Exact Solutions and Scalar Fields in Gravity - Instituto Avanzado de ...

Rotating equilibrium configurations in Einstein’s theory of gravitation 73
for the (complex) Ernst potential The metric can be calculated
from In the Newtonian limit, the Ernst potential becomes
The Newtonian boundary condition (11) on the disk has to be replaced
in Einstein’s theory by (see [1])
where denotes the Ernst potential in the corotating system and is
a real (negative) constant. The asymptotic condition has to be
replaced by at infinity. The Einsteinian solution depends on two
parameters as well. It turns out that a useful parameter combination is
given by
In terms of the two parameters µ and the solution is given by the
following expression [3]:
The upper integration limits and have to be determined from the
following Jacobi inversion problem:
where the functions and are given by