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

New Directions for the New Millennium 17
Our discovery of the generalized Alekseev equation permitted us to
identify a transformation group that is significantly larger than that associated
with Kinnersley–Chitre transformations. This involved, among
other things, showing that, under appropriately chosen premises, solving
the generalized Alekseev equation was equivalent to solving a certain
Fredholm equation of the second kind. The complete proof [8] is rather
lengthy.
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This proof is divided into six sections:
Statement of our generalized Geroch conjecture
Alekseev–type singular integral equation equivalent to our HHP
Fredholm integral equation equivalent to Alekseev-type equation
Existence and properties of the HHP solution
Derivatives of and H
Proof of our generalized Geroch conjecture
It should be emphasized that, in order to formulate properly a generalized
Geroch conjecture, years of work were required before we even
arrived at the formulation that we spelled out in Sec. 1. This started
with a complete analysis of the Weyl case, where all objects we used
could be worked out explicitly.
In developing this proof, we had to be very careful to specify the
domains of all the potentials we were employing, whereas in the proofs
that we developed for the Geroch conjecture itself in the early 1980s,
the assumption of an sufficed to guarantee analyticity.
Initially we employed in this more recent work quite general domains for
our but it turned out that nothing was gained thereby, and
we eventually elected to use certain diamond shaped regions bounded by
null lines. In certain cases these diamond shaped regions were truncated
when one reached the analog of the “axis” in the case of the
elliptic Ernst equation. In the case of the hyperbolic Ernst equation,
is where colliding wave solutions typically exhibit either horizons
or singular behavior.
We employed one of many linear systems for the Ernst equation. It
involved a 2 × 2 matrix field dependent upon the spectral parameter
The domain of this field was specified with equal care. Then we identified
a set of permissible and stated a homogeneous
Hilbert problem on arcs, analogous to the HHP that we had formulated
in 1980 on a closed contour to handle the far simpler problem we were