Hyperelliptic disk solution (real part)

Solutions to the Ernst equation in terms of multi-dimensional theta functions can describe the exterior gravitational field of stars and galaxies in thermodynamical equilibrium. Infinitesimally thin disks are discussed in astrophysics as models for disk-like galaxies. The video shows a family of disk solutions where the matter consists of two streams of counter-rotating dust. We use compactified coordinates to represent the whole spacetime. The video shows the transition between the Newtonian limit (flat spacetime) to the extreme relativistic limit where the central redshift in the disk diverges.  The real part of the Ernst potential can be seen above, the corresponding imaginary part in Hyperelliptic disk solution (imaginary part). The situation is equatorially symmetric (the real part is an even function in ζ ) and asymptotically flat (the real part tends to 1 at infinity). The potential is smooth in the exterior of the disk and just continous there. See LN 29/2006 for details.

J. Frauendiener and C. Klein