Variational principle for the Einstein-Vlasov equations

TL;DR

This paper develops a variational principle for the Einstein-Vlasov equations, providing a new theoretical foundation for describing Einstein spacetimes with collisionless matter through action formulations.

Contribution

It introduces the first variational derivation of the Einstein-Vlasov system, treating matter as a fluid on the tangent bundle and presenting actions in both Lagrangian and Eulerian frameworks.

Findings

Derived the action principles for Einstein-Vlasov equations.

Unified the treatment of matter as a fluid on phase space.

Provided a theoretical basis for further analytical and numerical studies.

Abstract

The Einstein-Vlasov equations govern Einstein spacetimes filled with matter which interacts only via gravitation. The matter, described by a distribution function on phase space, evolves under the collisionless Boltzmann equation, corresponding to the free geodesic motion of the particles, while the source of the gravitational field is given by the stress-energy tensor defined in terms of momenta of the distribution function. As no variational derivation of the Einstein-Vlasov system appears to exist in the literature, we here set out to fill this gap. In our approach we treat the matter as a generalized type of fluid, flowing in the tangent bundle instead of the spacetime. We present the actions for the Einstein-Vlasov system in both the Lagrangian and Eulerian pictures.