Spin-two fields and general covariance

It is widely believed that all ‘‘consistent’’ theories of a spin-two field coupled to matter or nonlinearly self-coupled must be generally covariant. The extent to which this statement is true is investigated here. We consider at the classical level nonlinear equations of motion for a field ${\gamma }_{\mathrm{ab}}$ in a flat background spacetime which are derived from a Lagrangian and which reduce, in linear order, to the equations of a spin-two field. In a perturbation expansion about ${\gamma }_{\mathrm{ab}}$=0, we argue that in order for all the linearized solutions to give rise to a one-parameter family of exact solutions, the exact equations of motion must satisfy a certain type of divergence identity. (This is our ‘‘consistency’’ condition.) When the equations of motion arise from an action principle as we assume, this divergence identity implies an infinitesimal gauge invariance of the action. However, our main result is the demonstration that only a very restricted class of candidate infinitesimal gauge symmetries can actually arise from an exact (i.e., finite) gauge symmetry, as is necessary to realize the theory. Under some assumptions concerning the number of derivatives which occur in terms appearing in the divergence identity, we prove that only two types of gauge invariance are possible: (i) normal spin-two gauge invariance and (ii) general covariance. Explicit examples of nonlinear field theories of type (i) are constructed. When coupling to matter is considered, the requirement that in linear order ${\gamma }_{\mathrm{ab}}$ couple directly to the stress-energy tensor of matter may eliminate possibility (i), but I have shown this only in special cases. A similar analysis of nonlinear generalizations of the equations for a collection of spin-one fields is given, and it is shown that under analogous assumptions, the only possible type of gauge invariance for the nonlinear theory is Yang-Mills gauge invariance with respect to an arbitrary Lie algebra.