Discussion of the Theta Formula for the Ernst Potential of the Rigidly Rotating Disk of Dust

Abstract

The exact global solution of the rigidly rotating disk of dust[1] is given in terms of ultraelliptic functions. Here we discuss the “theta formula” for the Ernst potential[2]. The space-time coordinates of the problem enter the arguments of these functions via ultraelliptic line integrals which are related to a Riemann surface.

The solution is reformulated so as to make it easier to handle and all integrals are transformed into definite real integrals. For the axis of symmetry and the plane of the disk these general formulae can be reduced to standard elliptic functions and elliptic integrals.