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

Self–gravitating stationary axisymmetric perfect fluids 7
condition (11) for the boundary value (16) is easily found to be
but (as was to be expected from the obviously inadequate Ansatz), (12)
is not satisfied:
on unless one of the following holds:
(a) (giving i.e. no fluid), or
(b) (static case).
Ignoring these two degenerate cases, there will be a discontinuity in the
gravitational force, expressed by (18); it would be interpreted as the
appearance of a surface layer at the fluid boundary, which artificially
constrains the rotating body to keep the spherical shape we had assumed.
By making virtue out of necessity, a similar behavior in the
general relativistic case is sometimes “explained” a posteriori by an interpretation
in terms of crusts, etc. But if we adopt a more conservative
point of view (namely, if we stick to the perfect fluid case we started
out with) such a situation is clearly undesirable and signals a failure in
satisfying all the conditions inherent to the problem. All this is a consequence
of the fact that we are not dealing with a Dirichlet-type problem
(nor even with a Cauchy-type one!), but with a free-boundary problem.
Unfortunately, the panoply of existing mathematical techniques for an
exact treatment of such problems is yet quite limited.
Finally, let us mention that various interesting rigorous results for
Newtonian configurations have been proven over the years. For lack of
space, we refrain from discussing them here and giving an exhaustive list
of references; please see [5] for further references, including those to deep
mathematical results that have appeared since the classic monograph [6]
was published.
3. INTERIOR SOLUTIONS WITH
DIFFERENTIAL ROTATION IN
GENERAL RELATIVITY
Clearly, all the difficulties associated with an exact treatment of rotating
perfect fluid configurations in the Newtonian approach are tremendously
increased in general relativity. As a matter of fact, it was only in
1968 that H.D. Wahlquist found for the first time an exact description
of the interior of a particular general relativistic axisymmetric perfect
fluid for the special case of stationary rigid rotation (vanishing shear of
the fluid) [7]. Several other particular interior solutions for barotropic