No consistent bimetric gravity?

Two-tensor, bimetric (or ‘‘f-g’’) gravity (BMG) was born as the fusion of the strong-interaction resonances of the 1960s (specifically that of a massive spin 2 field) with general relativity’s (GR) massless metric [1]. The idea was to join the two symmetrically by giving each its own Einstein-Hilbert action, then coupling the two resulting ‘‘spaces’’ through a nonderivative ‘‘mass term’’ that necessarily reduced the separate coordinate invariances to a single one, each field transforming as a tensor (and the mass term as a scalar density). It was soon realized that upon setting, by hand, one metric to be a background, BMG could be reduced to a single metric, but nongeometric, massive gravity (mGR) theory with explicit mass terms such that the linear limit yielded the free, Fierz-Pauli (FP) field [2]. These models enjoy the isometries of the background metric as symmetries (so are Lorentz invariant massive spin 2 theories in Minkowski backgrounds). However, interest was soon brought to a halt when they were found to propagate ghost modes at nonlinear level [3], even for generic mass terms which reduced linearly to FP (coupled to a massless graviton for BMG). The obstruction was the appearance of a sixth—necessarily ghost— massive ‘‘bulk’’ degree of freedom (d.o.f.). It was only some forty years later that this obstruction was averted [4] by the discovery of—exactly 3—mass terms for which only 5 massive d.o.f.’s remain, all nonghost (one of the preferred mass terms was actually proposed in [2] soon after [1], and had been explicitly exempted in the no-go list of [3]). This ‘‘no-ghost’’ result was quickly extended to BMG with the same class of mass terms [5]. These results opened an instant, nowmajor industry—see [6] for an early review. Philosophically, evasion of the classical mGR no-go theorem would be unsatisfactory, as it would demote GR from its (Yang-Mills-like) status as an isolated theory— without permitted massive neighbors—whose local symmetry cannot be broken by hand. However, absence of a sixth massive, ghostlike mode is only a necessary, but not sufficient, condition for the ultimate consistency of mGR. In fact, it was rapidly realized that these improved mGR models exhibited superluminality in their auxiliary sector and decoupling limit [7,8], and quite generally suffered (local) acausalities and tachyonic modes [9,10] whose interactions with (hitherto) normal matter entail catastrophic consequences [11–14]. Separately, detailed analysis of mGR’s cosmological properties showed it to exhibit ghost instabilities about its homogeneous solutions [15]. The one remedy that might have saved mGR also failed; it cannot be made partially massless (PM): recall, in background de Sitter (dS) spaces, there exist irreps of spin 2 with only 4, rather than the 5, d.o.f.’s of flat space, when the mass and are suitably tuned. Indeed, not only is the offending mode removed in PM, but the four remaining, helicity (2, 1) d.o.f.’s all propagate exactly on the dS light cone, further linking PM with causal behavior [16]. However, the PM limit for single metric, two derivative theories is subject to various no-go results: first, a study of PM-gauge invariant vertices gave a no-go result at quartic order [17]. An analysis of possible truncations of Weyl-squared, conformal gravity (CG)—whose spectrum consists of relatively ghost graviton and PM modes [18,19]—to its PM sector also yielded a negative result [19]. Finally, two independent groups [10,20] showed that PM limit of mGR does not exist. Indeed, the very (constraint) terms responsible for acausal mGR propagation [9] also imply the absence of a PM Bianchi identity and hence of any accompanying gauge symmetry [10]. At this point then, mGR is relegated at best to an effective theory of dubious physical relevance. On the other hand BMG, although at grave risk of suffering similar *[email protected] [email protected] [email protected] PHYSICAL REVIEW D 88, 081501(R) (2013) RAPID COMMUNICATIONS