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

8 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
fluids rotating rigidly have been subsequently found. Unfortunately,
none of them has been shown so far to possess a corresponding asymptotically
flat exterior solution, in such a way that the overdetermined
matching conditions (which in general relativity are the continuity of
the metric and of its first derivatives across the boundary of the object)
are satisfied. (Obviously, we are disregarding stationary axisymmetric
configurations with extra Killing vectors, such as the case of cylindrical
symmetry, which is irrelevant for astrophysical objects.)
The first stationary axisymmetric interior solution in general relativity
for a barotropic perfect fluid with differential (i.e. non-rigid) rotation
and only two Killing vectors appeared in 1990 [8]; the case considered
there was one of vanishing vorticity (irrotational case). Let me now
describe briefly the formalism we developed for finding that solution
(the formalism itself was published later in detail in a separate, larger
paper [9]).
An orthonormal tetrad of 1-forms, is introduced, with
the choice (where is the velocity 1-form of the fluid). The
presence of two commuting Killing fields and such that they can
be identified with and is assumed (where is a time
coordinate and a cyclic one, and denotes the contravariant expression
corresponding to the 1-form etc.). The flow is assumed to be
azimuthal: If we are interested in non-trivial situations, we
have to make sure (perhaps a posteriori, after some solution is found)
that we are not dealing with a static case, by checking that there does
not exist a timelike Killing vector such that For the reasons
given above, we find it preferable that there be no additional independent
Killing vectors (especially if the extra Killing field commutes with
and which could be interpreted as giving rise to a cylindrical or
some such physically undesirable symmetry). By a standard reduction,
and are chosen as lying in the subspace. Due to the
symmetries, the coordinates and are ignorable, and the coefficients
of the tetrad 1-forms (and of all the differential forms that appear in
the formulation) may only depend on two additional coordinates. The
energy-momentum tensor is taken to be where
is the spacetime metric, the velocity 1-form, the pressure, and
µ the energy density of the fluid. Hodge duality in the subspace
is defined by and It is an important feature
of the system under consideration that one can write and
where is the shear tensor and the vorticity
2-tensor; (“deformity”) and (“roticity”) are 1-forms which carry the
information for the shear and the vorticity, respectively. With a being
the acceleration of the fluid, a connection 1-form in the