Palatini Variation Research Articles

Palatini variation in generalized geometry and string effective actions

We develop the Palatini formalism within the framework of generalized Riemannian geometry of Courant algebroids. In this context, the Palatini variation of a generalized Einstein–Hilbert–Palatini action - formed using a generalized metric, a Courant algebroid connection (in contrary to the ordinary case, not necessarily a torsionless one) and a volume form - leads naturally to a proper notion of a generalized Levi-Civita connection and low-energy effective actions of string theory.

Gravitational duality, Palatini variation and boundary terms: a synopsis

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 , 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, where ES duality is not applicable, we find that both models are Weyl invariant and equivalent to a first order formulation of the bosonic string. We also discuss the form of the boundary terms needed for the variational principle to be well defined on manifolds with non-null boundaries. This requires a modification of the Gibbons–Hawking–York (GHY) boundary term for gravity. This modification also means that the dualities between metric and connection formulations are consistent and include the boundary terms.

Is Spacetime Non-metric?

If one assumes higher dimensions and that dimensional reduction from higher dimensions produces scalar-tensor theory and also that Palatini variation is the correct method of varying scalar-tensor theory then spacetime is nonmetric. Palatini variation of Jordan frame lagrangians gives an equation relating the dilaton to the object of non-metricity and hence the existence of the dilaton implies that the spacetime connection is more general than that given soley by the Christoffel symbol of general relativity. Transferring from Jordan to Einstein frame, which connection, lagrangian, field equations and stress conservation equations occur are discussed: it is found that the Jordan frame has more information, this can be expressed in several ways, the simplest is that the extra information corresponds to the function multiplying the Ricci scalar in the action. The Einstein frame has the advantages that stress conservation implies no currents and that the field equations are easier to work with. This is illustrated by application to Robertson-Walker spacetime.

From Brans-Dicke gravity to a geometrical scalar-tensor theory

We consider an approach to the Brans-Dicke theory of gravity in which the scalar field has a geometrical nature. By postulating the Palatini variation, we find out that the role played by the scalar field consists in turning the space-time geometry into a Weyl integrable manifold. This procedure leads to a scalar-tensor theory that differs from the original Brans-Dicke theory in many aspects and presents some new features.

Variational approach to gravitational theories with two independent connections

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 to the former ones only when no connections appear in the matter action.

Cosmology of Ricci-tensor-squared gravity in the Palatini variational approach

We consider the cosmology of the Ricci-tensor-squared gravity in the Palatini variational approach. The gravitational action of standard general relativity is modified by adding a function $f({R}^{ab}{R}_{ab})$ to the Einstein-Hilbert action, and the Palatini variation is used to derive the field equations. A general method of obtaining the background and first-order covariant and gauge-invariant perturbation equations is outlined. As an example, we consider the cosmological constraints on such theories arising from the supernova type Ia and cosmic microwave background observations. We find that the best fit to the data is a non-null leading-order correction to Einstein gravity, but the current data exhibit no significant preference over the concordance model. The growth of nonrelativistic matter density perturbations at late times is also analyzed, and we find that a scale-dependent (positive or negative) sound-speed-squared term generally appears in the growth equation for small-scale density perturbations. We also estimate the observational bound imposed by the matter power spectrum for the model with $f({\mathbf{R}}^{ab}{\mathbf{R}}_{ab})=\ensuremath{\alpha}({\mathbf{R}}^{ab}{\mathbf{R}}_{ab}{)}^{\ensuremath{\beta}}$ to be roughly $|\ensuremath{\beta}|\ensuremath{\lesssim}\mathcal{O}({10}^{\ensuremath{-}5})$ so long as the dark matter does not possess compensating anisotropic stresses.

The metric-affine formalism of f(R) gravity

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 a metric-affine theory of gravity, as will be discussed in this talk. The general formalism will be presented and several aspects of the theory will be covered, mainly focusing on the enriched phenomenology that such theories exhibit with respect to General Relativity, relevant not only to large scales (cosmology) but also to small scales (e.g. torsion).

1/Rcurvature corrections as the source of the cosmological acceleration

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 with the Einstein-Hilbert variation, which yields fourth order equations, the Palatini approach produces second order equations in the metric. The Lagrangian $L(R)=R\ensuremath{-}{\ensuremath{\alpha}}^{2}/R$ 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 alternative explanation for the cosmological acceleration.

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.

Palatini variational principle for an extended Einstein-Hilbert action

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.

Variational principle for theP(4) affine theory of gravitation and electromagnetism

We propose a Lagrangian for theP(4) theory of gravitation and electromagnetism which is a straightforward generalization of the Einstein Lagrangian. A constrained Palatini variation of this Lagrangian yields the geometrical Einstein-Maxwell affine field equations. We show that these results can be extended easily to include both electric and magnetic charges. Finally, we consider conservation laws arising from the invariance properties of the Lagrangian.

Palatini variation of curvature-squared action and gravitational collapse

It is shown that Palatini variation of a class of gravitational actions based on a quadratic generalization of the Einstein–Hilbert action results in a metric-incompatible theory of gravity but one that satisfies Birkhoff’s theorem. The usual fourth-order field equations are replaced by two second-order equations. Application of the field equations to a model of freely falling dust are discussed.

Palatini variation of a generalized Einstein Lagrangian

The Palatini variation of the Einstein Lagiangian is analyzed for a symmetric metric and for a more general sesquilinear (Hermitian) form. In the latter case the Lagrangian is no longer projectively invariant, and the connection is determined to bemetrical up to a real λ transformation. The Einstein-Straus equations emerge as a naturalgeometrical generalization of general relativity.