Boundstates in a Massless Kerr-Newman Potential


Description of Problem and Solution

The Kerr-Newman solution to the Einstein Maxwell equations describes the spacetime of a charged and rotating black hole. The metric depends on the total mass, M, charge, Q, and angular momentum per unit mass, a, of the black hole. If we take such a solution and allow the mass to tend to zero, whilst keeping Q and a finite, the resulting spacetime is flat, but contains an electromagnetic field. This electromagnetic field is given by the expression:

In the above, a = (0,0,a). This is a coulomb potential with the origin shifted to the imaginary point r = ia. The potential represents a charge and current distribution confined to an earthed superconducting disc at z = 0 and r < a. The charge density in the interior of the disc is negative, and tends to minus infinity as the edge of the disc is approached. However, there is an infinite positive charge on the ring r = a, which means that the overall charge on the disc is positive and equal to q. The current is negative everywhere in the interior of the disc and becomes infinite as the edge is approached. However, on the edge of the disc there is an infinite positive current which means the overall magnetic moment is positive and equal to qa. If the current was to be generated by the movement of the surface charge, the velocity of the charge would have to increase linearly outwards, and reach the velocity of light at the ring singularity. The electric and magnetic fields are illustrated here.

This field has both charge and a magnetic moment. Dirac's equation describes the behaviour of particles with spin in an electromagnetic field. Hence the solution of the Dirac equation for this potential describes a quantum mechanical system in which there are two interacting particles with magnetic moments. There are no exact solutions of this type known, and the solution is of interest for this reason. It is also the case that the massless Kerr-Newman potential described here is the significant part of the interaction that an electron orbiting a micro black hole would feel. It is likely that micro black holes were created in the early universe, and may still be present today. A solution for the bound states of such a system predicts the spectral lines that would be expected from these black hole atoms, which may therefore be distinguishable from normal matter. The solution has applications to both of these problems.

The use of the Kerr-Newman potential enables a solution to the problem, since Chandrasekhar demonstrated that the Dirac equation separated for the full, massive Kerr-Newman potential. This is still true in the massless limit, but the coordinates in which it separates are oblate spheroidal coordinates rather than spherical polars.

As a sideline to the solution of the Dirac equation, I also considered the same problem for the Schrodinger and Klein-Gordon equations. The former is non-relativistic and neither equation accounts for the spin of the particle. For this reason it does not describe interacting objects with magnetic moments. However, the solution of these equations was a useful step towards the full Dirac solution and provides a comparison. The solution in both cases is computed numerically by using a combination of infinite continued fractions and numerical integration. In the Schrodinger case, it is possible to regard the potential as a perturbed coulomb potential. The perturbation is in fact infinite at the origin, and conventional perturbation theory does not work properly, which is interesting. A more careful treatment of the problem gave predicted perturbations in excellent agreement with the numerical values.

The solution to the Dirac equation is still in progress. The equations have been reduced to a solvable form. The main problem in this case is the boundary condition to impose at the disc. In the Schrodinger and Klein-Gordon cases, there is a single wave function which describes the probability density of the electron. Continuity through the disc then provides a natural boundary condition and divides the solution space into even and odd modes. In the Dirac case the wavefunction is described by a four component spinor, and the boundary condition is more difficult. Continuity of the probability density through the disc will probably provide the best boundary condition. As a model of a black hole atom, it is not clear that this boundary condition is appropriate, since in that case the event horizon provides a natural innner boundary. But the solution may still be applicable, and it is certainly relevant to the first problem.

Pekeris considered the same problem with the motivation of a black hole atom solution, and indeed he computed the bound state eigenvalues. However, he imposed a boundary condition that the solution should be regular at the imaginary point r = i a. This is analogous to the condition of regularity at the origin in the standard coulomb solution. There is some motivation from this through Born-Infeld electromagnetism. However, it does seem a somewhat arbitrary condition, since the imaginary point is only added to write the electromagnetic field in a nice form and so it seems unphysical to use it to derive the real space solution. The continuity of probability through the disc is more physically motivated and should provde a more physical solution.


References

My Research

  • Gair, J R, 2001, Bound states in a Kerr-Newman potential. Essay submitted for a Rayleigh-Knight prize at the University of Cambridge. Describes the solution of the Schrodinger and Klein-Gordon equations. This essay will be reworked as two papers for submission, probably to Phys. Rev. A. The essay can be downloaded by clicking here.

    Other Useful References

  • Chandrasekhar, S, 1976, Proc. Roy. Soc. Lond. A 349 571. The paper in which Chandrasekhar separates the Dirac equation for the Kerr potential.

  • Page, D N, 1976, Phys. Rev. D 14 1509. Page generalises Chandrasekhar's result for the Kerr-Newman solution.

  • Newman, E T and Penrose, R, 1962, J. Math. Phys. 3 566. The tetrad formalism on which Chandrasekhar's work was based.

  • Kinnersley, W, 1969, J. Math. Phys. 10 1195. Kinnersley derives the specific tetrad relevant for the work on the Kerr-Newman solution.

  • Pekeris, C L, 1987, Phys. Rev. A 35 14. A description of the properties of the massless Kerr-Newman electromagnetic field.

  • Pekeris, C L and Frankowski, K, 1989, Phys. Rev. A 39 518. The solution for this problem with boundary condition of regularity at the imaginary origin.


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