Geometrically motivated hyperbolic coordinate conditions for numerical relativity: Analysis, issues and implementations

We study the implications of adopting hyperbolic driver coordinate conditions motivated by geometrical considerations. In particular, conditions that minimize the rate of change of the metric variables. We analyze the properties of the resulting system of equations and their effect when implementing excision techniques. We find that commonly used coordinate conditions lead to a characteristic structure at the excision surface where some modes are not of outflow-type with respect to any excision boundary chosen inside the horizon. Thus, boundary conditions are required for these modes. Unfortunately, the specification of these conditions is a delicate issue as the outflow modes involve both gauge and main variables. As an alternative to these driver equations, we examine conditions derived from extremizing a scalar constructed from Killing's equation and present specific numerical examples.Comment: 9 figure

Geometrical Hyperbolic Systems for General Relativity and Gauge Theories

The evolution equations of Einstein's theory and of Maxwell's theory---the latter used as a simple model to illustrate the former--- are written in gauge covariant first order symmetric hyperbolic form with only physically natural characteristic directions and speeds for the dynamical variables. Quantities representing gauge degrees of freedom [the spatial shift vector $\beta^{i}(t,x^{j})$ and the spatial scalar potential $\phi(t,x^{j})$, respectively] are not among the dynamical variables: the gauge and the physical quantities in the evolution equations are effectively decoupled. For example, the gauge quantities could be obtained as functions of $(t,x^{j})$ from subsidiary equations that are not part of the evolution equations. Propagation of certain (``radiative'') dynamical variables along the physical light cone is gauge invariant while the remaining dynamical variables are dragged along the axes orthogonal to the spacelike time slices by the propagating variables. We obtain these results by $(1)$ taking a further time derivative of the equation of motion of the canonical momentum, and $(2)$ adding a covariant spatial derivative of the momentum constraints of general relativity (Lagrange multiplier $\beta^{i}$) or of the Gauss's law constraint of electromagnetism (Lagrange multiplier $\phi$). General relativity also requires a harmonic time slicing condition or a specific generalization of it that brings in the Hamiltonian constraint when we pass to first order symmetric form. The dynamically propagating gravity fields straightforwardly determine the ``electric'' or ``tidal'' parts of the Riemann tensor.Comment: 24 pages, latex, no figure

Relativistic Lagrange Formulation

It is well-known that the equations for a simple fluid can be cast into what is called their Lagrange formulation. We introduce a notion of a generalized Lagrange formulation, which is applicable to a wide variety of systems of partial differential equations. These include numerous systems of physical interest, in particular, those for various material media in general relativity. There is proved a key theorem, to the effect that, if the original (Euler) system admits an initial-value formulation, then so does its generalized Lagrange formulation.Comment: 34 pages, no figures, accepted in J. Math. Phy

Einstein and Yang-Mills theories in hyperbolic form without gauge-fixing

The evolution of physical and gauge degrees of freedom in the Einstein and Yang-Mills theories are separated in a gauge-invariant manner. We show that the equations of motion of these theories can always be written in flux-conservative first-order symmetric hyperbolic form. This dynamical form is ideal for global analysis, analytic approximation methods such as gauge-invariant perturbation theory, and numerical solution.Comment: 12 pages, revtex3.0, no figure

Future complete spacetimes with spacelike isometry group and field sources

We extend to the Einstein Maxwell Higgs system results first obtained previously in collaboration with V. Moncrief for Einstein equations in vacuum.Comment: to appear in proceedings of the greek relativity meeting 200

A variational analysis of Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds

We establish new existence and non-existence results for positive solutions of the Einstein-scalar field Lichnerowicz equation on compact manifolds. This equation arises from the Hamiltonian constraint equation for the Einstein-scalar field system in general relativity. Our analysis introduces variational techniques, in the form of the mountain pass lemma, to the analysis of the Hamiltonian constraint equation, which has been previously studied by other methods.Comment: 15 page

Heat flow method to Lichnerowicz type equation on closed manifolds

In this paper, we establish existence results for positive solutions to the Lichnerowicz equation of the following type in closed manifolds -\Delta u=A(x)u^{-p}-B(x)u^{q},\quad in\quad M, where $p>1, q>0$, and $A(x)>0$, $B(x)\geq0$ are given smooth functions. Our analysis is based on the global existence of positive solutions to the following heat equation {ll} u_t-\Delta u=A(x)u^{-p}-B(x)u^{q},\quad in\quad M\times\mathbb{R}^{+}, u(x,0)=u_0,\quad in\quad M with the positive smooth initial data $u_0$.Comment: 10 page

The Well-posedness of the Null-Timelike Boundary Problem for Quasilinear Waves

The null-timelike initial-boundary value problem for a hyperbolic system of equations consists of the evolution of data given on an initial characteristic surface and on a timelike worldtube to produce a solution in the exterior of the worldtube. We establish the well-posedness of this problem for the evolution of a quasilinear scalar wave by means of energy estimates. The treatment is given in characteristic coordinates and thus provides a guide for developing stable finite difference algorithms. A new technique underlying the approach has potential application to other characteristic initial-boundary value problems.Comment: Version to appear in Class. Quantum Gra

The locally covariant Dirac field

We describe the free Dirac field in a four dimensional spacetime as a locally covariant quantum field theory in the sense of Brunetti, Fredenhagen and Verch, using a representation independent construction. The freedom in the geometric constructions involved can be encoded in terms of the cohomology of the category of spin spacetimes. If we restrict ourselves to the observable algebra the cohomological obstructions vanish and the theory is unique. We establish some basic properties of the theory and discuss the class of Hadamard states, filling some technical gaps in the literature. Finally we show that the relative Cauchy evolution yields commutators with the stress-energy-momentum tensor, as in the scalar field case.Comment: 36 pages; v2 minor changes, typos corrected, updated references and acknowledgement

M–SURGE: new software specifically designed for multistate capture–recapture models

M–SURGE, al igual que su compañero, el programa U–CARE, se ha escrito con el propósito específico de manejar modelos multiestado de captura–recaptura, lo que a su vez permite mitigar las dificultades inherentes a los mismos (especificación de los modelos, calidad de la convergencia, flexibilidad de parametrización, evaluación del ajuste). En su terreno, M–SURGE abarca una gama de modelos más extensa que un programa general, como el MARK (White & Burnham, 1999), al tiempo que resulta más accesible para el usuario que el MS–SURVIV (Hines, 1994). De entre las principales características del M–SURGE, cabe destacar una amplia gama de modelos y varias parametrizaciones: (1) M–SURGE abarca los modelos condicionales con probabilidad de recaptura según el estado actual (modelos tipo Arnason–Schwarz), y según el estado actual y previo (modelos tipo Jolly–movement). En ambos casos, es posible examinar los efectos dependientes de la edad y/o del tiempo, así como grupos múltiples. (2) Las probabilidades combinadas de supervivencia–transición pueden representarse como tales, o descomponerse en probabilidades de transición y supervivencia. (3) Por lo que respecta a las probabilidades de transición con el mismo estado de partida, el usuario puede elegir libremente la probabilidad que deberá calcularse por sustracción. Además de ser un programa muy accesible para el usuario, también debe subrayarse la facilidad con que permite construir modelos constreñidos utilizando un lenguaje interpretado denominado GEMACO. En este estudio desarrollamos y presentamos varios tipos de modelos multiestado.M–SURGE (along with its companion program U–CARE) has been written specifically to handle multistate capture–recapture models and to alleviate their inherent difficulties (model specification, quality of convergence, flexibility of parameterization, assessment of fit). In its domain, M–SURGE covers a broader range of models than a general program like MARK (White & Burnham, 1999), while being more user–friendly than MS–SURVIV (Hines, 1994). Among the main features of M–SURGE is a wide class of models and a variety of parameterizations: (1) M–SURGE covers conditional models with probability of recapture depending on the current state (Arnason–Schwarz type models) as well as on the current and previous state (Jolly–movement type models). In both cases, age and/or time–dependence and multiple groups can be considered. (2) Combined survival–transition probabilities can be represented as such or decomposed into transition and survival probabilities. (3) Among the transition probabilities with the same state of departure, the one to be computed by subtraction can be freely picked by the user. User–friendliness is enhanced by the easiness with which constrained models are built, using an interpreted language called GEMACO. Examples of various types of multistate models are developed and presented