Matter Models in General Relativity.
Abstract
The Cauchy problem (or, initial value problem) provides a setting for the
analysis of generic solutions to the Einstein field equations parametrised in
terms of the initial conditions. In particular, one is interested in showing
that the Einstein equations admit a well-posed initial value formulation.
The standard strategy to address this issue is to show that the Einstein
equations imply evolution equations that are on a hyperbolic form. This
has been done for the vacuum, dust and Einstein-Euler equations — each
treated separately. In the first part of this thesis, we use an orthonormal
frame approach to show that one can avoid the details of a specific Einstein
- matter model in the construction of a first order symmetric hyperbolic
system by introducing an auxiliary field. The frame is Fermi-Walker propagated
and coordinates are chosen such as to satisfy the Lagrange condition.
It is shown that the solution of the system established is a solution to
the Einstein Equations everywhere on the space time by propagation of
constraints. Our analysis covers the special cases of dust and perfect fluid,
and we also provide a discussion of self-gravitating elastic matter. In the
second part of the thesis, we study the conformal Einstein field equations
and show the future stability of N self-gravitating dust bodies in a space
time with positive cosmological constant. This is achieved in three parts.
First, we show that the choice of density function representing N dust
bodies — when ascribed as initial data on the conformal boundary (I+) —
is a solution to the conformal field equations. This result is obtained using
a theorem by H. Friedrich and the Fredholm alternative. We then show
stability for small conformal time for this data, which is equivalent to an
infinite physical time. Finally, using a theorem by Choquet-Bruhat, we
give sufficient conditions for the Einstein constraint equations to admit a
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solution representing N bodies of dust, and show that the geodesics are
future complete.
Authors
Normann, MikaelCollections
- Theses [4248]