Palatini Variational Principle Research Articles

How the non-metricity of the connection arises naturally in the classical theory of gravity

Spacetime geometry is described by two–a priori independent–geometric structures: the symmetric connection Γ and the metric tensor g. Metricity condition of Γ (i.e. ∇g = 0) is implied by the Palatini variational principle, but only when the matter fields belong to an exceptional class. In case of a generic matter field, Palatini implies non-metricity of Γ. Traditionally, instead of the (first order) Palatini principle, we use in this case the (second order) Hilbert principle, assuming metricity condition a priori. Unfortunately, the resulting right-hand side of the Einstein equations does not coincide with the matter energy-momentum tensor. We propose to treat seriously the Palatini-implied non-metric connection. The conventional Einstein’s theory, rewritten in terms of this object, acquires a much simpler and universal structure. This approach opens a room for the description of the large scale effects in General Relativity (dark matter?, dark energy?), without resorting to purely phenomenological terms in the Lagrangian of gravitational field. All theories discussed in this paper belong to the standard General Relativity Theory, the only non-standard element being their (much simpler) mathematical formulation. As a mathematical bonus, we propose a new formalism in the calculus of variations, because in case of hyperbolic field theories the standard approach leads to nonsense conclusions.

Interacting quintessence from new formalism of gravitoelectromagnetism formulated on a geometrical scalar–tensor gauge theory of gravity

Abstract We derive an interacting quintessence model on the framework of a recently introduced new class of geometrical scalar–tensor theories of gravity formulated on a Weyl-Integrable geometry, where the gravitational sector is described by both a scalar and a tensor metric field. By using a Palatini variational principle we construct a scalar–tensor action invariant under the Weyl symmetry group of the background geometry, which in the Einstein–Riemann frame leads to a gravitoelectromagnetic theory. We use the gauge freedom of the theory and the fact that the Weyl scalar field couples with matter fields to formulate an interacting quintessential model with a non-canonical kinetic term, where the quintessence field has a geometrical origin. Due to this non-canonicity we obtain that the mass of the quintessence field in the past epochs results to be small enough not to cause modifications in the baryon to photon ratio during nucleosynthesis.

Stability in quadratic torsion theories

We revisit the definition and some of the characteristics of quadratic theories of gravity with torsion. We start from a Lagrangian density quadratic in the curvature and torsion tensors. By assuming that General Relativity should be recovered when the torsion vanishes and investigating the behaviour of the vector and pseudo-vector torsion fields in the weak-gravity regime, we present a set of necessary conditions for the stability of these theories. Moreover, we explicitly obtain the gravitational field equations using the Palatini variational principle with the metricity condition implemented via a Lagrange multiplier.

Kaluza–Klein reduction of a quadratic curvature model

Palatini variational principle is implemented on a five dimensional quadratic curvature gravity model, rendering two sets of equations which can be interpreted as the field equations and the stress-energy tensor. Unification of gravity with electromagnetism and the scalar dilaton field is achieved through the Kaluza-Klein dimensional reduction mechanism. The reduced curvature invariant, field equations and the stress-energy tensor in four dimensional spacetime are obtained. The structure of the interactions among the constituent fields is exhibited in detail. It is shown that the Lorentz force naturally emerges from the reduced field equations and the equations of the standard Kaluza-Klein theory is demonstrated to be intrinsically contained in this model.

Palatini–Lovelock–Cartan gravity—Bianchi identities for stringy fluxes

A Palatini-type action for Einstein and Gauss–Bonnet gravity with non-trivial torsion is proposed. A 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-geometric Q- and R-fluxes which include derivative and curvature terms. Finally, the problem of relating torsional gravity to higher order corrections of the bosonic string-effective action is revisited.

Inflationary perturbations in Palatini generalized gravity

We examine the generation of primordial perturbations during an inflationary epoch in generalized 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. Nonlinear generalizations 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 $f(R)$ modifications does not result in suitable curvature perturbations, while scalar-tensor theories generate nearly scalar invariant curvature perturbations but no tensor modes.

The possibility of an accelerating cosmology in Rastall's theory

In an attempt to look for a viable mechanism leading to a present day accelerated expansion, we investigate the possibility that the observed cosmic speed up may be recovered in the framework of the Rastall's theory, relying on the non-conservativity of the stress-energy tensor, i.e. Tμv;μ ≠ 0. We derive the modified Friedmann equations and show that they correspond to Cardassian-like equations. We also show that, under suitable assumptions on the equation of state of the matter term sourcing the gravitational field, it is indeed possible to get an accelerated expansion, in agreement with the Hubble diagram of both Type Ia Supernovae (SNeIa) and Gamma Ray Bursts (GRBs). Unfortunately, to achieve such a result one has to postulate a matter density parameter larger than the typical ΩM ≊ 0.3 value inferred from cluster gas mass fraction data. As a further issue, we discuss the possibility to retrieve the Rastall's theory from a Palatini variational principle approach to f(R) gravity. However, such an attempt turns out to be unsuccessful.

The cosmic snap parameter in f(R) gravity

We derive the expression for the snap parameter in f(R) gravity. We use the Palatini variational principle to obtain the field equations and regard the Einstein conformal frame as physical. We predict the present-day value of the snap parameter for the particular case f(R) = R − const/R, which is the simplest f(R) model explaining the current acceleration of the universe.

Hydrostatic equilibrium equation and Newtonian limit of the singular f(R) gravity

We derive the hydrostatic equilibrium equation of a spherical star for any gravitational Lagrangian density of the form . The Palatini variational principle for the Helmholtz Lagrangian in the Einstein gauge is used to obtain the field equations in this gauge. The equilibrium hydrostatic equation is obtained and is used to study the Newtonian limit for . The same procedure is carried out for the more generally case giving a good Newtonian limit.

Interacting dark energy inf(R)gravity

The field equations in $f(R)$ gravity derived from the Palatini variational principle and formulated in the Einstein conformal frame yield a cosmological term which varies with time. Moreover, they break the conservation of the energy-momentum tensor for matter, generating the interaction between matter and dark energy. Unlike phenomenological models of interacting dark energy, $f(R)$ gravity derives such an interaction from a covariant Lagrangian which is a function of a relativistically invariant quantity (the curvature scalar $R$). We derive the expressions for the quantities describing this interaction in terms of an arbitrary function $f(R)$, and examine how the simplest phenomenological models of a variable cosmological constant are related to $f(R)$ gravity. Particularly, we show that $\ensuremath{\Lambda}{c}^{2}={H}^{2}(1\ensuremath{-}2q)$ for a flat, homogeneous and isotropic, pressureless universe. For the Lagrangian of form $R\ensuremath{-}1/R$, which is the simplest way of introducing current cosmic acceleration in $f(R)$ gravity, the predicted matter-dark energy interaction rate changes significantly in time, and its current value is relatively weak (on the order of 1% of ${H}_{0}$), in agreement with astronomical observations.

The nearly Newtonian regime in non-linear theories of gravity

The present paper reconsiders the Newtonian limit of models of modified gravity including higher order terms in the scalar curvature in the gravitational action. This was studied using the Palatini variational principle in Meng and Wang (Gen. Rel. Grav. 36, 1947 (2004)) and Dominguez and Barraco (Phys. Rev. D 70, 043505 (2004)) with contradicting results. Here a different approach is used, and problems in the previous attempts are pointed out. It is shown that models with negative powers of the scalar curvature, like the ones used to explain the present accelerated expansion, as well as their generalization which include positive powers, can give the correct Newtonian limit, as long as the coefficients of these powers are reasonably small. Some consequences of the performed analysis seem to raise doubts for the way the Newtonian limit was derived in the purely metric approach of fourth order gravity [Dick in Gen. Rel. Grav. 36, 217 (2004)]. Finally, we comment on a recent paper [Olmo in Phys. Rev. D 72, 083505 (2005)] in which the problem of the Newtonian limit of both the purely metric and the Palatini formalism is discussed, using the equivalent Brans–Dicke theory, and with which our results partly disagree.

The cosmic jerk parameter in f(R) gravity

Abstract We derive the expression for the jerk parameter in f ( R ) gravity. We use the Palatini variational principle and the field equations in the Einstein conformal gauge. For the particular case f ( R ) = R − α 2 3 R , the predicted value of the jerk parameter agrees with the SNLS SNIa and X-ray galaxy cluster distance data but does not with the SNIa gold sample data.

The present universe in the Einstein frame, metric-affine R + 1/R gravity

We study the present, flat isotropic universe in 1/R-modified gravity. We use the Palatini (metric-affine) variational principle and the Einstein (metric-compatible connected) conformal frame. We show that the energy density scaling deviates from the usual scaling for nonrelativistic matter, and the largest deviation occurs in the present epoch. We find that the current deceleration parameter derived from the apparent matter density parameter is consistent with observations. There is also a small overlap between the predicted and observed values for the redshift derivative of the deceleration parameter. The predicted redshift of the deceleration-to-acceleration transition agrees with that in the ΛCDM model but it is larger than the value estimated from SNIa observations.

A note on covariant conservation of energy–momentum in modified gravities

An explicit proof of the vanishing of the covariant divergence of the energy–momentum tensor in modified theories of gravity is presented. The gravitational action is written in arbitrary dimensions and allowed to depend nonlinearly on the curvature scalar and its couplings with a scalar field. Also the case of a function of the curvature scalar multiplying a matter Lagrangian is considered. The proof is given both in the metric and in the first-order formalism, i.e. under the Palatini variational principle. It is found that the covariant conservation of energy–momentum is built in to the field equations. This crucial result, called the generalized Bianchi identity, can also be deduced directly from the covariance of the extended gravitational action. Furthermore, in all of these cases, the freely falling world lines are determined by the field equations alone and turn out to be the geodesics associated with the metric compatible connection. The independent connection in the Palatini formulation of these generalized theories does not have a similar direct physical interpretation. However, in the conformal Einstein frame a certain bi-metricity emerges into the structure of these theories.

Unification of inflation and cosmic acceleration in the Palatini formalism

A modified model of gravity with additional positive and negative powers of the scalar curvature, $R$, in the gravitational action is studied. This is done using the Palatini variational principle. It is demonstrated that using such a model might prove useful to explain both the early time inflation and the late time cosmic acceleration without the need for any form of dark energy.

Cosmological perturbations in the Palatini formulation of modified gravity

Cosmology in extended theories of gravity is considered assuming the Palatini variational principle, for which the metric and connection are independent variables. The field equations are derived to linear order in perturbations about the homogeneous and isotropic but possibly spatially curved background. The results are presented in a unified form applicable to a broad class of gravity theories allowing arbitrary scalar–tensor couplings and nonlinear dependence on the Ricci scalar in the gravitational action. The gauge-ready formalism exploited here makes it possible to obtain the equations immediately in any of the commonly used gauges. Of the three type of perturbations, the main attention is on the scalar modes responsible for the cosmic large-scale structure. Evolution equations are derived for perturbations in a late universe filled with cold dark matter and accelerated by curvature corrections. Such corrections are found to induce effective pressure gradients which are problematical in the formation of large-scale structure. This is demonstrated by analytic solutions in a particular case. A physical equivalence between scalar–tensor theories in metric and in Palatini formalisms is pointed out.

Born-Infeld-Einstein theory with matter

The field equations associated with the Born-Infeld-Einstein action including matter are derived using a Palatini variational principle. Scalar, electromagnetic, and Dirac fields are considered. It is shown that an action can be chosen for the scalar field that produces field equations identical to the usual Einstein field equations minimally coupled to a scalar field. In the electromagnetic and Dirac cases the field equations reproduce the standard equations only to lowest order. The spherically symmetric electrovac equations are studied in detail. It is shown that the resulting Einstein equations correspond to gravity coupled to a modified Born-Infeld theory. It is also shown that point charges are not allowed. All particles must have a finite size. Mass terms for the fields are also considered.

Higher-order gravity theories and scalar–tensor theories

We generalize the known equivalence between higher-order gravity theories and scalar–tensor theories to a new class of theories. Specifically, in the context of a first-order or Palatini variational principle where the metric and connection are treated as independent variables, we consider theories for which the Lagrangian density is a function f of (i) the Ricci scalar computed from the metric, and (ii) a second Ricci scalar computed from the connection. We show that such theories can be written as tensor–multi-scalar theories with two scalar fields with the following features: (i) the two-dimensional σ-model metric that defines the kinetic energy terms for the scalar fields has constant, negative curvature; (ii) the coupling function determining the coupling to matter of the scalar fields is universal, independent of the choice of function f; and (iii) if both mass eigenstates are long range, then the Eddington post-Newtonian parameter γ has value 1/2. Therefore, in order to be compatible with solar system experiments at least one of the mass eigenstates must be short range.

Modified Friedmann equations in R−1-modified gravity

Recently, corrections to the Einstein–Hilbert action that become important at small curvature have been proposed. We discuss the first-order and second-order approximations to the field equations derived by the Palatini variational principle. We work out the first- and second-order modified Friedmann equations and present the upper redshift bounds when these approximations are valid. We show that the second-order effects can be neglected in the cosmological predictions involving only the Hubble parameter, e.g. the various cosmological distances, but the second-order effects cannot be neglected in the predictions involving the derivatives of the Hubble parameter. Furthermore, the modified Friedmann equations fit the SNe Ia data at an acceptable level.

Palatini variational principle for N-dimensional dilaton gravity

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 transformations.