Conformal Methods in General Relativity

Kroon, Juan A. Valiente

doi:10.1017/cbo9781139523950

Cited by 83 publications

(204 citation statements)

References 351 publications

(586 reference statements)

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“…This shows that the previous "uniform approach" to de Sitter is too strong a requirement. A similar conclusion can be obtained for the full Einstein equations by considering the freedom we have to prescribe data at I + for the Friedrich conformal field equations [12,13,17].…”

Section: Introductionsupporting

confidence: 75%

Cosmic No-Hair in Spherically Symmetric Black Hole Spacetimes

Costa

Natário

Oliveira

2019

Ann. Henri Poincaré

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We analyze in detail the geometry and dynamics of the cosmological region arising in spherically symmetric black hole solutions of the Einstein-Maxwell-scalar field system with a positive cosmological constant. More precisely, we solve, for such a system, a characteristic initial value problem with data emulating a dynamic cosmological horizon. Our assumptions are fairly weak, in that we only assume that the data approaches that of a subextremal Reissner-Nordström-de Sitter black hole, without imposing any rate of decay. We then show that the radius (of symmetry) blows up along any null ray parallel to the cosmological horizon ("near" i + ), in such a way that r = +∞ is, in an appropriate sense, a spacelike hypersurface. We also prove a version of the Cosmic No-Hair Conjecture by showing that in the past of any causal curve reaching infinity both the metric and the Riemann curvature tensor asymptote to those of a de Sitter spacetime. Finally, we discuss conditions under which all the previous results can be globalized.

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Section: Introductionsupporting

confidence: 75%

Cosmic No-Hair in Spherically Symmetric Black Hole Spacetimes

Costa

Natário

Oliveira

2019

Ann. Henri Poincaré

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“…Since the initial data can be chosen as arbitrary, regular functions at τ = 0, it is not clear how the individual components could be extracted. (If the above hypothesis is true, according to which the only entirely regular solutions in our domain 0 ≤ τ ≤ 1 and 0 ≤ ρ ≤ ρ max may be the test solution (18) and its multiples, then the "regular portion" of the initial data may correspond to initial data for this solution. But this was only a conjecture.…”

Section: Numerical Studies In the General Casementioning

confidence: 96%

“…as an exact solution that is regular in the entire domain 0 ≤ ρ ≤ ρ max , 0 ≤ τ ≤ 1. Now we try to reconstruct this solution from the corresponding initial data f (0, ρ) and f ,τ (0, ρ), which can be read off from (18). In particular, we are interested in the numerical error, which we compute as the largest absolute difference between the numerical approximation and the exact solution, max |f numerical − f exact |.…”

Section: A Simple Test Solutionmentioning

confidence: 99%

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Fully pseudospectral solution of the conformally invariant wave equation near the cylinder at spacelike infinity. II: Schwarzschild background

Frauendiener¹,

Hennig²

2017

Class. Quantum Grav.

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Abstract. It has recently been demonstrated (Class. Quantum Grav. 31, 085010, 2014) that the conformally invariant wave equation on a Minkowski background can be solved with a fully pseudospectral numerical method. In particular, it is possible to include spacelike infinity into the numerical domain, which is appropriately represented as a cylinder, and highly accurate numerical solutions can be obtained with a moderate number of gridpoints. In this paper, we generalise these considerations to the spherically-symmetric wave equation on a Schwarzschild background. In the Minkowski case, a logarithmic singularity at the future boundary is present at leading order, which can easily be removed to obtain completely regular solutions. An important new feature of the Schwarzschild background is that the corresponding solutions develop logarithmic singularities at infinitely many orders. This behaviour seems to be characteristic for massive space-times. In this sense this work is indicative of properties of the solutions of the Einstein equations near spatial infinity. The use of fully pseudospectral methods allows us to still obtain very accurate numerical solutions, and the convergence properties of the spectral approximations reveal details about the singular nature of the solutions on spacelike and null infinity. These results seem to be impossible to achieve with other current numerical methods. Moreover, we describe how to impose conditions on the asymptotic behaviour of initial data so that the leading-order logarithmic terms are avoided, which further improves the numerical accuracy.

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“…We recall here the basics of space spinors. The reader who wishes to have a deeper overview on space spinors can refer to [12] and [9,Chapter 4]. Let p ab be a 2-tensor; g admits the space-time spinor g AA ′ BB ′ representation:…”

Section: Introductionmentioning

confidence: 99%

Gluing for the constraints for higher spin fields

Joudioux

2017

Journal of Mathematical Physics

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This short note is a follow-up on the paper by Beig and Chru\'sciel regarding the use of potentials to perform a gluing and shielding of initial data for Maxwell fields and linearised gravity. Based on a work in collaboration with Andersson and B\"ackdahl, this gluing and shielding procedure is generalised to higher spin fields. The approach is based on a generalisation of the de Rham complex to higher spin fields providing a parametrization of the set of constraints, as well as standard elliptic theory to prove the existence of a potential for solutions to the constraints for higher spin fields

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Conformal Methods in General Relativity

Cited by 83 publications

References 351 publications

Cosmic No-Hair in Spherically Symmetric Black Hole Spacetimes

Cosmic No-Hair in Spherically Symmetric Black Hole Spacetimes

Fully pseudospectral solution of the conformally invariant wave equation near the cylinder at spacelike infinity. II: Schwarzschild background

Gluing for the constraints for higher spin fields

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