1307.1725 (Cornelius Rampf)

Frame dragging from the perspective of a relativistic Lagrangian perturbation theory    [PDF]

Cornelius Rampf

We identify the Lagrangian displacement field in terms of a passive coordinate/gauge transformation. We restrict to the gravitational evolution of an irrotational cold dark matter (CDM) component in an Einstein-de Sitter universe. We begin with the second-order solution of the Einstein equations in a synchronous coordinate system, and transform it to a Poissonian coordinate system. Explicitly, we connect the two metrics via a space-like displacement field and a time-like perturbation, and thus provide a relativistic generalisation from Lagrangian to Eulerian coordinates. We report the occurence of scalar and vector perturbations at second order in the (observer's) Poissonian coordinate system, and these perturbations set up non-linear constraints for the density and velocity field at initial time. The origin of the vector perturbation is due to a transverse component in the Lagrangian displacement field, which induces a frame dragging in the observer's frame. The transverse displacement field does not affect the matter density at second order but the (initial) velocity field of the CDM component. We derive the relativistic displacement and velocity field to set up the non-linear initial constraints, which are then needed e.g. for generating the initial conditions in N-body simulations.

View original: http://arxiv.org/abs/1307.1725