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Jeanphilippe Uzan - One of the best experts on this subject based on the ideXlab platform.
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on the trace free Einstein Equations as a viable alternative to general relativity
Classical and Quantum Gravity, 2011Co-Authors: George F R Ellis, Henk Van Elst, Jeff Murugan, Jeanphilippe UzanAbstract:The quantum field theoretical prediction for the vacuum energy density leads to a value for the effective cosmological constant that is incorrect by between 60 and 120 orders of magnitude. We review an old proposal of replacing Einstein’s field Equations by their trace-free part (the trace-free Einstein Equations), together with an independent assumption of energy–momentum conservation by matter fields. While this does not solve the fundamental issue of why the cosmological constant has the value that is observed cosmologically, it is indeed a viable theory that resolves the problem of the discrepancy between the vacuum energy density and the observed value of the cosmological constant. However, one has to check that, as well as preserving the standard cosmological Equations, this does not destroy other predictions, such as the junction conditions that underlie the use of standard stellar models. We confirm that no problems arise here: hence, the trace-free Einstein Equations are indeed viable for cosmological and astrophysical applications.
Simonetta Frittelli - One of the best experts on this subject based on the ideXlab platform.
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well posed first order reduction of the characteristic problem of the linearized Einstein Equations
Physical Review D, 2005Co-Authors: Simonetta FrittelliAbstract:A choice of first-order variables for the characteristic problem of the linearized Einstein Equations is found which casts the system into manifestly well-posed form. The concept of well-posedness for characteristic problems invoked is that there exists an \textit{a priori} estimate of the solution of the characteristic problem in terms of the data. The notion of manifest well-posedness consists of an algebraic criterion sufficient for the existence of the estimates, and is to characteristic problems as symmetric hyperbolicity is to Cauchy problems. Both notions have been made precise elsewere.
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Einstein boundary conditions in relation to constraint propagation for the initial boundary value problem of the Einstein Equations
Physical Review D, 2004Co-Authors: Simonetta Frittelli, Roberto GomezAbstract:We show how the use of the normal projection of the Einstein tensor as a set of boundary conditions relates to the propagation of the constraints, for two representations of the Einstein Equations with vanishing shift vector: the Arnowitt-Deser-Misner formulation, which is ill posed, and the Einstein-Christoffel formulation, which is symmetric hyperbolic. Essentially, the components of the normal projection of the Einstein tensor that act as nontrivial boundary conditions are linear combinations of the evolution Equations with the constraints that are not preserved at the boundary, in both cases. In the process, the relationship of the normal projection of the Einstein tensor to the recently introduced ``constraint-preserving'' boundary conditions becomes apparent.
Olindo Zanotti - One of the best experts on this subject based on the ideXlab platform.
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conformal and covariant z4 formulation of the Einstein Equations strongly hyperbolic first order reduction and solution with discontinuous galerkin schemes
Physical Review D, 2018Co-Authors: Michael Dumbser, Federico Guercilena, Luciano Rezzolla, Sven Koppel, Olindo ZanottiAbstract:We present a strongly hyperbolic first-order formulation of the Einstein Equations based on the conformal and covariant Z4 system (CCZ4) with constraint-violation damping, which we refer to as FO-CCZ4. As CCZ4, this formulation combines the advantages of a conformal and traceless formulation, with the suppression of constraint violations given by the damping terms, but being first order in time and space, it is particularly suited for a discontinuous Galerkin (DG) implementation. The strongly hyperbolic first-order formulation has been obtained by making careful use of first and second-order ordering constraints. A proof of strong hyperbolicity is given for a selected choice of standard gauges via an analytical computation of the entire eigenstructure of the FO-CCZ4 system. The resulting governing partial differential Equations system is written in non-conservative form and requires the evolution of 58 unknowns. A key feature of our formulation is that the first-order CCZ4 system decouples into a set of pure ordinary differential Equations and a reduced hyperbolic system of partial differential Equations that contains only linearly degenerate fields. We implement FO-CCZ4 in a high-order path-conservative arbitrary-high-order-method-using-derivatives (ADER)-DG scheme with adaptive mesh refinement and local time-stepping, supplemented with a third-order ADER-WENO subcell finite-volume limiter in order to deal with singularities arising with black holes. We validate the correctness of the formulation through a series of standard tests in vacuum, performed in one, two and three spatial dimensions, and also present preliminary results on the evolution of binary black-hole systems. To the best of our knowledge, these are the first successful three-dimensional simulations of moving punctures carried out with high-order DG schemes using a first-order formulation of the Einstein Equations.
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a strongly hyperbolic first order ccz4 formulation of the Einstein Equations and its solution with discontinuous galerkin schemes
arXiv: General Relativity and Quantum Cosmology, 2017Co-Authors: Michael Dumbser, Federico Guercilena, Sven Koeppel, Luciano Rezzolla, Olindo ZanottiAbstract:We present a strongly hyperbolic first-order formulation of the Einstein Equations based on the conformal and covariant Z4 system (CCZ4) with constraint-violation damping, which we refer to as FO-CCZ4. As CCZ4, this formulation combines the advantages of a conformal and traceless formulation, with the suppression of constraint violations given by the damping terms, but being first order in time and space, it is particularly suited for a discontinuous Galerkin (DG) implementation. The strongly hyperbolic first-order formulation has been obtained by making careful use of first and second-order ordering constraints. A proof of strong hyperbolicity is given for a selected choice of gauges via an analytical computation of the entire eigenstructure of the FO-CCZ4 system. The resulting governing partial differential Equations system is written in non-conservative form and requires the evolution of 58 unknowns. A key feature of our formulation is that the first-order CCZ4 system decouples into a set of pure ordinary differential Equations and a reduced hyperbolic system of partial differential Equations that contains only linearly degenerate fields. We implement FO-CCZ4 in a high-order path-conservative arbitrary-high-order-method-using-derivatives (ADER)-DG scheme with adaptive mesh refinement and local time-stepping, supplemented with a third-order ADER-WENO subcell finite-volume limiter in order to deal with singularities arising with black holes. We validate the correctness of the formulation through a series of standard tests in vacuum, performed in one, two and three spatial dimensions, and also present preliminary results on the evolution of binary black-hole systems. To the best of our knowledge, these are the first successful three-dimensional simulations of moving punctures carried out with high-order DG schemes using a first-order formulation of the Einstein Equations.
Cecile Huneau - One of the best experts on this subject based on the ideXlab platform.
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stability in exponential time of minkowski space time with a translation space like killing field
Annals of PDE, 2016Co-Authors: Cecile HuneauAbstract:In this paper, we prove the nonlinear stability in exponential time of Minkowki space-time with a translation space-like Killing field. In the presence of such a symmetry, the \(3\,+\,1\) vacuum Einstein Equations reduce to the \(2+1\) Einstein Equations with a scalar field. We work in generalised wave coordinates. In this gauge the Einstein Equations can be written as a system of quasilinear quadratic wave Equations. The main difficulty in this paper is due to the decay in \(\frac{1}{\sqrt{t}}\) of free solutions to the wave equation in 2 dimensions, which is weaker than in 3 dimensions. As in [21], we have to rely on the particular structure of the Einstein Equations in wave coordinates. We also have to carefully choose the behaviour of our metric in the exterior region to enforce convergence to Minkowski space-time at time-like infinity.
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constraint Equations for 3 1 vacuum Einstein Equations with a translational space like killing field in the asymptotically flat case
Annales Henri Poincaré, 2016Co-Authors: Cecile HuneauAbstract:We solve the Einstein constraint Equations for a 3 + 1- dimensional vacuum space–time with a space-like translational Killing field. The presence of a space-like translational Killing field allows for a reduction of the 3 + 1-dimensional problem to a 2 + 1-dimensional one. Vacuum Einstein Equations with a space-like translational Killing field have been studied by Choquet-Bruhat and Moncrief in the compact case. In the case where an additional rotational symmetry is added, the problem has a long history. In this paper we consider the asymptotically flat case. This corresponds to solving a nonlinear elliptic system on \({\mathbb{R}^2}\) . The main difficulty in that case is due to the delicate inversion of the Laplacian on \({\mathbb{R}^2}\) . In particular, we have to work in the non-constant mean curvature setting, which enforces us to consider the intricate coupling of the Einstein constraint Equations.
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stability in exponential time of minkowski space time with a translation space like killing field
arXiv: Analysis of PDEs, 2014Co-Authors: Cecile HuneauAbstract:In this paper, we prove the nonlinear stability in exponential time of Minkowki space-time with a translation space-like Killing field. In the presence of such a symmetry, the 3 + 1 vacuum Einstein Equations reduce to the 2 + 1 Einstein Equations with a scalar field. We work in generalised wave coordinates. In this gauge Einstein Equations can be written as a system of quasilinear quadratic wave Equations. The main difficulty in this paper is due to the decay in $\frac{1}{\sqrt{t}}$ of free solutions to the wave equation in 2 dimensions, which is weaker than in 3 dimensions. We have to rely on the particular structure of Einstein Equations in wave coordinates. We also have to carefully choose the behaviour of our metric in the exterior region to enforce convergence to Minkowski space-time at time-like infinity.
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constraint Equations for 3 1 vacuum Einstein Equations with a translational space like killing field in the asymptotically flat case ii
arXiv: Analysis of PDEs, 2013Co-Authors: Cecile HuneauAbstract:We solve the constraint Equations for a vacuum space-time with a translational space-like Killing field satisfying the vacuum Einstein Equations. Vacuum Einstein Equations with a translational space-like Killing field have been studied by Choquet-Bruhat and Moncrief in the compact case, and by Ashtekar, Bicak and Schmidt in the case where an additional spherical symmetry is added. In this paper we consider the asymptotically flat case. This corresponds to solving a nonlinear elliptic system on R2. The main difficulty in that case is due to the delicate inversion of the Laplacian on R2.
Roberto Gomez - One of the best experts on this subject based on the ideXlab platform.
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Einstein boundary conditions in relation to constraint propagation for the initial boundary value problem of the Einstein Equations
Physical Review D, 2004Co-Authors: Simonetta Frittelli, Roberto GomezAbstract:We show how the use of the normal projection of the Einstein tensor as a set of boundary conditions relates to the propagation of the constraints, for two representations of the Einstein Equations with vanishing shift vector: the Arnowitt-Deser-Misner formulation, which is ill posed, and the Einstein-Christoffel formulation, which is symmetric hyperbolic. Essentially, the components of the normal projection of the Einstein tensor that act as nontrivial boundary conditions are linear combinations of the evolution Equations with the constraints that are not preserved at the boundary, in both cases. In the process, the relationship of the normal projection of the Einstein tensor to the recently introduced ``constraint-preserving'' boundary conditions becomes apparent.