Seminar

# The problem of existence of static and electrostatic solutions of the Einstein equations in arbitrary topology

#### General Relativity, Geometry and Analysis: beyond the first 100 years after Einstein

#### 10 September 10:00 - 11:00

#### Martin Reiris - Universidad de la República Uruguay

Vacuum static black hole solutions of the Einstein equations are central in General Relativity and important in geometry. A recent classification [1]showed that S1-symmetric black holes can be only of three kinds: Schwarzschild, Boosts, or Myers/Korotkin-Nicolai, each family having its distinct topological type. We will explain that, while Schwarzschild's topology admits charged static S1-symmetric black holes, those of the Boosts and Myers/Korotkin-Nicolai do not. Black holes in such topologies cannot hold a charge. The proof of this peculiar fact is done by contradiction, by first transforming the supposedly existent charged static solution into a vacuum stationary solution by means of a Kramer-Neugebauer transform, then proving decay estimates for the fields using techniques a la Bakry-\'Emery, and finally using them for showing that, on the original electrostatic solution, too much electrostatic energy would concentrate at infinity making the lapse converge asymptotically to zero and thus reaching an impossibility. The type of result as well as the techniques appear to be new. [1] Reiris M. arXiv:1904.1216

Organizers

Lars Andersson

Max Planck Institute for Gravitational Physics (Albert Einstein Institute)

Mattias Dahl

KTH Royal Institute of Technology

Philippe G. LeFloch

Sorbonne University

Richard Schoen

University of California, Irvine