After the original discovery of the Kerr metric, Newman and Janis showed that this solution could be “derived” by making an elementary complex transformation to the Schwarzschild solution. The same method was then used to obtain a new stationary axisymmetric solution to Einstein's field equations now known as the Kerr–Newman metric, representing a rotating massive charged black hole. However no clear reason has ever been given as to why the Newman–Janis algorithm works, many physicist considering it to be an ad hoc procedure or “fluke” and not worthy of further investigation. Contrary to this belief this paper shows why the Newman–Janis algorithm is successful in obtaining the Kerr–Newman metric by removing some of the ambiguities present in the original derivation. Finally we show that the only perfect fluid generated by the Newman–Janis algorithm is the (vacuum) Kerr metric and that the only Petrov typed D solution to the Einstein–Maxwell equations is the Kerr–Newman metric.

Uniqueness of the Newman-Janis algorithm in generating the Kerr-Newman metric

Suppose that we havethe unit Euclidean ball in Rn and construct new bodies using three op- erations — linear transformations, closure in the radial metric, and multiplicative summation defined by kxkK+0L = p kxkKkxkL. We prove that in dimension 3 this procedure gives all origin-symmetric convex bodies, while this is no longer true in dimensions 4 and higher. We introduce the concept of embedding of a normed space in L0 that naturally extends the corresponding properties of Lp-spaces with p 6 0, and show that the procedure described above gives exactly the unit balls of subspaces of L0 in everydimension. We provideFourier analytic and geometric characterizations of spaces embedding in L0, and prove several facts confirming the place of L0 in the scale of Lp-spaces.

The geometry of $L_0$

The set of pure states of an ordinary quantum-mechanical system has the structure of a Kahler manifold. As in classical mechanics, a vector field, called Hamiltonian, on this manifold is associated to the system. The equations induced by quantum-mechanical Hamiltonian vector fields are Schrodinger equations. A complete characterization of these vector fields is given and a quantum Liouville theorem is obtained.

Hamiltonian vector fields in quantum mechanics

This is the first of a series of papers on complex spaces and their use in complex relativity. The basic aim is to develop the theory of complex relativity but only insofar as it helps in obtaining, and understanding, real solutions of Einstein's vacuum equations as slices of complex solutions. In this first paper, specific aims of the whole series are first presented. The basic equations and key entities, which are used later, are presented. The basic relativistic language used is that of Newman and Penrose. Included is a discussion of a number of important properties which arise in the development of the basic equations and key concepts, these properties being mainly ones which are not apparent in standard real formulations.

Complex relativity and real solutions. I: Introduction

Theoretical and experimental studies of Berry and Pancharatnam phases are reviewed. Basic elements of differential geometry necessary for understanding the topological nature of these phases are presented. The basic theory analysed by Berry in relation to magnetic monopoles is presented. The theory is generalized to nonadiabatic processes and to noncyclic Pancharatnam phases. Different systems are discussed including polarization optics, n-level atomic systems, neutron interferometry and molecular topological phases.

https://iopscience.iop.org/article/10.1088/1464-4266/6/4/R01/pdf

Berry and Pancharatnam Topological Phases of Atomic and Optical Systems

Flaherty, E. J., Hermitian and Kählerian Geometry in Relativity, Berlin‐Heidelberg‐New York. Springer‐Verlag. 1976. VIII, 365 S., DM 32,–. US $ 13.20. (Lecture Notes in Physics 46)

H. G. Schöpf

Papers

Flaherty, E. J., Hermitian and Kählerian Geometry in Relativity, Berlin‐Heidelberg‐New York. Springer‐Verlag. 1976. VIII, 365 S., DM 32,–. US $ 13.20. (Lecture Notes in Physics 46)