Reformulating Scalar-Tensor Field Theories as Scalar-Scalar Field Theories Using a Novel Geometry

Abstract

In this paper I shall show how the notions of Finsler geometry can be used to construct a similar geometry using a scalar field, f, on the cotangent bundle of a differentiable manifold M. This will enable me to use the second vertical derivatives of f, along with the differential of a scalar field phi on M, to construct a Lorentzian metric on M that depends upon phi. I refer to a field theory based upon a manifold with such a Lorentzian structure as a scalar-scalar field theory. We shall study such a field theory when f is chosen so that the resultant metric on M has the form of a Friedmann-Lemaitre-Robertson-Walker metric, and the Lagrangian has a particularly simple form. It will be shown that the scalar-scalar theory determined by this Lagrangian can generate self-inflating universes, which can be pieced together to form multiverses with non-Hausdorff topologies, in which the global time function multifurcates at t=0. Some of the universes in these multiverses begin explosively, and then settle down to a period of much quieter accelerated expansion, which can be followed by a collapse to its original pre-expansion state. I conclude the paper with a discussion of how probabilities can be assigned to the various universes of a multiverse. This is accomplished by using the action of the universes, with universes having action closer to zero being more likely than universes with large positive values for their action. In order to assure that universe models similar to our own universe are likely to exist I found it necessary to introduce a second scalar field on M, and to modify the original Lagrangian. In the end my theory has three scalar fields, two on the manifold M and one on the cotangent bundle of M.