On covariant expansion of the gravitational Stückelberg trick

A new approach to expanding the "Stückelbergized" 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 "Stückelbergized" 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.