Variational principle for the Einstein-Vlasov equations

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.