Absence of a Vainshtein radius in torsion bigravity

Abstract

It was pointed out long ago by Vainshtein [Phys. Lett. 39B, 393 (1972)] that the weak-field perturbation expansion of generic theories (of the nonlinear Fierz-Pauli type) involving massive spin-2 excitations breaks down below a certain distance around a material source ("Vainshtein radius"), scaling as some inverse power of the spin-2 mass m2, i.e., some positive power of the range m −1 2 . Here we prove that this conclusion does not apply in a generalized Einstein-Cartan theory (called "torsion bigravity") whose spectrum is made (like that of bimetric gravity) of a massless spin-2 excitation and a massive spin-2 one. Working within a static spherically symmetric ansatz, we prove, by reformulating the field equations in terms of new variables, that one can construct an all-order weak-field perturbative expansion where no denominators involving m2 ever appear in the region r ≪ m −1 2 . In particular, we show how the formal large-range limit, m2 → 0, leads to a well-defined, finite perturbation expansion, whose all-order structure is discussed in some detail.

Heterogeneous oxidative desulfurization catalysed by titanium grafted mesoporous silica nanoparticles containing tethered hydrophobic ionic liquid: A dual activation mechanism

“…In the presence of 23 titanocene dichloride, the signal attributed to tetrafluoroborate counter ion appears also down shielded at  -152.55 ppm which highlights the existence of some type of interaction between the organometallic complex and the counter anion. In fact, metallocene Ti(IV) fluoride complexes can be prepared from the corresponding chloride or alkyl congeners by halogen exchange reaction with AgBF 4 or BF 3 [33] Finally, when the 19 F-NMR spectrum of the mixture titanocene-DBT-ILBF 4 is recorded the signal appears downfield shifted at  -152.52 ppm. These results reflect the existence of a synergic effect in the interaction of DBT and titanocene dichloride with the tetrafluoroborate anion in the ionic liquid and allow us to propose the establishment of a three compounds interaction system (see Fig.…”