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

Self–gravitating stationary axisymmetric perfect fluids 5
Note that equations (5)-(7) are equivalent to their following first integral:
where and are constants.
The rigidly rotating case is characterized by vanishing shear and expansion.
Such conditions reduce here to which in the
present case is equivalent to where a prime denotes differenti-
ation with respect to In other words, we have for
rigid rotation. Another interesting case is that of an irrotational rotation
hence, where
is a constant. In the irrotational case, which blows up
at the axis of rotation; however, one should bear in mind that regular
toroidal-like irrotational configurations may be possible, as the rotation
axis is outside the fluid in that case.
For positivity reasons, not all velocity profiles are allowed for a particular
equation of state; for instance, in the case of dust (pressureless
fluid), we have from (9): By substituting this
value of in (8), we obtain which excludes
differential rotations with if the natural requirement
is made. In particular, an irrotational dust is not allowed.
Finally, let us say a word about the matching conditions. Outside the
fluid, assuming a surrounding vacuum
to be solved reduce to
the equations
for the gravitational potential U. The matching conditions are derived
from the physical requirement that the body forces (in the present case,
the gravitational force) be continuous across the boundary:
The surface forces (let us call them t) should also be continuous across
the boundary. If where n is the normal to the boundary and
the stress tensor, the following has to hold:
As for vacuum, and for a perfect fluid, the three
conditions (13) reduce in the perfect fluid case to the single condition