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16 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
2. THE QUEST FOR THE GEROCH GROUP
AND ITS GENERALIZATIONS
In 1972 Robert Geroch [2] speculated that there might exist a transformation
group acting upon an infinite hierarchy of complex potentials,
using which one might be able to transform any one stationary axisymmetric
spacetime into any other such spacetime. William Kinnersley and
D. M. Chitre [3] painstakingly worked out the detailed structure of the
infinitesimal elements of Geroch’s group (actually an electrovac generalization
of Geroch’s group). By summing series they were able to deduce
the Kerr metric in several ways. Similarly they were able to generalize
certain solutions of Tomimatsu and Sato. Their most important contribution,
however, was that they demonstrated convincingly that Geroch
was on the right track.
A number of people, among whom were Isidore Hauser and the author
[4], observed that the so–called “Ernst equations” were associated with
a “linear system.” With this observation the mathematical machinery
of Riemann–Hilbert problems could be brought to bear upon vacuum
and electrovac spacetimes with two commuting Killing vectors. This had
ramifications both for practical calculation and for development of proofs
that such Geroch groups actually existed in a well–defined mathematical
sense.
In the early 1980’s, Hauser and Ernst succeeded in using the homogeneous
Hilbert problem (HHP) formalism to prove Geroch’s conjecture
and its electrovac generalization [5]. This involved showing that, under
appropriately chosen premises, solving the HHP was equivalent to
solving a certain Fredholm integral equation of the second kind. Development
of the proof was facilitated by the fact that solutions of the
elliptic Ernst equation are analytic if they are
From the Hauser–Ernst homogeneous Hilbert problem formalism for
stationary axisymmetric fields was derived the integral equation formalism
of Nail Sibgatullin [6], which has been used very effectively by
Vladimir Manko here at CINVESTAV to generate a large number of
exact stationary axisymmetric solutions.
Georgii Alekseev [7] observed that if one admits solutions that exist
nowhere on the symmetry axis, the Kinnersley–Chitre transformations
do not tell the whole story. He found that a certain nonsingular integral
equation describes the larger picture. Much later, Isidore Hauser and
I found in our study of spacetimes admitting two commuting spacelike
Killing vectors that the Alekseev equation was again germane. In this
case, one can have spacetimes that are even without being analytic,
so it is truly amazing that an Alekseev type equation remains applicable.