J an 2 00 6 Constrained dynamics of universally coupled massive spin 2 - spin 0 gravities

The 2-parameter family of massive variants of Einstein’s gravity (on a Minkowski background) found by Ogievetsky and Polubarinov by excluding lower spins can also be derived using universal coupling. A Dirac-Bergmann constrained dynamics analysis seems not to have been presented for these theories, the Freund-Maheshwari-Schonberg special case, or any other massive gravity beyond the linear level treated by Marzban, Whiting and van Dam. Here the Dirac-Bergmann apparatus is applied to these theories. A few remarks are made on the question of positive energy. Being bimetric, massive gravities have a causality puzzle, but it appears soluble by the introduction and judicious use of gauge freedom. 1. Universal Coupling and Massive Gravity: Historical Sketch Although the field approach to Einstein’s equations is sometimes contrasted with the geometrical approach to gravitation (supposedly Einstein’s), in fact core ideas of the field approach, such as the gravitation-electromagnetism analogy, universal coupling to a combined matter-gravity energy-momentum complex as source, and conservation (in the sense of an ordinary divergence) of energy-momentum due to the gravitational field equations alone, were employed by Einstein in the 1910s [1–4] in his quest for the gravitational field equations. The field approach enjoyed a revival in the 1950s-70s [5], especially in works by Kraichnan [6], Gupta [7], Feynman [8], and Deser [9]. In the 1960s V. I. Ogievetsky and I. V. Polubarinov (OP) derived a 2-parameter family of Poincaré-symmetric massive Einstein’s equations. Universal coupling was not used, but the spin 1 degrees of freedom were removed from the interacting theory to avoid negative energies and one spin 0 to preserve locality [10]. Thus spin 2 and one spin 0 degrees of freedom remained. Independently, Freund, Maheshwari and Schonberg (FMS) derived perhaps the most attractive member of the OP family using universal coupling with a mass term included [11]. Previously the author and W. C. Schieve showed that two one-parameter subfamilies of OP theories are universally coupled [12]. Recently (unpublished) the author showed that all OP theories are universally coupled. Empirically, massive spin 2-spin 0 gravity matches GR except for strong fields or large distances [13, 14]. The spin 0, which is repulsive, theoretically can have a mass anywhere between 0 and ∞, including both endpoints, so its phenomenology has some flexibility [14]. Two further questions arise: positive energy and causality. In the late 1930s Pauli and Fierz, working to linear order, argued that massive spin 2 theories ought to have no spin 0, because the latter was of the wrong sign and so threatened positive energy. While OP seem to have ignored this problem for their nonlinear theories, FMS suggested that subtle effects might render the wrong-sign spin 0 harmless. In 1970 the van Dam-Veltman-Zakharov discontinuity was derived: at linear level, massive spin 2 gravity with no spin 0 disagrees in the massless limit with GTR and, more seriously, with experiment [15, 16]. Boulware and Deser argued [17] that massive gravity was a doomed enterprise because every theory suffered from either violation of positive energy (the spin 2-spin 0 case) or empirical refutation (the spin 2 case), or both. This argument was widely but not quite universally accepted for over two decades. Since the mid-1990s both the empirical inadequacy of spin 2 theories [18] and the negative energy instability of spin 2-spin 0 theories [13, 14] have been discussed anew. The former question is of little relevance here (except perhaps to the infinitely massive spin 0 case). About the latter question I have little to say here, though elsewhere [12] I have suggested that nonperturbative features of the Hamiltonian should not be ignored and might be of some help. In any case, the spin 2-spin 0 theories discussed here are viable only if the negative energy worry can be handled. While no one has proven that these theories are stable, recently Visser and Babak and Grishchuk have suggested that they might be [13, 14]. The issue of causality, which was not considered until the last two decades, will be discussed at the end. 2. Ogievetsky-Polubarinov Massive Gravities Some mass terms are better motivated than others. The OP mass terms are well motivated. The author’s recent rederivation of the OP massive gravities using universal coupling (extending that in [12] is based on the metrical rather than canonical stress tensor, along the lines of Kraichnan [6] and Deser [9] rather than Gupta [7] and FMS [11], so formal general covariance is achieved with a flat metric tensor (under arbitrary coordinate transformations) ημν . (By contrast, OP used Cartesian coordinates with imaginary time.) The OP theories are best expressed using tensor densities of arbitrary real weight. The flat metric’s weight −l covariant concomitant is η̃μν = √−η ημν ; its inverse, the weight l contravariant concomitant, is η̃ = √−η η . The densitized curved metric g̃μν and its inverse g̃ μν are analogously defined in terms of gμν . In addition to the density weight l, the OP theories are parametrized by another parameter n (l 6= 12 , n 6= 0). The two metrics are connected by the equation g̃ = η̃ + λγ̃ , where γ̃ is the (weight l) gravitational potential and λ = − √ 32πG. The parameter n gives the power (in the sense of a binomial series) to which g̃ is raised: [g̃ ] = η̃ + nλγ̃ + n(n− 1) 2! λγ̃η̃αργ̃ ρν + . . . Thus, for example, [g̃ ]2 = g̃η̃αρg̃ ρν and [g̃ ]−1 = η̃g̃αρη̃ ρν . In case this series diverges, these arbitrary real powers can be defined using a generalized eigenvalue formalism based on the Segré classification of gμν with respect to ημν [19]. The action for the S for the OP theories, once stripped of parts that contribute nothing to the equations of motion, is