2,169 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
Symmetries of distributional domain wall geometries
Generalizing the Lie derivative of smooth tensor fields to
distribution-valued tensors, we examine the Killing symmetries and the
collineations of the curvature tensors of some distributional domain wall
geometries. The chosen geometries are rigorously the distributional thin wall
limit of self gravitating scalar field configurations representing thick domain
walls and the permanence and/or the rising of symmetries in the limit process
is studied. We show that, for all the thin wall spacetimes considered, the
symmetries of the distributional curvature tensors turns out to be the Killing
symmetries of the pullback of the metric tensor to the surface where the
singular part of these tensors is supported. Remarkably enough, for the
non-reflection symmetric domain wall studied, these Killing symmetries are not
necessarily symmetries of the ambient spacetime on both sides of the wall
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
Motion of Isolated bodies
It is shown that sufficiently smooth initial data for the Einstein-dust or
the Einstein-Maxwell-dust equations with non-negative density of compact
support develop into solutions representing isolated bodies in the sense that
the matter field has spatially compact support and is embedded in an exterior
vacuum solution
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
Hamiltonian Time Evolution for General Relativity
Hamiltonian time evolution in terms of an explicit parameter time is derived
for general relativity, even when the constraints are not satisfied, from the
Arnowitt-Deser-Misner-Teitelboim-Ashtekar action in which the slicing density
is freely specified while the lapse is not.
The constraint ``algebra'' becomes a well-posed evolution system for the
constraints; this system is the twice-contracted Bianchi identity when
. The Hamiltonian constraint is an initial value constraint which
determines and hence , given .Comment: 4 pages, revtex, to appear in Phys. Rev. Let
Conformal ``thin sandwich'' data for the initial-value problem of general relativity
The initial-value problem is posed by giving a conformal three-metric on each
of two nearby spacelike hypersurfaces, their proper-time separation up to a
multiplier to be determined, and the mean (extrinsic) curvature of one slice.
The resulting equations have the {\it same} elliptic form as does the
one-hypersurface formulation. The metrical roots of this form are revealed by a
conformal ``thin sandwich'' viewpoint coupled with the transformation
properties of the lapse function.Comment: 7 pages, RevTe
General structure of the solutions of the Hamiltonian constraints of gravity
A general framework for the solutions of the constraints of pure gravity is
constructed. It provides with well defined mathematical criteria to classify
their solutions in four classes. Complete families of solutions are obtained in
some cases. A starting point for the systematic study of the solutions of
Einstein gravity is suggested.Comment: 17 pages, LaTeX, submitted to International J. of Geom. Meth. in
Modern Physics. Added comments in the last sectio
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