Non-uniqueness in conformal formulations of the Einstein constraints

Walsh, Darragh

doi:10.1088/0264-9381/24/8/002

Cited by 31 publications

(58 citation statements)

References 21 publications

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“…Above ω ¼ 0.01625M −1 ⊙ , the initial data code fails to converge. The steepening of the χ vs ω curve is reminiscent of features related to the nonuniqueness of solutions of the extended conformal thin sandwich equations [45,[50][51][52], and it is possible that our failure to find solutions originates in an analogous breakdown of the uniqueness of solutions of the constraint equations. For S-.5z, we achieve an accuracy of ∼10 −7 , whereas for S.4z and S.4x, the accuracy is ∼10 −4 due to finite L. While the focus of our investigation lies on rotating NS, we note that for ω ¼ 0 our data sets reduce to the standard formalism for irrotational NS.…”

Section: Quasilocal Spinmentioning

confidence: 99%

Binary neutron stars with arbitrary spins in numerical relativity

et al. 2015

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We present a code to construct initial data for binary neutron star systems in which the stars are rotating. Our code, based on a formalism developed by Tichy, allows for arbitrary rotation axes of the neutron stars and is able to achieve rotation rates near rotational breakup. We compute the neutron star angular momentum through quasilocal angular momentum integrals. When constructing irrotational binary neutron stars, we find a very small residual dimensionless spin of ∼2 × 10 −4 . Evolutions of rotating neutron star binaries show that the magnitude of the stars' angular momentum is conserved, and that the spin and orbit precession of the stars is well described by post-Newtonian approximation. We demonstrate that orbital eccentricity of the binary neutron stars can be controlled to ∼0.1%. The neutron stars show quasinormal mode oscillations at an amplitude which increases with the rotation rate of the stars.

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Section: Quasilocal Spinmentioning

confidence: 99%

Binary neutron stars with arbitrary spins in numerical relativity

et al. 2015

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“…Therefore, assessing wheter or not the scalar equations (15) and (16) present the good signs for the application of a maximum principle is an important step for understanding the uniqueness properties of the whole system. However, as pointed out in the previous section, the CFC equations for the conformal factor and the lapse possess the wrong signs in the quadratic extrinsic curvature terms (once everything is expressed in terms of the lapse and the shift).…”

Section: The New Scheme In the Conformally Flat Casementioning

confidence: 99%

Improved constrained scheme for the Einstein equations: An approach to the uniqueness issue

et al. 2009

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Uniqueness problems in the elliptic sector of constrained formulations of Einstein equations have a dramatic effect on the physical validity of some numerical solutions, for instance, when calculating the spacetime of very compact stars or nascent black holes. The fully constrained formulation (FCF) proposed by Bonazzola, Gourgoulhon, Grandclément, and Novak is one of these formulations. It contains, as a particular case, the approximation of the conformal flatness condition (CFC) which, in the last ten years, has been used in many astrophysical applications. The elliptic part of the FCF basically shares the same differential operators as the elliptic equations in CFC scheme. We present here a reformulation of the elliptic sector of CFC that has the fundamental property of overcoming the local uniqueness problems. The correct behavior of our new formulation is confirmed by means of a battery of numerical simulations. Finally, we extend these ideas to FCF, complementing the mathematical analysis carried out in previous studies.

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“…In section 2 we study the solution branches of the constant density star found in [2], [19] and [9]. We prove that the lower branch of weak-field solutions is stable whereas the upper branch is unstable.…”

Section: Introductionmentioning

confidence: 93%

“…For the LS projection operators and splittings of the range and domain we follow the notation of [12] section 1.4-1.7, which is equivalent to [19]. In particular we have that 15) where N denotes the kernel space and R the range of the linearisation.…”

Section: Stability Of Solutions Via Liapunov-schmidt Reductionmentioning

confidence: 99%

“…One motivation for that study was to examine the uniqueness of solutions in the far-fromconstant mean curvature regime now that existence results for this case exist (see [8], [14]). He found interesting non-existence and non-uniqueness results showing that the constraints are ill-posed beyond the non-uniqueness that is introduced when one couples the lapse fixing equation to the four constraint equations, as in the extended conformal thin sandwich (XCTS) formulation (see [15], [2], [19]) and some constrained evolution schemes (see [16], [6] for a resolution of this scaling problem).…”

Section: Introductionmentioning

confidence: 99%

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On the stability of solutions of the Lichnerowicz–York equation

Walsh¹

2013

Class. Quantum Grav.

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We study the stability of solution branches for the Lichnerowicz-York equation at moment of time symmetry with constant unscaled energy density. We prove that the weakfield lower branch of solutions is stable whilst the upper branch of strong-field solutions is unstable. The existence of unstable solutions is interesting since a theorem by Sattinger proves that the sub-super solution monotone iteration method only gives stable solutions.

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Non-uniqueness in conformal formulations of the Einstein constraints

Cited by 31 publications

References 21 publications

Binary neutron stars with arbitrary spins in numerical relativity

Binary neutron stars with arbitrary spins in numerical relativity

Improved constrained scheme for the Einstein equations: An approach to the uniqueness issue

On the stability of solutions of the Lichnerowicz–York equation

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