1,730 research outputs found
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
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
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
Cones of material response functions in 1D and anisotropic linear viscoelasticity
Viscoelastic materials have non-negative relaxation spectra. This property
implies that viscoelastic response functions satisfy certain necessary and
sufficient conditions. It is shown that these conditions can be expressed in
terms of each viscoelastic response function ranging over a cone. The elements
of each cone are completely characterized by an integral representation. The
1:1 correspondences between the viscoelastic response functions are expressed
in terms of cone-preserving mappings and their inverses. The theory covers
scalar and tensor-valued viscoelastic response functionsComment: submitted to Proc. Roy. Soc.
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
Local 4/5-Law and Energy Dissipation Anomaly in Turbulence
A strong local form of the ``4/3-law'' in turbulent flow has been proved
recently by Duchon and Robert for a triple moment of velocity increments
averaged over both a bounded spacetime region and separation vector directions,
and for energy dissipation averaged over the same spacetime region. Under
precisely stated hypotheses, the two are proved to be proportional, by a
constant 4/3, and to appear as a nonnegative defect measure in the local energy
balance of singular (distributional) solutions of the incompressible Euler
equations. Here we prove that the energy defect measure can be represented also
by a triple moment of purely longitudinal velocity increments and by a mixed
moment with one longitudinal and two tranverse velocity increments. Thus, we
prove that the traditional 4/5- and 4/15-laws of Kolmogorov hold in the same
local sense as demonstrated for the 4/3-law by Duchon-Robert.Comment: 14 page
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
The Epstein-Glaser approach to pQFT: graphs and Hopf algebras
The paper aims at investigating perturbative quantum field theory (pQFT) in
the approach of Epstein and Glaser (EG) and, in particular, its formulation in
the language of graphs and Hopf algebras (HAs). Various HAs are encountered,
each one associated with a special combination of physical concepts such as
normalization, localization, pseudo-unitarity, causality and an associated
regularization, and renormalization. The algebraic structures, representing the
perturbative expansion of the S-matrix, are imposed on the operator-valued
distributions which are equipped with appropriate graph indices. Translation
invariance ensures the algebras to be analytically well-defined and graded
total symmetry allows to formulate bialgebras. The algebraic results are given
embedded in the physical framework, which covers the two recent EG versions by
Fredenhagen and Scharf that differ with respect to the concrete recursive
implementation of causality. Besides, the ultraviolet divergences occuring in
Feynman's representation are mathematically reasoned. As a final result, the
change of the renormalization scheme in the EG framework is modeled via a HA
which can be seen as the EG-analog of Kreimer's HA.Comment: 52 pages, 5 figure
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