# Palatini frames in scalar–tensor theories of gravity

## Abstract

A new systematic approach extending the notion of frames to the Palatini scalar–tensor theories of gravity in various dimensions \(n>2\) is proposed. We impose frame transformation induced by the group action which includes almost-geodesic and conformal transformations. We characterize theories invariant with respect to these transformations dividing them up into solution-equivalent subclasses (group orbits). To this end, invariant characteristics have been introduced. The formalism provides new frames incorporating non-metricity that lead to re-definition of Jordan frames. The case of Palatini *F*(*R*)-gravity is considered in more detail.

## 1 Introduction

Despite many theoretical and experimental triumphs [1], including recent detection of gravitational waves [2], general relativity is not considered a fundamental theory describing gravitational interactions; see e.g. [3, 4, 5, 6, 7, 8]. Based on our current understanding of the workings of Nature, a few arguments for modifying it can be given. First of all, GR cannot be satisfactorily quantized, as attempts to renormalize it have been futile. Secondly, it is not a low-energy limit of theories regarded as fundamental, such as bosonic string theories [9], where dilaton fields couple non-minimally to the spacetime curvature. Another problem concerns the \(\Lambda \)CDM model: it is customary to consider that the value of \(\Lambda \) being responsible for the current acceleration of the expansion of the Universe is usually incomprehensibly small (120 order of magnitude smaller) when compared to the value predicted by quantum field theory. In fact, more realistic estimations taking into account Pauli–Zeldovich cancellation effect, quantum field theory in curved background or supersymmetry, make this discrepancy not so drastic (for more discussion see [10, 11, 12]).

As far as the mathematical reasons for modifying the Einstein’s gravity are concerned, we can take the so-called Palatini formalism into consideration. In the standard gravity, the underlying assumption of geometric structures defined on spacetime is that the affine connection is the Levi-Civita connection of the metric. In the Palatini approach, however, we consider these two objects as unrelated, since there is no reason whatsoever we should impose a relation between them a priori. In case of Einstein gravity, introducing Palatini formalism does not affect the resulting field equations in any way; however, in case of more complicated theories, such as scalar–tensor or *F*(*R*) theories of gravity, both approaches usually give different results, describing different physics. Palatini formalism has been investigated especially in the context of cosmological applications [13, 14, 15, 16, 17, 18, 19, 20, 21].

Scalar–tensor (S–T) theories of gravity are a very promising modification of the Einstein gravity. In these theories, a scalar field is non-minimally coupled to the curvature scalar [22]. Historically, the prototype of all contemporary scalar–tensor theories was the Brans–Dicke theory [23]. An interesting feature of the scalar–tensor theories of gravity is their equivalence to the *F*(*R*) theories, which basically means that the latter can be analyzed using the “mathematical machinery” developed for the former [24]. The reason why the scalar–tensor theories deserve some attention is that they can be successfully used to build credible models for cosmic inflation [25] (utilizing the equivalence between the scalar–tensor and *F*(*R*) theories of gravity) and dark energy [26].

Hitherto, the scalar–tensor theories of gravity have been considered mostly in a purely metric approach [13, 22, 26, 27, 28, 29, 30] and the possible effects of adopting the Palatini approach have been analyzed somewhat less commonly

[31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54]. So far, general conditions for a correct formulation of the scalar–tensor theories have been analyzed [34]. Change of formalism from metric to Palatini applied to S–T theories has been investigated in the context of cosmology, to analyze the problem of cosmological constant [35], quintessence – to show that equation of state in the Palatini formalism can cross the phantom divide line [36], and inflation, where it was discovered that in the Palatini approach [37, 38, 39, 40, 41, 42, 43, 44, 45], inflationary epoch is naturally provided [37, 38, 39, 40], and almost scale-invariant curvature perturbations are generated with no tensor modes [46]. Some authors generalized scalar–tensor theories and allowed non-minimal derivative coupling as well [47, 48, 49, 50, 51, 52]. In such theories, one makes extensive use of so-called “disformal transformations”. It was shown that for a special choice of parameters characterizing the theory, adopting Palatini approach allows one to avoid Ostrogradski ghosts [47].^{1} Also, vector-Horndeski theories were analyzed with the metric structure decoupled from the affine structure. It was proven that in the Palatini formalism, there exist cosmological solutions which can pass through singularities [53].

The main goal of this paper is to introduce the general theory of scalar–tensor gravity analyzed in the Palatini approach and to develop mathematical formalism enabling us to analyze any S–T theory in a (conformally) frame-independent manner. The outline of this paper^{2} goes as follows: in the first part, postulated action functional will be presented, and equations of motion derived. Next, modified conformal transformations in the Palatini approach will be introduced in order to allow the connection to transform independently of the metric tensor. A solution of the equation resulting from varying with respect to the independent connection will be inspected. Then, following the procedure carried out in [26] (see also [27, 29]), invariant quantities defined for the Palatini S–T theory will be obtained. The results will be applied to an analysis of *F*(*R*) Palatini gravity. In the last part, general conditions on the possible equivalence between a given S–T theory and some *F*(*R*) gravity will be discussed. For reader’s convenience, some supplementary material is collected in four Appendices.

## 2 Action functional and equations of motion

The main idea behind the Palatini approach is the following: we no longer consider metric tensor and linear connection to be dependent on each other. This approach was originally analyzed by Einstein [56], but then was attributed to an Italian mathematician Attilio Palatini [57, 58]. In this approach, one decouples causal structure of spacetime from its affine structure (which determines geodesics followed by particles). In practical terms, Palatini formalism amounts to varying the action functional with respect to both the metric tensor and the torsionless (i.e. symmetric) affine connection, resulting in two sets of field equations. One of these sets establishes a relation between the metric and the connection. There is no particular reason to apply the Palatini variation to the standard Einstein–Hilbert action, as in that case the independent connection turns out to be Levi-Civita with respect to the metric tensor, i.e. related to the metric by the standard formula: \(\Gamma ^\alpha _{\mu \nu }=\frac{1}{2}g^{\alpha \beta }(\partial _\mu g_{\beta \nu }+\partial _\nu g_{\beta \mu }-\partial _\beta g_{\mu \nu })\). However, in case of more complicated theories, such as *F*(*R*) theories of gravity, where the curvature scalar in the Einstein–Hilbert action is replaced by a function of it, both approaches give physically incompatible results, leading to different field equations describing different physics in the presence of matter sources. Instead, in the vacuum case, the Einstein equations enriched by adding cosmological constant are still valid [59, 60].

*M*is

*n*-dimensional \(n>2\) manifold

^{3}equipped with a torsion-free (\(\equiv \) symmetric) connection \(\Gamma =\Gamma _{\mu \nu }^\alpha =\Gamma _{\nu \mu }^\alpha \) and a metric tensor \(g=g_{\mu \nu }\), possibly of the Lorentzian signature. The affine connection is used to build the Riemann curvature tensor:

Utilizing the Palatini approach, we want now to write down the most general action functional for scalar–tensor theories, which is consistent with some class of transformations (see explanations below and Appendix B). The action should contain a scalar field \(\Phi \) – or a function thereof – non-minimally coupled to the curvature defined above and possibly to the matter fields. Furthermore, one must include also a kinetic term rendering the scalar field dynamic, and a self-interaction potential of the field. Presence of additional terms resulting from the approach we adopt, absent in the metric version of the theory, cannot be excluded.

*M*. They provide, together with the dynamical variables \((\Gamma , g,\Phi )\), the so-called frame for the action (3). A change of frame is governed by a consistent action which will be introduced later on. Some of these coefficients have exactly the same meaning as their metric counterparts (cf. Appendix A), i.e. \({\mathcal {A}}\) describes coupling between curvature and the field, \({\mathcal {B}}\) is the kinetic coupling, \({\mathcal {V}}\) is the potential of self-interaction of the scalar field, while non-zero \(\alpha \) means that the action functional features an anomalous coupling between the scalar and matter fields \(\chi \). One requires \({\mathcal {A}}\) be non-negative, otherwise, gravity would be rendered a repulsive force. The coefficients \({\mathcal {C}}_1\) and \({\mathcal {C}}_2\) do not have a clear interpretation yet. Their inclusion in the functional is a direct consequence of the Palatini approach we adopted; they do not appear in the metric S–T theory.

The form of the action functional follows necessarily from our requirement that the action remain form-invariant under conformal and almost-geodesic transformations, accompanied by a re-parametrization of the scalar field. This condition states that if one changes the metric tensor, the connection and the scalar field according to the transformation relations given below (we shall call such transformation “changing the frame”, and the choice of particular metric, connection and scalar field – “(conformal) frame”), solutions to the field equations are mapped into corresponding solutions obtained in the transformed frame.

Equation (5b) is called a generalized almost-geodesic transformation of type \(\pi _3\); the word “almost” suggests that one needs to distinguish between the transformation (5b) and a transformation which genuinely preserves geodesics on the space-time (see Appendix D). In fact, if the function \(\gamma _3\) was equal zero, one would have precisely the geodesic transformation of the affine connection. The new connection preserves also the light cones, leaving the causal structure of spacetime unchanged. If all functions \(\gamma _i\) were equal, one would recover standard conformal transformation formulae, identical to the case when the connection is Levi-Civita with respect to the metric tensor. One can also think of the transformation as Weyl transformation, i.e. without assuming that the connection is metric; in particular setting \(\gamma _1\ne \gamma _2=\gamma _3\).

Since the connection can be always solved in terms of the metric and the scalar field, there are no additional physical degrees of freedom carried by it. The connection always turns out to be an auxiliary field [64].

The relation (11) is defined by two functions, which in general (except the case mentioned above) are not equal. One can identify them as the functions \({\check{\gamma }}_2\) and \({\check{\gamma }}_3\) relating affine connections of two different frames. Frame, in which the theory turns out to be fully metric, can be obtained by plugging back the connection (11) in the action functional (3). Such a change of frame should not affect the form of action functional (otherwise solutions of equations of motion in one frame would not be mapped to solution in another frame, which would contradict one of our basic assumptions), and the coefficients \(\{{\mathcal {A}},{\mathcal {B}},{\mathcal {C}}_1,{\mathcal {C}}_2,{\mathcal {V}},\alpha \}\) will change in a way that preserves the functional form of the action. Exact transformation relations will be presented in the next section.

Because the transformation (5b) depends on two independent parameters, one cannot in general end up in a frame in which the initial independent connection is Levi-Civita with respect to *some* metric tensor, as the transformation of the metric is governed by a single function \({\check{\gamma }}_1\). However, if \({\mathcal {C}}_1={\mathcal {C}}_2\), then it is possible to transform the metric tensor in such a way that the initial independent connection becomes a Levi-Civita connection of the transformed, new metric.

## 3 Transformation formulae

*n*dimensions, requiring the transformations be defined by Eqs. (5a)–(5c), the formulae relating Riemann tensors of two different conformal frames are the following:

As we can see, some of the transformation relations involve nothing but a simple multiplication of the “old” coefficients by a factor related to the transformation of the metric tensor. These relations do not depend on the approach we adopt – they retain the same form regardless of whether we work within metric or Palatini formalism. However, coefficients \({\mathcal {C}}_1, {\mathcal {C}}_2\) and \({\mathcal {B}}\) transform in a more complicated way depending on whether the theory is metric or not. The transformation relations preserve the sign of the \({\mathcal {A}}\) coefficient. Similarly, if \({\mathcal {B}}\) is subject to a scalar field re-parametrization only, then its sign does not change as well. By the same token, if the potential \({\mathcal {V}}\) vanishes in one frame, it cannot emerge in any other.

Due to our freedom of choice of three functions \(\{\gamma _1,\gamma _2,\gamma _3\}\) and re-parametrization of the scalar field \(\Phi =\check{f}({\bar{\Phi }})\), it is always possible to fix four of the above six coefficients. We shall call such fixing “choosing a frame”, as it was mentioned before. If we specify the remaining two functions, we choose a theory. For example, the four functions \(\{\gamma _1,\gamma _2,\gamma _3,f\}\) can be chosen in such a way that four coefficients \(\{{\mathcal {B}},{\mathcal {C}}_1,{\mathcal {C}}_2,\alpha \}\) vanish, simplifying the calculations. Results obtained in a given frame can be always “translated” to another frame if the two frames can be related by a conformal transformation accompanied by a re-parametrization of the scalar field. It must be also noted that increased number of functions used to change the frame (from two in scalar–tensor theory in the metric approach – see Appendix A – to four in case of the Palatini formalism) result in additional coefficients appearing in the action functional. However, analogously to the metric case, despite the fact we are able to fix four of them, we are always left with two functions, defining the particular theory.

Conformal and generalized almost-geodesic transformation establish a mathematical equivalence of two frames. On the physical ground, they may constitute two very different theories [65, 66, 67, 68, 69, 70, 71, 72, 73]. The multitude of equivalent theories poses a problem of identifying frames which can be related by the transformations given by Eqs. (5a)–(5c). Such frames may bear no resemblance to one another and yet, be two different manifestations of the same theory, but written using different variables. This situation suggests that it would be desirable to formulate the general scalar–tensor theory in a frame-independent way, fully analogous to the way GR circumvents the problem of deciding upon the “right” coordinate system to describe physical phenomena by resorting to the language of tensors, allowing one to write equations in a covariant manner. In case of scalar–tensor gravity in the Palatini approach, we decided to follow on [26] and find invariant quantities built from coefficients \(\{{\mathcal {A}},\ldots ,\alpha \}\), metric and connection, whose values are independent of the choice of frame – just like, for instance, value of \(R^\alpha _{\,\,\mu \beta \nu }R_{\alpha }^{\,\,\mu \beta \nu }\) does not depend on our choice of coordinate frame. This analogy, however, should not be taken too seriously, as general covariance in case of GR is a consequence of the fact that our description of Nature should not depend on an artificial construct of coordinate frame, whereas such invariance of physical laws is not present when changing conformal frames. For example, geodesic curves, due to covariant formulation of geodesic equations, are the same in every coordinate frame; on the other hand, if the mapping (5b) is applied, geodesics are not preserved (unless \(\gamma _3=0\)), thus leading to emergence of an unobserved “fifth force”, causing particles to deviate from their standard trajectories, see e.g. [74] for application to explaining galaxy rotational curves.

## 4 Invariant quantities and their applications

^{4}:

### 4.1 Integral invariants

^{5}:

^{6}can be written as:

## 5 Einstein and Jordan frames, and their invariant generalizations

So far, we have been using terms “Jordan/Einstein frame” without defining it in an unambiguous way. As it is widely known, the notion of a (conformal) frame has been applied to an analysis of the S–T theories primarily in the metric approach. It is straightforward to extend the concepts of Einstein and Jordan frames to Palatini theory as well. We define the former in the following way:

### Definition 5.1

The *Einstein frame in the Palatini theory* is characterized by specific values of four out of six arbitrary functions \(\{{\mathcal {A}},\ldots ,\alpha \}\): \({\mathcal {A}}=1,\,{\mathcal {B}}=\epsilon _\text {Palatini},\,{\mathcal {C}}_1={\mathcal {C}}_2=0.\)

It follows from the very definition that there are three types of Einstein frames, depending on the value of the parameter \(\epsilon _\text {Palatini}\), which cannot transform each other by a diffeomorphism.^{7} In the simplest case \(\gamma _1=\gamma _2=\gamma _3=0\) its values can be identified with the signature of \({{{\mathcal {B}}}}\), i.e. \(\epsilon _\text {Platini}=\text {sign}({{{\mathcal {B}}}})\). In fact, Einstein frames can be labelled as a triple \((\epsilon _\text {Palatini}, {{{\mathcal {V}}}},\alpha )\). They include the original Einstein–Hilbert–Palatini action as a particular case: \(\epsilon _\text {Palatini}={{{\mathcal {V}}}}=\alpha =0\). One should notice that the frames with \(\epsilon _\text {Palatini}=0\) are singular in the following sense: scalar field re-definition by an arbitrary diffeomorphism \(f\in \mathtt {Diff}^{}({\mathbb {R}})\) transforms one Einstein frame into another (within the same orbit) without changing the value of \(\epsilon _\text {Palatini}=0\). This is not the case for \(\epsilon _\text {Palatini}=\pm 1\): such frames are not preserved under diffeomorphisms. In the Einstein frame, the choice \(\epsilon _\text {Palatini}=+1\) suggests that the scalar field has positive energy, whereas for \(\epsilon _\text {Palatini}=-1\), the theory features a ghost^{8} [22].

Because the transformations (5a) and (5b) act in a self-consistent way, any theory has a mathematically equivalent Einstein frame representation. Therefore, all possible scalar–tensor theories in the Palatini approach can be also labelled by the triple \((\epsilon _\text {Palatini}, {\mathcal {V}}, \alpha )\) in the Einstein frame.

More generally, one can show (cf. (29b)) that the theory written in the Einstein frame becomes effectively metric.

The Jordan frame is defined as follows:

### Definition 5.2

The *Jordan frame in the Palatini theory* is characterized by specific values of four out of the six arbitrary functions \(\{{\mathcal {A}},\ldots ,\alpha \}\): \({\mathcal {A}}=\Psi ,\,{\mathcal {C}}_1={\mathcal {C}}_2=\alpha =0\).

Therefore, the Jordan frame can be described by two functions \(({{{\mathcal {B}}}}, {\mathcal {U}})\). In the Jordan frame, there is no coupling between the scalar field and matter; the field – or a function of it, but it can always be re-defined appropriately – is coupled directly to the curvature. We impose no conditions on the kinetic coupling \({\mathcal {B}}\) and the potential \({\mathcal {U}}\). It can be shown, varying the action expressed in the Jordan frame w.r.t. all dynamical variables, that the curvature scalar is in fact built from a metric conformally related to the initial one. Thence, the Jordan frame in the Palatini approach is in fact almost identical to its metric counterpart, except for a difference in the kinetic coupling. This difference is simply a Brans–Dicke term \(\frac{\omega }{\Psi }\), where \(\omega \) is a constant and depends on the number of dimensions. This term shall be given explicitly later on when considering the invariant generalizations of the Jordan frame.

Let us notice that if the invariant \({\mathcal {I}}^{n}_E\) vanishes, the scalar field has no dynamics, as the kinetic term is not present in the Lagrangian. In this case, the invariant \({\mathcal {I}}^{n}_2\) can be thought of as a function of the invariant \({\mathcal {I}}^{n}_1\) (the case in which \({\mathcal {I}}^{n}_E=0\) and \({\mathcal {I}}^{n}_2=0\) will not be considered, as such a theory is ill-posed). Regardless of which invariant will play the role of the scalar field, at the level of field equation the relation between the scalar field and the remaining fields will be purely algebraic, so that no additional physical degree of freedom will correspond to the extra scalar field included in the action. Since the transformation group acts always in a self-consistent way, this property must hold for all conformally related frames, for which \({\mathcal {I}}^{n}_E=0\). This is the case when \(\epsilon _\text {Palatini}=0\) in the Einstein frame, thence all theories located on its orbit have no additional physical degree of freedom due to the presence of the scalar field. Moreover, at the level of the action functional, a given theory may look as if it featured a dynamical scalar field (e.g. when \({\mathcal {B}}\ne 0\), \({\mathcal {C}}_1\ne 0\) and \({\mathcal {C}}_2\ne 0\)) but in fact it would be just an artifact of poorly chosen independent variables (metric and connection).

As it can be seen, it is possible to find out a short cut passage from the complicated general action functional given by (3) to a surprisingly simple and familiar form written above without using the group transformation rules. In the new frame, the scalar field is coupled only to matter part of the Lagrangian, which means that the Principle of Equivalence does not hold any more. The gravitational part is now free of terms \({\mathcal {C}}_1\) and \({\mathcal {C}}_2\), which were difficult to handle due to their coupling to the non-metricity tensors. Also, the kinetic coupling \({\mathcal {B}}\) is now equal to \(\epsilon _{\text {Palatini}}\), leading to a further simplification of the field equations.

### 5.1 Scalar–tensor extension of \(F({\hat{R}})\) gravity

By means of a simple transformation, it can be shown that \(F({\hat{R}})\) gravity is equivalent to special cases of [15], both in the metric and Palatini approach.^{9} This is achieved by a simple trick, as presented in the Appendix C. In fact, the metric *F*(*R*) is equivalent to the Brans–Dicke (BD) theory with \(\omega _{BD}=0\) (no kinetic term), while the Palatini \(F({\hat{R}})\) is equivalent to the Brans–Dicke theory with \(\omega _{BD}=-\frac{n-1}{n-2}\) (with potential added to the Lagrangian in both cases and in *n* dimensions). However, we may invert the problem and ask whether a given scalar–tensor gravity is equivalent to some \(F({{\hat{R}}})\) theory (in mathematical, not physical sense). Answering this question might be much easier thanks to the introduction of invariant quantities, which are the same for different theories related to each other via conformal transformation. In order to find out whether two arbitrary theories can be linked by a transformation, we need to calculate the invariants and compare them. In this chapter, we will focus on \(F({\hat{R}})\) gravity and discuss conditions for equivalence with an S–T theory. First, let us introduce the notion of Brans–Dicke theory in Palatini approach, which is a particular case of the Jordan frame (cf. Definition 5.2.)

### Definition 5.3

*Brans–Dicke theory in Palatini approach*is given by the following action functional expressed in the Jordan frame:

^{10}

^{11}

## 6 Conclusions

In this paper, we have combined two frequently used ways of altering general relativity, Palatini variation and addition of a scalar field non-minimally coupled to the curvature, into a single theory of gravity. Our motivation for considering such coalescence of modifications of classical gravity was the lack of formalism of invariants defined for Palatini approach in S–T theories. Although the prevalent approach to the analysis of S–T theories is the metric one, the Palatini formalism has many interesting features to offer.

In the course of the paper, we placed special emphasis on the notion of conformal and almost-geodesic transformations, as it allows us to establish – under well-defined and strict conditions – mathematical equivalence between two different conformal frames. We did not aim to take a stand on the issue of which frame is the physical one; the main purpose of this paper was to obtain solution-equivalent classes of frames and introduce proper language enabling one to analyze the theory in a frame-independent manner. The first step to creating such language was to recognize that in case of the Palatini approach, one must transform the metric and the connection independently. Decoupling of metric from affine structure of spacetime influenced the action functional defined for a general S–T theory, devised to preserve its form under conformal change, enforcing us to add special terms linear in scalar field derivatives. These terms do not have any clear interpretation yet.

We singled out two frames most commonly used in the literature – Jordan and Einstein. Quantities behaving as invariants on the orbits of the two frames were also introduced and the role they play when comparing equivalent theories was discussed. In general, the theory possesses three degrees of freedom: one introduced by the scalar field, and the remaining two being a property of the metric. However, the independent scalar field turns out to be an auxiliary field in case the invariant \({\mathcal {I}}^n_E\) vanishes; then, the theory has only two degrees of freedom.

It was discovered that there exists a subclass of conformal frames with \({\mathcal {C}}_1={\mathcal {C}}_2=0\) fully analogous to the metric frames. In such frames, the (initially independent) connection is always Levi-Civita with respect to a metric \({\bar{g}}\) conformally related to the initial metric *g*. This class is invariant under the action of the subgroup \(\gamma _2=\gamma _3=0\).

If a given theory has the same \(\{{\mathcal {A}},{\mathcal {B}},{\mathcal {V}},\alpha \}\) functions both in the metric and Palatini approach, the latter one can be brought to the metric form using the property discussed above. The only difference between such two theories will be the exact form of the kinetic coupling \({\mathcal {B}}\); in the metric formalism resulting from a prior Palatini frame, the coupling will take on the form \({\mathcal {B}}-\frac{n-1}{n-2}\frac{1}{\Phi }\). This fact allowed us to establish a correspondence between the Brans–Dicke theories in the metric and Palatini formalism.

It was also shown that for an arbitrary S–T theory in the Palatini approach there always exists a unique transformation defined for the connection such that it renders the theory effectively metric. This useful property allows us to analyze a specific theory within the metric formalism.

Finally, \(F({\hat{R}})\) theories were analyzed using the language of invariants. We made use of the well-established equivalence of these theories to S–T gravity – to the Brans–Dicke theory, to be precise. Invariants made it possible for us to address an issue of the relation between S–T and \(F({\hat{R}})\), namely, we identified cases in which those two theories could be related by the transformation (5a)–(5c), meaning that they are mathematically equivalent. It was discovered that the coefficients \(\{{\mathcal {A}},{\mathcal {B}},{\mathcal {C}}_1,{\mathcal {C}}_2,{\mathcal {V}},\alpha \}\), which characterize a specific S–T theory, must fulfil certain relations (given by (35)) in order for the theory to be equivalent to \(F({\hat{R}})\) gravity in the Palatini approach. Furthermore, because the metric and the Palatini formalisms always give two non-equivalent theories, if a given scalar–tensor theory results from some *F*(*R*) theory, it cannot simultaneously be derived from both the metric and the Palatini *F*(*R*).

The main aim of this paper was to introduce a new class of scalar–tensor theories of gravity and analyze some of its mathematical properties. Due to its introductory nature, it focuses on the formal aspects of the theory, with a special emphasis put on self-consistency conditions, and lacks direct physical applications. Also, due to adopting the Palatini approach and adding more degrees of freedom into the theory, it will be straightforward to include torsion and/or disformal transformations in order to investigate theirs impact on self-consistency of the theory. Analysis of real-world phenomena will be carried out in the forthcoming papers. In order to find out whether the predictions of the theory are in agreement with experiment, we plan on computing the post-Newtonian parameters in the first place. Furthermore, topics to be covered in the future works will include cosmological applications (cf. [20, 21]), F(R) theories with non-minimal curvature coupling (see e.g. [17, 19]), the appearance of ghosts and tachyons.

## Footnotes

- 1.
It should be noted that the disformal transformations can be combined together with the conformal transformations considered in the present paper, see e.g. [48].

- 2.
This is an extension of the results obtained initially in [55].

- 3.
- 4.
In [26], this invariant is defined as \({\mathcal {I}}_1(\Phi )=\frac{e^{2\alpha (\Phi )}}{{\mathcal {A}}(\Phi )}\) (in four dimensions).

- 5.
This is integral invariant, which is determined up to arbitrary integration constant. The choice of the sign ± in (23) has to ensure positivity of the expression inside the square root.

- 6.
From now on, all invariants shall be written without the superscript denoting the number of dimensions if \(n=4\).

- 7.
However, it can be changed by making use of disformal transformations [47].

- 8.
In the metric case, when one considers weak-field approximation, due to the presence of non-minimal coupling, the negative value of the parameter \(\epsilon _\text {Palatini}\) does not necessarily mean that the physical, interacting field is a ghost, even if the initial field \(\Phi \) is [22].

- 9.
In this section \({{\hat{R}}}\) denotes, for short cut, Palatini–Ricci scalar, i.e. \({{\hat{R}}}= R(g,\Gamma )\equiv g^{\mu \nu }R_{\mu \nu }(\Gamma )\).

- 10.
The sign “−” corresponds to \(\omega _\text {Palatini}<0\).

- 11.
- 12.
This implies that the Levi-Civita connection undergoes the Weyl transformation \({{\bar{\Gamma }}}^\alpha _{\mu \nu }=\Gamma ^\alpha _{\mu \nu }+2 \delta ^\alpha _{(\mu }\partial _{\nu )}\gamma _2(\Phi )-g_{\mu \nu } g^{\alpha \beta }\partial _\beta \gamma _2(\Phi ) \).

- 13.
Since \(f^\prime \ne 0\) one can also consider a subgroup \(f^\prime >0\).

- 14.
More general action with the gradient field \(\partial _\alpha \Phi \) replaced by an arbitrary one form will be considered elsewhere.

- 15.
One should stress that Palatini

*F*(*R*)-gravity is not dynamically equivalent to metric one with the same function*F*(*R*). - 16.
One can observe that the trivial, i.e. constant, potential \(U(\Phi )\) corresponds to the linear Lagrangian \(F( R)=R-2\Lambda \). More generally, for a given

*F*the potential \(U_F\) is a (singular) solution of the Clairaut’s differential equation: \(U_F(\Phi )=\Phi \frac{d U_F}{d\Phi } -F(\frac{d U_F}{d\Phi })\). - 17.
It also corresponds to the Palatini Brans–Dicke theory, in a sense of Definition 5.3, with \(\omega _\text {Palatini}=0\).

## Notes

### Acknowledgements

We are grateful to Ulf Lindström for helpful comments concerning his earlier papers on the subject. This research was supported by Polish National Science Center (NCN), Project UMO-2017/27/B/ST2/01902.

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