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

12 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
It is an interesting question which one of the two discussed gauges (or
perhaps an intermediate one?) is more convenient for problems such as
the matching between interior and exterior solutions.
5. SOME GEOMETRIC FEATURES OF A
RIGIDLY ROTATING PERFECT FLUID
INTERIOR SOLUTION
It is well known that the analog of some standard Newtonian results
for rotating perfect fluids, such as the existence of an equatorial symmetry
plane, the spherical shape of a static configuration, or the fact that
straight lines parallel to the rotation axis intersect the two-dimensional
surface of the body on at most two points, are not yet fully proven in
general relativity. In order to get some intuition in this new regime, it
is worthwhile to analize some of the salient features of particular relativistic
solutions.
We are fortunate to have one such (rigidly rotating) interior solution,
due to D. Kramer [13] , whose main quantities can be expressed in
comparatively simple explicit analytical terms. Recently, we carried out
an analysis of some geometric properties of that solution [14]. It had
been already pointed out [13] that the body is oblate. By computing
the Gauss-Bonnet integral on the bounding 2-surface, we can conclude
that the object is topologically a sphere.
One interesting fact is that the Gaussian curvature of the 2-surface
becomes negative at the equator (and near it) for high enough rotation
rates, and in fact tends to an infinite negative value at the equator for
rotation rates approaching the maximum one allowed by the solution.
The length of parallels increases monotonically from pole to equator for
a certain range of the rotation parameter, but it presents a maximum
away from the equator when sufficiently high rotation rates are considered;
when the terminal rotation speed is reached, the length along the
equator vanishes. Our intuition with Euclidean geometry would lead us
to believe that a fission of the body takes place for the maximal speed
of rotation; but a more detailed investigation of the three-dimensional
geometry within the body (for constant time) shows that the geodesic
distance between a point in the boundary and the axis of rotation increases
as the point moves from the pole to the equator along a meridian
(for any rotation rate). Geodesic distances from the center to a point
in a meridian exhibit a similar monotonically increasing behavior, when
the point varies from one pole to the equator (all these conclusions are
made technically possible by the fact that in the Kramer solution the
geodesics can be expressed in terms of quadratures) [14]. Finally, it can