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

TL;DR

This work develops a purely geometrical scalar-tensor theory on Lyra manifolds by promoting the Lyra scale function $\phi$ to a field and integrating it into a generalized Brans-Dicke–Gauss–Bonnet framework. It shows that the symmetry group of gravity in Lyra geometry includes local transformations of length units, with the Lyra function acting as a conformal factor that locally fixes measurement units; a Lyra transformation to a frame with globally fixed units recovers General Relativity, while the corresponding Lyra frames yield BD and Einstein–Gauss–Bonnet theories as geometrized variants. A key result is that the Lyra scale-function source term satisfies $\Omega = -T$ for a wide class of matter, including imperfect fluids, Maxwell fields, and scalar fields, linking curvature to the matter content in a gauge-invariant way. The paper also clarifies the Jordan–Einstein frame conundrum in Lyra geometry, demonstrating that these frames are merely different representations of the same theory, and that energy-momentum conservation and geodesic motion are most transparently expressed in the Lyra framework. Overall, Lyra geometry provides a natural setting to geometrize scalar-tensor gravity, preserve classical dynamics across frames, and connect with familiar 4D BD–GB theories through frame transformations, with several promising avenues for future work in disformal and quantum extensions.

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.