  We apply the Faddeev-Jackiw method to the Hamiltonian analysis of the
massless spin-two field. As expected, the reduced Hamiltonian contains only the
traceless-transverse tensor, while some, but not all of the non-propagating
components are determined by the constraints of the theory. In particular, it
is concluded that no gauge choice can be imposed on the fields such that only
the propagating modes remain in the theory, meaning that for the spin-2 field
there is no direct analogue to the Coulomb gauge of electromagnetism.
Implications for General Relativity are discussed.
