Academic literature on the topic 'Einstein constraint system'

We study different dimensional fluids inspired by noncommutative geometry which admit conformal Killing vector (CKV). The solutions of the Einstein field equations were examined specifically for five different set of spacetime. We calculate the active gravitational mass and impose stability conditions of the fluid sphere. The analysis thus carried out immediately indicates that at four dimension only one can get a stable configuration for any spherically symmetric stellar system and any other dimension, lower or higher, becomes untenable as far as the stability of a system is concerned.

The recent evolution of the observational technics and the development of new tools in cosmology and gravitation have a significant impact on the study of the cosmological models. In particular, the qualitative and numerical methods used in dynamical system and elsewhere, enable the resolution of some difficult problems and allow the analysis of different cosmological models even with a limited number of symmetries. On the other hand, following Einstein point of view the manifold [Formula: see text] and the metric should be built simultaneously when solving Einstein’s equation [Formula: see text]. From this point of view, the only kinematic condition imposed is that at each point of space–time, the tangent space is endowed with a metric (which is a Minkowski metric in the physical case of pseudo-Riemannian manifolds and an Euclidean one in the Riemannian analogous problem). Then the field [Formula: see text] describes the way these metrics depend on the point in a smooth way and the Einstein equation is the “dynamical” constraint on [Formula: see text]. So, we have to imagine an infinite continuous family of copies of the same Minkowski or Euclidean space and to find a way to sew together these infinitesimal pieces into a manifold, by respecting Einstein’s equation. Thus, Einstein field equations do not fix once and for all the global topology. [Formula: see text] Given this freedom in the topology of the space–time manifold, a question arises as to how free the choice of these topologies may be and how one may hope to determine them, which in turn is intimately related to the observational consequences of the space–time possessing nontrivial topologies. Therefore, in this paper we will use a different qualitative dynamical methods to determine the actual topology of the space–time.

For the measurement of two-pion Bose–Einstein correlations (BEC) in reactions with [Formula: see text] final state particles, a new event-mixing technique is proposed. The new technique introduces a new mixing constraint, namely energy hierarchy correspondence (EHC) cut, which requires that two bosons being swapped should be equal in energy hierarchy in their original events. Numerical tests are performed to check the validity of the new mixing method. Compared to the previous proposed mixing technique (He et al., Chin. Phys. C 42 (2018) 074004), this new method has smaller systematic uncertainties, improving the accuracy of fit BEC parameters.

We construct non-compactness examples for the fully coupled Einstein–Lichnerowicz constraint system in the focusing case. The construction is obtained by combining pointwise a priori asymptotic analysis techniques, finite-dimensional reductions and a fixed-point argument. More precisely, we perform a fixed-point procedure on the remainders of the expected blow-up decomposition. The argument consists of an involved finite-dimensional reduction coupled with a ping-pong method. To overcome the non-variational structure of the system, we work with remainders which belong to strong function spaces and not merely to energy spaces. Performing both the ping-pong argument and the finite-dimensional reduction therefore heavily relies on the a priori pointwise asymptotic techniques of the [Formula: see text] theory.

In General Relativity, the Einstein field equations allow us to study the evolution of a spacelike 3-manifold, provided that its metric and extrinsic curvature satisfy a system of geometric constraint equations. The so-called Einstein constraint equations, arise as a consequence of the fact that the 3-manifold in question is necessarily a submanifold of the spacetime its evolution defines. This thesis is devoted to a study of the structure of the Einstein constraint system in the special case when the spacelike 3-manifold also satisfies the quasispherical ansatz of Bartnik [B93]. We make no mention of the generality of this gauge; the extent to which the quasispherical ansatz applies remains an open problem. After imposing the quasispherical gauge, we give an argument to show that the resulting Einstein constraint system may be viewed as a coupled system of partial differential equations for the parameters describing the metric and second fundamental form. The hencenamed quasisperical Einstein constraint system, consists of a parabolic equation, a first order elliptic system and (essentially) a system of ordinary differential equations. The question of existence of solutions to this system naturally arises and we provide a partial answer to this question. We give conditions on the initial data and prescribable fields under which we may conclude that the quasispherical Einstein constraint system is uniquley solvable, at least in a region surrounding the unit sphere. The proof of this fact is centred on a linear iterative system of partial differential equations, which also consist of a parabolic equation, a first order elliptic system and a system of ordinary differential equations. We prove that this linear system consistently defines a sequence, and show via a contraction mapping argument, that this sequence must converge to a fixed point of the iteration. The iteration, however, has been specifically designed so that any fixed point of the iteration coincides with a solution of the quasispherical Einstein constraints. The contraction mapping argument mentioned above, relies heavily on a priori estimates for the solutions of linear parabolic equations. We generalise and extend known results 111 concerning parabolic equations to establish special a priori estimates which relate a useful property: the L2-Sobolev regularity of the solution of a parabolic equation is greater than that of the coefficients of the elliptic operator, provided that the initial data is sufficiently regular. This 'smoothing' property of linear parabolic equations along with several estimates from elliptic and ordinary differential equation theory form the crucial ingredients needed in the proof of the existence of a fixed point of the iteration. We begin in chapter one by giving a brief review of the extensive literature concerning the initial value problem in General Relativity. We go on, after mentioning two of the traditional methods for constructing spacetime initial data, to introduce the notion of a quasispherical foliation of a 3-manifold and present the Einstein constraint system written in terms of this gauge. In chapter two we introduce the various inequalities and tracts of analysis we will make use of in subsequent chapters. In particular we define the, perhaps not so familiar, complex differential operator 9 (edth) of Newman and Penrose. In chapter three we develop the appropriate Sobolev-regularity theory for linear parabolic equations required to deal with the quasispherical initial data constraint equations. We include a result due to Polden [P] here, with a corrected proof. This result was essential for deriving the results contained in the later chapters of [P], and it is for this reason we include the result. We don't make use of it explicitly when considering the quasispherical Einstein constraints, but the ideas employed are similar to those we use to tackle the problem of existence for the quasispherical constraints. Chapter four is concerned with the local existence of quasispherical initial data. We firstly consider the question of existence and uniqueness when the mean curvature of the 3-manifold is prescribed, then after introducing the notion of polar curvature, we also present another quasispherical constraint system in which we consider the polar curvature as prescribed. We prove local existence and uniqueness results for both of these alternate formulations of the quasispherical constraints. This thesis was typeset using LATEXwith the package amssymb.

Corvino, Corvino and Schoen, Chruściel and Delay have shown the existence of a large class of asymptotically flat vacuum initial data for Einstein's field equations which are static or stationary in a neighborhood of space-like infinity, yet quite general in the interior. The proof relies on some abstract, non-constructive arguments which makes it difficult to calculate such data numerically by using similar arguments. A quasilinear elliptic system of equations is presented of which we expect that it can be used to construct vacuum initial data which are asymptotically flat, time-reflection symmetric, and asymptotic to static data up to a prescribed order at space-like infinity. A perturbation argument is used to show the existence of solutions. It is valid when the order at which the solutions approach staticity is restricted to a certain range. Difficulties appear when trying to improve this result to show the existence of solutions that are asymptotically static at higher order. The problems arise from the lack of surjectivity of a certain operator. Some tensor decompositions in asymptotically flat manifolds exhibit some of the difficulties encountered above. The Helmholtz decomposition, which plays a role in the preparation of initial data for the Maxwell equations, is discussed as a model problem. A method to circumvent the difficulties that arise when fast decay rates are required is discussed. This is done in a way that opens the possibility to perform numerical computations. The insights from the analysis of the Helmholtz decomposition are applied to the York decomposition, which is related to that part of the quasilinear system which gives rise to the difficulties. For this decomposition analogous results are obtained. It turns out, however, that in this case the presence of symmetries of the underlying metric leads to certain complications. The question, whether the results obtained so far can be used again to show by a perturbation argument the existence of vacuum initial data which approach static solutions at infinity at any given order, thus remains open. The answer requires further analysis and perhaps new methods.
Corvino, Corvino und Schoen als auch Chruściel und Delay haben die Existenz einer grossen Klasse asymptotisch flacher Anfangsdaten für Einsteins Vakuumfeldgleichungen gezeigt, die in einer Umgebung des raumartig Unendlichen statisch oder stationär aber im Inneren der Anfangshyperfläche sehr allgemein sind. Der Beweis beruht zum Teil auf abstrakten, nicht konstruktiven Argumenten, die Schwierigkeiten bereiten, wenn derartige Daten numerisch berechnet werden sollen. In der Arbeit wird ein quasilineares elliptisches Gleichungssystem vorgestellt, von dem wir annehmen, dass es geeignet ist, asymptotisch flache Vakuumanfangsdaten zu berechnen, die zeitreflektionssymmetrisch sind und im raumartig Unendlichen in einer vorgeschriebenen Ordnung asymptotisch zu statischen Daten sind. Mit einem Störungsargument wird ein Existenzsatz bewiesen, der gilt, solange die Ordnung, in welcher die Lösungen asymptotisch statische Lösungen approximieren, in einem gewissen eingeschränkten Bereich liegt. Versucht man, den Gültigkeitsbereich des Satzes zu erweitern, treten Schwierigkeiten auf. Diese hängen damit zusammen, dass ein gewisser Operator nicht mehr surjektiv ist. In einigen Tensorzerlegungen auf asymptotisch flachen Räumen treten ähnliche Probleme auf, wie die oben erwähnten. Die Helmholtzzerlegung, die bei der Bereitstellung von Anfangsdaten für die Maxwellgleichungen eine Rolle spielt, wird als ein Modellfall diskutiert. Es wird eine Methode angegeben, die es erlaubt, die Schwierigkeiten zu umgehen, die auftreten, wenn ein schnelles Abfallverhalten des gesuchten Vektorfeldes im raumartig Unendlichen gefordert wird. Diese Methode gestattet es, solche Felder auch numerisch zu berechnen. Die Einsichten aus der Analyse der Helmholtzzerlegung werden dann auf die Yorkzerlegung angewandt, die in den Teil des quasilinearen Systems eingeht, der Anlass zu den genannten Schwierigkeiten gibt. Für diese Zerlegung ergeben sich analoge Resultate. Es treten allerdings Schwierigkeiten auf, wenn die zu Grunde liegende Metrik Symmetrien aufweist. Die Frage, ob die Ergebnisse, die soweit erhalten wurden, in einem Störungsargument verwendet werden können um die Existenz von Vakuumdaten zu zeigen, die im räumlich Unendlichen in jeder Ordnung statische Daten approximieren, bleibt daher offen. Die Antwort erfordert eine weitergehende Untersuchung und möglicherweise auch neue Methoden.

The notion of the formless found a lasting definition in Documents, the dissident Surrealist magazine led by Georges Bataille, Carl Einstein and Michel Leiris from 1929 to 1931. In an unassuming short entry for its ‘Dictionnaire’, Bataille presents the informe emphatically not as a system or a structure, but as ‘un terme servant à déclasser’; yet neither the disruptive impulse of the 'Dictionnaire', nor the more recent exhibitions it has generated, can avoid a measure of taxonomic organisation (L'Informe: mode d'emploi, 1996; Undercover Surrealism, 2006). In the realm of poetry, free verse has eroded the boundaries of the poetic, but its freedom from formal constraints is limited too; as Jay Parini (2008) contends, ‘formless poetry does not really exist, as poets inevitably create patterns in language that replicate forms of experience.’ Through a small number of case studies, this chapter will consider the legacy of Bataille’s definition while assessing the ongoing tension between form and its undoing in textual and visual art of the twenty-first century.

Statistical mechanics is clearly mechanics (classical, quantum, special or general relativistic, or any other) plus the theory of probabilities, as is well known. It is our understanding, however, that it is more than that. It is also the adoption of a specific entropic functional, which will, in some sense, adequately shortcut the vast, and for most practical purposes useless, detailed microscopic mechanical information on the system. It is, in particular, through this functional that the connection with thermodynamics and its macroscopic laws will be established. This particular functional is determined by the specific type (or geometry) of occupation of the phase space (or Hilbert space or analogous space). This geometrical structure depends in turn not only on the microscopic dynamics that the system obeys, but also on the initial conditions at which the system is placed at t = 0. In colloquial terms, we could say that the microscopic dynamics determine where the system is allowed to live, whereas the initial conditions determin where it likes to live within the allowed region. This viewpoint is consistent with Einstein's perspective on classical statistical mechanics, and especially with his criticism [82, 92] of the celebrated Boltzmann principle However, the problem is that, up to now, no systematic manner exists for univocally determining the entropic functional to be used, given the dynamics and the initial conditions. The optimization of this entropy under the physically appropriate constraints is expected to provide the correct probability distribution for the microscopic states of the macroscopic stationary state of the system. Boltzmann, then complemented by Gibbs, proposed the celebrated form which is the foundation of standard statistical mechanics.

This paper presents an algorithm for modeling the dynamics of multi-rigid body systems in generalized topologies including topologies with closed kinematic loops that may or may not be coupled together. The algorithm uses a hierarchic assembly disassembly process in parallel implementation and a recursive assembly disassembly process in serial implementation to achieve highly efficient simulation turn-around times. The kinematic constraints are imposed using the formalism of kinematic joints that are modeled using motion spaces and their orthogonal complements. A mixed set of coordinates are used viz. absolute coordinates to develop the equations of motion and internal or relative coordinates to impose the constraints. The equations of motion are posed in terms of operational inertias to capture the nonlinear coupling between the dynamics of the individual components of the system. Einstein’s notation is used to explain the generality of the approach. Constraint impositions at the acceleration, velocity and configuration levels are discussed. The application of this algorithm in modeling a complex micro robot with multiple coupled closed loops is discussed.