Towards modified bimetric theories within non-product spectral geometry

Abstract

We discuss class of doubled geometry models with diagonal metrics. Based on the analysis of known examples we formulate a hypothesis that supports treating them as modified bimetric gravity theories. Certain steps towards the generic case are then performed. I. INTRODUCTIONThe description of gravity in terms of geometric objects is the cornerstone of Einstein's General Relativity and leads to an intriguing possibility of geometrizing all of the fundamental interactions. One of the existing proposal is based on the noncommutative geometry [1] -a framework that puts on equal footing both the metric structure of manifolds, Yang-Mills-type theories and also the Higgs mechanism. The spectral description of manifolds [2] can be generalized into other than classical geometries like discrete spaces and their products with manifolds. The latter one leads to the definition of the so-called almost-commutative geometries that were successfully applied to the description of gauge theories [3][4][5]. Appropriate choice of the finite space allows e.g. for the formulation of the noncommutative Standard Model of Particle Physics. In this case the finite geometry is build on the matrix algebra C ⊕ H ⊕ M 3 (C) whose choice is dictated by the gauge group of the model [9].Yet another model of this type, but much more simpler, is the one studied by Connes and Lott [10], where the finite algebra is just C ⊕ C and corresponds to the two points. In this case, the product space can be thought of as M × Z 2 , that is, we have two copies of the same manifold. One can further generalize this geometry and can allow for two distinct metrics on these two sheets [14]. Such a doubled geometry is beyond the usual almostcommutative framework and therefore is of a non-product type. Since the spectral action principle applied to a single copy produces the Hilbert-Einstein action, the natural question of the form of an action functional for this non-product type of geometries arises. The answer for generic choices of metrics is not known yet, but in the case of the Friedmann-Lemaître-Robertson-Walker (FRLW) type of Euclidean metrics this was done analitycally [13,14], and the stability of certain solutions was also analysed [13]. It was demonstrated therein that the interaction between the metrics resembles features characteristic to bimetric gravity models [7,8]. Despite numerous similarities, certain significant differences are also present.In particular, the interaction potential for bimetric model is a polynomial one, while for the two-sheeted model it is a rational function. Further similarities and differences for generic metrics were recently analysed in [19], where yet another interpretation of this model in terms of interacting branes was proposed.In this note we discuss yet another class of models beyond the FLRW framework. We illustrate the generally claimed features on the simplified example -the so-called Hopf model.In this case the interaction potential has nontrivial logarithmic terms but it still possesses bimetric gravity characterist...