Global solutions of the Einstein–Maxwell equations in higher dimensions

Abstract

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 .

Topological Censorship for Kaluza–Klein Space-Times

“…In this appendix we wish to point out that sufficiently small data vacuum spacetimes constructed using the Lindblad-Rodnianski method [24], as generalised by Loizelet to higher dimensions [25,26] (compare [5]), contain past inwards trapped, closed to the future (in a sense which should be made clear by what is said below), timelike hypersurfaces. This is irrelevant as far as the topological implications of our analysis are concerned, as in this case the space-time manifold is R n+1 anyway, but it illustrates the fact that such hypersurfaces can arise in vacuum space-times which are not necessarily stationary.…”

Asymptotic behavior of Maxwell fields in higher dimensions

“…We will generally study the r-dependence (for r → ∞) of the leading field components only, under the assumption that this is power-like (some comments on certain subleading terms will however be necessary to arrive, e.g., at (3)). It will thus not be necessary to assume that the field admits a power-series expansion in 1/r (however, the existence of full Einstein-Maxell solutions with that property is proven in [19], in the case of even dimensions). More technical assumptions will be explained in section 2, where certain results of [14] needed in this work will also be summarized.…”

The Evolution Problem in General Relativity

“…This has now been extended in various directions in [31]. The original result [46] was extended to the Maxwell case in the Ph.D. thesis of Zipser [147].…”