Abstract
Black holes (BH) are perhaps the most interesting prediction of General Relativity (GR).They arise as simple static or stationary metric solutions to the Einstein field equations of the geometry outside a massive object. Their simplicity derives, in part, from the fact that any BH can be described solely by three parameters, mass, global charge, and spin, irrespective of the details of their formation. This statement is somewhat formally encoded in Bekenstein’s so-called "no-hair" theorem. However, certain caveats to this theorem have been recently exploited in the context of an EBH in work completed by Aretakis and others. This thesis seeks to study the existence and nature of Aretakis charge and its potentially observable imprint at a finite distance from the horizon (Ori coefficient) in near-extremal black hole backgrounds. Specifically, the time evolution of horizon penetrating scalar and gravitational perturbations is considered, with compact support on near-extremal Reissner-Nordström and Kerr. This is accomplished by numerically solving the Teukolsky equation and determining the Aretakis charge values on the horizon and at a finite distance from the black hole. This work proves that these values are no longer strictly conserved in the nonextremal case; however, their decay rate can be arbitrarily slow as the black hole approaches extremality allowing for the possibility of their observation as a transient hair.