Non-existence of time-periodic vacuum spacetimes

TL;DR

This paper proves that smooth, asymptotically flat vacuum solutions to Einstein's equations that are periodic in time must actually be stationary near infinity, removing previous analyticity assumptions.

Contribution

It establishes the non-existence of non-stationary, time-periodic vacuum spacetimes under physically relevant regularity conditions, extending previous results by removing analyticity requirements.

Findings

Time-periodic vacuum solutions are necessarily stationary near infinity.

The proof uses Carleman estimates to extend a candidate Killing vector field.

The result applies under regularity assumptions at the initial data level.

Abstract

We prove that smooth asymptotically flat solutions to the Einstein vacuum equations which are assumed to be periodic in time, are in fact stationary in a neighborhood of infinity. 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 [Bicak, Scholtz, and Tod; 2010]. 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].