Independent variation of the metric and connection in the Einstein-Hilbert action, called the Palatini variation, is generally taken to be equivalent to the usual formulation of general relativity in which only the metric is varied. However, when an abelian symmetry is allowed for the connection, the Palatini variation leads to an integrable Weyl geometry, not Riemannian. We derive this result using two possible metric/connection pairs: (1) the metric and general coordinate connection and (2) the solder form and local Lorentz spin connection of Poincarè gauge theory. Both lead to the same conclusion. Finally, we relate our work to other treatments in the literature.
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 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…
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…
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…
The Einstein field equations can be derived in $n$ dimensions ($n>2$) by the variations of the Palatini action. The Killing reduction of 5-dimensional Palatini action is studied on the assumption that pentads and Lorentz connections are preserved by the Killing vector field. A Palatini formalism of 4-dimensional action for gravity coupled to a vector field and a scalar field is obtained, which gives exactly the same fields equations in Kaluza-Klein theory.
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.
An abelian version of standard general relativity in the Cartan-Palatini gauge-like formulation in four dimensions has been introduced. Traditional canonical analysis utilizing similarities to the akin Husain-Kuchar SU(2) version of gravity has been performed. The model has been next quantized in the canonical path-integral Faddeev-Popov formalism yielding abelian BF theory.
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…
We consider $f(R)$ gravity and Born-Infeld-Einstein (BIE) gravity in formulations where the metric and connection are treated independently and integrate out the metric to find the corresponding models solely in terms of the connection, the archetypical treatment being that of Eddington-Schrödinger (ES) duality between cosmological Einstein and Eddington theories. For dimensions $D\ne2$, we find that this requires $f(R)$ to have a specific form which makes the model Weyl invariant, and that its Eddington reduction is then equivalent to that of BIE with certain parameters. For $D=2$ dimensions,…
We consider the possible existence of gravitationally bound stringlike objects in the framework of the generalized hybrid metric-Palatini gravity theory, in which the gravitational action is represented by an arbitrary function of the Ricci and of the Palatini scalars, respectively. The theory admits an equivalent scalar-tensor representation in terms of two independent scalar fields. Assuming cylindrical symmetry, and the boost invariance of the metric, we obtain the gravitational field equations that describe cosmic stringlike structures in the theory. The physical and geometrical properties…
Starting from a knowledge of certain identities in the Lagrangian description, the diffeomorphism transformations of metric and connection are obtained for both the second order (metric) and the first order (Palatini) formulations of gravity. The transformation laws of the connection and the metric are derived independently in the Palatini formulation in contrast to the metric formulation where the gauge variation of the connection is deduced from the gauge variation of the metric.
With a scalar field non-minimally coupled to curvature, the underlying geometry and variational principle of gravity - metric or Palatini - becomes important and makes a difference, as the field dynamics and observational predictions generally depend on this choice. In the present paper we describe a classification principle which encompasses both metric and Palatini models of inflation, employing the fact that inflationary observables can be neatly expressed in terms of certain quantities which remain invariant under conformal transformations and scalar field redefinitions. This allows us to …
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 i…
Palatini (or metric-affine) theories of gravity are characterized by having {\it a priori} independent metric and affine structures. The theories built in this framework have their field equations obtained as independent variations of the action with respect to the metric, the affine connection, and the other fields. In this work we consider the issue of singularity-removal in several members of a family of theoretically consistent and observationally viable subclass of them, built as contractions of the Ricci tensor with the metric (Ricci-based gravities or RBGs, for short). Several types of …
We shall use the variational decomposition technique in order to calculate equations of motion and Noether energy-momentum complex for some classes of non-linear gravitational Lagrangians within the first-order (Palatini) formalism. In particular, a complex space-time appears as a solution of our variational problem.
Affine gravity and the Palatini formalism contribute both to produce a simple and unique formula for calculating charges at spatial and null infinity for Lovelock type Lagrangians whose variational derivatives do not depend on second-order derivatives of the field components. The method is based on the covariant generalization due to Julia and Silva of the Regge-Teitelboim procedure that was used to define properly the mass in the classical formulation of Einstein's theory of gravity. Numerous applications reproduce standard results obtained by other secure but mostly specialized methods. As a…
We study the field equations of extensions of General Relativity formulated within a metric-affine formalism setting torsion to zero (Palatini approach). We find that different (second-order) dynamical equations arise depending on whether torsion is set to zero i) a priori or ii) a posteriori, i.e., before or after considering variations of the action. Considering a generic family of Ricci-squared theories, we show that in both cases the connection can be decomposed as the sum of a Levi-Civita connection and terms depending on a vector field. However, while in case i) this vector field is rela…
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 consider the cosmologies that arise in a subclass of f(R) gravity with f(R)=R+μ^{2n+2}/(-R)^{n} and -1<n<0 in the metric (as opposed to the Palatini) variational approach to deriving the gravitational field equations. The calculations of the isotropic and homogeneous cosmological models are undertaken in the Jordan frame and at both the background and the perturbation levels. For the former, we also discuss the connection to the Einstein frame in which the extra degree of freedom in the theory is associated with a scalar field sharing some of the properties of a 'chameleon' field. For the l…
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 fiel…
Corrections to Einstein's equations that become important at small curvatures are considered. The field equations are derived using a Palatini variation in which the connection and metric are varied independently. In contrast to the Einstein-Hilbert variation, which yields fourth order equations, the Palatini approach produces second order equations in the metric. The Lagrangian $L(R)=R-α^2/R$ is examined and it is shown that it leads to equations whose solutions approach a de Sitter universe at late times. Thus, the inclusion of 1/R curvature terms in the gravitational action offers an altern…
The field equations associated with the Born-Infeld-Einstein action are derived using the Palatini variational technique. In this approach the metric and connection are varied independently and the Ricci tensor is generally not symmetric. For sufficiently small curvatures the resulting field equations can be divided into two sets. One set, involving the antisymmetric part of the Ricci tensor $R_{\stackrel{μν}{\vee}}$, consists of the field equation for a massive vector field. The other set consists of the Einstein field equations with an energy momentum tensor for the vector field plus additio…
A new variational approach for general relativity and modified theories of gravity is presented. In addition to the metric tensor, two independent affine connections enter the action as dynamical variables. In the matter action the dependence upon one of the connections is left completely unspecified. When the variation is applied to the Einstein-Hilbert action the Einstein field equations are recovered. However when applied to $f(R)$ and Scalar-Tensor theories, it yields gravitational field equations which differ from their equivalents obtained with a metric or Palatini variation and reduce t…
Two toy models are considered within the framework of noncommutative differential geometry. In the first one, the Einstein action of the Levi-Civita connection is computed for the algebra of matrix valued functions on a torus. It is shown that, assuming some constraints on the metric, this action splits into a classical-like, a quantum-like and a mixed term. In the second model, an analogue of the Palatini method of variation is applied to obtain critical points of the Einstein action functional for $M\sb 4(R)$. It is pointed out that a solution to the Palatini variational problem is not neces…
In this work, we study cosmological and astrophysical applications of the recently proposed generalized hybrid metric-Palatini gravity theory, which combines features of both the metric and the Palatini approaches to the variational method in $f\left(R\right)$ gravity. This theory arises as a natural generalization of the hybrid metric-Palatini gravity which has been proven to be the first theory to unify the cosmic acceleration with the solar system constraints, without resource to the chameleon mechanism. In the cosmological point of view, we show using reconstruction methods that the usual …
The usual interpretation of Weyl geometry is modified in two senses. First, both the additive Weyl connection and its variation are treated as (1, 2) tensors under the action of Weyl covariant derivative. Second, a modified covariant derivative operator is introduced which still preserves the tensor structure of the theory. With its help, the Riemann tensor in Weyl geometry can be written in a more compact form. We justify this modification in detail from several aspects and obtain some insights along the way. By introducing some new transformation rules for the variation of tensors under the …
A generalization of General Relativity is studied. The standard Einstein-Hilbert action is considered in the Palatini formalism, where the connection and the metric are independent variables, and the connection is not symmetric. As a result of variation with respect to the metric Einstein equations are obtained. A variation with respect to the connection leads to an arbitrariness in the determination of connection, i.e. the presence of gauge invariance. Then a matter in a form of point particle which interacts with field of connection is introduced. Also the action is complemented by a kinetic…
It is shown that the first order (Palatini) variational principle for a generic nonlinear metric-affine Lagrangian depending on the (symmetrized) Ricci square invariant leads to an almost-product Einstein structure or to an almost-complex anti-Hermitian Einstein structure on a manifold. It is proved that a real anti-Hermitian metric on a complex manifold satisfies the Kähler condition on the same manifold treated as a real manifold if and only if the metric is the real part of a holomorphic metric. A characterisation of anti-Kähler Einstein manifolds and almost-product Einstein manifolds is ob…
We outline unimodular conformal and projective relativity (UCPR), an extension of unimodular relativity in which the conformal and projective structures play central roles. Under $SL(4,\mathbb{R})$ symmetry group, the pseudo-Riemannian metric naturally decomposes into a four-volume element and a conformal metric; and the affine connection decomposes into a one-form and a trace-free projective connection. In UCPR, these four space-time structures are treated as independent fields that have clear physical interpretations. A Palatini-type variational principle for the usual general relativity Lag…
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 …
Recently a class of alternative theories of gravity which goes under the name f(R) gravity, has received considerable attention, mainly due to its interesting applications in cosmology. However, the phenomenology of such theories is not only relevant to cosmological scales, especially when it is treated within the framework of the so called Palatini variation, an independent variation with respect to the metric and the connection, which is not considered a priori to be the Levi-Civita connection of the metric. If this connection has its standard geometrical meaning the resulting theory will be…
