  Special gravity refers to interacting theories of massless gravitons in
Minkowski space-time which are invariant under the abelian gauge invariance
$h_{ab}\rightarrow h_{ab}+\partial_{(a}\chi_{b)}$ only. In this article we
determine the most general form of special gravity free of Ostrogradski ghosts,
meaning its equation of motion is of at most second order. Together with the
recent works, this result could be helpful in formulating proofs of General
Relativity as the unique physical theory of self-interacting massless
gravitons. We also study how to construct gauge invariant couplings to matter
fields.
