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

Self–gravitating stationary axisymmetric perfect fluids 11
Now is no longer related to the velocity of a fluid, so one is
free to redefine and allowing for a hyperbolic rotation. It is an
interesting technical point that, under this additional gauge, it is always
possible to make either or vanish (the integrability conditions
for doing so are satisfied) [9]. Let us briefly examine the case
(for obvious reasons, we shall call it the rigid-rotation gauge.) We can
choose functions and such that
( and can be interpreted as
quasi-cylindrical coordinates, while is the twist potential). The Ernst
potential [11] is simply which satisfies the Ernst equation
The relation among the Ernst
formulation and the 1-forms discussed in the previous section is given
by
Let us note in passing that the symmetry inherent in the expressions
given above for the 1-forms and their duals can be implemented by
means of the Kramer-Neugebauer transformation [12]:
A more unconventional formulation for the vacuum case may be obtained
by means of the alternative “irrotational” gauge [9]: Now we have
and we find
for appropriate and In analogy with the Ernst
formulation, we can introduce which satisfies
The analog of (37) is
while a discrete symmetry transformation, analogous to the Kramer-
Neugebauer one, is now [9]
By examining the equations for the rigid-rotation and irrotational gauges,
it is seen that both pictures can be related by the transformation [9]
or, in terms of functions rather than 1-forms,
where denotes the function associated with the roticity, while is
the corresponding one for the deformity.