Frame dragging and Eulerian frames in general relativity

The physical interpretation of cold dark matter perturbations is clarified by associating Bertschinger's Poisson gauge with a Eulerian frame of reference. We obtain such an association by using a Lagrangian approach to relativistic cosmological structure formation. Explicitly, we begin with the second-order solution of the Einstein equations in a synchronous/comoving coordinate system, which defines the Lagrangian frame, and transform it to a Poissonian coordinate system. The generating vector of this coordinate/gauge transformation is found to be the relativistic displacement field. The metric perturbations in the Poissonian coordinate system contain known results from standard/Eulerian Newtonian perturbation theory but contain also purely relativistic corrections. On subhorizon scales, these relativistic corrections are dominated by the Newtonian bulk part. These corrections, however, set up nonlinear (initial) constraints for the density and for the velocity that become important on scales close to the horizon. Furthermore, we report the occurrence of a transverse component in the displacement field and find that it induces a nonlinear frame dragging as seen in the Eulerian frame, which is subdominant at late times and subhorizon scales. Finally, we find two other gauges that can be associated with a Eulerian frame. We argue that the Poisson gauge is to be preferred because it comes with the simplest physical interpretation.