  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.
