On asymptotically flat solutions of Einstein's equations periodic in time: I. Vacuum and electrovacuum solutions

Abstract

By an argument similar to that of Gibbons and Stewart [10], but in a different coordinate system and less restrictive gauge, we show that any weaklyasymptotically-simple, analytic vacuum or electrovacuum solutions of the Einstein equations which are periodic in time are necessarily stationary.

Non-existence of time-periodic vacuum space-times

“…In fact, time-periodicity is here exploited as in the proof of Prop. 3.2 above, and as in the argument given in [BST10a]. In complete analogy to the above, we prove…”

“…The proof in [Gal84] is an application of a splitting theorem in Lorentzian geometry which relies on the dominant energy condition (and is thus applicable in the presence of matter fields). In the asymptotically flat setting, in particular for asymptotically simple space-times, all approaches to this problem [Pap57,Pap58a,Pap58b,GS84,BST10a,BST10b] seek to relate the time-periodicity property to an analysis at null infinity. (An exception is the paper [Daf03] of Dafermos who used the event horizons instead, to show that in spherical symmetry time-periodic black hole space-times coupled to suitable matter models are static.…”

“…Our proof follows the two-step strategy of [BST10a]: In Section 3 we first derive the existence of a "candidate" Killing vector field which satisfies the Killing equation to all orders. We take special care to perform this analysis in the actual space-time; c.f.…”

“…Our result applies under physically relevant regularity assumptions purely at the level of the initial data. In particular, our work removes the assumption of analyticity up to null infinity in [Bičák, Scholtz, and Tod;. The proof relies on extending a suitably constructed "candidate" Killing vector field from null infinity, via Carleman-type estimates obtained in [Alexakis, Schlue, Shao;2013].…”

Massive graviton geons

“…This time-dependent self-gravitating object is called a gravitational geon, short for gravitational-electromagnetic entity, which was introduced by Wheeler [5]. Although a gravitational geon can be constructed in asymptotically flat spacetimes [6,7], the geon is not exactly periodic in time [8][9][10][11][12][13][14][15] and it has a finite lifetime. However, the instability of geons could be remedied in asymptotically anti-de Sitter (AdS) spacetimes since the AdS boundary can confine particles and waves as in a box.…”