Entropy of Self-Gravitating Systems from Holst's Lagrangian

We shall prove here that conservation laws from Holst's Lagrangian, often used in LQG, do not agree with the corresponding conservation laws in standard GR. Nevertheless, these differences vanish on-shell, i.e. along solutions, so that they eventually define the same classical conserved quantities. Accordingly, they define in particular the same entropy of solutions, and the standard law S=A/4 is reproduced for systems described by Holst's Lagragian. This provides the classical support to the computation usually done in LQG for the entropy of black holes which is in turn used to fix the Barbero-Immirzi parameter.Comment: 4 pages, no figures; just acknowledgments change

ADM Pseudotensors, Conserved Quantities and Covariant Conservation Laws in General Relativity

The ADM formalism is reviewed and techniques for decomposing generic components of metric, connection and curvature are obtained. These techniques will turn out to be enough to decompose not only Einstein equations but also covariant conservation laws. Then a number of independent sets of hypotheses that are sufficient (though non-necessary) to obtain standard ADM quantities (and Hamiltonian) from covariant conservation laws are considered. This determines explicitely the range in which standard techniques are equivalent to covariant conserved quantities. The Schwarzschild metric in different coordinates is then considered, showing how the standard ADM quantities fail dramatically in non-Cartesian coordinates or even worse when asymptotically flatness is not manifest; while, in view of their covariance, covariant conservation laws give the correct result in all cases.Comment: 40 page

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

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 page

Extended Loop Quantum Gravity

We discuss constraint structure of extended theories of gravitation (also known as f(R) theories) in the vacuum selfdual formulation introduced in ref. [1].Comment: 7 pages, few typos correcte

Extended Theories of Gravitation

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.Comment: Workshop Variational principles and conservation laws in General Relativity, Torino, June 24-25, 2015in memory of Mauro Francavigli

The Cauchy problem in General Relativity: An algebraic characterization

In this paper we shall analyse the structure of the Cauchy Problem (CP briefly) for General Relativity (GR briefly) by applying the theory of first order symmetric hyperbolic systems