Exterior Solution Research Articles

An elementary introduction to the construction and the analysis of perfectly matched layers for time domain wave propagation

As soon as people have been interested in the numerical simulation of wave propagation phenomena, particularly in the time domain, they had to face the delicate question of realizing “transparent” artificial boundaries. Indeed, many problems of this type are posed in a unbounded domain (or in a domain which is very large with respect to a zone of interest in which one wishes to observe the solution): one thus has to bound artificially the computational domain in such a way that the non physical boundaries are “transparent” for the solution. Due to the crucial importance of this subject, but also because of its difficulty, this question has aroused the curiosity from many researchers, first from engineers, next from applied mathematicians. It would be much too ambitious to give a complete panorama of a research that began almost forty years ago (see however the article by T. Hagstrm [20] which constitutes a quite complete review of the domain) but one can now measure how much the many ingenious ideas proposed by the one or the others have permitted to accomplish impressive progress in this matter. Nowadays, if one restricts oneself to linear wave propagation problems, one can consider that, even though some questions remain open, satisfactory solutions are available for a large number of applications, in particular in the “canonical case” I will have in mind throughout this paper, namely the case where one wishes to reduce numerical computations to a convex bounded domain, assuming that the exterior domain is homogeneous and source free. Recently, in particular for the applications in acoustics and electromagnetism, important progress has been achieved in the use of “exact” boundary conditions associated to homogeneous exterior domains. One usually introduces a boundary operator called “Dirichlet to Neumann” (DtN) operator that can be represented by using separation of variables in the exterior domain (in 3D, one will use typically spherical harmonics [19], which requires a spherical artificial boundary) or through the coupling with an integral representation formula of the exterior solution (which relies on the explicit knowledge of the Green’s function of the problem) [4].

Spacetime Junctions and the Collapse to Black Holes in Higher Dimensions

We review recent results about the modelling of gravitational collapse to black holes in higher dimensions. The models are constructed through the junction of two exact solutions of the Einstein field equations: an interior collapsing fluid solution and a vacuum exterior solution. The vacuum exterior solutions are either static or containing gravitational waves. We then review the global geometrical properties of the matched solutions which, besides black holes, may include the existence of naked singularities and wormholes. In the case of radiating exteriors, we show that the data at the boundary can be chosen to be, in some sense, arbitrarily close to the data for the Schwarzschild-Tangherlini solution.

Release characteristic and stability of curcumin incorporated in β-chitin non-woven fibrous sheet using Tween 20 as an emulsifier

β-Chitin sheets containing curcumin—a naturally occurring substance that possesses several advantages biological properties including antioxidant, antimicrobial, and anti-inflammatory activities—were fabricated by the paper-making process using a water-based system. Using the chitin matrix consisting of small fibers could give rise to a large surface area that could improve the diffusion of solvent and reagent, so as to make the material suitable for use as support and carrier for drugs. Tween 20 was used as an emulsifier to improve water solubility of the curcumin. The change in surface morphology of the fabricated chitin sheets after curcumin loading was indicated by scanning electron microscope (SEM). A rough surface consisting of fibrous chitin could be seen both on the neat chitin sheets and the curcumin-loaded chitin sheets, while the increase in the curcumin content led the sheets to show an occurrence of spots. Investigation of the release behavior of the curcumin loaded into the chitin sheet was carried out by the total immersion method in an acetate buffer solution, pH 5.5, at 37°C (simulating human skin). It was found that the amounts of loaded curcumin affected the release characteristics of the curcumin from the chitin sheet as a function of releasing time. In addition, the Tween 20 played an important role in the release ability of the curcumin to an exterior solution and in the stability of the curcumin present in the chitin sheet. It could be postulated that the water solubility, release ability, and stability of the curcumin incorporated into the β-chitin sheets was improved by the inclusion of curcumin into the cores of the Tween 20 micelles and the β-chitin non-woven fibrous sheet containing curcumin could be a promising candidate for would care materials.

A geometric representation for visualizing relativistic length and time measurement

The simplest solution of Einstein's field equations is Schwarzschild solution. This solution is not able to describe for any nonspherical shaped objects. Some stars and galaxies are in ellipsoidal. Consequently, the gravitational field around these objects should be different in compare with spherical form. This paper is considering a new line element so that we are able to construct not only spherical objects, but also we are able to explain an ellipsoidal object too. This new line element is more accurate and complete than Schwarzschild line element. In this research we see that Schwarzschild line element and its solution is only a part of whole work, which we have done. For more consideration we applied this metric to an arbitrary object in the next step. Moreover, we used this line element for solution of a planetary orbit of an ellipsoid planet by using Einstein’s field equations. These equations used for the exterior solution of an ellipsoidal celestial object.

Electrical Sensitivity and Mechanical Properties of Fast Responsive PAMPS–PAA‐PVA T–IPN Hydrogels

AbstractMacroporous ternary‐interpenetrating polymer network (M‐T‐IPN) hydrogels composed of poly(acrylic acid) (PAA), poly(2‐acrylamido‐2‐methyl propyl sulfonic acid) (PAMPS), and poly(vinyl acohol) (PVA) were synthesized by two steps of polymerization using PEG6000 as a pore‐forming agent. By selectively washing away the PVA component, a macroporous structure was formed homogeneously in the hydrogels. The M‐T‐IPN hydrogels exhibited excellent compressive strength, and the maximum compressive strength was about 12.0 MPa. The bending behaviors of the M‐T‐IPN hydrogels in an electric field were investigated. When deionized water swollen samples were placed between a pair of electrodes in NaCl solutions, the M‐T‐IPN hydrogels showed a significant bending behavior upon the application of an electric field. The hydrogels bend toward the cathode when the concentration of polyanions in interior hydrogels is higher than the concentration of NaCl in exterior solutions. In contrast, the hydrogels bend toward the anode when the concentration of polyanions in interior hydrogels is lower than the concentration of NaCl in exterior solutions. © 2011 Wiley Periodicals, Inc. Adv Polym Techn 32: E20–E31, 2013; View this article online at wileyonlinelibrary.com. DOI 10.1002/adv.20266

Static solutions of Einstein's equations with cylindrical symmetry

In analogy with the standard derivation of the Schwarzschild solution, we find all static, cylindrically symmetric solutions of the Einstein field equations for vacuum. These include not only the well-known cone solution, which is locally flat, but others in which the metric coefficients are powers of the radial coordinate and the spacetime is curved. These solutions appear in the literature, but in different forms, corresponding to different definitions of the radial coordinate. Because all the vacuum solutions are singular on the axis, we attempt to match them to ‘interior’ solutions with nonvanishing energy density and pressure. In addition to the well-known ‘cosmic string’ solution joining on to the cone, we find some numerical solutions that join on to the other exterior solutions.

New model of relativistic slowly rotating neutron stars with surface layer crust: application to giant glitches of Vela Pulsar

Introducing a surface layer of matter on the edge of a neutron star in slow rigid rotation, we analyze, from an intrinsic point of view, the junction conditions that must be satisfied between the interior and exterior solutions of the Einstein equations. In our model the core-crust transition pressure arise as an essential parameter in the description of a configuration. As an application of this formalism, we describe giant glitches of the Vela pulsar as a result of variations in the transition pressure, finding that these small changes are compatible with the expected temperature variations of the inner crust during glitch time

Some static spherically symmetric interior solutions of f (R) gravity

We have found some new exact static spherically symmetric interior solutions of metric f (R) gravitational theories describing the equilibrium configuration of a star. Then the solution is matched to the exterior solution and thus gives a complete description of a star in R 1-n/2 theory.

Self–similar and charged spheres in the free–streaming approximation

We evolve nonadiabatic charged spherical distributions of matter. Dissipation is described by the free-streaming approximation. We match a self-similar interior solution with the Reissner-Nordstr\"om-Vaidya exterior solution. The transport mechanism is decisive to the fate of the gravitational collapse. Almost a half of the total initial mass is radiated away. The transport mechanism determines the way in which the electric charge is redistributed.

Exact dark energy star solutions

Adopting the phantom (ghost) scalar field description of dark energy, we construct a general class of exact interior solutions describing mixed relativistic stars containing both ordinary matter and dark energy in different proportions. The exterior solution that continuously matches the interior solutions is found. Exact solutions describing extremal configuration with zero ordinary matter pressure are also constructed.

Pulsating Bead-Based Assay

In recent years, there has been a growing interest in using porous microbeads such as agarose beads as solid supports to bind target molecules from complex fluid samples. Porous beads have large surface area to volume ratios and high receptor concentrations, and they facilitate relatively high sensitivity detection and multiplexing. Unfortunately, to take full advantage of the porous beads' attributes, long incubation times are needed due to the relatively slow mass transfer of target molecules from the exterior solution into the beads' interior. To accelerate the mass transfer process, we propose a novel assay in which functionalized porous beads are periodically compressed and expanded. Preliminary experiments were carried out to compare the performance of the pulsating beads with that of conventional, nonpulsating beads. These experiments indicate that the pulsating beads significantly accelerate binding rates with minimal increase in nonspecific binding. Thus, pulsing has the potential of significantly reducing assay time.

Behavior of elliptical objects in general theory of relativity

The simplest solution to Einstein's field equations is the Schwarzschild solution. This solution is not able to describe any non-spherical shaped objects. Some stars and galaxies are ellipsoidal. Consequently, the gravitational field around these objects should be different in comparison with the spherical form. This paper is considering a new line element so that we are able to construct not only spherical objects but also we are able to explain an ellipsoidal object too. This new line element is more accurate and complete than the Schwarzschild line element. In this research, we see that the Schwarzschild line element and its solution is only a part of the whole work, which we have done. For more consideration, we applied this metric to an arbitrary object in the next step. Moreover, we used this line element for the solution of a planetary orbit of an ellipsoid planet by using Einstein’s field equations. These equations used for the exterior solution of an ellipsoidal celestial object.

Approximate Kerr-like interior and exterior solutions for a very slowly rotating star of perfect fluid

Approximate Kerr- like interior and exterior solutions for a very slowly rotating star of perfect fluid is presented here which is valid for a very slowly rotating and hence very slightly oblate star of perfect fluid rotating with a very small constant angular velocity. Corresponding approximate exterior metric represents approximate Kerr exterior metric for a perfect fluid. The aim here is to obtain Kerr-like approximate exterior and interior solutions for a very slowly rotating star of perfect fluid simply following standard procedure used to obtain Schwarzschild solutions. Key words: Slowly rotating oblate star, approximate Kerr- like metric, exterior interior solution.

A local characterization for static charged black holes

We obtain a purely local characterization that singles out the Majumdar–Papapetrou class, the near-horizon Bertotti–Robinson geometry and the Reissner–Nordström exterior solution, together with its plane and hyperbolic counterparts, among the static electrovacuum spacetimes. These five classes are found to form the whole set of static Einstein–Maxwell fields without sources and conformally flat space of orbits, that is, the conformastat electrovacuum spacetimes. The main part of the proof consists in showing that a functional relationship between the gravitational and electromagnetic potentials must always exist. The classification procedure also provides an improved characterization of Majumdar–Papapetrou, by only requiring a conformally flat space of orbits with a vanishing Ricci scalar of the usual conveniently rescaled 3-metric. A simple global consideration allows us to state that the asymptotically flat subset of the Majumdar–Papapetrou class and the Reissner–Nordström exterior solution are the only asymptotically flat conformastat electrovacuum spacetimes.

Nonadiabatic charged spherical evolution in the postquasistatic approximation

We apply the postquasistatic approximation, an iterative method for the evolution of self-gravitating spheres of matter, to study the evolution of dissipative and electrically charged distributions in General Relativity. We evolve nonadiabatic distributions assuming an equation of state that accounts for the anisotropy induced by the electric charge. Dissipation is described by streaming out or diffusion approximations. We match the interior solution, in noncomoving coordinates, with the Vaidya-Reissner-Nordstr\"om exterior solution. Two models are considered: i) a Schwarzschild-like shell in the diffusion limit; ii) a Schwarzschild-like interior in the free streaming limit. These toy models tell us something about the nature of the dissipative and electrically charged collapse. Diffusion stabilizes the gravitational collapse producing a spherical shell whose contraction is halted in a short characteristic hydrodynamic time. The streaming out radiation provides a more efficient mechanism for emission of energy, redistributing the electric charge on the whole sphere, while the distribution collapses indefinitely with a longer hydrodynamic time scale.

A two-mass expanding exact space-time solution

In order to understand how locally static configurations around gravitationally bound bodies can be embedded in an expanding universe, we investigate the solutions of general relativity describing a space-time whose spatial sections have the topology of a 3-sphere with two identical masses at the poles. We show that Israel junction conditions imply that two spherically symmetric static regions around the masses cannot be glued together. If one is interested in an exterior solution, this prevents the geometry around the masses to be of the Schwarzschild type and leads to the introduction of a cosmological constant. The study of the extension of the Kottler space-time shows that there exists a non-static solution consisting of two static regions surrounding the masses that match a Kantowski-Sachs expanding region on the cosmological horizon. The comparison with a Swiss-Cheese construction is also discussed.

Functional interactions between membrane-bound transporters and membranes

One key role of many cellular membranes is to hold a transmembrane electrochemical ion gradient that stores free energy, which is used, for example, to generate ATP or to drive transmembrane transport processes. In mitochondria and many bacteria, the gradient is maintained by proton-transport proteins that are part of the respiratory (electron-transport) chain. Even though our understanding of the structure and function of these proteins has increased significantly, very little is known about the specific role of functional protein-membrane and membrane-mediated protein-protein interactions. Here, we have investigated the effect of membrane incorporation on proton-transfer reactions within the membrane-bound proton pump cytochrome c oxidase. The results show that the membrane acts to accelerate proton transfer into the enzyme's catalytic site and indicate that the intramolecular proton pathway is wired via specific amino acid residues to the two-dimensional space defined by the membrane surface. We conclude that the membrane not only acts as a passive barrier insulating the interior of the cell from the exterior solution, but also as a component of the energy-conversion machinery.

Shear viscosity in the postquasistatic approximation

We apply the postquasistatic approximation, an iterative method for the evolution of self-gravitating spheres of matter, to study the evolution of anisotropic nonadiabatic radiating and dissipative distributions in general relativity. Dissipation is described by viscosity and free-streaming radiation, assuming an equation of state to model anisotropy induced by the shear viscosity. We match the interior solution, in noncomoving coordinates, with the Vaidya exterior solution. Two simple models are presented, based on the Schwarzschild and Tolman VI solutions, in the nonadiabatic and adiabatic limit. In both cases, the eventual collapse or expansion of the distribution is mainly controlled by the anisotropy induced by the viscosity.

A rotating cylinder in an asymptotically locally anti-de Sitter background

A family of exact solutions is presented which represents a rigidly rotating cylinder of dust in a background with a negative cosmological constant. The interior of the infinite cylinder is described by the Gödel solution. An exact solution for the exterior region is found which depends both on the rotation of the interior and on its radius. For values of these parameters less than a certain limit, the exterior solution is shown to be locally isomorphic to the Linet–Tian solution. For values larger than another limit, it is shown that the exterior solution extends into a region which contains closed timelike curves. At large distances from the source, the spacetime is shown to be asymptotic locally to anti-de Sitter space.

Bondian frames to couple matter with radiation

A study is presented for the non linear evolution of a self gravitating distribution of matter coupled to a massless scalar field. The characteristic formulation for numerical relativity is used to follow the evolution by a sequence of light cones open to the future. Bondian frames are used to endow physical meaning to the matter variables and to the massless scalar field. Asymptotic approaches to the origin and to infinity are achieved; at the boundary surface interior and exterior solutions are matched guaranteeing the Darmois–Lichnerowicz conditions. To show how the scheme works some numerical models are discussed. We exemplify evolving scalar waves on the following fixed backgrounds: (a) an atmosphere between the boundary surface of an incompressible mixtured fluid and infinity; (b) a polytropic distribution matched to a Schwarzschild exterior; (c) a Schwarzschild–Schwarzschild spacetime. The conservation of energy, the Newman–Penrose constant preservation and other expected features are observed.