327 research outputs found
On the Newtonian Limit of General Relativity
We find a choice of variables for the 3+1 formulation of general relativity
which casts the evolution equations into (flux-conservative)
symmetric-hyperbolic first order form for arbitrary lapse and shift, for the
first time. We redefine the lapse function in terms of the determinant of the
3-metric and a free function U which embodies the lapse freedom. By rescaling
the variables with appropriate factors of 1/c, the system is shown to have a
smooth Newtonian limit when the redefined lapse U and the shift are fixed by
means of elliptic equations to be satisfied on each time slice. We give a
prescription for the choice of appropriate initial data with controlled
extra-radiation content, based on the theory of problems with different
time-scales. Our results are local, in the sense that we are not concerned with
the treatment of asymptotic regions. On the other hand, this local theory is
all what is needed for most problems of practical numerical computation.Comment: 16 pages, uses REVTe
Well-posed forms of the 3+1 conformally-decomposed Einstein equations
We show that well-posed, conformally-decomposed formulations of the 3+1
Einstein equations can be obtained by densitizing the lapse and by combining
the constraints with the evolution equations. We compute the characteristics
structure and verify the constraint propagation of these new well-posed
formulations. In these formulations, the trace of the extrinsic curvature and
the determinant of the 3-metric are singled out from the rest of the dynamical
variables, but are evolved as part of the well-posed evolution system. The only
free functions are the lapse density and the shift vector. We find that there
is a 3-parameter freedom in formulating these equations in a well-posed manner,
and that part of the parameter space found consists of formulations with causal
characteristics, namely, characteristics that lie only within the lightcone. In
particular there is a 1-parameter family of systems whose characteristics are
either normal to the slicing or lie along the lightcone of the evolving metric.Comment: 22 page
Nonlinear electrodynamics as a symmetric hyperbolic system
Nonlinear theories generalizing Maxwell's electromagnetism and arising from a
Lagrangian formalism have dispersion relations in which propagation planes
factor into null planes corresponding to two effective metrics which depend on
the point-wise values of the electromagnetic field. These effective Lorentzian
metrics share the null (generically two) directions of the electromagnetic
field. We show that, the theory is symmetric hyperbolic if and only if the
cones these metrics give rise to have a non-empty intersection. Namely that
there exist families of symmetrizers in the sense of Geroch which are positive
definite for all covectors in the interior of the cones intersection. Thus, for
these theories, the initial value problem is well-posed. We illustrate the
power of this approach with several nonlinear models of physical interest such
as Born-Infeld, Gauss-Bonnet and Euler-Heisenberg
Einstein's Equations with Asymptotically Stable Constraint Propagation
We introduce a proposal to modify Einstein's equations by embedding them in a
larger symmetric hyperbolic system. The additional dynamical variables of the
modified system are essentially first integrals of the original constraints.
The extended system of equations reproduces the usual dynamics on the
constraint surface of general relativity, and therefore naturally includes the
solutions to Einstein gravity. The main feature of this extended system is
that, at least for a linearized version of it, the constraint surface is an
attractor of the time evolution. This feature suggests that this system may be
a useful alternative to Einstein's equations when obtaining numerical solutions
to full, non-linear gravity.Comment: 23 pages, submitted to JMP, added reference for section
On Existence of Static Metric Extensions in General Relativity
Motivated by problems related to quasi-local mass in general relativity, we
study the static metric extension conjecture proposed by R. Bartnik
\cite{Bartnik_energy}. We show that, for any metric on that is
close enough to the Euclidean metric and has reflection invariant boundary
data, there always exists an asymptotically flat and scalar flat {\em static}
metric extension in such that it satisfies Bartnik's
geometric boundary condition \cite{Bartnik_energy} on .Comment: 20 page
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