Vlasov equation and collisionless hydrodynamics adapted to curved spacetime

TL;DR

This paper derives a modified Vlasov equation and collisionless hydrodynamics equations that incorporate both electromagnetic forces and general relativistic effects in curved spacetime, providing a comprehensive framework for plasma dynamics in gravitational fields.

Contribution

It presents a formal Hamiltonian derivation of the Vlasov equation and hydrodynamics equations adapted to arbitrary curved spacetime, including effects of gravity and extrinsic curvature.

Findings

Derived the Vlasov equation in curved spacetime with gravitational and extrinsic curvature effects

Obtained collisionless hydrodynamics equations in three-vector form for arbitrary spatial metrics

Unified electromagnetic and gravitational influences in plasma modeling

Abstract

The modification of the Vlasov equation, in its standard form describing a charged particle distribution in the six-dimensional phase space, is derived explicitly within a formal Hamiltonian approach for arbitrarily curved spacetime. The equation accounts simultaneously for the Lorentz force and the effects of general relativity, with the latter appearing as the gravity force and an additional force due to the extrinsic curvature of spatial hypersurfaces. For an arbitrary spatial metric, the equations of collisionless hydrodynamics are also obtained in the usual three-vector form.