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

## TL;DR

This paper explores how disformal and other field redefinitions using differential forms affect scalar-tensor theories, revealing invariances, classifying orbits, and connecting various gravity theories.

## Contribution

It provides a systematic analysis of disformal transformations in scalar-tensor theories using differential forms, identifying invariances and new classes of transformations.

## Key findings

- Horndeski's theory maps onto itself under certain disformal transformations.
- Invariant combinations of Lagrangians depend only on first derivatives of the scalar.
- New extended disformal transformations connecting different gravity theories are identified.

## 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.