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Black Hole Theory Lecture 12: The Kerr-Newman Metric and Spinning Black Holes

In this lecture, I discuss the physical and geometric properties of spinning black holes. First, I compute the Ricci tensors for axisymmetric spacetimes using differential forms and computer algebra systems. However, solving these equations is difficult, as they are coupled and nonlinear. As a result, I show a more efficient algorithm, namely the tetrad formalism and the Newman-Janis algorithm. The idea is to use a seed metric and convert a spherically symmetric metric into an axisymmetric one. Subsequently, I talked about important surfaces for Kerr-Newman black holes and showed an animation of a cross-section of a spinning black hole depicting the evolution of the surfaces with spin. Then, I showed that spinning black holes have a ring singularity or a "ringularity" by computing Kretschmann scalar and showing the singularity region in oblate spheroidal coordinates. Finally, I sketched a proof of black hole uniqueness theorems by showing that there can be no scalar fields for static, asymptotically flat spacetimes. Disclaimer: At 22:58, I wrote the Kerr-Newman metric in Boyer-Lindquist coordinates slightly incorrectly. On the second term, there should be an additional factor of $\rho^2$. Sorry for this mistake! For the Mathematica code: https://github.com/ThePolyphysicsProj... Chapters: 00:00 - Introduction 00:29 - Outline of lecture 01:45 - Summary of last lecture 08:03 - Differential forms and computer algebra systems 08:45 - Axisymmetric metrics 09:18 - Orthonormal bases and Cartan's formula 10:52 - Motivation behind computer algebra systems 11:59 - Mathematica demonstration 18:47 - Tetrad formalism and null tetrads 20:11 - The hydrogen atoms of astrophysics! 20:32 - The Janis-Newman algorithm 21:24 - Computing the uu- and ur- components 22:24 - Kerr-Newman metric in advanced null coordinates 22:47 - Boyer-Lindquist coordinates and astrophysics 23:32 - The Penrose process 24:00 - Discussion of Hawking's are theorem 24:26 - The ergosphere 24:58 - The Cauchy horizons 25:31 - Black hole cross-section animation 26:17 - Kretschmann scalar of Kerr-Newman black holes 27:15 - Ring singularities or "Ringularities" 28:37 - Geometry of Kerr-Newman metric in oblate spheroidal coordinates 29:26 - Discussion of black hole uniqueness theorems 29:48 - A restricted proof of coupled scalar fields 35:01 - Physical implication of the proof 35:42 - Content of next lecture