Search

We investigate how the cosmological equation of state can be used for scrutinizing extended theories of gravity, in particular, the Palatini $$f({{\mathcal {R}}})$$ f ( R ) gravity. Specifically, the approach consists, at first, in investigating the effective equation of state produced by a given model. Then, the inverse problem can also be considered in view of determining which models are compatible with a given effective equation of state. We consider and solve some cases and show that, for example, power-law models are (the only models) capable of transforming barotropic Equations of State into effective barotropic ones. Moreover, the form of equation of state is preserved (only) for $$f({{\mathcal {R}}})={{\mathcal {R}}}$$ f ( R ) = R , as expected. In this perspective, modified Equations of State are a feature capable of distinguishing extended gravity with respect to general relativity. We investigate also quadratic and non-homogeneous effective Equations of State showing, in particular, that they contain the Starobinsky model and other ones.

We define spin frames, with the aim of extending spin structures from the category of (pseudo-)Riemannian manifolds to the category of spin manifolds with a fixed signature on them, though with no selected metric structure. Because of this softer requirements, transformations allowed by spin frames are more general than usual spin transformations and they usually do not preserve the induced metric structures. We study how these new transformations affect connections both on the spin bundle and on the frame bundle and how this reflects on the Dirac equations., Comment: 23 pages

We compare Mercury’s precession test in standard general relativity, Brans–Dicke theories (BD), and Palatini$$f({\mathcal {R}})$$f(R)-theories. We avoid post-Newtonian approximation and compute exact precession in these theories. We show that the well-known mathematical equivalence between Palatini$$f({\mathcal {R}})$$f(R)-theories and a specific subset of BD theories does not extend to a really physical equivalence among theories since equivalent models still allow a different incompatible precession for Mercury depending on the solution one chooses. As a result one cannot use BD equivalence to rule out Palatini$$f({\mathcal {R}})$$f(R)-theories. On the contrary, we directly discuss that Palatini$$f({\mathcal {R}})$$f(R)-theories can (and specific models do) easily pass Solar System tests as Mercury’s precession.

We discuss the design for a discrete, immediate, simple relativistic positioning system (rPS) which is potentially able of self-positioning (up to isometries) and operating without calibration or ground control assistance. The design is discussed in dimension two on spacetime (i.e. one spatial dimension plus one time dimension), in Minkowski and Schwarzschild solutions, as well as in dimension three (i.e. two spatial dimensions plus one time dimension) in Minkowski. The system works without calibration, clock synchronizations, or a priori knowledge about the motion of clocks, it is able to self-diagnose hypotheses break down (for example, if one clock temporarily becomes not-freely falling, or the gravitational field changes) and it is automatically back and operational when the assumed conditions are restored. In the Schwarzschild case, we show that the system can also best fit the gravitational mass of the source of the gravitational field and stress that no weak field assumptions are made anywhere. In particular, the rPS we propose can work in a region close to the horizon since it does not use approximations or PPN expansions. More generally, the rPS can be adapted as detectors for the gravitational field and we shall briefly discuss their role in testing different theoretical settings for gravity. In fact, rPS is a natural candidate for a canonical method to extract observables out of a gravitational theory, an activity also known as designing experiments to test gravity., Comment: 24 pages, 6 figures

We propose a covariant and geometric framework to introduce space distances as they are used by astronomers. In particular, we extend the definition of space distances from the one used between events to non-test-bodies with horizons and singularities so that the definition extends through the horizons and it matches the protocol used to measure them. The definition we propose can be used in standard General Relativity although it extends directly to Weyl geometries to encompass a number of modified theories, extended theories in particular., 12 pages, 4 figures

In a Palatini $f(\mathcal{R})$-model, we define chonodynamical effects due to the choice of atomic clocks as standard reference clocks and we develop a formalism able to quantitatively separate them from the usual effective dark sources one has in extended theories. We apply the formalism to Hubble drift and briefly discuss the issue about the physical frame. In particular, we argue that there is no physical frame in the sense one does different things in different frames and that, in a sense, is the physical characteristic of extended gravity. As an example, we discuss how Jordan frame may be well suited to discuss cosmology, though it fails within the solar system., 18 pages, 4 figures (some references added, some typos corrected)

We consider the cosmological models based on Palatini f(R)-theory for the function f(R)=aR-2bR^2-3c/R, which, when only dust visible matter is considered, is called dune cosmology in view of the shape of the function f(R(a)) (being a the scale factor). We discuss about the meaning of solving the model, and interpret it according to Ehlers-Pirani-Schild framework as defining a Weyl geometry on spacetime. Accordingly, we extend the definitions of luminosity distance, proper distance, and redshift to Weyl geometries and fit the values of parameters to SNIa data. Since the theoretical prediction is model-dependent, we argue that the it is affected by an extra choice, namely a model for atomic clocks, which, in principle, produces observable effects. To the best of our knowledge, these effects have not being considered in the literature before., Comment: 24 pages, 14 figures

We shall discuss equivalence of frames in Palatini f(ℛ)-theories at action level. A Palatini formulation of Brans-Dicke theories (equivalent to the purely metric ones) will also be discussed.

After reviewing the hole argument and its relations with initial value problem and general covariance, we shall discuss how much freedom one has to define the physical state of a system in a generally covariant (or gauge covariant) field theory. We shall show that in gauge covariant theories (and generally covariant theories with a a compact space) one has no freedom and one is forced to declare as physically equivalent two configurations which differ by a gauge transformation (or by a global spacetime diffeomorphism), as it is usually prescribed. On the contrary, when space is not compact, the result proven for the compact case does not hold true and one may have different options to define physically equivalent configurations, still preserving determinism., 12 pages

We show how Conformal Gravity (CG) has to satisfy a fine-tuning condition to describe the rotation curves of disk galaxies without the aid of dark matter. Interpreting CG as a gauge natural theory yields conservation laws and their associated superpotentials without ambiguities. We consider the light deflection of a point-like lens and impose that the two Schwarzschild-like metrics with and without the lens are identical at infinite distances from the lens. The energy conservation law implies that the parameter $\gamma$ in the linear term of the metric has to vanish, otherwise the two metrics are physically inaccessible from each other. This linear term is responsible to mimic the role of dark matter in disk galaxies and gravitational lensing systems. Our analysis shows that removing the need of dark matter with CG thus relies on a fine-tuning condition on $\gamma$. We also illustrate why the results of previous investigations of gravitational lensing in CG largely disagree. These discrepancies derive from the erroneous use of the deflection angle definition adopted in General Relativity, where the vacuum solution is asymptotically flat, unlike CG. In addition, the lens mass is identified with various combinations of the metric parameters. However, these identifications are arbitrary, because the mass is not a conformally invariant quantity, unlike the conserved charge associated to the energy conservation law. Based on this conservation law and by removing the fine-tuning condition on $\gamma$, i.e. by setting $\gamma=0$, the energy difference between the metric with the point-like lens and the metric without it defines a conformally invariant quantity that can in principle be used for (1) a proper derivation of light deflection in CG, and (2) the identification of the lens mass with a function of the parameters $\beta$ and $k$ of the Schwarzschild-like metric., Comment: 16 pages, 1 figure. Revised version according to the referees comments. The results reported in the original version remain unchanged

A direct and non-trivial link between Padmanabhan's entropy used in emergent gravity and standard GR action is established. To do that, Augmented Variational Principles (AVP) will be used. We shall discuss how this link accounts for the details of the variation of Padmanabhan's action based on gravitational entropy. It will also clarify the role of the background metric and its non-dynamical role., 19 pages, references added. Some typos and equations (35-37) corrected

We shall introduce the principal symbol for Euler-Lagrange operators and use them to charac- terise well-posed initial value problems. We shall clarify how constraints can arise in Lagrangian covariant theories by extending the standard treatment in GR. Finally, we sketch a quantization procedure based on what done in LQG., Some improvements

We shall review conformal gravity as a gauge natural theory and discuss the consequences of Weyl covariance on the definition of physical states., Comment: 7 pages, Workshop Variational principles and conservation laws in General Relativity, Torino, June 24-25, 2015in memory of Mauro Francaviglia

In this paper we shall review the equivalence between Palatini$-f(\mathcal R)$ theories and Brans- Dicke (BD) theories at the level of action principles. We shall define the Helmholtz Lagrangian associated to Palatini$-f(\mathcal R)$ theory and we will define some transformations which will be useful to recover Einstein frame and Brans-Dicke frame. We shall see an explicit example of matter field and we will discuss how the conformal factor affects the physical quantities., Workshop Variational principles and conservation laws in General Relativity, Torino, June 24-25, 2015in memory of Mauro Francaviglia

We shall present and analyze two examples of extended theories of gravitation in Palatini formalism with matter that couples to the connection. This will show that the class of Further Extended Theories of Gravitation introduced in ref. [1] does not trivially reduce to f(R) models. It will also produce an example of theory that on-shell endowes spacetime with a non-trivial Weyl geometry where the connection is not induced by the metric structure (though it is compatible with it in the sense of Ehlers-Pirani-Schild; see ref. [2])., Some refs added

We review the Lagrangian formulation of Noether symmetries (as well as "generalized Noether symmetries") in the framework of Calculus of Variations in Jet Bundles, with a special attention to so-called "Natural Theories" and "Gauge-Natural Theories", that include all relevant Field Theories and physical applications (from Mechanics to General Relativity, to Gauge Theories, Supersymmetric Theories, Spinors and so on). It is discussed how the use of Poincare'-Cartan forms and decompositions of natural (or gauge-natural) variational operators give rise to notions such as "generators of Noether symmetries", energy and reduced energy flow, Bianchi identities, weak and strong conservation laws, covariant conservation laws, Hamiltonian-like conservation laws (such as, e.g., so-called ADM laws in General Relativity) with emphasis on the physical interpretation of the quantities calculated in specific cases (energy, angular momentum, entropy, etc.). A few substantially new and very recent applications/examples are presented to better show the power of the methods introduced: one in Classical Mechanics (definition of strong conservation laws in a frame-independent setting and a discussion on the way in which conserved quantities depend on the choice of an observer); one in Classical Field Theories (energy and entropy in General Relativity, in its standard formulation, in its spin-frame formulation, in its first order formulation "`a la Palatini" and in its extensions to Non-Linear Gravity Theories); one in Quantum Field Theories (applications to conservation laws in Loop Quantum Gravity via spin connections and Barbero-Immirzi connections)., Comment: 27 pages

We shall review different approaches to the entropy of self-gravitating systems inGeneral Relativity. Then we shall discuss in detail the macroscopic approach based on a la `Clausius point of view. Recent developments will be reviewed discussing the aims as well asthe assumptions which the framework is based on.

Different notions of mechanical energy and conservation laws are considered and discussed from the viewpoint of transformation laws with respect to changes of reference. Here, in particular, we shall consider and investigate the behavior of suitable conservation laws in non-inertial frames.

We investigate the covariant formulation of Chern-Simons theories in a general odd dimension which can be obtained by introducing a vacuum connection field as a reference. Field equations, Noether currents and superpotentials are computed so that results are easily compared with the well-known results in dimension 3. Finally we use this covariant formulation of Chern-Simons theories to investigate their relation with topological BF theories.

In the framework of Digital Art applied to Scientific Visualization we have realized a short video together with an interactive multimedia aimed at visualizing and diffusing the main concepts of Special Relativity. This note makes a short presentation of it.

In this paper we shall analyse the structure of the Cauchy problem for general relativity by applying the theory of first-order symmetric hyperbolic systems. The role of harmonic coordinates will be discussed.

We introduce the natural lift of spacetime diffeomorphisms for conformal gravity and discuss the physical equivalence between the natural and gauge natural structure of the theory. Accordingly, we argue that conformal transformations must be introduced as gauge transformations (affecting fields but not spacetime point) and then discuss special structures implied by the splitting of the conformal group., Comment: 11 pages, comments are welcome

Variational principles for field theories where variations of fields are restricted along a parametrization are considered. In particular, gauge-natural parametrized variational problems are defined as those in which both the Lagrangian and the parametrization are gauge covariant and some further conditions are satisfied in order to formulate a Nöther theorem that links horizontal and gauge symmetries to the relative conservation laws (generalizing what Fernández, García and Rodrigo did in some recent papers). The case of vakonomic constraints in field theory is also studied within the framework of parametrized variational problems, defining and comparing two different concepts of criticality of a section, one arising directly from the vakonomic schema, the other making use of an adapted parametrization. The general theory is then applied to the case of hydrodynamics of a charged fluid coupled with its gravitational and electromagnetic field. A variational formulation including conserved currents and superpotentials is given that turns out to be computationally much easier than the standard one.

It is known that superpotentials of so-called higher-order theories of gravitation and Einstein theories with a cosmological constant are different in the "Palatini" formulation. However, these theories share a number of solutions, the energy of which should be defined out of superpotentials. We shall check that using augmented variational principles, recently introduced by us, the energy of a solution is defined regardless of the theory, and hence regardless of the superpotential used.

We investigate the covariant formulation of Chern–Simons theories in a general odd dimension which can be obtained by introducing a vacuum connection field as a reference. Field equations, Nöther currents and superpotentials are computed so that results are easily compared with the well-known results in dimension 3. Finally we use this covariant formulation of Chern–Simons theories to investigate their relation with topological BF theories.

We shall review some results about the globality of Chern–Simons theory in any odd dimension. The relation between Chern–Simons theories and some recent results about conservation laws are also considered in detail.

We review the classical notion of spin structure on a (pseudo)-Rienannian manifold and appropriately extend it to be applied to manifolds without any a priori fixed metric structure. The relation with soldering forms is discussed in detail and so-called Ashtekar variables for four-dimensional Euclidean GR are introduced as a leading application.

We consider the thermodynamic properties of $(d+1)$-dimensional spacetimes with NUT charges. Such spacetimes are asymptotically locally anti de Sitter (or flat), with non-trivial topology in their spatial sections, and can have fixed point sets of the Euclidean time symmetry that are either $(d-1)$-dimensional (called "bolts") or of lower dimensionality (pure "NUTs"). We compute the free energy, conserved mass, and entropy for 4, 6, 8 and 10 dimensions for each, using both Noether charge methods and the AdS/CFT-inspired counterterm approach. We then generalize these results to arbitrary dimensionality. We find in $4k+2$ dimensions that there are no regions in parameter space in the pure NUT case for which the entropy and specific heat are both positive, and so all such spacetimes are thermodynamically unstable. For the pure NUT case in $4k$ dimensions a region of stability exists in parameter space that decreases in size with increasing dimensionality. All bolt cases have some region of parameter space for which thermodynamic stability can be realized., Comment: Latex, 11 figures, 37 pages, minor latex errors corrected

We shall show equivalence between Palatini-$f(\calR)$ theories and Brans-Dicke (BD) theories at the level of action principles in generic dimension with generic matter coupling. We do that by introducing the Helmholtz Lagrangian associated to Palatini-$f(\calR)$ theory and then performing frame transformations in order to recover Einstein frame and Brans-Dicke frame. This clarifies the relation among different formulations and the transformations among different frames. Additionally, it defines a formulation {\it a l\'a Palatini} for the Brans-Dicke theory which is dynamically equivalent to metric BD (unlike the standard Palatini-formulation of metric BD theory which are {\it not} dynamically equivalent). In conclusion we discuss interpretation of extended theories of gravitation and perspectives.

We discuss in a critical way the physical foundations of geometric structure of relativistic theories of gravity by the so-called Ehlers-Pirani-Schild formalism. This approach provides a natural interpretation of the observables showing how relate them to General Relativity and to a large class of Extended Theories of Gravity. In particular we show that, in such a formalism, geodesic and causal structures of space-time can be safely disentangled allowing a correct analysis in view of observations and experiment. As specific case, we take into account the case of f (R) gravity.

We shall consider possible potentials emerging in (purely metric) f(R)-theories for the conformal scalar field. We shall discuss possible approaches to determine models with specific potentials and show that some potentials qualitatively similar to the typical Higgs potentials are allowed.

The second order symmetry operators that commute with the Dirac operator with external vector, scalar and pseudo-scalar potentials are computed on a general two-dimensional spin-manifold. It is shown that the operator is defined in terms of Killing vectors, valence two Killing tensors and scalar fields defined on the background manifold. The commuting operator that arises from a non-trivial Killing tensor is determined with respect to the associated system of Liouville coordinates and compared to the the second order operator that arises from that obtained from the unique separation scheme associated with such operators. It shown by the study of several examples that the operators arising from these two approaches coincide., Comment: 21 pages

It is known that the entropy for a singular spacetime metric can be calculated in the framework of classical field theories by applying Noether's theorem to stationary solutions of Einstein's field equations, integrating a suitable form on a trapping surface for the singularity. When the Kerr solution is considered, two different horizons appear. The physical entropy for the system is well known to be related to the outer horizon. We investigate here which is the meaning of the entropy calculated (via first principle of black hole thermodynamics) on the inner horizon. We show that this entropy, which was earlier interpreted as a sort of "spin entropy" of the black hole, admits in fact an interpretation as a quantity associated to a conserved charge which is related to the rotational degrees of freedom of the system.

A formalism suited to deal with Bosonic matter represented by tensor densities coupled with a gauge field and gravitational field is developed. Nother conserved currents and superpotentials are found and the relation between the canonical and the Hilbert stress tensors are investigated in the general case. The particular cases of gauge theories with scalar matter fields and of purely geometrical theories with arbitrary matter are recovered from this general formalism; as an example of application we consider scalar densities and the vector matter fields.

Extended theories of gravitation are naturally singled out by an analysis in- spired by the Ehelers-Pirani-Schild framework. In this framework the structure of space- time is described by a Weyl geometry which is enforced by dynamics. Standard General Relativity is just one possible theory within the class of extended theories of gravita- tion. Also all Palatini f (R) theories are shown to be extended theories of gravitation. This more general setting allows a more general interpretation scheme and more general possible couplings between gravity and matter. The definitions and constructions of ex- tended theories will be reviewed. A general interpretation scheme will be considered for extended theories and some examples will be considered.

We discuss in a critical way the physical foundations of geometric structure of relativistic theories of gravity by the so-called Ehlers-Pirani-Schild formalism. This approach provides a natural interpretation of the observables showing how relate them to General Relativity and to a large class of Extended Theories of Gravity. In particular we show that, in such a formalism, geodesic and causal structures of space-time can be safely disentangled allowing a correct analysis in view of observations and experiment. As specific case, we take into account the case of f(R) gravity., 11 pages, 2 figures

We shall investigate the consequences of non-trivial Weyl geometries on conservation laws of a fluid. In particular we shall obtain a set of properties which allow to obtain in this generalized setting the standard relation between conservation of the energy-momentum tensor and number of particles., Comment: 8 pages

In view of Ehlers-Pirani-Schild formalism, since 1972 Weyl geometries should be considered to be the most appropriate and complete framework to represent (relativistic) gravitational fields. We shall here show that in any given Lorentzian spacetime (M,g) that admits global timelike vector fields any such vector field u determines an essentially unique Weyl geometry ([g], \Gamma) such that u is \Gamma-geodesic (i.e. parallel with respect to \Gamma)., Comment: 8 pages, reference added

It is shown that when in a higher order variational principle one fixes fields at the boundary leaving the field derivatives unconstrained, then the variational principle (in particular the solution space) is not invariant with respect to the addition of boundary terms to the action, as it happens instead when the correct procedure is applied. Examples are considered to show how leaving derivatives of fields unconstrained affects the physical interpretation of the model. This is justified in particularl by the need of clarifying the issue for the purpose of applications to relativistic gravitational theories, where a bit of confusion still exists. On the contrary, as it is well known for variational principles of order k, if one fixes variables together with their derivatives (up to order k-1) on the boundary then boundary terms leave solution space invariant., Comment: 7 pages

A signature independent formalism is created and utilized to determine the general second-order symmetry operators for Dirac's equation on two-dimensional Lorentzian spin manifolds. The formalism is used to characterize the orthonormal frames and metrics that permit the solution of Dirac's equation by separation of variables in the case where a second-order symmetry operator underlies the separation. Separation of variables in complex variables on two-dimensional Minkowski space is also considered., Comment: This paper is dedicated to Professor Willard Miller, Jr. on the occasion of his retirement from the School of Mathematics at the University of Minnesota