arXiv subjects

Juan A. Valiente Kroon

Publications and source records attributed to Juan A. Valiente Kroon.

At least 37 records · Page 2Linked to original sources

arXiv2020

Improved existence for the characteristic initial value problem with the conformal Einstein field equations

We adapt Luk's analysis of the characteristic initial value problem in General Relativity to the asymptotic characteristic problem for the conformal Einstein field equations to demonstrate the local existence of solutions in a neighbourhood of the set on which the data are given. In particular, we obtain existence of solutions along a narrow rectangle along null infinity which, in turn, corresponds to an infinite domain in the asymptotic region of the physical spacetime. This result generalises work by Kánnár on the local existence of solutions to the characteristic initial value problem by means of Rendall's reduction strategy. In analysing the conformal Einstein equations we make use of the Newman-Penrose formalism and a gauge due to J. Stewart.

gr-qc↗

arXiv2020

The Conformal Einstein Field Equations with Massless Vlasov Matter

We prove the stability of de Sitter space-time as a solution to the Einstein-Vlasov system with massless particles. The semi-global stability of Minkowski space-time is also addressed. The proof relies on conformal techniques, namely Friedrich's conformal Einstein field equations. We exploit the conformal invariance of the massless Vlasov equation on the cotangent bundle and adapt Kato's local existence theorem for symmetric hyperbolic systems to prove a long enough time of existence for solutions of the evolution system implied by the Vlasov equation and the conformal Einstein field equations.

gr-qc↗

arXiv2019

Revisiting the characteristic initial value problem for the vacuum Einstein field equations

Using the Newman-Penrose formalism we study the characteristic initial value problem in vacuum General Relativity. We work in a gauge suggested by Stewart, and following the strategy taken in the work of Luk, demonstrate local existence of solutions in a neighbourhood of the set on which data are given. These data are given on intersecting null hypersurfaces. Existence near their intersection is achieved by combining the observation that the field equations are symmetric hyperbolic in this gauge with the results of Rendall. To obtain existence all the way along the null-hypersurfaces themselves, a bootstrap argument involving the Newman-Penrose variables is performed.

gr-qc↗

arXiv2019

Conformal wave equations for the Einstein-tracefree matter system

Inspired by a similar analysis for the vacuum conformal Einstein field equations by Paetz [Ann. H. Poincaré 16, 2059 (2015)], in this article we show how to construct a system of quasilinear wave equations for the geometric fields associated to the conformal Einstein field equations coupled to matter models whose energy-momentum tensor has vanishing trace. In this case, the equation of conservation for the energy-momentum tensor is conformally invariant. Our analysis includes the construction of a subsidiary evolution system which allows to prove the propagation of the constraints. We discuss how the underlying structure behind these systems of equations is the integrability conditions satisfied by the conformal field equations. The main result of our analysis is that both the evolution and subsidiary equations for the geometric part of the conformal Einstein-tracefree matter field equations close without the need of any further assumption on the matter models other than the vanishing of the trace of the energy-momentum tensor. Our work is supplemented by an analysis of the evolution and subsidiary equations associated to three basic tracefree matter models: the conformally invariant scalar field, the Maxwell field and the Yang-Mills field. As an application we provide a global existence and stability result for de Sitter-like spacetimes. In particular, the result for the conformally coupled scalar field is new in the literature.

gr-qc↗

arXiv2019

Construction of anti-de Sitter-like spacetimes using the metric conformal Einstein field equations: the tracefree matter case

Using a metric conformal formulation of the Einstein equations, we develop a construction of 4-dimensional anti-de Sitter-like spacetimes coupled to tracefree matter models. Our strategy relies on the formulation of an initial-boundary problem for a system of quasilinear wave equations for various conformal fields by exploiting the conformal and coordinate gauges. By analysing the conformal constraints we show a systematic procedure to prescribe initial and boundary data. This analysis is complemented by the propagation of the constraints, showing that a solution to the wave equations implies a solution to the Einstein field equations. In addition, we study three explicit tracefree matter models: the conformally invariant scalar field, the Maxwell field and the Yang-Mills field. For each one of these we identify the basic data required to couple them to the system of wave equations. As our main result, we establish the local existence and uniqueness of solutions for the evolution system in a neighbourhood around the corner, provided compatibility conditions for the initial and boundary data are imposed up to a certain order.

gr-qc↗

arXiv2018

Construction of anti-de Sitter-like spacetimes using the metric conformal Einstein field equations: the vacuum case

We make use of the metric version of the conformal Einstein field equations to construct anti-de Sitter-like spacetimes by means of a suitably posed initial-boundary value problem. The evolution system associated to this initial-boundary value problem consists of a set of conformal wave equations for a number of conformal fields and the conformal metric. This formulation makes use of generalised wave coordinates and allows the free specification of the Ricci scalar of the conformal metric via a conformal gauge source function. We consider Dirichlet boundary conditions for the evolution equations at the conformal boundary and show that these boundary conditions can, in turn, be constructed from the 3-dimensional Lorentzian metric of the conformal boundary and a linear combination of the incoming and outgoing radiation as measured by certain components of the Weyl tensor. To show that a solution to the conformal evolution equations implies a solution to the Einstein field equations we also provide a discussion of the propagation of the constraints for this initial-boundary value problem. The existence of local solutions to the initial-boundary value problem in a neighbourhood of the corner where the initial hypersurface and the conformal boundary intersect is subject to compatibility conditions between the initial and boundary data. The construction described is amenable to numerical implementation and should allow the systematic exploration of boundary conditions.

gr-qc↗

arXiv2018

Killing boundary data for anti-de Sitter-like spacetimes

Given an initial-boundary value problem for an anti-de Sitter-like spacetime, we analyse conditions on the conformal boundary ensuring the existence of Killing vectors in the spacetime arising from this problem. This analysis makes use of a system of conformal wave equations describing the propagation of the Killing equation first considered by Paetz. We identify an obstruction tensor constructed from Killing vector candidate and the Cotton tensor of the conformal boundary whose vanishing is a necessary condition for the existence of Killing vectors in the spacetime. This obstruction tensor vanishes if the conformal boundary is conformally flat.

gr-qc↗

arXiv2018

Spectral methods for the spin-2 equation near the cylinder at spatial infinity

We solve, numerically, the massless spin-2 equations, written in terms of a gauge based on the properties of conformal geodesics, in a neighbourhood of spatial infinity using spectral methods in both space and time. This strategy allows us to compute the solutions to these equations up to the critical sets where null infinity intersects with spatial infinity. Moreover, we use the convergence rates of the numerical solutions to read-off their regularity properties.

gr-qc↗

arXiv2017

Conformal geodesics in spherically symmetric vacuum spacetimes with Cosmological constant

An analysis of conformal geodesics in the Schwarzschild-de Sitter and Schwarzschild-anti de Sitter families of spacetimes is given. For both families of spacetimes we show that initial data on a spacelike hypersurface can be given such that the congruence of conformal geodesics arising from this data cover the whole maximal extension of canonical conformal representations of the spacetimes without forming caustic points. For the Schwarzschild-de Sitter family, the resulting congruence can be used to obtain global conformal Gaussian systems of coordinates of the conformal representation. In the case of the Schwarzschild-anti de Sitter family, the natural parameter of the curves only covers a restricted time span so that these global conformal Gaussian systems do not exist.

gr-qc↗

arXiv2017

The Einstein-Friedrich-nonlinear scalar field system and the stability of scalar field Cosmologies

A frame representation is used to derive a first order quasi-linear symmetric hyperbolic system for a scalar field minimally coupled to gravity. This procedure is inspired by similar evolution equations introduced by Friedrich to study the Einstein-Euler system. The resulting evolution system is used to show that small nonlinear perturbations of expanding Friedman-Lemaître-Robertson-Walker backgrounds, with scalar field potentials satisfying certain future asymptotic conditions, decay exponentially to zero, in synchronous time.

gr-qc↗

arXiv2017

Polyhomogeneous expansions from time symmetric initial data

We make use of Friedrich's construction of the cylinder at spatial infinity to relate the logarithmic terms appearing in asymptotic expansions of components of the Weyl tensor to the freely specifiable parts of time symmetric initial data sets for the Einstein field equations. Our analysis is based on the assumption that a particular type of formal expansions near the cylinder at spatial infinity corresponds to the leading terms of actual solutions to the Einstein field equations. In particular, we show that if the Bach tensor of the initial conformal metric does not vanish at the point at infinity then the most singular component of the Weyl tensor decays near null infinity as $O(\tilde{r}^{-3}\ln \tilde{r})$ so that spacetime will not peel. We also provide necessary conditions on the initial data which should lead to a peeling spacetime. Finally, we show how to construct global spacetimes which are candidates for non-peeling polyhomogeneous) asymptotics.

gr-qc↗

arXiv2017

Dain's invariant on non-time symmetric initial data sets

We extend Dain's construction of a geometric invariant characterising static initial data sets for the vacuum Einstein field equations to situations with a non-vanishing extrinsic curvature. This invariant gives a measure of how much the initial data sets deviates from stationarity. In particular, it vanishes if and only if the initial data set is stationary. Thus, the invariant provides a quantification of the amount of gravitational radiation contained in the initial data set.

gr-qc↗

arXiv2016

A geometric invariant characterising initial data for the Kerr-Newman spacetime

We describe the construction of a geometric invariant characterising initial data for the Kerr-Newman spacetime. This geometric invariant vanishes if and only if the initial data set corresponds to exact Kerr-Newman initial data, and so characterises this type of data. We first illustrate the characterisation of the Kerr-Newman spacetime in terms of Killing spinors. The space spinor formalism is then used to obtain a set of four independent conditions on an initial Cauchy hypersurface that guarantee the existence of a Killing spinor on the development of the initial data. Following a similar analysis in the vacuum case, we study the properties of solutions to the approximate Killing spinor equation and use them to construct the geometric invariant.

gr-qc↗

arXiv2016

Killing spinors as a characterisation of rotating black hole spacetimes

We investigate the implications of the existence of Killing spinors in a spacetime. In particular, we show that in vacuum and electrovacuum a Killing spinor, along with some assumptions on the associated Killing vector in an asymptotic region, guarantees that the spacetime is locally isometric to the Kerr or Kerr-Newman solutions. We show that the characterisation of these spacetimes in terms of Killing spinors is an alternative expression of characterisation results of Mars (Kerr) and Wong (Kerr-Newman) involving restrictions on the Weyl curvature and matter content.

gr-qc↗

arXiv2015

On the locally rotationally symmetric Einstein-Maxwell perfect fluid

We examine the stability of an Einstein-Maxwell perfect fluid configuration with a privileged direction of symmetry by means of a $1+1+2$-tetrad formalism. We use this formalism to cast, in a quasi linear symmetric hyperbolic form the equations describing the evolution of the system. This hyperbolic reduction is used to discuss the stability of solutions of the linear perturbation. By restricting the analysis to isotropic fluid configurations, we made use of a constant electrical conductivity coefficient for the fluid (plasma), and the nonlinear stability for the case of an infinitely conducting plasma is also considered. As a result of this analysis we provide a complete classification and characterization of various stable and unstable configurations. We found in particular that in many cases the stability conditions is strongly determined by the constitutive equations by means of the square of the velocity of sound and the electric conductivity, and a threshold for the emergence of the instability appears in both contracting and expanding systems.

astro-ph.HE↗

arXiv2015

A formalism for the calculus of variations with spinors

We develop a frame and dyad gauge-independent formalism for the calculus of variations of functionals involving spinorial objects. As part of this formalism we define a modified variation operator which absorbs frame and spin dyad gauge terms. This formalism is applicable to both the standard spacetime (i.e. SL(2,C)) 2-spinors as well as to space (i.e. SU(2,C)) 2-spinors. We compute expressions for the variations of the connection and the curvature spinors.

gr-qc↗

arXiv2013

On the conformal structure of the extremal Reissner-Nordström spacetime

We analyse various conformal properties of the extremal Reissner-Nordström spacetime. In particular, we obtain conformal representations of the neighbourhoods of spatial infinity, timelike infinity and the cylindrical end ---the so-called cylinders at spatial infinity and at the horizon, respectively--- which are regular with respect to the conformal Einstein field equations and their associated initial data sets. We discuss possible implications of these constructions for the propagation of test fields and non-linear perturbations of the gravitational field close to the horizon.

gr-qc↗

arXiv2013

On the evolution equations for a self-gravitating charged scalar field

We consider a complex scalar field minimally coupled to gravity and to a U(1) gauge symmetry and we construct of a first order symmetric hyperbolic evolution system for the Einstein-Maxwell-Klein-Gordon system. Our analysis is based on a 1+3 tetrad formalism which makes use of the components of the Weyl tensor as one of the unknowns. In order to ensure the symmetric hyperbolicity of the evolution equations, implied by the Bianchi identity, we introduce a tensor of rank 3 corresponding to the covariant derivative of the Faraday tensor, and two tensors of rank 2 for the covariant derivative of the vector potential and the scalar field.

gr-qc↗