Note on the form of the metric for an isolated vortex in general relativity

Calling a metric semidiagonal if it has a single off-diagonal element g t, we show that the stationary interior solution for a cylindrically symmetrical perfect fluid possessing an angular momentum cannot have a semidiagonal metric, unless the motion of the fluid particles is purely rotational around the axis of symmetry. A discussion is given of the relativistic spherical vortex in flat space, with a view of seeking a solution for such an isolated vortex in which gravitation is not neglected.

Solution of the coupled Einstein-Maxwell equations in oblate spheroidal coordinates

Using Ernst's theory of complex potentials, a solution of the coupled Einstein-Maxwell equations in oblate spheroidal coordinates is obtained for a source possessing mass, electric charge, and angular momentum. The density of the electromagnetic energy is evaluated. Explicit expressions are derived for the components of the Ricci tensor Rmunu, and of the electromagnetic energy momentum tensor Emunu, as well as for Enumu.

Solution of Dirac's equation in Reissner-Nordström geometry

The nature of the singularity at the origin of the solution of Dirac's equation in Reissner-Nordström geometry is different from that in flat space. Here, both independent solutions are regular at the origin, and both give a convergent normalization integral. The choice between the two solutions is made by requiring that the variational integral, from which the differential equation can be derived, be convergent. The gravitational shift of the ground state of hydrogen in Reissner-Nordström geometry is estimated to be only of order 10(-36) MH.

Note on four-particle wave functions

Dougall's expansion is used in the expression for the volume element in tetrahedral coordinates, thereby lifting the spherical triangle restriction on the angles at the tetrahedral vertex. Application is made to the evaluation of matrix elements for the ground state of three-electron atoms.