On covariant expansion of the gravitational St\"{u}ckelberg trick

TL;DR

This paper introduces a covariant method for expanding the St"{u}ckelbergized fiducial metric by embedding a 4D space into a 5D Minkowski bulk, linking Goldstone modes with curvature and matching previous Riemann Normal Coordinates results.

Contribution

It develops a covariant expansion technique for the St"{u}ckelbergized metric using a 5D embedding approach, providing a new perspective and confirming consistency with prior RNC methods.

Findings

The covariant expansion matches Riemann Normal Coordinates results after field redefinition.

The approach expresses the fiducial metric in terms of 4D Goldstone modes and curvature.

The method offers a geometric interpretation via 5D embedding.

Abstract

A new approach to expanding the "St\"{u}ckelbergized" fiducial metric in a covariant manner is developed. The idea is to consider the curved 4-dimensional space as a codimension-one hypersurface embedded in a 5-dimensional Minkowski bulk, in which the 5-dimensional Goldstone modes can be defined as usual. After solving one constraint among the five 5-dimensional Goldstone modes and projecting onto the 4-dimensional hypersurface, we are able to express the "St\"{u}ckelbergized" fiducial metric in terms of the 4-dimensional Goldstone modes as well as 4-dimensional curvature quantities. We also compared the results with expressions got using the Riemann Normal Coordinates (RNC) in Gao et al [Phys. Rev. D90, 124073 (2014)] and find that, after a simple field redefinition, results got in two approaches exactly coincide.