1,461 research outputs found
A scalar hyperbolic equation with GR-type non-linearity
We study a scalar hyperbolic partial differential equation with non-linear
terms similar to those of the equations of general relativity. The equation has
a number of non-trivial analytical solutions whose existence rely on a delicate
balance between linear and non-linear terms. We formulate two classes of
second-order accurate central-difference schemes, CFLN and MOL, for numerical
integration of this equation. Solutions produced by the schemes converge to
exact solutions at any fixed time when numerical resolution is increased.
However, in certain cases integration becomes asymptotically unstable when
is increased and resolution is kept fixed. This behavior is caused by subtle
changes in the balance between linear and non-linear terms when the equation is
discretized. Changes in the balance occur without violating second-order
accuracy of discretization. We thus demonstrate that a second-order accuracy
and convergence at finite do not guarantee a correct asymptotic behavior
and long-term numerical stability.
Accuracy and stability of integration are greatly improved by an exponential
transformation of the unknown variable.Comment: submitted to Class. Quantum Gra
Mixed Hyperbolic - Second-Order Parabolic Formulations of General Relativity
Two new formulations of general relativity are introduced. The first one is a
parabolization of the Arnowitt, Deser, Misner (ADM) formulation and is derived
by addition of combinations of the constraints and their derivatives to the
right-hand-side of the ADM evolution equations. The desirable property of this
modification is that it turns the surface of constraints into a local attractor
because the constraint propagation equations become second-order parabolic
independently of the gauge conditions employed. This system may be classified
as mixed hyperbolic - second-order parabolic. The second formulation is a
parabolization of the Kidder, Scheel, Teukolsky formulation and is a manifestly
mixed strongly hyperbolic - second-order parabolic set of equations, bearing
thus resemblance to the compressible Navier-Stokes equations. As a first test,
a stability analysis of flat space is carried out and it is shown that the
first modification exponentially damps and smoothes all constraint violating
modes. These systems provide a new basis for constructing schemes for long-term
and stable numerical integration of the Einstein field equations.Comment: 19 pages, two column, references added, two proofs of well-posedness
added, content changed to agree with submitted version to PR
Quantum singularities in (2+1) dimensional matter coupled black hole spacetimes
Quantum singularities considered in the 3D BTZ spacetime by Pitelli and
Letelier (Phys. Rev. D77: 124030, 2008) is extended to charged BTZ and 3D
Einstein-Maxwell-dilaton gravity spacetimes. The occurence of naked
singularities in the Einstein-Maxwell extension of the BTZ spacetime both in
linear and non-linear electrodynamics as well as in the
Einstein-Maxwell-dilaton gravity spacetimes are analysed with the quantum test
fields obeying the Klein-Gordon and Dirac equations. We show that with the
inclusion of the matter fields; the conical geometry near r=0 is removed and
restricted classes of solutions are admitted for the Klein-Gordon and Dirac
equations. Hence, the classical central singularity at r=0 turns out to be
quantum mechanically singular for quantum particles obeying Klein-Gordon
equation but nonsingular for fermions obeying Dirac equation. Explicit
calculations reveal that the occurrence of the timelike naked singularities in
the considered spacetimes do not violate the cosmic censorship hypothesis as
far as the Dirac fields are concerned. The role of horizons that clothes the
singularity in the black hole cases is replaced by repulsive potential barrier
against the propagation of Dirac fields.Comment: 13 pages, 1 figure. Final version, to appear in PR
Orthogonality relations in Quantum Tomography
Quantum estimation of the operators of a system is investigated by analyzing
its Liouville space of operators. In this way it is possible to easily derive
some general characterization for the sets of observables (i.e. the possible
quorums) that are measured for the quantum estimation. In particular we analyze
the reconstruction of operators of spin systems.Comment: 10 pages, 2 figure
Conditions for strictly purity-decreasing quantum Markovian dynamics
The purity, Tr(rho^2), measures how pure or mixed a quantum state rho is. It
is well known that quantum dynamical semigroups that preserve the identity
operator (which we refer to as unital) are strictly purity-decreasing
transformations. Here we provide an almost complete characterization of the
class of strictly purity-decreasing quantum dynamical semigroups. We show that
in the case of finite-dimensional Hilbert spaces a dynamical semigroup is
strictly purity-decreasing if and only if it is unital, while in the infinite
dimensional case, unitality is only sufficient.Comment: 4 pages, no figures. Contribution to the special issue "Real-time
dynamics in complex quantum systems" of Chemical Physics in honor of Phil
Pechukas. v2: Simplified proof of theorem 1 and validity conditions clarifie
Low energy dynamics of spinor condensates
We present a derivation of the low energy Lagrangian governing the dynamics
of the spin degrees of freedom in a spinor Bose condensate, for any phase in
which the average magnetization vanishes. This includes all phases found within
mean-field treatments except for the ferromagnet, for which the low energy
dynamics has been discussed previously. The Lagrangian takes the form of a
sigma model for the rotation matrix describing the local orientation of the
spin state of the gas
Symmetries of Discontinuous Flows and the Dual Rankine-Hugoniot Conditions in Fluid Dynamics
It has recently been shown that the maximal kinematical invariance group of
polytropic fluids, for smooth subsonic flows, is the semidirect product of
SL(2,R) and the static Galilei group G. This result purports to offer a
theoretical explanation for an intriguing similarity, that was recently
observed, between a supernova explosion and a plasma implosion. In this paper
we extend this result to discuss the symmetries of discontinuous flows, which
further validates the explanation by taking into account shock waves, which are
the driving force behind both the explosion and implosion. This is accomplished
by constructing a new set of Rankine-Hugoniot conditions, which follow from
Noether's conservation laws. The new set is dual to the standard
Rankine-Hugoniot conditions and is related to them through the SL(2,R)
transformations. The entropy condition, that the shock needs to satisfy for
physical reasons, is also seen to remain invariant under the transformations.Comment: 14 pages, 1 figur
On the iterated Crank-Nicolson for hyperbolic and parabolic equations in numerical relativity
The iterated Crank-Nicolson is a predictor-corrector algorithm commonly used
in numerical relativity for the solution of both hyperbolic and parabolic
partial differential equations. We here extend the recent work on the stability
of this scheme for hyperbolic equations by investigating the properties when
the average between the predicted and corrected values is made with unequal
weights and when the scheme is applied to a parabolic equation. We also propose
a variant of the scheme in which the coefficients in the averages are swapped
between two corrections leading to systematically larger amplification factors
and to a smaller numerical dispersion.Comment: 7 pages, 3 figure
Compact oscillons in the signum-Gordon model
We present explicit solutions of the signum-Gordon scalar field equation
which have finite energy and are periodic in time. Such oscillons have a
strictly finite size. They do not emit radiation.Comment: 12 pages, 4 figure
C+O detonations in thermonuclear supernovae: Interaction with previously burned material
In the context of explosion models for Type Ia Supernovae, we present one-
and two-dimensional simulations of fully resolved detonation fronts in
degenerate C+O White Dwarf matter including clumps of previously burned
material. The ability of detonations to survive the passage through sheets of
nuclear ashes is tested as a function of the width and composition of the ash
region. We show that detonation fronts are quenched by microscopically thin
obstacles with little sensitivity to the exact ash composition. Front-tracking
models for detonations in macroscopic explosion simulations need to include
this effect in order to predict the amount of unburned material in delayed
detonation scenarios.Comment: 6 pages, 9 figures, uses isotope.sty, accepted for publication in A&
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