  We present a systematic exposition of the Lagrangian field theory for the
massive spin-two field generated in higher-derivative gravity. It has been
noticed by various authors that this nonlinear field overcomes the well known
inconsistency of the theory for a linear massive spin-two field interacting
with Einstein's gravity. Starting from a Lagrangian quadratically depending on
the Ricci tensor of the metric, we explore the two possible second-order
pictures usually called "(Helmholtz-)Jordan frame" and "Einstein frame". In
spite of their mathematical equivalence, the two frames have different
structural properties: in Einstein frame, the spin-two field is minimally
coupled to gravity, while in the other frame it is necessarily coupled to the
curvature, without a separate kinetic term. We prove that the theory admits a
unique and linearly stable ground state solution, and that the equations of
motion are consistent, showing that these results can be obtained independently
in either frame. The full equations of motion and the energy-momentum tensor
for the spin--two field in Einstein frame are given, and a simple but
nontrivial exact solution to these equations is found. The comparison of the
energy-momentum tensors for the spin-two field in the two frames suggests that
the Einstein frame is physically more acceptable. We point out that the
energy-momentum tensor generated by the Lagrangian of the linearized theory is
unrelated to the corresponding tensor of the full theory. It is then argued
that the ghost-like nature of the nonlinear spin-two field, found long ago in
the linear approximation, may not be so harmful to classical stability issues,
as has been expected.
