S. Mercadante

From Dark Energy & Dark Matter to Dark Metric

We present a new approach to the mathematical objects of General Relativity in terms of which a generic f(R)-gravity theory gravitation is written in a first-order (à la Palatini) formalism, and introduce the concept of Dark Metric which could bypass the emergence of disturbing concepts as Dark Energy and Dark Matter. These issues are related to the fact that General Relativity could not be the definitive theory of Gravitation due to several shortcomings that come out both from theoretical and experimental viewpoints. At large scales, the attempts to match it with the recent observational data lead to invoke Dark Energy and Dark Matter as the bulk components of the cosmic fluid. Since no final evidence, at fundamental level, exists for such ingredients, it could be useful to reconsider the gravitational sector in order to see if suitable extensions of General Relativity could solve the shortcomings present at infrared scales.

MATTER LAGRANGIANS COUPLED WITH CONNECTIONS

We shall consider here extended theories of gravitation in the metric-affine formalism with matter coupled directly to the connection. A sufficiently general procedure will be exhibited to solve the resulting field equation associated to the connection. As special cases one has the no-coupling case (which is standard in f(R) literature) as well as the cases already analyzed in [1].

Noether Symmetries and Covariant Conservation Laws in Classical, Relativistic and Quantum Physics

We review the Lagrangian formulation of (generalised) 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, etc.). It is discussed how the use of Poincar´e–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-calledADMlaws 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).

COVARIANT FORMULATION OF CHERN–SIMONS 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.

First Order Extended Gravity and the Dark Side of the Universe: the General Theory

General Relativity is not the definitive theory of Gravitation due to several shortcomings which are coming out both from theoretical and experimental viewpoints. At large scales (astro‐physical and cosmological scales) the attempts to match it with the today observational data lead to invoke Dark Energy and Dark Matter as the bulk components of the cosmic fluid. Since no final evidence, at fundamental level, exists for such ingredients, it is clear that General Relativity presents shortcomings at infrared scales. On the other hand, the attempts to formulate theories more general than the Einstein one give rise to mathematical difficulties that need workarounds which, in turn, generate problems from the interpretative viewpoint. We present here a completely new approach to the mathematical objects in terms of which a theory of Gravitation may be written in a first‐order (a la Palatini) formalism, and introduce the concept of Dark Metric which could completely bypass the introduction of disturbing concepts as Dark Energy and Dark Matter.

COVARIANT LAGRANGIAN FORMALISM FOR CHERN-SIMONS THEORIES

We review the global formalism for Chern–Simons theories. Superpotentials are computed and the relation between the global Chern–Simons Lagrangian and the augmented variational principle is investigated.

Higgs-like field from extended theory of gravitation

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.

Obituary of Professor Mauro Francaviglia

Mauro Francaviglia was born in Torino, on 22nd June 1953 and passed away in Cosenza, on 24th June 2013. He graduated, magna cum laude, in Mathematics at the University of Torino in 1975. He held the position of Full Professor of Mathematical Physics at the University of Torino since 1980, when he was 27 years old. He was considered an extremely valid researcher and he was one of the best Italian Mathematical Physicists of his generation. Mauro started his research activity in General Relativity, as a classical relativist, and his scientific interests covered a wide range of topics, including the application of Differential Geometry in Mathematical Physics, Classical Mechanics, Field Theories, Calculus of Variations, Symmetries and Conservation Laws, Quantization, Thermodynamics and, in late years, Extended Theories of Gravitation. Author of over 250 papers, three monographs and 11 encyclopedia long entries, Editor of 19 volumes of Proceedings and Director of two CIME courses. He organized 20 national and international conferences, among which several national conferences in General Relativity and the World Conference GR14 (Florence, 1995). He has been the Director of several national and international research projects and he held over 300 invited lectures at various institutions in Italy and abroad. He supervised several successful Ph.D. students. Mauro has been a member of the Scientific Council of GNFM-CNR (1980–1996), a co-founder (1984) and Managing Editor of “Journal of Geometry and Physics”, a co-founder (1990) and President (1990–1996 and 2008–2012) of the Italian Society for General Relativity and Gravitation (SIGRAV). He was a life member of the

First order Extended Gravity and the Dark side of the Universe: matching with observations

The so called f(R)‐gravity could be, in principle, able to explain the accelerated expansion of the Universe without adding unknown forms of dark energy/dark matter but, more simply, extending the General Relativity by generic functions of the Ricci scalar. However, a part several phenomenological models, there is no final f(R)‐theory capable of fitting all the observations and addressing all the issues related to the presence of dark energy and dark matter. Astrophysical observations are pointing out huge amounts of “dark matter” and “dark energy” needed to explain the observed large scale structures and cosmic accelerating expansion. Up to now, no experimental evidence has been found, at fundamental level, to explain such mysterious components. The problem could be completely reversed considering dark matter and dark energy as “shortcomings” of General Relativity and considering the concept of Dark Metric introduced in the companion paper of these Proceedings by the same authors.

Geometric Entropy of Self-Gravitating Systems

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.