extended by including the remaining diffeomorphisms, giving full diffeomorphism invariance, or by appending Weyl transformations, yielding “WTDiff” invariance [@Alvarez]. Contrary to popular folklore, however, neither extension is required for a consistent nonlinear theory; Wald [@Wald] and Heiderich and Unruh [@Heiderich] have explicitly constructed consistent non-covariant models of interacting massless spin two fields, and the latter contain explicit transverse diffeomorphism invariance.[^2]

If one makes the much stronger assumption that the field is sourced by its own stress-energy tensor, then, as noted above, general relativity (and thus full diffeomorphism invariance) will emerge [@Deser; @Deser2], although for the exceptional choice of a WTDiff-invariant linearized action one may instead obtain unimodular gravity [@Alvarez; @Blas; @Alvarez2]. The Weinberg-Witten theorem, described below, leads to a similar conclusion, that a pure spin two field with a conserved source must normally be a gauge theory.

If Lorentz invariance is not exact, the Fierz-Pauli action is no longer unique. The problem then becomes more difficult to analyze, although there are some models that appear to avoid both ghosts and the vDVZ discontinuity [@Pilo]. But similar issues of extra “gauge” modes appear in other settings, such as lattice models [@Dittrich3]. A basic lesson is that the linear approximation may be 