G.C. Williams

Matrix operator symmetries of the Dirac equation and separation of variables

The set of all matrix‐valued first‐order differential operators that commute with the Dirac equation in n‐dimensional complex Euclidean space is computed. In four dimensions it is shown that all matrix‐valued second‐order differential operators that commute with the Dirac operator in four dimensions are obtained as products of first‐order operators that commute with the Dirac operator. Finally some additional coordinate systems for which the Dirac equation in Minkowski space can be solved by separation of variables are presented. These new systems are comparable to the separation in oblate spheroidal coordinates discussed by Chandrasekhar [S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford U.P., Oxford, 1983)].

Superintegrability in three-dimensional Euclidean space

Potentials for which the corresponding Schrodinger equation is maximally superintegrable in three-dimensional Euclidean space are studied. The quadratic algebra which is associated with each of these potentials is constructed and the bound state wave functions are computed in the separable coordinates.

On superintegrable symmetry-breaking potentials in N-dimensional Euclidean space

We give a graphical prescription for obtaining and characterizing all separable coordinates for which the Schr?dinger equation admits separable solutions for one of the superintegrable potentials Here xn+1 is a distinguished Cartesian variable. The algebra of second-order symmetries of the resulting Schr?dinger equation is given and, for the first potential, the closure relations of the corresponding quadratic algebra. These potentials are particularly interesting because they occur in all dimensions n ? 1, the separation of variables problem is highly nontrivial for them, and many other potentials are limiting cases.

Killing–Yano tensors and variable separation in Kerr geometry

A complete analysis of the free‐field massless spin‐s equations (s=0, (1)/(2) ,1) in Kerr geometry is given. It is shown that in each case the separation constants occurring in the solutions obtained from a potential function can be characterized in an invariant way. This invariant characterization is given in terms of the Killing–Yano tensor admitted by Kerr geometry.

Recent advances in the use of separation of variables methods in general relativity

This review article is a guide to work that uses the method of separation of variables for problems that occur in general relativity. The main emphasis is on recent progress in the solution of important systems of equations such as Dirac’s equation, Maxwell’s equations and the gravitational perturbation (or spin 2) equations. Recent advances and established results for these equations in Kerr black hole and Robertson—Walker space-time backgrounds form the central theme of the discussion. These two important physical examples also illustrate some of the difficulties in a theory of solution by separation of variables methods for systems of equations. Other aspects of this subject such as solutions of the Rarita-Schwinger equation (spin 32) and the role of generalized Hertz potentials are also discussed.

Teukolsky–Starobinsky identities for arbitrary spin

The Teukolsky–Starobinsky identities are proven for arbitrary spin s. A pair of covariant equations are given that admit solutions in terms of Teukolsky functions for general s. The method of proof is shown to extend to the general class of space‐times considered by Torres del Castillo [J. Math. Phys. 29, 2078 (1988)].

Intrinsic Characterization of the Separation Constant for Spin One and Gravitational Perturbations in Kerr Geometry

In this article the characterization of the separation constant for the spin one and gravitational perturbations in Kerr geometry are given. The characterization is given for the corresponding Hertz potentials and induced for the corresponding vector and metric perturbations using the results of Cohen and Kegeles.

Symmetry operators and separation of variables for spin-wave equations in oblate spheroidal coordinates

A family of second‐order differential operators that characterize the solution of the massless spin s field equations, obtained via separation of variables in oblate spheroidal coordinates and using a null tetrad is found. The first two members of the family also characterize the separable solutions in the Kerr space‐time. It is also shown that these operators are symmetry operators of the field equations in empty space‐times whenever the space‐time admits a second‐order Killing–Yano tensor.

Separability of Wave Equations

The method of separation of variables has proved a useful tool with which to address various aspects of the physics of black holes. In this article we will review the progress made using this method in solving various systems of equations in black hole space-time backgrounds. The equations of important physical interest that we shall discuss consist primarily of the linear perturbation equations for various fields. In particular: Maxwell’s equations, Dirac’s equation and the Weyl neutrino equation and the gravitational perturbation equations. We will also discuss generalised Hertz potentials, the Rarita-Schwinger equation and the properties of Teukolsky functions.

Electromagnetic waves in Kerr geometry

The analogues of electric and magnetic multipoles for electromagnetic waves in a Kerr geometry are derived and given an operator characterization.