Note on Covariant St\"uckelberg Formalism and Absence of Boulware-Deser
  Ghost in Bi-gravity
  The covariant St\"{u}ckelberg formalism is applied to bi-gravity in order to
revisit the issue of absence of the Boulware-Deser (BD) ghost. We first confirm
that the leading order action in the decoupling limit for helicity-2 modes of
metrics and helicity-0 mode of St\"{u}ckelberg perturbations does not lead to
higher time derivative in equations of motion, which suggests the absence of
the BD ghost. By extending this method, we reconfirm that the BD ghost does not
appear for arbitrary order of the perturbations at the decoupling limit in
bi-gravity.
