In the context of weak-field metric-affine (i.e. Palatini) gravity near Minkowski spacetime, we compute the particle spectra in the simultaneous presence of all independent contractions quadratic in Ricci-type tensors. Apart from the full metric-affine geometry, we study kinematic limits with vanishing torsion (i.e. a symmetric connection) and vanishing non-metricity (i.e. a metric connection, which is physically indistinguishable from Poincaré gauge theory at the level of the particle spectrum). We present a detailed report on how spin-parity projection operators can be used to derive systematically and unambiguously the character of the propagating states. The unitarity constraints derived from the requirements of tachyon- and ghost-freedom are obtained. We show that, even in the presence of all Ricci-type operators, only a narrow selection of viable theories emerges by a tuning.
We analyse what happens when the Horndeski Lagrangian is varied within the Palatini approach by considering the metric and connection as independent variables. Assuming the connection to be torsionless, there can be infinitely many metric-affine versions $L_{\rm P}$ of the original Lagrangian which differ from each other by terms proportional to the non-metricity tensor. After integrating out the connection, each $L_{\rm P}$ defines a metric theory, which can either belong to the original Horndeski family, or it can be of a more general DHOST type, or it shows the Ostrogradsky ghost. We analys…
We present a covariant multisymplectic formulation for the Einstein-Palatini (or Metric-Affine) model of General Relativity (without energy-matter sources). As it is described by a first-order affine Lagrangian (in the derivatives of the fields), it is singular and, hence, this is a gauge field theory with constraints. These constraints are obtained after applying a constraint algorithm to the field equations, both in the Lagrangian and the Hamiltonian formalisms. In order to do this, the covariant field equations must be written in a suitable geometrical way, using integrable distributions wh…
As is known from studies of gravity models in the Palatini formalism, there exist two inequivalent definitions of the generalized Ricci tensor in terms of the generalized curvature namely, $\widetilde{R}_{μν}=R^ρ_{μρν}$ and $R^μ_ν=g^{αβ}R^μ_{ανβ}$. A deep formal investigation of theories with lagrangians of the form $L=L(\widetilde{R}_{(μν)})$ was initiated in [4]. In that work, the authors leave the connection free, and find out that the torsion only appears as a projective mode. This agrees with the widely employed condition of vanishing torsion in these theories as a simple gauge choice. In…
In the present paper, the 3+1 decomposition of the spacetime onto hypersurface(s) is analysed and established for theories within the Palatini formalism by considering a general function of the Ricci scalar in the gravitational action. The corresponding Gauss-Codazzi relations are obtained and the boundary term that has to be subtracted in the gravitational action is easily deduced. Then, these relations are applied to the so-called ADM decomposition to describe the foliation of the spacetime onto hypersurfaces of constant time within these theories. Finally, the junction conditions are also o…
Wormholes are hypothetical shortcuts in spacetime that in General Relativity unavoidably violate all of the pointwise energy conditions. In this paper, we consider several wormhole spacetimes that, as opposed to the standard \emph{designer} procedure frequently employed in the literature, arise directly from gravitational actions including additional terms resulting from contractions of the Ricci tensor with the metric, and which are formulated assuming independence between metric and connection (Palatini approach). We reinterpret such wormhole solutions under the prism of General Relativity a…
An introduction to extended theories of gravity formulated in metric-affine (or Palatini) spaces is presented. Focusing on spherically symmetric configurations with electric fields, we will see that in these theories the central singularity present in General Relativity is generically replaced by a wormhole structure. The resulting space-time becomes geodesically complete and, therefore, can be regarded as non-singular. We illustrate these properties considering two different models, namely, a quadratic $f(R)$ theory and a Born-Infeld like gravity theory.
It has been recently claimed in [arXiv:2211.12327] that an action of the Einstein-Palatini form plus a torsionless Pontryagin term (multiplied by a constant) represents a counterexample to the conclusions of [arXiv:0705.1879], namely, that Lovelock gravity is the only case in which the metric and metric-affine formulations of gravity are equivalent. However, given that the Pontryagin term (multiplied by a constant) can be written as a total D-divergence, it is a textbook matter to realise that the addition of such (or any other) D-divergence only affects at the boundary, leaving invariant the …
We study reheating of inflationary models with general non-minimal coupling $K(ϕ)R$ with $K(ϕ)\sim \sqrt{V(ϕ)}$ where $R$ is the Ricci scalar and $V$ is the inflaton potential. In particular, when we take the monomial potential $K(ϕ) \propto ϕ^m$ with $m \in \mathbb{Z}_+$, we provide general analytic expressions for cosmological observables. We consider a wide range of non-minimal coupling $ ξ\in [0, \infty) $ in metric and Palatini formalisms and derive the predictions for cosmological observables and the reheating temperature taking a general equation of state parameter $ w_\text{reh} $.
The generalized hybrid metric-Palatini gravity is a theory of gravitation that has an action composed of a Lagrangian $f(R,\cal R)$, where $f$ is a function of the metric Ricci scalar $R$ and a new Ricci scalar $\cal R$ formed from a Palatini connection, plus a matter Lagrangian. This theory can be rewritten by trading the new geometric degrees of freedom of $f(R,\cal R)$ into two scalar fields, $φ$ and $ψ$, yielding an equivalent scalar-tensor theory. Given a spacetime theory, the next step is to find solutions. To construct solutions it is often necessary to know the junction conditions betw…
We calculate the tree-level amplitudes for electrically neutral $2\to2$ scattering for the Standard Model Higgs doublet non-minimally coupled to the Ricci scalar. We consider both the metric and the Palatini formulation of gravity. We find the partial wave unitarity limit for a general background field value. Our results are in agreement with previous work. In the electroweak vacuum, tree-level unitarity is violated at $\sim M_\text{Planck}/ξ$ in the metric formulation, and at $\sim M_\text{Planck}/\sqrtξ$ in the Palatini formulation. In the inflationary large field background, the unitarity l…
The hybrid metric-Palatini theory of gravity (HMPG), proposed in 2012 by T. Harko et al., is known to successfully describe both local (solar-system) and cosmological observations. We discuss static, spherically symmetric vacuum solutions of HMPG with the aid of its scalar-tensor representation. This scalar-tensor theory coincides with general relativity with a conformally coupled scalar field (which can be canonical or phantom), therefore the known solutions of this theory are re-interpreted in terms of HMPG. In particular, in the case of zero scalar field potential $V (ϕ)$, such that both Ri…
In this paper we prove that the $k$-th order metric-affine Lovelock Lagrangian is not a total derivative in the critical dimension $n=2k$ in the presence of non-trivial non-metricity. We use a bottom-up approach, starting with the study of the simplest cases, Einstein-Palatini in two dimensions and Gauss-Bonnet-Palatini in four dimensions, and focus then on the critical Lovelock Lagrangian of arbitrary order. The two-dimensional Einstein-Palatini case is solved completely and the most general solution is provided. For the Gauss-Bonnet case, we first give a particular configuration that violate…
Slowly rotating charged BTZ black hole solutions in Palatini Chern-Simons gravity
Instituto de Física Corpuscular (IFIC), CSIC‐Universitat de València（粒子物理研究所 (IFIC)，西班牙瓦伦西亚大学）；Universidade Federal do Ceará (UFC)（塞阿拉联邦大学）；Federal University of Rio Grande do Norte（北里奥格兰德联邦大学）；International Institute of Physics, Federal University of Rio Grande do Norte（北里奥格兰德联邦大学国际物理研究所）；Universidade Federal da Paraíba（帕拉伊巴联邦大学）
The Palatini gravitational action is enlarged by an arbitrary function $f(X)$ of the determinants of the Ricci tensor and the metric, $X=|\textbf{det}.R|/|\textbf{det}.g|$. The resulting Ricci-determinant theory exhibits novel deviations from general relativity. We study a particular realization where the extension is characterized by the square-root of the Ricci-determinant, $f(X)=λ_\text{Edd}\sqrt{X}$, which corresponds to the famous Eddington action. We analyze the obtained equations for a perfect fluid source and show that the affine connection can be solved in terms of the energy density …
We reveal the existence of a certain hidden symmetry in general ghost-free scalar-tensor theories which can only be seen when generalizing the geometry of the spacetime from Riemannian. For this purpose, we study scalar-tensor theories in the metric-affine (Palatini) formalism of gravity, which we call scalar-metric-affine theories for short, where the metric and the connection are independent. We show that the projective symmetry, a local symmetry under a shift of the connection, can provide a ghost-free structure of scalar-metric-affine theories. The ghostly sector of the second-order deriva…
The so-called hybrid metric-Palatini theory of gravity (HMPG), proposed in 2012 by T. Harko et al., is known to successfully describe both local (solar-system) and cosmological observations. This paper gives a complete description of static, spherically vacuum solutions of HMPG in the simplest case where its scalar-tensor representation has a zero scalar field potential $V(ϕ)$, and both Riemannian ($R$) and Palatini ($\cal R$) Ricci scalars are zero. Such a scalar-tensor theory coincides with general relativity with a phantom conformally coupled scalar field as a source of gravity. Generic asy…
We consider the Palatini formulation of $f(R,T)$ gravity theory, in which a nonminimal coupling between the Ricci scalar and the trace of the energy-momentum tensor is introduced, by considering the metric and the affine connection as independent field variables. The field equations and the equations of motion for massive test particles are derived, and we show that the independent connection can be expressed as the Levi-Civita connection of an auxiliary, energy-momentum trace dependent metric, related to the physical metric by a conformal transformation. Similarly to the metric case, the fiel…
We generalize the algorithm that establishes the correspondence between metric-affine Eddington-inspired Born-Infeld (EiBI) gravity and General Relativity (GR) to any bosonic matter sector. Along the way, a polished version of the proof of existence of a metric compatible frame associated to any metric-affine Ricci-Based Gravity (RBG) is presented. Particular attention is given to the problem of generating solutions of a RBG from the GR conterpart, providing a general recipe for the EiBI case. This extends previous results obtained for specific matter that included anisotropic fluids, scalar f…
We analytically derive the covariant form of the Riemann (curvature) tensor for homogeneous Metric-Affine Cosmologies. That is, we present, in a Cosmological setting, the most general covariant form of the full Riemann tensor including also its non-Riemannian pieces which are associated to spacetime torsion and non-metricity. Having done so we also compute a list of the curvature tensor by-products such as Ricci tensor, homothetic curvature, Ricci scalar, Einstein tensor etc. Finally we derive the generalized Metric-Affine version of the usual Gauss-Bonnet density in this background and demons…
Wormhole solutions in a generalized hybrid metric-Palatini matter theory, given by a gravitational Lagrangian $f\left(R,\cal{R}\right)$, where $R$ is the metric Ricci scalar, and $\mathcal{R}$ is a Palatini scalar curvature defined in terms of an independent connection, and a matter Lagrangian, are found. The solutions are worked in the scalar-tensor representation of the theory, where the Palatini field is traded for two scalars, $φ$ and $ψ$, and the gravitational term $R$ is maintained. The main interest in the solutions found is that the matter field obeys the null energy condition (NEC) ev…
We investigate cosmological perturbations of scalar-tensor theories in Palatini formalism. First we introduce an action where the Ricci scalar is conformally coupled to a function of a scalar field and its kinetic term and there is also a k-essence term consisting of the scalar and its kinetic term. This action has three frames that are equivalent to one another: the original Jordan frame, the Einstein frame where the metric is redefined, and the Riemann frame where the connection is redefined. For the first time in the literature, we calculate the quadratic action and the sound speed of scala…
We show that the space of solutions of a wide family of Ricci-based metric-affine theories of gravity can be put into correspondence with the space of solutions of general relativity (GR). This allows us to use well-established methods and results from GR to explore new gravitational physics beyond it.
We study the most general solution for affine connections that are compatible with the variational principle in the Palatini formalism for the Einstein-Hilbert action (with possible minimally coupled matter terms). We find that there is a family of solutions generalising the Levi-Civita connection, characterised by an arbitrary, non-dynamical vector field ${\cal A}_μ$. We discuss the mathematical properties and the physical implications of this family and argue that, although there is a clear mathematical difference between these new Palatini connections and the Levi-Civita one, both unparamet…
We study non-trivial (i.e. non-Levi-Civita) connections in metric-affine Lovelock theories. First we study the projective invariance of general Lovelock actions and show that all connections constructed by acting with a projective transformation of the Levi-Civita connection are allowed solutions, albeit physically equivalent to Levi-Civita. We then show that the (non-integrable) Weyl connection is also a solution for the specific case of the four-dimensional metric-affine Gauss-Bonnet action, for arbritrary vector fields. The existence of this solution is related to a two-vector family of tra…
We extend the correspondence between metric-affine Ricci-Based Gravity theories and General Relativity (GR) to the case in which the matter sector is represented by linear and nonlinear electromagnetic fields. This complements previous studies focused on fluids and scalar fields. We establish the general algorithm that relates the matter fields in the GR and RBG frames and consider some applications. In particular, we find that the so-called Eddington-inspired Born-Infeld gravity theory coupled to Maxwell electromagnetism is in direct correspondence with GR coupled to Born-Infeld electromagnet…
We classify the metric-affine theories of gravitation, in which the metric and the connections are treated as independent variables, by use of several constraints on the connections. Assuming the Einstein-Hilbert action, we find that the equations for the distortion tensor (torsion and non-metricity) become algebraic, which means that those variables are not dynamical. As a result, we can rewrite the basic equations in the form of Riemannian geometry. Although all classified models recover the Einstein gravity in the Palatini formalism (in which we assume there is no coupling between matter an…
Starting from an affinely connected space, we consider a model of gravity whose fundamental field is the connection. We build up the action using as sole premise the invariance under diffeomorphisms, and study the consequences of a cosmological ansatz for the affine connection in the torsion-free sector. Although the model is built without requiring a metric, we show that the nondegenerated Ricci curvature of the affine connection can be interpreted as an \emph{emergent} metric on the manifold. We show that there exists a parametrization in which the \((r,φ)\)-restriction of the geodesics coin…
We provide a comprehensive overview of metric-affine geometries with spherical symmetry, which may be used in order to solve the field equations for generic gravity theories which employ these geometries as their field variables. We discuss the most general class of such geometries, which we display both in the metric-Palatini formulation and in the tetrad / spin connection formulation, and show its characteristic properties: torsion, curvature and nonmetricity. We then use these properties to derive a classification of all possible subclasses of spherically symmetric metric-affine geometries,…
