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

18 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
considering back then. Finally, we stated our new generalized Geroch
conjecture, which involved an unspecified subset of the set of permissible
Identifying this subset only came in Sec. 6 of our paper, where we
showed that the members of the subset could be determined in terms of
a simple differentiabilty criterion imposed on the Abel transforms of the
initial data on two intersecting null lines.
Before we ever got to this stage in the proof, it was necessary, in Sec.
2, to derive an Alekseev–type singular integral equation, and then show
that, making appropriate differentiability assumptions, the HHP and
this Alekseev–type were equivalent. Then, in Sec. 3, we derived from
the Alekseev–type equation a Fredholm integral equation of the second
kind, and showed that, making appropriate differentiability assumptions,
the Alekseev–type equation and this Fredholm equation were equivalent.
In Sec. 4 we employed the Fredholm alternative theorem to prove the
existence and uniqueness of a solution of the Fredholm equation, the
Alekseev–type equation and the HHP. In Sec. 5 we studied the partial
derivatives of the various potentials that are constructed from the HHP
solution.
In Sec. 6, we found that we could identify the subset of permissible
by demanding at least differentiability for the generalized
Abel transforms of the initial data, We then showed that the HHP
solution was a member of the selected subset of The generalized
Geroch group was thus identified.
Finally, we see that the optimism born of Geroch’s speculation of
1972 was well-founded, not only for the elliptic Ernst equation but for
the hyperbolic Ernst equation as well. I am pleased that this work could
be completed during the lifetime of my 77 year old colleague, Isidore
Hauser, who has contributed so much to proving mathematically that
which everyone else seems to think needs no proof other than a little
hand–waving.
3. THE PERFECT FLUID JOINING
PROBLEM
During the last two years he and I have developed, using our homogeneous
Hilbert problem formalism, a straightforward procedure [9] for
determining the axis values of the complex that describes a
not necessarily flat vacuum metric joined smoothly across a zero pressure
surface to a given perfect fluid (or other type of) interior solution.
Of course, once the axis values are known, one can use Sibgatullin’s formalism
to generate the complete exterior field, the study of which may