Scalar-tensor theories in the Lyra geometry: Invariance under local transformations of length units and the Jordan-Einstein frame conundrum

Abstract: The Lyra geometry provides an interesting approach to develop purely geometrical scalar-tensor theories due to the natural presence of the Lyra scale function. This paper explores further the scale function source term to construct a theory on Lyra manifolds which contains proper generalizations of both Brans-Dicke gravity and the Einstein-Gauss-Bonnet scalar-tensor theory. It is shown that the symmetry group of gravitational theories on the Lyra geometry comprises not only coordinate transformations but also local transformations of length units, so that the Lyra function is the conformal factor which locally fixes the unit of length. By performing a Lyra transformation to a frame in which the unit of length is globally fixed, it is shown that General Relativity is obtained from the Lyra Scalar-Tensor Theory (LyST). Through the same procedure, even in the presence of matter fields, it is found that Brans-Dicke gravity and the Einstein-Gauss-Bonnet scalar-tensor theory are obtained from their Lyra counterparts. It is argued that this approach is consistent with the Mach-Dicke principle, since the strength of gravity in Brans-Dicke-Lyra is controlled by the scale function. It might be possible that any known scalar-tensor theory can be naturally geometrized by considering a particular Lyra frame, for which the scalar field is the function which locally controls the unit of length. The Jordan-Einstein frame conundrum is also assessed from the perspective of Lyra transformations, it is shown that the Lyra geometry makes explicit that the two frames are only different representations of the same theory, so that in the Einstein frame the unit of length varies locally. The Lyra formalism is then shown to be better suited for exploring scalar-tensor gravity, since in its well-defined structure the conservation of the energy-momentum tensor and geodesic motion are assured in the Einstein frame.