Omar E. Ortiz

Strongly hyperbolic second order Einstein's evolution equations

BSSN-type evolution equations are discussed. The name refers to the Baumgarte, Shapiro, Shibata, and Nakamura version of the Einstein evolution equations, without introducing the conformal-traceless decomposition but keeping the three connection functions and including a densitized lapse. It is proved that a pseudodifferential first order reduction of these equations is strongly hyperbolic. In the same way, densitized Arnowitt-Deser-Misner evolution equations are found to be weakly hyperbolic. In both cases, the positive densitized lapse function and the spacelike shift vector are arbitrary given fields. This first order pseudodifferential reduction adds no extra equations to the system and so no extra constraints.

Some Mathematical and Numerical Questions Connected with First and Second Order Time-Dependent Systems of Partial Differential Equations

There is a tendency to write the equations of general relativity as a first order symmetric system of time dependent partial differential equations. However, for numerical reasons, it might be advantageous to use a second order formulation like one obtained from the ADM equations. We shall discuss the wellposedness of the Cauchy problem for such systems and their advantage in numerical calculations.

The behavior of hyperbolic heat equations’ solutions near their parabolic limits

Standard energy methods are used to study the relation between the solutions of one parameter families of hyperbolic systems of equations describing heat propagation near their parabolic limits, which for these cases are the usual diffusive heat equation. In the linear case it is proven that given any solution to the hyperbolic equations there is always a solution to the diffusion equation which after a short time stays very close to it for all times. The separation between these solutions depends on the square of the ratio between the assumed very short decay time appearing in Cattaneo’s relation and the usual characteristic smoothing time (initial data dependent) of the limiting diffusive equation. The techniques used in the linear case can be readily used for nonlinear equations. As an example we consider the theories of heat propagation introduced by Coleman, Fabrizio, and Owen, and prove that near a solution to the limiting diffusive equation there is always a solution to the nonlinear hyperbolic equations for a time which usually is much longer than the decay time of the corresponding Cattaneo relation. An alternative derivation of the heat theories of divergence type, which are consistent with thermodynamic principles, is given as an appendix.

Global existence and exponential decay for hyperbolic dissipative relativistic fluid theories

We consider dissipative relativistic fluid theories on a fixed flat, globally hyperbolic, Lorentzian manifold (R×T3,gab). We prove that for all initial data in a small enough neighborhood of the constant equilibrium states (in an appropriate Sobolev norm), the solutions evolve smoothly in time forever and decay exponentially to some, in general undetermined, constant equilibrium state. To prove this, three conditions are imposed on these theories. The first condition requires the system of equations to be symmetric hyperbolic, a fundamental requisite to have a well posed and physically consistent initial value formulation. For the flat space-times considered here the equilibrium states are constant, which is used in the proof. The second condition is a generic consequence of the entropy law, and is imposed on the non-principal part of the equations. The third condition is imposed on the principal part of the equations and it implies that the dissipation affects all the fields of the theory. With these req...

Numerical evidences for the angular momentum-mass inequality for multiple axially symmetric black holes

We present numerical evidences for the validity of the inequality between the total mass and the total angular momentum for multiple axially symmetric (nonstationary) black holes. We use a parabolic heat flow to solve numerically the stationary axially symmetric Einstein equations. As a by-product of our method, we also give numerical evidences that there are no regular solutions of Einstein equations that describe two extreme, axially symmetric black holes in equilibrium.

Inequality between size and charge in spherical symmetry

We prove that for a spherically symmetric charged body two times the radius is always strictly greater than the charge of the body. We also prove that this inequality is sharp. Finally, we discuss the physical implications of this geometrical inequality and present numerical examples that illustrate this theorem.

Size, angular momentum and mass for objects

Fil: Anglada, Pablo Ruben. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Cordoba. Instituto de Fisica Enrique Gaviola. Universidad Nacional de Cordoba. Instituto de Fisica Enrique Gaviola; Argentina. Universidad Nacional de Cordoba. Facultad de Matematica, Astronomia y Fisica. Seccion Fisica. Grupo de Relatividad y Gravitacion; Argentina

Stability of nonconservative hyperbolic systems and relativistic dissipative fluids

A stability theorem for general quasi-linear symmetric hyperbolic systems (not necessarily conservation laws) is proved in this work. The key assumption is the “stability eigenvalue condition,” which requires all the eigenvalues of the constant coefficient system symbol to have negative real part for nonzero Fourier frequency, decaying no faster than |ω|2 when |ω|→0. The decay of the solution to zero, as time grows to infinity, is proved when the space dimension is bigger than or equal to 3. As an application of the general theorem, stability is proved for the equations describing relativistic dissipative fluids.

Exponential Decay Rates in Quasi-Linear Hyperbolic Heat Conduction

We study different exponential decay rates that appear in quasi-linear symmetric hyperbolic systems describing heat conduction models in the context of extended thermodynamics. In normal conditions they are described using two different time scales. We show, after a study of the Cauchy problem for these systems, that for initial data close enough to equilibrium, the solutions exist globally in time and decay exponentially with the shortest relaxation time to the classical (Fouriers theory) solutions; then they continue decaying exponentially to equilibrium with the larger decaying time.