# Ernst equation

The Ernst equation

\mathfrak{R}\mathcal{E}\Delta\mathcal{E} = (\nabla\mathcal{E})^2

is equivalent to the stationary axisymmetric Einstein equations in vacuum. Solutions describe the gravitational field of black holes (correspond to solitonic solutions) and stars and galaxies in thermodynamical equilibrium.

#### 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) (below). 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.

#### Hyperelliptic disk solution (imaginary part)

The video shows the imaginary part of the Ernst potential for a family of disk solutions where the matter consists of two streams of counter-rotating dust (the real part can be seen in Hyperelliptic disk solution (real part)). We show the transition between the Newtonian limit (flat spacetime) to the extreme relativistic limit where the central redshift diverges. The situation is equatorially symmetric (the imaginary part is an odd function in 𝜁 ) and asymptotically flat (the imaginary part tends to 0 at infinity). The potential is smooth in the exterior of the disk and has a jump there.

### References

On the Ernst equation, see LN 29/2006 and references therein. On the Hyperelliptic disk solution (imaginary part), see nlin/0512065.