Existence of axially symmetric static solutions of the Einstein-Vlasov system

TL;DR

This paper proves the existence of static, axially symmetric solutions to the Einstein-Vlasov system, extending known spherically symmetric models to include axial symmetry in general relativity.

Contribution

It demonstrates the existence of non-vacuum, axially symmetric static solutions to Einstein-Vlasov equations using perturbation techniques from spherically symmetric states.

Findings

Existence of axially symmetric static solutions proven.

Solutions are asymptotically flat and non-vacuum.

Constructed via implicit function theorem from spherically symmetric states.

Abstract

We prove the existence of static, asymptotically flat non-vacuum spacetimes with axial symmetry where the matter is modeled as a collisionless gas. The axially symmetric solutions of the resulting Einstein-Vlasov system are obtained via the implicit function theorem by perturbing off a suitable spherically symmetric steady state of the Vlasov-Poisson system.