Asymptotically Simple Solutions of the Vacuum Einstein Equations in Even Dimensions

We consider the Einstein-Maxwell equations in space-dimension n. We point out that the Lindblad-Rodnianski stability proof applies to those equations whatever the space-dimension n 3. In even spacetime dimension n + 1 6, we use the standard conformal method on a Minkowski background to give a simple proof that the maximal globally hyperbolic development of initial data sets which are sufficiently close to the data for Minkowski spacetime and which are Schwarzschildian outside of a compact set lead to geodesically complete spacetimes, with a complete Scri, with a smooth conformal structure, and with the gravitational field approaching the Minkowski metric along null directions at least as fast as r −(n−1)/2 .

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“…7 ) We will use this fact as in previous works [1,5,9], but here with the Schwarzschild metric in wave coordinates.…”

Section: Initial Datamentioning

confidence: 99%

Global solutions of the Einstein–Maxwell equations in higher dimensions

Choquet–Bruhat

Chruściel

Loizelet

2006

Class. Quantum Grav.

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“…Proof: Since X(0) is timelike we have g(γ(0), X(0)) = 0, and the result follows from (1). ✷ Next, let κ be a smooth timelike curve in M , note thatκ has no zeros.…”

Section: Preliminariesmentioning

confidence: 99%

Conformal boundary extensions of Lorentzian manifolds

Chruściel¹

2010

J. Differential Geom.

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We study the question of local and global uniqueness of completions, based on null geodesics, of Lorentzian manifolds. We show local uniqueness of such boundary extensions. We give a necessary and sufficient condition for existence of unique maximal completions. The condition is verified in several situations of interest. This leads to existence and uniqueness of maximal spacelike conformal boundaries, of maximal strongly causal boundaries, as well as uniqueness of conformal boundary extensions for asymptotically simple space-times. Examples of applications include the definition of mass, or the classification of inequivalent extensions across a Cauchy horizon of the Taub space-time.

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“…Methods known in principle 2 show that, in harmonic space-time coordinates in the asymptotically flat region, and whatever n ≥ 3, both u andg ij have a full asymptotic expansion in terms of powers of ln r and inverse powers of r. Solutions without ln r terms are of special interest, because the associated space-times contain smoothly compactifiable hyperboloidal hypersurfaces. In even space-time dimension initial data sets containing such asymptotic regions, when close enough to Minkowskian data, lead to asymptotically simple spacetimes [1,12]. It has been shown by Beig and Simon that logarithmic terms can always be gotten rid of by a change of coordinates when space dimension equals three [4,16].…”

Section: Static Vacuum Metricsmentioning

confidence: 96%