In General Relativity, the Einstein field equations allow us to study the evolution of a
spacelike 3-manifold, provided that its metric and extrinsic curvature satisfy a system of
geometric constraint equations. The so-called Einstein constraint equations, arise as a
consequence of the fact that the 3-manifold in question is necessarily a submanifold of
the spacetime its evolution defines.
This thesis is devoted to a study of the structure of the Einstein constraint system in
the special case when the spacelike 3-manifold also satisfies the quasispherical ansatz of
Bartnik [B93]. We make no mention of the generality of this gauge; the extent to which
the quasispherical ansatz applies remains an open problem.
After imposing the quasispherical gauge, we give an argument to show that the resulting
Einstein constraint system may be viewed as a coupled system of partial differential
equations for the parameters describing the metric and second fundamental form. The
hencenamed quasisperical Einstein constraint system, consists of a parabolic equation, a
first order elliptic system and (essentially) a system of ordinary differential equations.
The question of existence of solutions to this system naturally arises and we provide a
partial answer to this question. We give conditions on the initial data and prescribable
fields under which we may conclude that the quasispherical Einstein constraint system is
uniquley solvable, at least in a region surrounding the unit sphere.
The proof of this fact is centred on a linear iterative system of partial differential equations,
which also consist of a parabolic equation, a first order elliptic system and a system of
ordinary differential equations. We prove that this linear system consistently defines a
sequence, and show via a contraction mapping argument, that this sequence must converge
to a fixed point of the iteration. The iteration, however, has been specifically designed
so that any fixed point of the iteration coincides with a solution of the quasispherical
Einstein constraints.
The contraction mapping argument mentioned above, relies heavily on a priori estimates
for the solutions of linear parabolic equations. We generalise and extend known results
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concerning parabolic equations to establish special a priori estimates which relate a useful
property: the L2-Sobolev regularity of the solution of a parabolic equation is greater
than that of the coefficients of the elliptic operator, provided that the initial data is
sufficiently regular. This 'smoothing' property of linear parabolic equations along with
several estimates from elliptic and ordinary differential equation theory form the crucial
ingredients needed in the proof of the existence of a fixed point of the iteration.
We begin in chapter one by giving a brief review of the extensive literature concerning
the initial value problem in General Relativity. We go on, after mentioning two of the
traditional methods for constructing spacetime initial data, to introduce the notion of a
quasispherical foliation of a 3-manifold and present the Einstein constraint system written
in terms of this gauge.
In chapter two we introduce the various inequalities and tracts of analysis we will make use
of in subsequent chapters. In particular we define the, perhaps not so familiar, complex
differential operator 9 (edth) of Newman and Penrose.
In chapter three we develop the appropriate Sobolev-regularity theory for linear parabolic
equations required to deal with the quasispherical initial data constraint equations. We
include a result due to Polden [P] here, with a corrected proof. This result was essential
for deriving the results contained in the later chapters of [P], and it is for this reason we
include the result. We don't make use of it explicitly when considering the quasispherical
Einstein constraints, but the ideas employed are similar to those we use to tackle the
problem of existence for the quasispherical constraints.
Chapter four is concerned with the local existence of quasispherical initial data. We
firstly consider the question of existence and uniqueness when the mean curvature of
the 3-manifold is prescribed, then after introducing the notion of polar curvature, we also
present another quasispherical constraint system in which we consider the polar curvature
as prescribed. We prove local existence and uniqueness results for both of these alternate
formulations of the quasispherical constraints.
This thesis was typeset using LATEXwith the package amssymb.
| Identifer | oai:union.ndltd.org:ADTP/219325 |
| Date | January 2001 |
| Creators | Sharples, Jason, n/a |
| Publisher | University of Canberra. Mathematics &Statistics |
| Source Sets | Australiasian Digital Theses Program |
| Language | English |
| Detected Language | English |
| Rights | ), Copyright Jason Sharples |
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