1,792 research outputs found
Geometrical Hyperbolic Systems for General Relativity and Gauge Theories
The evolution equations of Einstein's theory and of Maxwell's theory---the
latter used as a simple model to illustrate the former--- are written in gauge
covariant first order symmetric hyperbolic form with only physically natural
characteristic directions and speeds for the dynamical variables. Quantities
representing gauge degrees of freedom [the spatial shift vector
and the spatial scalar potential ,
respectively] are not among the dynamical variables: the gauge and the physical
quantities in the evolution equations are effectively decoupled. For example,
the gauge quantities could be obtained as functions of from
subsidiary equations that are not part of the evolution equations. Propagation
of certain (``radiative'') dynamical variables along the physical light cone is
gauge invariant while the remaining dynamical variables are dragged along the
axes orthogonal to the spacelike time slices by the propagating variables. We
obtain these results by taking a further time derivative of the equation
of motion of the canonical momentum, and adding a covariant spatial
derivative of the momentum constraints of general relativity (Lagrange
multiplier ) or of the Gauss's law constraint of electromagnetism
(Lagrange multiplier ). General relativity also requires a harmonic time
slicing condition or a specific generalization of it that brings in the
Hamiltonian constraint when we pass to first order symmetric form. The
dynamically propagating gravity fields straightforwardly determine the
``electric'' or ``tidal'' parts of the Riemann tensor.Comment: 24 pages, latex, no figure
A rigidity theorem for nonvacuum initial data
In this note we prove a theorem on non-vacuum initial data for general
relativity. The result presents a ``rigidity phenomenon'' for the extrinsic
curvature, caused by the non-positive scalar curvature.
More precisely, we state that in the case of asymptotically flat non-vacuum
initial data if the metric has everywhere non-positive scalar curvature then
the extrinsic curvature cannot be compactly supported.Comment: This is an extended and published version: LaTex, 10 pages, no
figure
Einstein and Yang-Mills theories in hyperbolic form without gauge-fixing
The evolution of physical and gauge degrees of freedom in the Einstein and
Yang-Mills theories are separated in a gauge-invariant manner. We show that the
equations of motion of these theories can always be written in
flux-conservative first-order symmetric hyperbolic form. This dynamical form is
ideal for global analysis, analytic approximation methods such as
gauge-invariant perturbation theory, and numerical solution.Comment: 12 pages, revtex3.0, no figure
Cosmological spacetimes not covered by a constant mean curvature slicing
We show that there exist maximal globally hyperbolic solutions of the
Einstein-dust equations which admit a constant mean curvature Cauchy surface,
but are not covered by a constant mean curvature foliation.Comment: 11 page
Geometrically motivated hyperbolic coordinate conditions for numerical relativity: Analysis, issues and implementations
We study the implications of adopting hyperbolic driver coordinate conditions
motivated by geometrical considerations. In particular, conditions that
minimize the rate of change of the metric variables. We analyze the properties
of the resulting system of equations and their effect when implementing
excision techniques. We find that commonly used coordinate conditions lead to a
characteristic structure at the excision surface where some modes are not of
outflow-type with respect to any excision boundary chosen inside the horizon.
Thus, boundary conditions are required for these modes. Unfortunately, the
specification of these conditions is a delicate issue as the outflow modes
involve both gauge and main variables. As an alternative to these driver
equations, we examine conditions derived from extremizing a scalar constructed
from Killing's equation and present specific numerical examples.Comment: 9 figure
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
The constraint equations for the Einstein-scalar field system on compact manifolds
We study the constraint equations for the Einstein-scalar field system on
compact manifolds. Using the conformal method we reformulate these equations as
a determined system of nonlinear partial differential equations. By introducing
a new conformal invariant, which is sensitive to the presence of the initial
data for the scalar field, we are able to divide the set of free conformal data
into subclasses depending on the possible signs for the coefficients of terms
in the resulting Einstein-scalar field Lichnerowicz equation. For many of these
subclasses we determine whether or not a solution exists. In contrast to other
well studied field theories, there are certain cases, depending on the mean
curvature and the potential of the scalar field, for which we are unable to
resolve the question of existence of a solution. We consider this system in
such generality so as to include the vacuum constraint equations with an
arbitrary cosmological constant, the Yamabe equation and even (all cases of)
the prescribed scalar curvature problem as special cases.Comment: Minor changes, final version. To appear: Classical and Quantum
Gravit
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.
A model problem for conformal parameterizations of the Einstein constraint equations
We investigate the possibility that the conformal and conformal thin sandwich
(CTS) methods can be used to parameterize the set of solutions of the vacuum
Einstein constraint equations. To this end we develop a model problem obtained
by taking the quotient of certain symmetric data on conformally flat tori.
Specializing the model problem to a three-parameter family of conformal data we
observe a number of new phenomena for the conformal and CTS methods. Within
this family, we obtain a general existence theorem so long as the mean
curvature does not change sign. When the mean curvature changes sign, we find
that for certain data solutions exist if and only if the transverse-traceless
tensor is sufficiently small. When such solutions exist, there are generically
more than one. Moreover, the theory for mean curvatures changing sign is shown
to be extremely sensitive with respect to the value of a coupling constant in
the Einstein constraint equations.Comment: 40 pages, 4 figure
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