Transforming gravity: from derivative couplings to matter to second-order scalar-tensor theories beyond the Horndeski Lagrangian

TL;DR

This work investigates scalar–tensor gravity with derivative couplings between a scalar and matter mediated by an effective metric, classifying these couplings as conformal, disformal, or extended disformal. By analyzing frame transformations via the Jacobian and its eigenstructure, the authors show when inverse mappings exist between frames and how energy–momentum tensors transform. They demonstrate that, although general disformal mappings can induce higher-derivative terms in the Jordan frame, contracting with a Jacobian eigentensor yields implicit constraints that reduce the dynamics to second order, revealing a loophole in Horndeski's theorem. This framework extends the landscape of Ostrogradski-stable scalar–tensor theories beyond Horndeski and provides a systematic method to study derivative couplings and their implications for cosmology and gravity. The results connect derivative conformal and disformal transformations to a generalized kinetic mixing that preserves second-order dynamics, offering new perspectives on frame equivalence and potential quantum considerations.

Abstract

We study the structure of scalar-tensor theories of gravity based on derivative couplings between the scalar and the matter degrees of freedom introduced through an effective metric. Such interactions are classified by their tensor structure into conformal (scalar), disformal (vector) and extended disformal (traceless tensor), as well as by the derivative order of the scalar field. Relations limited to first derivatives of the field ensure second order equations of motion in the Einstein frame and hence the absence of Ostrogradski ghost degrees of freedom. The existence of a mapping to the Jordan frame is not trivial in the general case, and can be addressed using the Jacobian of the frame transformation through its eigenvalues and eigentensors. These objects also appear in the study of different aspects of such theories, including the metric and field redefinition transformation of the path integral in the quantum mechanical description. Although sane in the Einstein frame, generic disformally coupled theories are described by higher order equations of motion in the Jordan frame. This apparent contradiction is solved by the use of a hidden constraint: the contraction of the metric equations with a Jacobian eigentensor provides a constraint relation for the higher field derivatives, which allows one to express the dynamical equations in a second order form. This signals a loophole in Horndeski's theorem and allows one to enlarge the set of scalar-tensor theories which are Ostrogradski-stable. The transformed Gauss-Bonnet terms are also discussed for the simplest conformal and disformal relations.

Figures (2)

• Figure 1: Inverse transformation $g_{\mu\nu}=A(\tilde{X})\tilde{g}_{\mu\nu}$ for $\tilde{g}_{\mu\nu}=e^{X/M^4}g_{\mu\nu}$ (left) and $\tilde{g}_{\mu\nu}=e^{-X^2/(2M^8)}g_{\mu\nu}$ (right). The inverse conformal factor $A(\tilde{X})$ is obtained implicitly through Eq. (\ref{['inv_conf1']}), and can be multi-valuated for certain ranges of $\tilde{X}$, giving rise to two branches characterized by the kinetic eigenvalue $\lambda_K$, given by Eq. (\ref{['conf_eigen2']}). The branches with positive (solid blue) and the negative (dotted blue) values of $\lambda_K$ meet or end at singular points, in which either one of the Jacobian eigenvalues (\ref{['conf_eigen1']}, \ref{['conf_eigen2']}) vanishes. This is seen explicitly in the case of the exponential function (left), which becomes bi-valued for $\tilde{X}/M^4>0$. Both branches meet at $\tilde{X}/M^4 = e^{-1}$ (corresponding to $X/M^4=1$) for which the kinetic eigenvalue becomes zero. The singular point $A(\tilde{X})=0$ corresponds to $C(X)\to \infty$. The gray shaded region, $\tilde{X}/M^4 \ge e^{-1}$, is forbidden. In the Gaussian case (right) both eigenvalues are always positive. Therefore there are no singular points and the two branches are not connected.
• Figure 2: Inverse map for $\tilde{g}_{\mu\nu} = \exp(X/M^4)g_{\mu\nu} + {1\over 2} X^3/M^{16}\phi_{,\mu}\phi_{,\nu}$ given by Eq. (\ref{['inv_disf1']}) with the inverse conformal function $A(\tilde{X})$ on the left panel and the inverse disformal function $B(\tilde{X})$ on the right. Both functions admit four branches with positive (solid) and negative (dotted) values of $C(X)-2D(X)X$, due to the fact that $\tilde{X}(X)$ has multiple poles. The inverse conformal factor of $\tilde{g}_{\mu\nu} = \exp(X/M^4)g_{\mu\nu}$ (left panel of figure \ref{['fig_inverseConfExp']}) is shown for comparison (blue dash-dotted). A branch with $C(X)-2D(X)X>0$ exists for large values of $X$, but is difficult to plot because the inverse conformal factor tends to zero rapidly.