Frame dragging and Eulerian frames in General Relativity

TL;DR

This paper clarifies the physical interpretation of dark matter perturbations in cosmology by linking different coordinate systems in General Relativity, revealing relativistic effects and frame dragging phenomena relevant on large scales.

Contribution

It introduces a relativistic displacement field connecting Lagrangian and Eulerian frames, highlighting relativistic corrections and frame dragging in cosmological perturbation theory.

Findings

Relativistic displacement field induces non-linear constraints near the horizon.

Transverse component causes non-linear frame dragging effects.

Poisson gauge offers the simplest physical interpretation.

Abstract

The physical interpretation of cold dark matter perturbations is clarified by associating Bertschinger's Poisson gauge with a Eulerian/observer's 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 sub-horizon scales these relativistic corrections are dominated by the Newtonian bulk part. These corrections however set up non-linear constraints for the density and for the velocity which become important on scales close to the horizon. Furthermore, we report the occurence of a transverse component in the displacement field, and find that it induces a non-linear frame dragging as seen in the observer's frame, which is sub-dominant at late-times and sub-horizon scales. Finally, we find two other gauges which 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.