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Habilitation à diriger des recherches

Approche algébro-géométrique aux équations d'Einstein et d'Einstein-Maxwell : cas stationnaire avec symétrie axiale

Abstract : This work presents a discussion of mathematical and physical
aspects of solutions to the stationary axisymmetric Einstein and
Einstein-Maxwell equations in vacuum generated with
methods from algebraic geometry. Since the equations in this case
are equivalent to the completely integrable Ernst equations,
Riemann-Hilbert methods can be used to generate solutions to
boundary value problems. We show here that Riemann-Hilbert
problems with analytic jump data can be solved on Riemann
surfaces. If the surfaces are non-compact, the existence of
solutions is proven with the help of fiber bundle theory. The surfaces
are compact if the jump data are rational functions which allows
for an explicit solution of the Riemann-Hilbert problem in terms
of Korotkin's theta functional solutions. With the help of
Fay's identity, all components of the corresponding metric can be
given in terms of hyperelliptic theta functions. We discuss the
possible singularities of these solutions, and we identify a
subclass which is regular in the exterior of a contour that can
represent the surface of a matter distribution. As an astrophysical
example we consider the case of dust disks which are used as
models for certain galaxies and the matter in accretion disks
around black holes. The solutions in terms of theta functions
and their derivatives are related via algebraic relations
on a given Riemann surface. These relations determine which
classes of boundary value problems can be solved on this surface. We establish these relations in order to describe
dust disks. We give the explicit
solution for a disk with two counter-rotating dust components.
This solution contains a static Morgan and Morgan disk and
the disk with one component in rigid rotation as limiting cases.
We discuss the metric, limiting cases, multipole moments and the
energy momentum tensor. The theta functions are evaluated
numerically with the help of pseudo-spectral methods. The case of
static black holes with an annular ring is also discussed. We
give an existence and uniqueness proof for the solutions by
applying a theorem of Poole. Approximate solutions are presented
for the annular case, explicit solutions are given for the case of
a ring stretching to infinity. In the stationary case we show that
solutions on partially degenerate Riemann surfaces describe
infinite annular disks around a Kerr-type horizon. We discuss the
degeneration of a surface of genus 2 in detail where the solutions
can be given in terms of elementary functions. For the
Einstein-Maxwell equations, we construct hyperelliptic solutions
with charge. We use the symmetry of the equations to construct
these solutions by applying a Harrison transformation to solutions
without electromagnetic fields. Counter-rotating disks with
charge are discussed.
Document type :
Habilitation à diriger des recherches
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Contributor : Christian Klein <>
Submitted on : Thursday, March 1, 2007 - 9:19:17 PM
Last modification on : Thursday, December 10, 2020 - 11:09:03 AM
Long-term archiving on: : Wednesday, April 7, 2010 - 1:13:37 AM



  • HAL Id : tel-00134376, version 1


Christian Klein. Approche algébro-géométrique aux équations d'Einstein et d'Einstein-Maxwell : cas stationnaire avec symétrie axiale. Physique mathématique [math-ph]. Université Pierre et Marie Curie - Paris VI, 2002. ⟨tel-00134376⟩



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