Universally Coupled Massive Gravity, III: dRGT-Maheshwari Pure Spin-$2$, Ogievetsky-Polubarinov and Arbitrary Mass Terms

Einstein's equations were derived for a free massless spin-$2$ field using universal coupling in the 1950-70s by various authors; total stress-energy including gravity's served as a source for linear free field equations. A massive variant was likewise derived in the late 1960s by Freund, Maheshwari and Schonberg, and thought to be unique. How broad is universal coupling? In the last decade four $1$-parameter families of massive spin-$2$ theories (contravariant, covariant, tetrad, and cotetrad of almost any density weights) have been derived using universal coupling. The (co)tetrad derivations included 2 of the 3 pure spin-$2$ theories due to de Rham, Gabadadze, and Tolley; those two theories first appeared in the $2$-parameter Ogievetsky-Polubarinov family (1965), which developed the symmetric square root of the metric as a nonlinear group realization. One of the two theories was identified as pure spin-$2$ by Maheshwari in 1971-2, thus evading the Boulware-Deser-Tyutin-Fradkin ghost by the time it was announced. Unlike the previous 4 families, this paper permits _nonlinear_ field redefinitions to build the effective metric. By not insisting in advance on knowing the observable significance of the graviton potential to all orders, one finds that an _arbitrary_ graviton mass term can be derived using universal coupling. The arbitrariness of a universally coupled mass/self-interaction term contrasts sharply with the uniqueness of the Einstein kinetic term. One might have hoped to use universal coupling as a tie-breaking criterion for choosing among theories that are equally satisfactory on more crucial grounds (such as lacking ghosts and having a smooth massless limit). But the ubiquity of universal coupling implies that the criterion doesn't favor any particular theories among those with the Einstein kinetic term.