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

4 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
with a compact boundary are the Maclaurin ellipsoids in the stationary
axisymmetric case, and the Jacobi, Dedekind, and Riemann ellipsoids
in the (in general) non-axisymmetric case [1, 2]. It is remarkable that
the Jacobi ellipsoids, triaxial figures rotating around a principal axis
as a whole (rigidly) with constant angular velocity, can be considered
formally as stationary, due to the fact that the equations (1)-(3) can
be transformed to the rotating frame anchored to the principal axes so
that does not appear explicitly; obviously, the analog of such nonaxisymmetric
configurations in general relativity (whatever they might
be) would not be stationary, as they would emit gravitational radiation.
Another interesting stationary axisymmetric configuration is a torus-like
figure rotating around its symmetry axis. It was shown to exist (among
other highly interesting results) in a very important paper of Poincaré
[3], and was further analyzed in [4] by means of power series expansions.
In what follows, we shall concentrate on the stationary, axisymmetric
case with azimuthal flow of equations (1)-(4); the is taken to be
the symmetry axis. By imposing and (where
denotes the Lie derivative in the direction of the vectorfield
and being the azimuthal angle), one finds
where is the angular velocity,
depending in general on the position within the fluid (possible differential
rotation). Taking this into account, one has and
From these and the integrability conditions for the set of
three scalar equations arising from (2), one gets (taking into
account that and are functionally dependent, according to (4)), so
that the angular velocity depends only on the distance to the rotation
axis: where we have defined It is easy to see that
the continuity equation (1) is automatically satisfied under the assumed
symmetries (which also require and, in particular, the
fluid flow has vanishing expansion. Equations (l)-(3) now reduce to the
following ones: