamics and beyond. Extremal Kerr black holes, in which the spin parameter of the rotating black hole is at a maximum without breaching certain postulates, has especially been of interest to the research community due to its sensitivity to higher order corrections to Einstein's theory of relativity. In this paper, we review some unresolved open questions regarding the discontinuity of certain properties of the Kerr black hole as it approaches the extreme limit. These include the form of the innermost stable circular orbit (ISCO), the nature of vanishing entropy, and the disappearance of the trapped surface as a Kerr black hole approaches extremality. We conclude that current theories are likely not suitable for studies of extremal black holes, and we posit that further development in theory may likely require a quantum approach to gravity. G=c=1 and Einstein summation convention are assumed throughout this paper.