A general $f(\mathcal{R})$ gravitational theory is considered within the Palatini formalism. By applying the variational principle and the usual conditions on the boundary, we show explicitly that a surface term remains such that as in their metric-compatible counterparts, an additional surface term has to be added in the gravitational action, which plays a fundamental role when calculating the entropy of the black hole.
In this paper, we explicitly prove the presymplectic forms of the Palatini and Ashtekar gravity to be zero along gauge orbits of the Lorentz and diffeomorphism groups, which ensures the diffeomorphism invariance of these theories.
We formulate the generalized Tolman-Oppenheimer-Volkoff equations for the $f(\hat{R})$ Palatini gravity in the case of static and spherical symmetric geometry. We also show that a neutron star is a stable system independently of the form of the functional $f(\hat{R})$
In this paper we generalize the off-shell Abbott-Deser-Tekin (ADT) conserved charge formalism to Palatini theory of gravity with torsion and non-metricity. Our construction is based on the coordinate formalism and the independent dynamic fields are the metric and the affine connection. For a general Palatini theory of gravity, which is diffeomorphism invariant up to a boundary term, we obtain the most general expression for off-shell ADT potential. As explicit examples, we derive the off-shell ADT potentials for Einstein-Hilbert action, the most general $L(g_{μν}, R^λ{}_{ναμ}, T^λ{}_{αβ}, Q_{α…
A Palatini-type action for Einstein and Gauss-Bonnet gravity with non-trivial torsion is proposed. Three-form flux is incorporated via a deformation of the Riemann tensor, and consistency of the Palatini variational principle requires the flux to be covariantly constant and to satisfy a Jacobi identity. Studying gravity actions of third order in the curvature leads to a conjecture about general Palatini-Lovelock-Cartan gravity. We point out potential relations to string-theoretic Bianchi identities and, using the Schouten-Nijenhuis bracket, derive a set of Bianchi identities for the non-geomet…
We consider cosmological scalar field evolving under exponential potential of the Non-minimal Derivative Coupling (NMDC) gravity model in Palatini formalism. Slow-roll regime is assumed. GR and metric formalism NMDC cases are compared in this study. Phase portraits show that Palatini NMDC effect restricts acceleration phase into smaller region in the phase space. NMDC effect of the Palatini case suppresses expansion rate than that of the GR while the metric NMDC enhances rate of expansion of the GR case.
A variational principle was recently suggested by Goenner, where an independent metric generates the spacetime connection. It is pointed out here that the resulting theory is equivalent to the usual Palatini theory. However, a bimetric reformulation of the variational principle leads to theories which are physically distinct from both the metric and the metric-affine ones, even for the Einstein-Hilbert action. They are obtained at a decoupling limit of C-theories, which contain also other viable generalizations of the Palatini theories.
A class of scalar-tensor theories (STT) including a non-metricity that unifies metric, Palatini and hybrid metric-Palatini gravitational actions with non-minimal interaction is proposed and investigated from the point of view of their consistency with generalized conformal transformations. It is shown that every such theory can be represented on-shell by a purely metric STT possessing the same solutions for a metric and a scalar field. A set of generalized invariants is also proposed. This extends the formalism previously introduced in \cite{kozak2019}. We then apply the formalism to Starobins…
A variational principle is suggested within Riemannnian geometry, in which an auxiliary metric and the Levi Civita connection are varied independently. The auxiliary metric plays the role of a Lagrange multiplier and introduces non-minimal coupling of matter to the curvature scalar. The field equations are 2nd order PDEs and easier to handle than those following from the so-called Palatini method. Moreover, in contrast to the latter method. no gradients of the matter variables appear. In cosmological modeling, the physics resulting from the new variational principle will differ from the modeli…
We consider a Palatini variation on a generalized Einstein-Hilbert action. We find that the Hilbert constraint, that the connection equals the Christoffel symbol, arises only as a special case of this general action, while for particular values of the coefficients of this generalized action, the connection is completely unconstrained. We discuss the relationship between this situation and that usually encountered in the Palatini formulation.
We compare the metric and the Palatini formalism to obtain the Einstein equations in the presence of higher-order curvature corrections that consist of contractions of the Riemann tensor, but not of its derivatives. We find that there is a class of theories for which the two formalisms are equivalent. This class contains the Palatini version of Lovelock theory, but also more Lagrangians that are not Lovelock, but respect certain symmetries. For the general case, we find that imposing the Levi-Civita connection as an Ansatz, the Palatini formalism is contained within the metric formalism, in th…
We formulate quantum field theories of massive fields of arbitrary spins. The presence of both physical and fake particles, organized into multiplets, makes it possible to fulfill the requirements of locality, unitarity and renormalizability at the same time. The theories admit cubic and quartic self-interactions and can be coupled to quantum gravity and gauge fields. The simplest irreducible bosonic and fermionic multiplets are made of towers of alternating physical and fake particles. Their mass spectrum is constrained by RG invariant relations and depends on just one or two masses. The fixe…
We consider a Palatini variation on a general $N$-Dimensional second order, torsion-free dilaton gravity action and determine the resulting equations of motion. Consistency is checked by considering the restraint imposed due to invariance of the matter action under simple coordinate transformations, and the special case of N=2 is examined. We also examine a sub-class of theories whereby a Palatini variation dynamically coincides with that of the "ordinary" Hilbert variational principle; in particular we examine a generalized Brans-Dicke theory and the associated role of conformal transformatio…
The scrutiny of black hole perturbations and its application to testing gravity theories have been a very crucial frontier in modern physics. In this paper we study the quasi-normal modes (QNM) of massless scalar perturbations for various charged black holes in the Palatini-type theories of gravity. Specifically, we consider the Palatini $f(R)$ theory coupled with Born-Infeld nonlinear electrodynamics and the Eddington-inspired-Born-Infeld gravity (EiBI) coupled with Maxwell electromagnetic fields. Special attention is paid to Einstein-Born-Infeld black holes, EiBI charged black holes and Born…
It is known that the metric and Palatini formalisms of gravity theories have their own interesting features but also suffer from some different drawbacks. Recently, a novel gravity theory called hybrid metric-Palatini gravity was put forward to cure or improve their individual deficiencies. The action of this gravity theory is a hybrid combination of the usual Einstein-Hilbert action and a $f(\mathcal{R})$ term constructed by the Palatini formalism. Interestingly, it seems that the existence of a light and long-range scalar field in this gravity may modify the cosmological and galactic dynamic…
We derive the gravitational energy-momentum pseudo-tensor $τ^μ_{\phantomμν}$ in both Palatini and metric approaches to $f(R)$ gravity. We then obtain the related cosmological gravitational energy density. Considering a flat Friedmann-Lemaître-Robertson-Walker spacetime, the energy density complex of matter and gravitation vanishes in the metric approach, but results non-vanishing in the Palatini formalism. This feature could be relevant in order to physically discriminate between the two approaches.
In a classical theory of gravity, the Barbero-Immirzi parameter ($η$) appears as a topological coupling constant through the Lagrangian density containing the Hilbert-Palatini term and the Nieh-Yan invariant. In a quantum framework, the topological interpretation of $η$ can be captured through a rescaling of the wavefunctional representing the Hilbert-Palatini theory, as in the case of the QCD vacuum angle. However, such a rescaling cannot be realized for pure gravity within the standard (Dirac) quantization procedure where the second-class constraints of Hilbert-Palatini theory are eliminated…
In the context of modified gravity, we point out how the Palatini version of these theories is singled out as a very special case corresponding to the unique fixed point of a transformation involving a special conformal rescaling of the metric. This mathematical peculiarity signals deeply rooted problems which make the theory unphysical.
We reconsider the problem of $f(R)$ theories of gravity coupled to Born-Infeld theory of electrodynamics formulated in a Palatini approach, where metric and connection are independent fields. By studying electrovacuum configurations in a static and spherically symmetric space-time, we find solutions which reduce to their Reissner-Nordström counterparts at large distances but undergo important non-perturbative modifications close to the center. Our new analysis reveals that the point-like singularity is replaced by a finite-size wormhole structure, which provides a geodesically complete and thu…
We consider the Palatini formalism of the Born-Infeld gravity. In the Palatini formalism, the propagating mode is only graviton, whose situation is different from that in the metric formalism. We discuss about the FRW cosmology by using an effective potential. Especially we consider the condition that the bouncing could occur. We also give some speculations about the black hole formation
We show that an invariant an universal length scale can be consistently introduced in a generally covariant theory through the gravitational sector using the Palatini approach. The resulting theory is able to capture different aspects of quantum gravity phenomenology in a single framework. In particular, it is found that in this theory field excitations propagating with different energy-densities perceive different background metrics, which is a fundamental characteristic of the DSR and Rainbow Gravity approaches. We illustrate these properties with a particular gravitational model and explici…
Non-vacuum static spherically-symmetric solutions in Palatini f(R) gravity are examined. It is shown that for generic choices of f(R), there are commonly-used equations of state for which no satisfactory physical solution of the field equations can be found within this framework, apart from in the special case of General Relativity, casting doubt on whether Palatini f(R) gravity can be considered as giving viable alternatives to General Relativity.
We study isotropic and anisotropic (Bianchi I) cosmologies in Palatini $f(R)$ and $f(R,R_{μν}R^{μν})$ theories of gravity and consider the existence of non-singular bouncing solutions in the early universe. We find that all $f(R)$ models with isotropic bouncing solutions develop shear singularities in the anisotropic case. On the contrary, the simple quadratic model $R+a R^2/R_P+R_{μν}R^{μν}/R_P$ exhibits regular bouncing solutions in both isotropic and anisotropic cases for a wide range of equations of state, including dust (for $a<0$) and radiation (for arbitrary $a$). It thus represents a p…
We introduce a new approach to modified gravity which generalizes the recently proposed hybrid metric-Palatini gravity. The gravitational action is taken to depend on a general function of both the metric and Palatini curvature scalars. The dynamical equivalence with a non-minimally coupled bi-scalar field gravitational theory is proved. The evolution of cosmological solutions is studied using dynamical systems techniques.
By establishing that Palatini formulation of $L(R)$ gravity is equivalent to $ω=-3/2$ Brans-Dicke theory, we show that energy-momentum tensor is covariantly conserved in this type of modified gravity theory.
We use the collision-free Boltzmann equation in Palatini $f({\mathcal{R}})$ gravity to derive the virial theorem within the context of the Palatini approach. It is shown that the virial mass is proportional to certain geometrical terms appearing in the Einstein field equations which contribute to gravitational energy and that such geometric mass can be attributed to the virial mass discrepancy in cluster of galaxies. We then derive the velocity dispersion relation for clusters followed by the metric tensor components inside the cluster as well as the $f({\mathcal{R}})$ lagrangian in terms of t…
We consider extensions of the Einstein-Hilbert Lagrangian to a general functional of metric and Riemann curvature tensor. A given such Lagrangian describes two different theories depending on considering connection and metric (Palatini formulation), or only the metric (metric formulation) as independent dynamical degrees of freedom. Equivalence of the Palatini and metric formulations at the level of equations of motion, which as we will argue is a manifestation of the Equivalence Principle, is the physical criterion that restricts form of the Lagrangians of modified gravity theories. We prove …
The equivalence between metric and Palatini formalisms in f(R)-gravity can be achieved in the general context of theories with divergence free current. This equivalence is a necessary result of a symmetry which is included in a particular conservation equation of the current. In fact the conservation equation, by an appropriate redefinition of the introduced auxiliary field, may be encoded in a massless scalar field equation.
Pure $R^2$ gravity has been shown to be equivalent to Einstein gravity with non-zero cosmological constant and a massless scalar field. We show that the Palatini formulation of pure $R^2$ gravity is equivalent to Einstein gravity with non-zero cosmological constant as before but with no massless scalar field. This is an important new development because the massless scalar field is not readily identifiable with any known particle in nature or unknown particles like cold dark matter which are expected to be massive. We then include a non-minimally coupled Higgs field as well as fermions to disc…
In this paper, we review modified $f(R)$ theories of gravity in Palatini formalism. In this framework, , we use the Raychaudhuri's equation along with the requirement that the gravity is attractive, which holds for any geometrical theory of gravity to discuss the energy conditions. Then, to derive these conditions, we obtain an expression for effective pressure and energy density by considering FLRW metric. Energy conditions derived in Palatini version of $f(R)$ Gravity differ from those derived in GR. We will see that the WEC (weak energy condition) derived in Palatini formalism has exactly t…
We investigate thermodynamics of the apparent horizon in $f(R)$ gravity in the Palatini formalism with non-equilibrium and equilibrium descriptions. We demonstrate that it is more transparent to understand the horizon entropy in the equilibrium framework than that in the non-equilibrium one. Furthermore, we show that the second law of thermodynamics can be explicitly verified in both phantom and non-phantom phases for the same temperature of the universe outside and inside the apparent horizon.
We consider several physical scenarios where black holes within classical gravity theories including $R^2$ and Ricci-squared corrections and formulated à la Palatini can be analytically studied.
We study the renormalization flow of Hilbert-Palatini gravity to lowest non-trivial order. We find evidence for an asymptotically safe high-energy completion based on the existence of an ultraviolet fixed point similar to the Reuter fixed point of quantum Einstein gravity. In order to manage the quantization of the large number of independent degrees of freedom in terms of the metric as well as the connection, we use an on-shell reduction scheme: for this, we quantize all degrees of freedom beyond Einstein gravity at a given order that remain after using the equations of motion at the precedin…
Using a dynamical system approach we study the cosmological phase space of the generalized hybrid metric-Palatini gravity theory, characterized by the function $f\left(R,\mathcal R\right)$, where $R$ is the metric scalar curvature and $\mathcal R$ the Palatini scalar curvature of the spacetime. We formulate the propagation equations of the suitable dimensionless variables that describe FLRW universes as an autonomous system. The fixed points are obtained for four different forms of the function $f\left(R,\mathcal R\right)$, and the behavior of the cosmic scale factor $a(t)$ is computed. We sho…
We derive the dynamical equations for a non-local gravity model in the Palatini formalism and we discuss some of the properties of this model. We have shown that, in some specific cases, the vacuum solutions of general relativity are also vacuum solutions of the non-local model, so we conclude that, at least in this case, the singularities of Einstein's gravity are not removed.
In this paper, the hybrid metric-Palatini gravity is an approach to modified gravity in which is added to the usual Einstein-Hilbert action a supplementary term containing a Palatini-type correction of the form $f({\cal R},T)$. Here, ${\cal R}$ is the Palatini curvature scalar, which is constructed from an independent connection and $T$ is the trace of the energy-momentum tensor. This theory describes a non-minimal coupling between matter and geometry. The modified Einstein field equations in this hybrid metric-Palatini approach are obtained. Then, it is investigated whether this modified theo…
We examine static spherically symmetric polytropic spheres in Palatini f(R) gravity and show that no regular solutions to the field equations exist for physically relevant cases such as a monatomic isentropic gas or a degenerate electron gas, thus casting doubt on the validity of Palatini f(R) gravity as an alternative to General Relativity.
We study nonlinear gravity theories in both the metric and the Palatini (metric-affine) formalisms. The nonlinear character of the gravity lagrangian in the metric formalism causes the appearance of a scalar source of matter in Einstein's equations that can be interpreted as a quintessence field. However, in the Palatini case no new energy sources appear, though the equations of motion get modified in such a way that usual matter can lead to repulsive gravity at very low densities. Thus, the Palatini formalism could provide a mechanism to explain the recent acceleration of the universe without…
We investigate in detail a family of quintessential inflation models in the context of $R+R^2$ Palatini modified gravity. We find that successful inflation and quintessence are obtained with an inflaton scalar potential that is approximately quadratic in inflation and inverse quartic in quintessence. We show that corrections for the kination period due to Palatini modified gravity are subdominant, while the setup does not challenge constraints on modified gravity from solar system observations and microscopic experiments, in contrast to the metric case. We obtain concrete predictions regarding…
We consider the Palatini formalism of gravity with cosmological constant $Λ$ coupled to a scalar field $ϕ$ in $n$-dimensions. The $n$-dimensional Einstein equations with $Λ$ can be derived by the variation of the coupled Palatini action provided $n>2$. The Hamiltonian analysis of the coupled action is carried out by a $1+(n-1)$ decomposition of the spacetime. It turns out that both Palatini action and Hilbert action lead to the same geometric dynamics in the presence of $Λ$ and $ϕ$. While, the $n$-dimensional Palatini action could not give a connection dynamics formalism directly.
The question of the correspondence between the number of degrees of freedom and the number of polarization states in hybrid metric-Palatini f(R)-gravity is considered. It is shown that the scalar transverse breathing and longitudinal modes are a mixture of one polarization state. The speed of gravitational waves propagation in hybrid f(R)-gravity is calculated.
We will briefly discuss the necessary conditions for stability of polytropies in $f(\hat R)$ Palatini gravity and the differences with the General Relativity ones.
By using the equivalence between metric and Palatini f(R) (or "modified") gravities with omega=0, -3/2 Brans-Dicke theories, it is shown that the Ehlers-Geren-Sachs theorem of general relativity is extended to modified gravity. In the case of metric f(R) gravity previously studied, this agrees with previous literature.
The Palatini $f(R,T)$ gravity theory is considered. The standard Einstein-Hilbert action is replaced by an arbitrary function of the Ricci scalar $R$ and of the trace $T$ of the energy-momentum tensor. In the Palatini approach, the Ricci scalar is a function of the metric and the connection. These two quantities, metric and connection, are taken as independents variables. Then, it is examined whether Palatini $f(R,T)$ gravity theory allows solutions in which lead to violation of causality. The Gödel and Gödel-type space-times are considered. In addition, a critical radius, which permits to exa…
We consider compact binary systems in f(R) gravity theories in the Palatini approach and calculate the post-Newtonian parameters to the 1.5PN order using the method of Direct Integration of the Relaxed Einstein equations (DIRE). The Palatini-type modifications of gravity can be formulated as Einsteins gravity with modified response to matter sources, and it is shown in detail how to treat these correctly within the DIRE formalism. Our results explicitly confirm the expectation that for binary black holes the new effects can be absorbed into redefinitions of the binary masses, rendering such sy…
We study the classical Euclidean wormholes in the context of extended theories of gravity. With no loss of generality, we use the dynamical equivalence between $f(\tilde{R})$ gravity and scalar-tensor theories to construct a point-like Lagrangian in the flat FRW space time. We first show the dynamical equivalence between Palatini $f(\tilde{R})$ gravity and the Brans-Dicke theory with self-interacting potential, and then show the dynamical equivalence between the Brans-Dicke theory with self-interacting potential and the minimally coupled O'Hanlon theory. We show the existence of new Euclidean …
We show that, within modified gravity, the non-linear nature of the field equations implies that the usual naive averaging procedure (replacing the microscopic energy-momentum by its cosmological average) is invalid. We discuss then how the averaging should be performed correctly and show that, as a consequence, at classical level the physical masses and geodesics of particles, cosmology and astrophysics in Palatini modified gravity theories are all indistinguishable from the results of general relativity plus a cosmological constant. Palatini gravity is however a different theory from general…
In Palatini $f(R)$ gravity, the parameters of the Schwarzschild - de Sitter solution as well as the whole interior solutions of compact objects are expected to change when compared to general relativity. We solve the Palatini field equations numerically in the case of the models $f(R) = R + αR^2$ and $f(R) = R - μ^4/R$, and using the equation of state of Fermi gas. We show how the density profiles and the prediction for the maximum masses of white dwarfs (the Chandrasekhar limit) and neutron stars (the Tolman-Oppenheimer-Volkoff limit) are altered, and thereby conclude that observations on com…
The well-formulation and the well-posedness of the Cauchy problem is discussed for {\it hybrid metric-Palatini gravity}, a recently proposed modified gravitational theory consisting of adding to the Einstein-Hilbert Lagrangian an $f(R)$ term constructed {\it à la} Palatini. The theory can be recast as a scalar-tensor one predicting the existence of a light long-range scalar field that evades the local Solar System tests and is able to modify galactic and cosmological dynamics, leading to the late-time cosmic acceleration. In this work, adopting generalized harmonic coordinates, we show that th…
We prove the equivalence in the covariant phase space of the metric and connection formulations for Palatini gravity, with nonmetricity and torsion, on a spacetime manifold with boundary. To this end, we will rely on the cohomological approach provided by the relative bicomplex framework. Finally, we discuss some of the physical implications derived from this equivalence in the context of singularity identification through curvature invariants.
This paper aims to analyze the generalization of Lema\^ıtre-Tolman-Bondi (LTB) spacetime for dissipative dust under the influence of Palatini $f(R)$ gravity. We explore the modified field equations, kinematical variables, and mass function in this scenario. We construct Bianchi identities using conservation and differential equations for shear, expansion, and curvature scalar in the background of Palatini $f(R)$ gravity. We calculated the scalar functions coming from the orthogonal decomposition of the Riemann tensor in this framework. These scalar functions known as structure scalars have bee…
We examine the question as to whether the Palatini f(R) gravity theories permit space-times in which the causality is violated. We show that every perfect-fluid Gödel-type solution of Palatini f(R) gravity with density $ρ$ and pressure $p$ that satisfy the weak energy condition $ρ+p \geq 0$ is necessarily isometric to the Gödel geometry, demonstrating therefore that these theories present causal anomalies in the form of closed time-like curves. This result extends a theorem on Gödel-type models to the framework of Palatini f(R) gravity theory. We concretely examine the Gödel-type perfect-fluid…
In this paper by deriving the Modified Friedmann equation in the Palatini formulation of $R^2$ gravity, first we discuss the problem of whether in Palatini formulation an additional $R^2$ term in Einstein's General Relativity action can drive an inflation. We show that the Palatini formulation of $R^2$ gravity cannot lead to the gravity-driven inflation as in the metric formalism. If considering no zero radiation and matter energy densities, we obtain that only under rather restrictive assumption about the radiation and matter energy densities there will be a mild power-law inflation $a(t)\sim…
We examine the generation of primordial perturbations during an inflationary epoch in generalised theories of gravity when the equations of motion are derived using the Palatini variational principle. Both f(R) and Scalar-Tensor theories are considered and we compare our results with those obtained under the conventional metric formalism. Non-linear generalisations of the action lead to different theories under the two variational choices and we obtain distinct results for scalar and tensor spectral indices and their ratio. We find the following general result; inflation driven solely by f(R) …
We review the recent literature on modified theories of gravity in the Palatini approach. After discussing the motivations that lead to consider alternatives to Einstein's theory and to treat the metric and the connection as independent objects, we review several topics that have been recently studied within this framework. In particular, we provide an in-depth analysis of the cosmic speedup problem, laboratory and solar systems tests, the structure of stellar objects, the Cauchy problem, and bouncing cosmologies. We also discuss the importance of going beyond the f(R) models to capture other …
In the frame work of Palatini approach to modified $f(R)$ theories of gravity, we attempt to investigate the thermodynamic behavior of time-dependent wormhole geometries. By considering an evolving wormhole spacetime, we find the field equations in Palatini $f(R)$ gravity. The energy density and pressure are redefined in order to satisfy the continuity equation. To discuss the thermodynamic behavior, an expression for the variation of total entropy is obtained. Then, we extend our discussions to apparent and event horizons of evolving wormholes. We employ the radius of apparent and event horiz…
