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

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 $t$ when numerical resolution is increased. However, in certain cases integration becomes asymptotically unstable when $t$ 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 $t$ 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

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

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

Analysis and optimization of a free-electron laser with an irregular waveguide

Using a time-dependent approach the analysis and optimization of a planar FEL-amplifier with an axial magnetic field and an irregular waveguide is performed. By applying methods of nonlinear dynamics three-dimensional equations of motion and the excitation equation are partly integrated in an analytical way. As a result, a self-consistent reduced model of the FEL is built in special phase space. The reduced model is the generalization of the Colson-Bonifacio model and takes into account the intricate dynamics of electrons in the pump magnetic field and the intramode scattering in the irregular waveguide. The reduced model and concepts of evolutionary computation are used to find optimal waveguide profiles. The numerical simulation of the original non-simplified model is performed to check the effectiveness of found optimal profiles. The FEL parameters are chosen to be close to the parameters of the experiment (S. Cheng et al. IEEE Trans. Plasma Sci. 1996, vol. 24, p. 750), in which a sheet electron beam with the moderate thickness interacts with the TE01 mode of a rectangular waveguide. The results strongly indicate that one can improve the efficiency by a factor of five or six if the FEL operates in the magnetoresonance regime and if the irregular waveguide with the optimized profile is used

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

Nothing but Relativity, Redux

Here we show how spacetime transformations consistent with the principle of relativity can be derived without an explicit assumption of the constancy of the speed of light, without gedanken experiments involving light rays, and without an assumption of differentiability, or even continuity, for the spacetime mapping. Hence, these historic results could have been derived centuries ago, even before the advent of calculus. This raises an interesting question: Could Galileo have derived Einsteinian relativity

Hydrodynamic singularities and clustering in a freely cooling inelastic gas

We employ hydrodynamic equations to follow the clustering instability of a freely cooling dilute gas of inelastically colliding spheres into a well-developed nonlinear regime. We simplify the problem by dealing with a one-dimensional coarse-grained flow. We observe that at a late stage of the instability the shear stress becomes negligibly small, and the gas flows solely by inertia. As a result the flow formally develops a finite time singularity, as the velocity gradient and the gas density diverge at some location. We argue that flow by inertia represents a generic intermediate asymptotic of unstable free cooling of dilute inelastic gases.Comment: 4 pages, 4 figure