Field redefinitions in theories beyond Einstein gravity using the language of differential forms

Abstract

We study the role of field redefinitions in general scalar-tensor theories. In particular, we first focus on the class of field redefinitions linear in the spin-2 field and involving derivatives of the spin-0 mode, generically known as disformal transformations. We start by defining the action of a disformal transformation in the tangent space. Then, we take advantage of the great economy of means of the language of differential forms to compute the full transformation of Horndeski's theory under general disformal transformations. We obtain that Horndeski's action maps onto itself modulo a reduced set of non-Horndeski Lagrangians. These new Lagrangians are found to be invariant under disformal transformation that depend only in the first derivatives of the scalar. Moreover, these combinations of Lagrangians precisely appear when expressing in our basis the constraints of the recently proposed Extended Scalar-Tensor (EST) theories. These results allow us to classify the different orbits of scalar-tensor theories invariant under particular disformal transformations, namely the special disformal, kinetic disformal and disformal Horndeski orbits. In addition, we consider generalizations of this framework. We find that there are possible well-defined extended disformal transformations that have not been considered in the literature. However, they generically cannot link Horndeski theory with EST theories. Finally, we study further generalizations in which extra fields with different spin are included. These field redefinitions can be used to connect different gravity theories such as multi-scalar-tensor theories, generalized Proca theories and bi-gravity. We discuss how the formalism of differential forms could be useful for future developments in these lines.PACS numbers: 04.50. Kd, 98.80.Cq, 95.36.+x,

Disformal Transformations in Scalar–Torsion Gravity

“…Our results show that the studies of disformal transformations in scalar-curvature gravity [13,14] can be generalized also to scalar-torsion theories. In this work we have undertaken a first step in this direction, by focusing on general disformal transformations only, and by explicitly constructing an invariant class of actions by applying these transformations to the torsion tensor.…”

“…While in the most general class of metric-affine theories the metric and connection may be transformed independently, leading to different notions of invariance under such transformations [9], assuming a more specific geometry limits the possible transformations. In the scalar-curvature class, where the affine connection is fixed as the curvature-free, metric compatible Levi-Civita connection, this leads to the well-known possibility of conformal transformations [10], or the more general class of disformal transformations [11,12] and its extensions [13,14]. The latter are of particular interest, as they connect classes of gravity theories with second order field equations, such as the well-known Horndeski class [15][16][17], to such higher derivative order theories which are healthy in the sense that they avoid Ostrogradsky instabilities due to the presence of constraints arising from degeneracies in their Lagrangians [18][19][20][21].…”

Novel matter coupling in general relativity via canonical transformation

“…The conformal transformation and its generalization, the disformal transformation [2], are the powerful tools to connect two different theories. The Horndeski theory, which is the most general scalar-tensor theory with the equation of motion with at most second derivatives [3][4][5][6][7], is transformed into the beyond Horndeski theories [8,9] via the disformal transformation [10][11][12][13][14][15][16]. Although the transformed theory is equivalent to the original theory [17], one should take care that the matter field couple with which metric tensor.…”

General invertible transformation and physical degrees of freedom

“…*13 To the best of our knowledge, it remains an open question whether these DHOST theories are closed under generic disformal transformations. *14 Apart from this line of research, the authors of Ref [31]. specified all the theories obtained via invertible disformal transformations from the Horndeski class in the language of differential forms.…”