  There exists a large class of generally covariant metric Lagrangians that
contain only local terms and describe two propagating degrees of freedom.
Trivial examples can be be obtained by applying a local field redefinition to
the Lagrangian of general relativity, but we show that the class of two
propagating degrees of freedom Lagrangians is much larger. Thus, we exhibit a
large family of non-local field redefinitions that map the Einstein-Hilbert
Lagrangian into ones containing only local terms. These redefinitions have
origin in the topological shift symmetry of BF theory, to which GR is related
in Plebanski formulation, and can be computed order by order as expansions in
powers of the Riemann curvature. At its lowest non-trivial order such a field
redefinition produces the (Riemann)^3 invariant that arises as the two-loop
quantum gravity counterterm. Possible implications for quantum gravity are
discussed.
