Scale Transformation,Modified Gravity,and Brans-Dicke Theory

A model of Einstein-Hilbert action subject to the scale transformation is studied. By introducing a dilaton field as a means of scale transformation a new action is obtained whose Einstein field equations are consistent with traceless matter with non-vanishing modified terms together with dynamical cosmological and gravitational coupling terms. The obtained modified Einstein equations are neither those in f(R) metric formalism nor the ones in f(ℛ) Palatini formalism, whereas the modified source terms are formally equivalent to those of f(R)=\frac12R²f({\mathcal{R}})=\frac{1}{2}{\mathcal{R}}^{2} gravity in Palatini formalism. The correspondence between the present model, the modified gravity theory, and Brans-Dicke theory with w = -\frac32\omega=-\frac{3}{2} is explicitly shown, provided the dilaton field is condensated to its vacuum state.     

Measurement of the space–time interval in modified gravity theories in Palatini formalism

There are strong motivations that lead cosmologists to consider alternatives to Einstein’s theory of gravity in Palatini formalism. In addition, there are two distinguishable local frames in this formalism in which one of them is local inertial frame and the equivalence principle is satisfied. Different features of speed of light such as the causal structure constant, electromagnetic and gravitational wave velocities and Einstein velocity will not coincide in this local inertial frame for extended gravity theories in Palatini formalism. On the other hand, both the measurement of time and exchange of a signal between the distant points are required to determine spatial distances. In a particular situation where these aspects of the speed of light do not coincide, the distance determination will become more demanding because light will follow a time-like geodesic of the metric. In modified gravity theories in Palatini approach, theories involve a varying speed of photon. Therefore these kinds of theories must be based on some other technique of measuring spatial distances than radar or some terms should be corrected in the line element in the proposed model. We found out we should consider a coefficient which is proportional to energy density in each era, in the line element in order to be able to use radar for measuring distance in modified gravity theories in Palatini formalism. Analysis of some observational data will be affected by considering this coefficient in the line element.     

Inflation with non-minimal coupling: Metric vs. Palatini formulations

We analyze non-minimally coupled scalar field theories in metric (second-order) and Palatini (first-order) formalisms in a comparative fashion. After contrasting them in a general setup, we specialize to inflation and find that the two formalisms differ in their predictions for various cosmological parameters. The main reason is that dependencies on the non-minimal coupling parameter are different in the two formalisms. For successful inflation, the Palatini approach prefers a much larger value for the non-minimal coupling parameter than the Metric approach. Unlike the Metric formalism, in Palatini, the inflaton stays well below the Planck scale whereby providing a natural inflationary epoch.     

Energy and momentum from the Palatini formalism

I derive from the Palatini formalism, in which metric and affinity are varied independently, an energy-momentum complex qualitatively different in form from the usual energy-momentum representations of general relativity. A similar procedure can be carried out for electrodynamics, illuminating by analogy the structure of the gravitational Lagrangian. The new energy density vanishes for all static vacuum solutions of the Einstein equations, and the radiated energy from an isolated system in an asymptotically flat space in general diverges. These facts suggest that the formalism could be used to express Mach's principle.     

自对偶引力的局部对称性与约束

在Ashtekar形式下，广义相对论的相空间被嵌入到复SU（2）Yang-Mils理论的相空间里．将一般场论中分析局部对称性与约束的方法推广到复的场论，从自对偶Palatini形式的位形空间构造出Ashtekar形式的相空间，进而讨论了位形空间上的局部对称性与相空间上的约束的关系． 关键词：     

Matter Loops Corrected Modified Gravity in Palatini Formulation

Recently, corrections to the standard Einstein-Hilbert action were proposed to explain the current cosmic acceleration in stead of introducing dark energy. In the Palatini formulation of those modified gravity models, there is an important observation due to Arkani-Hamed: matter loops will give rise to a correction to the modified gravity action proportional to the Ricci scalar of the metric. In the presence of such a term, we show that the current forms of modified gravity models in Palatini formulation, specifically, the 1/R gravity and ln R gravity, will have phantoms. Then we study the possible instabilities due to the presence of phantom fields. We show that the strong instability in the metric formulation of 1/R gravity indicated by Dolgov and Kawasaki will not appear and the decay timescales for the phantom fields may be long enough for the theories to make sense as effective field theory . On the other hand, if we change the sign of the modification terms to eliminate the phantoms, some other inconsistencies will arise for the various versions of the modified gravity models. Finally, we comment on the universal property of the Palatini formulation of the matter loops corrected modified gravity models and its implications.     

Metric-affine f(R) theories of gravity

General Relativity assumes that spacetime is fully described by the metric alone. An alternative is the so called Palatini formalism where the metric and the connections are taken as independent quantities. The metric-affine theory of gravity has attracted considerable attention recently, since it was shown that within this framework some cosmological models, based on some generalized gravitational actions, can account for the current accelerated expansion of the universe. However we think that metric-affine gravity deserves much more attention than that related to cosmological applications and so we consider here metric-affine gravity theories in which the gravitational action is a general function of the scalar curvature while the matter action is allowed to depend also on the connection which is not a priori symmetric. This general treatment will allow us to address several open issues such as: the relation between metric-affine f(R) gravity and General Relativity (in vacuum as well as in the presence of matter), the implications of the dependence (or independence) of the matter action on the connections, the origin and role of torsion and the viability of the minimal-coupling principle.     

Violation of the equivalence principle in modified theories of gravity

We show that the metric in f(R) theories of gravity in Palatini formalism can be solved as the product of a rank-two tensor times a scalar function which is very sensitive to the local energy-momentum densities. This local dependence of the metric generates new gravitationally-induced microscopic interactions, which eventually would lead to self-accelerated test body trajectories. These facts make very unlikely the viability of Palatini f(R) models designed to change the late-time cosmic evolution.     

A new Lagrangian for the vacuum einstein equations and its tetrad form

We give a modification of the Palatini Lagrangian for the free gravitational field that yields the vanishing of the torsion as a result of the field equations and requires only the assumption of the symmetry of the metric. We transcribe this Lagrangian into the tetrad formalism and show how the tetrad form of the Einstein field equations follows from it. Some remarks on possible generalization to a theory with nonvanishing torsion in the presence of matter conclude the paper.An earlier version of the results of this paper are found in [6].On leave from the Department of Physics, Boston University, Boston, Massachusetts.     

Palatini form of 1/R gravity

It has been suggested that the Universe's recent acceleration is due to a contribution to the gravitational action proportional to the reciprocal of the Ricci scalar. Although the original version of this theory disagrees with solar system observations, a modified Palatini version, in which the metric and connection are treated as independent variables, has been suggested as a viable model of the cosmic acceleration. We show that this theory is equivalent to a scalar-tensor theory in which the scalar field kinetic energy term is absent from the action. Integrating out the scalar field gives rise to additional interactions among the matter fields of the standard model of particle physics at an energy scale of order 10(-3) eV (the geometric mean of the Hubble and the Planck scales), and so the theory is excluded by, for example, electron-electron scattering experiments.     

Conformal aspects of the Palatini approach in Extended Theories of Gravity

The debate on the physical relevance of conformal transformations can be faced by taking the Palatini approach into account gravitational theories. We show that conformal transformations are not only a mathematical tool to disentangle gravitational and matter degrees of freedom (passing from the Jordan frame to the Einstein frame) but they acquire a physical meaning considering the bi-metric structure of Palatini approach which allows to distinguish between spacetime structure and geodesic structure. These facts are relevant at least at cosmological scales, while at small scale (i.e. in the spacetime regions relevant for observations) the conformal factor is slowly varying and its effects are not relevant. Examples of higher-order and non-minimally coupled theories are worked out and relevant cosmological solutions in Einstein frame and Jordan frame are discussed showing that also the interpretation of cosmological observations can drastically change depending on the adopted frame.     

The Chaotic Mixmaster and the Suppression of Chaos in Scalar–Tensor Cosmologies

We show that there is a threshold in energy for the onset of chaos in cosmology for the Universe described as a dynamical system derived from the Einstein equations of General Relativity (GR). In the case of the mixmaster model (homogeneous and anisotropic cosmology with a Bianchi IX metric), the chaos occurs precisely at the prescribed necessary value H _vac=0 of the GR for the energy of the Universe while the system is found to be regular for H<0 and chaotic for H>0 with respect to its pure vacuum part. In the case of generalized scalar tensor theories within the Bianchi IX model, we show using the ADM formalism and a conformal transformation that the energy of the dynamical system as compared to vacuum lies below the zero energy threshold. The system is thus not exhibiting chaos and the conclusion still holds in the presence of ordinary matter as well. The suppression of chaos occurs in a similar way for stiff matter alone.     

The gravity Lagrangian according to solar system experiments

In this work we show that the gravity Lagrangian at relatively low curvatures in both metric and Palatini formalisms is a bounded function that can only depart from the linearity within the limits defined by well-known functions. We obtain those functions by analyzing a set of inequalities that any theory must satisfy in order to be compatible with laboratory and solar system observational constraints. This result implies that the recently suggested f(R)gravity theories with nonlinear terms that dominate at low curvatures are incompatible with observations and, therefore, cannot represent a valid mechanism to justify the cosmic speedup.     

Quadratic,Higgs and hilltop potentials in Palatini gravity

In this work, we study the theory of inflation with the non-minimally coupled quadratic, standard model Higgs, and hilltop potentials, through ξφ~2R term in Palatini gravity. We first analyze observational parameters of the Palatini quadratic potential as functions of ξ for the high-N scenario. In addition to this, taking into account that the inflaton field f has a non-zero vacuum expectation value v after inflation, we display observational parameters of well-known symmetry-breaking potentials. The types of potentials considered are the Higgs potential and its generalizations, namely hilltop potentials in the Palatini formalism for the high-N scenario and the low-N scenario. We calculate inflationary parameters for the Palatini Higgs potential as functions of v for different ξ values, where inflaton values are both φv and φv during inflation, as well as calculating observational parameters of the Palatini Higgs potential in the induced gravity limit for high-N scenario. We illustrate differences between the Higgs potential's effect on ξ versus hilltop potentials, which agree with the observations for the inflaton values for φv and ξ, in which v1 for both these high and low N scenarios. For each considered potential, we also display n_s-r values fitted to the current data given by the Keck Array/BICEP2 and Planck collaborations.     

Symmetric Variations of the Metric and Extrema of the Action for Pure Gravity

Symmetries of generalized gravitational actions, yielding field equations which typically involve at most second-order derivatives of the metric, are considered. The field equations for several different higher-derivative theories in the first-order formalism are derived, and variations of a generic set of higher-order curvature terms appearing in string effective actions are studied. It is shown that there often exists a particular set of solutions to the field equations of pure gravity theories, consisting of different combinations of curvature tensors, which satisfies the vacuum equations with cosmological constant. Implications of generalized symmetries of the field equations derived from the superstring effective action for the cosmological constant problem are discussed.     

Geometrodynamics vs. connection dynamics

The purpose of this review is to describe in some detail the mathematical relationship between geometrodynamics and connection dynamics in the context of the classical theories of 2+1 and 3+1 gravity. We analyze the standard Einstein-Hilbert theory (in any spacetime dimension), the Palatini and Chern-Simons theories in 2+1 dimensions, and the Palatini and self-dual theories in 3+1 dimensions. We also couple varions matter fields to these theories and briefly describe a pure spin-connection formulation of 3+1 gravity. We derive the Euler-Lagrange equations of motion from an action principle and perform a Legendre transform to obtain a Hamiltonian formulation of each theory. Since constraints are present in all these theories, we construct constraint functions and analyze their Poisson bracket algebra. We demonstrate, whenever possible, equivalences between the theories.     

Covariant conservation laws from the palatini formalism

Covariant conservation laws in the Palatini formalism are derived. The result indicates that the gravitational part of conserved charges in general relativity should be calculated from a combination of Komar's strongly conserved current and the Einstein tensor. This implies that the set of complete diffeomorphism charges of a gravitating system consisting of scalar matter is described by Komar's vector density, and that the identification of gravitational energy and momentum reduces to two choices: a choice of relative weights of the contributions resulting from Komar's current and from the Einstein tensor, and a choice of preferred vector fields in space-time. A proposal is made which yields energy and momentum as scalars under diffeomorphisms and as a Lorentz vector in tangent space. Furthermore, the result can be used to identify covariant conservation laws holding separately for the matter contributions to diffeomorphism charges.     

Higher order mean field expansions for lattice gauge theories

The leading contribution to the free energy of lattice gauge theories is evaluated in the mean field expansion to the two-loop level. The methods are general but we only deal with theU(1) case in this paper. The corrections improve the agreement with Monte Carlo calculations. We show that in order to obtain a satisfactory formalism it is necessary to include a new redundant parameter, γ, in the mean field expansion. For γ→0 we recover the usual mean field expansion whereas for γ→∞ we obtain the weak coupling expansion. Thus γ measures the amount of resummation that is done by the mean field formalism.     

Energy-momentum complex of gravitational field in the Palatini formalism

It is shown that Murphy's energy-momentum complex of the gravitational field, derived from the Hilbert Lagrangian by use of the Palatini formalism, is identical to the complex derived from the same Lagrangian in a standard way by Mitskievic. The explicitly tensorial formulation of conservation laws in general relativity is eflectively used and some properties of the complex in question are discussed in connection with Murphy's article.     

Scale invariance and spontaneous symmetry breaking

Spontaneous breaking of gauge symmetries is studied in theories with nonlinearly realized scale invariance. The classically sliding vacuum expectation values are fixed through quantum corrections. The anomaly of the dilatation current determines the vacuum energy density as well as the dilaton mass. The coupling of gravity to matter is modified in such a way that the cosmological constant vanishes.