Geometry and symmetry structures in two-time gravity

Two-time (2T) gravity in d+2 dimensions predicts 1T general relativity in d dimensions, augmented with a local scale symmetry known as the Weyl symmetry in 1T field theory. The emerging general relativity comes with a number of constraints, particularly on scalar fields and their interactions in 1T field theory. These constraints, detailed in this paper, are footprints of 2T gravity and could be a basis for testing 2T physics. Some of the conceptually interesting consequences of the accidental Weyl symmetry include that the gravitational constant emerges from vacuum values of the dilaton and other Higgs-type scalars and that it changes after every cosmic phase transition (inflation, grand unification, electroweak phase transition, etc.). We show that this consequential Weyl symmetry in d dimensions originates from coordinate reparametrization, not from scale transformations, in the d+2 spacetime of 2T gravity. To recognize this structure we develop in detail the geometrical structures, curvatures, symmetries, etc. of the d+2 spacetime which is restricted by a homothety condition derived from the action of 2T gravity. Observers that live in d dimensions perceive general relativity and all degrees of freedom as shadows of their counterparts in d+2 dimensions. Kaluza-Klein type modes are removed by gauge symmetries and constraints that follow from the 2T-gravity action. However some analogs to Kaluza-Klein modes, which we call prolongations of the shadows into the higher dimensions, remain but they are completely determined, up to gauge freedom, by the shadows in d dimensions.