ghost at those orders. As a result, this Boulware–Deser ghost would be present around, for example, highly inhomogeneous backgrounds. This is problematic because a linearized theory of gravity, like Fierz–Pauli, is well-defined on its own but cannot interact with matter, as the coupling breaks diffeomorphism invariance. This must be remedied by adding new terms at higher and higher orders, ad infinitum. For a massless graviton, this process converges and the end result is well-known: one simply arrives at general relativity. This is the meaning of the statement that general relativity is the unique theory (up to conditions on dimensionality, locality, etc.) of a massless spin-2 field. In order for massive gravity to actually describe gravity, i.e., a massive spin-2 field coupling to matter and thereby mediating the gravitational force, a nonlinear completion must similarly be obtained. The Boulware–Deser ghost presents a serious obstacle to such an endeavor. The vast majority of theories of massive and interacting spin-2 fields will suffer from this ghost and therefore not be viable. In fact, until 2010 it was widely believed that all Lorentz-invariant massive gravity theories possessed the Boulware–Deser ghost. Ghost-free massive gravity In 2010 a breakthrough was achieved when de Rham, Gabadadze, and Tolley constructed, order by order, a theory of massive gravity with coefficients tuned to avoid the Boulware–Deser ghost by packaging all ghostly (i.e., higher-derivative) operators into total derivatives which do not contribute to the equations of motion. The complete absence of the Boulware–Deser ghost, to all orders and beyond the decoupling limit, was subsequently proven by Fawad Hassan and Rachel Rosen. The action for the ghost-free de Rham–Gabadadze–Tolley (dRGT) massive gravity is given by or, equivalently, The ingredients require some explanation. As in standard general relativity, there is an Einstein–Hilbert kinetic term proportional to the Ricci scalar and a minimal coupling to the matter Lagrangian , with representing all of the matter fields, such as those of the Standard Model. The new piece is a mass term, or interaction potential, constructed carefully to avoid the Boulware–Deser ghost, with an interaction strength which is (if the nonzero are ) closely related to the mass of the graviton. Th principle of gauge-invariance renders redundant expressions in any field theory provided with its corresponding gauge(s). For example, in the massive spin-1 Proca action, the massive part in the Lagrangian breaks the gauge-invariance. However, the invariance is restored by introducing the transformations: The same