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

10 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
The irrotational solution presented in [8] was found by imposing
the Ansatz
A word is in order concerning that choice. It was inspired by the fact
that, for a Newtonian fluid rotating rigidly, one has (with the
appropriate definitions of and in the underlying Euclidean metric);
it seemed then natural to impose (34) as a simplifying assumption in
the irrotational case, with the role of taken over by Although this
Ansatz did work as an aid in finding the solution, we should bear in mind,
however, that is satisfied due to the fact that in the Newtonian
case the vorticity lines are straight lines, whereas in a relativistic fluid
one would expect a dynamical (multipole-like) distribution for or
In order to arrive at the solution in [8], a further Ansatz of a more
mathematical nature was imposed; the end result was that all the quantities
of the solution, including the metric, can be written in terms of
a single function ( means here a coordinate!), which satisfies the
second-order ordinary differential equation
where is a constant, and subindices denote ordinary derivatives. Equation
(35) can be transformed to a first-order one, which turns out to be
an Abel equation of the first kind. The (non-constant) angular velocity
distribution for the rotating fluid is given by
where and are constants. The equation of state is that of stiff matter,
and the solution is of Petrov type I. The surface area of the body
is finite. The solution admits no further Killing
vectors.
A few other differentially rotating interior solutions have subsequently
been found [10]. Needless to say, none of the differentially rotating
interior solutions with no extra Killing fields has yet been shown to admit
a matching asymptotically flat exterior vacuum field, a fact similar to
that mentioned above for rigidly rotating solutions.
4. VACUUM AS A “PERFECT FLUID”. THE
“RIGID ROTATION” AND THE
“IRROTATIONAL” GAUGES
Equations (19)-(32) still hold when (and (33) becomes
a trivial identity.) One important difference, however, is the following.