603 research outputs found
Constraints and evolution in cosmology
We review some old and new results about strict and non strict hyperbolic
formulations of the Einstein equations.Comment: To appear in the proceedings of the first Aegean summer school in
General Relativity, S. Cotsakis ed. Springer Lecture Notes in Physic
Relativistic Lagrange Formulation
It is well-known that the equations for a simple fluid can be cast into what
is called their Lagrange formulation. We introduce a notion of a generalized
Lagrange formulation, which is applicable to a wide variety of systems of
partial differential equations. These include numerous systems of physical
interest, in particular, those for various material media in general
relativity. There is proved a key theorem, to the effect that, if the original
(Euler) system admits an initial-value formulation, then so does its
generalized Lagrange formulation.Comment: 34 pages, no figures, accepted in J. Math. Phy
Proof of the Thin Sandwich Conjecture
We prove that the Thin Sandwich Conjecture in general relativity is valid,
provided that the data satisfy certain geometric
conditions. These conditions define an open set in the class of possible data,
but are not generically satisfied. The implications for the ``superspace''
picture of the Einstein evolution equations are discussed.Comment: 8 page
Potential for ill-posedness in several 2nd-order formulations of the Einstein equations
Second-order formulations of the 3+1 Einstein equations obtained by
eliminating the extrinsic curvature in terms of the time derivative of the
metric are examined with the aim of establishing whether they are well posed,
in cases of somewhat wide interest, such as ADM, BSSN and generalized
Einstein-Christoffel. The criterion for well-posedness of second-order systems
employed is due to Kreiss and Ortiz. By this criterion, none of the three cases
are strongly hyperbolic, but some of them are weakly hyperbolic, which means
that they may yet be well posed but only under very restrictive conditions for
the terms of order lower than second in the equations (which are not studied
here). As a result, intuitive transferences of the property of well-posedness
from first-order reductions of the Einstein equations to their originating
second-order versions are unwarranted if not false.Comment: v1:6 pages; v2:7 pages, discussion extended, to appear in Phys. Rev.
D; v3: typos corrected, published versio
The Cauchy problem on a characteristic cone for the Einstein equations in arbitrary dimensions
We derive explicit formulae for a set of constraints for the Einstein
equations on a null hypersurface, in arbitrary dimensions. We solve these
constraints and show that they provide necessary and sufficient conditions so
that a spacetime solution of the Cauchy problem on a characteristic cone for
the hyperbolic system of the reduced Einstein equations in wave-map gauge also
satisfies the full Einstein equations. We prove a geometric uniqueness theorem
for this Cauchy problem in the vacuum case.Comment: 83 pages, 1 figur
On two theorems for flat, affine group schemes over a discrete valuation ring
We include short and elementary proofs of two theorems characterizing
reductive group schemes over a discrete valuation ring, in a slightly more
general context.Comment: 10 pages. To appear in C. E. J.
Existence and uniqueness of Bowen-York Trumpets
We prove the existence of initial data sets which possess an asymptotically
flat and an asymptotically cylindrical end. Such geometries are known as
trumpets in the community of numerical relativists.Comment: This corresponds to the published version in Class. Quantum Grav. 28
(2011) 24500
The Well-posedness of the Null-Timelike Boundary Problem for Quasilinear Waves
The null-timelike initial-boundary value problem for a hyperbolic system of
equations consists of the evolution of data given on an initial characteristic
surface and on a timelike worldtube to produce a solution in the exterior of
the worldtube. We establish the well-posedness of this problem for the
evolution of a quasilinear scalar wave by means of energy estimates. The
treatment is given in characteristic coordinates and thus provides a guide for
developing stable finite difference algorithms. A new technique underlying the
approach has potential application to other characteristic initial-boundary
value problems.Comment: Version to appear in Class. Quantum Gra
Differential Forms and Wave Equations for General Relativity
Recently, Choquet-Bruhat and York and Abrahams, Anderson, Choquet-Bruhat, and
York (AACY) have cast the 3+1 evolution equations of general relativity in
gauge-covariant and causal ``first-order symmetric hyperbolic form,'' thereby
cleanly separating physical from gauge degrees of freedom in the Cauchy problem
for general relativity. A key ingredient in their construction is a certain
wave equation which governs the light-speed propagation of the extrinsic
curvature tensor. Along a similar line, we construct a related wave equation
which, as the key equation in a system, describes vacuum general relativity.
Whereas the approach of AACY is based on tensor-index methods, the present
formulation is written solely in the language of differential forms. Our
approach starts with Sparling's tetrad-dependent differential forms, and our
wave equation governs the propagation of Sparling's 2-form, which in the
``time-gauge'' is built linearly from the ``extrinsic curvature 1-form.'' The
tensor-index version of our wave equation describes the propagation of (what is
essentially) the Arnowitt-Deser-Misner gravitational momentum.Comment: REVTeX, 26 pages, no figures, 1 macr
Modifying the Einstein Equations off the Constraint Hypersuface
A new technique is presented for modifying the Einstein evolution equations
off the constraint hypersurface. With this approach the evolution equations for
the constraints can be specified freely. The equations of motion for the
gravitational field variables are modified by the addition of terms that are
linear and nonlocal in the constraints. These terms are obtained from solutions
of the linearized Einstein constraints.Comment: 4 pages, 1 figure, uses REVTe
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