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

6 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
The analytical expression for the boundary in the stationary axisymmetric
case with azimuthal flow is thus and on that
surface (11) and (12) should hold. Obviously, the dust case is special, in
that (14) does not characterize its boundary (14) is identically satisfied
throughout space!); in that case, the boundary is given by the surface(s)
where (11) and (12) are mutually consistent.
2. DIRICHLET BOUNDARY PROBLEM VS.
FREE-BOUNDARY PROBLEM
Part of the technical difficulties associated with the problem of finding
equilibrium configurations for self-gravitating fluids is due to the overdetermined
character of conditions (11) and (12); in fact, the boundary is
determined by the consistency of (11), (12), and (14). The classical
equilibrium figures we have mentioned in the previous section were all
obtained by means of Ansätze that turn out to be consistent (an important
role is played by the fact that the Newtonian potential for an
ellipsoidal body can be obtained in closed form in the case of constant
density objects, and the resulting potential does not depend on the motion
of the body, i.e. there are no Newtonian gravitomagnetic effects.)
In that sense, the free-boundary problem is not really solved ab initio
for those examples, but happy coincidences and insights are exploited.
In order to illustrate the fact that wild guesses are doomed to failure,
let us consider the following naive attempt at getting a rigidly rotating
configuration for a homogeneous (i.e. constant density) body. The
integral (9) gives
and substituting (15) in (8), we get Assume now
the (intuitively wrong!) Ansatz with By
integrating the previous equation for we find
where and are constants; if we set for regularity
at the pressure can be expressed as where
denotes the pressure at By substituting back in (15), we get
where is a constant.
Let be the surface Then, and
the boundary value of U is
where is the appropriate Legendre polynomial, and the usual spherical
coordinate. The exterior gravitational potential satisfying (10) and