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Lorenzo Fatibene

Publications and source records attributed to Lorenzo Fatibene.

At least 19 recordsLinked to original sources
arXiv2026

Towards General Relativity as a generalized Yang-Mills theory

As an application of the generalized principal bundle theory to covariant Lagrangian field theories, we aim at the development of an instance of generalized gauge theories, with the prospect of a unifying language for Yang-Mills theories and General Relativity. After reviewing the basic definitions of Lie group fiber bundles and generalized principal bundles, we provide horizontal lift and local characterizations of Lie group fiber bundle connections and generalized principal connections. Subsequently, we consider in the framework of classical field theories the kinematics and dynamics sides of generalized principal connections, which lead to a proposed notion of generalized Yang-Mills theories. Finally, we show how vector bundles are examples of generalized principal bundles and that a generalized principal connection on a vector bundle is an affine connection given in terms of basic soldering forms. We are able to recover, under appropriate assumptions, the Vielbein formulation of (vacuum) General Relativity in this setting, hinting at a (generalized) gauge theory of gravity.

math-ph↗
arXiv2026

A Lagrangian framework for canonical analysis for the Holst model with $β= 0$

We perform a canonical analysis of the Holst model for General Relativity, within the framework laid out in arXiv:2401.07307 and arXiv:2010.07725, distinguishing our approach by setting the Barbero parameter to $β=0$ and leaving the lapse and shift functions unconstrained. The $β= 0$ choice is of particular interest because it is viable across all dimensions, providing a necessary foundation for extending the Loop Quantum Gravity formalism beyond $3+1$ dimensions. Through field decomposition and the projection of the field equations, we derive a system of 37 equations (10 differential constraints, 21 algebraic constraints, and 6 evolution equations) exactly matching the 37 field components to be determined. Moreover, leaving the gauge unfixed reveals that three equations, which are typically identically satisfied under normal evolution, are actually differential constraints whose triviality depends on specific gauge choices. The resulting framework remains fully consistent with the standard $3+1$ decomposition of the Einstein equations without requiring any constraints on the lapse and shift functions.

gr-qc↗
arXiv2026

A constructive approach to generalized principal connections

We address the recently introduced notions of generalized principal bundle and generalized principal connection by keeping track of global geometric properties through local coordinate transformation laws. This approach leads us to introduce generalized principal bundle coordinates and to find their transformation laws. Besides, we show that any Lie group fiber bundle (and hence, in particular, any vector bundle) is a generalized principal bundle and we give a proof of the fact that any Lie group fiber bundle with connected typical fiber is an associated bundle to a suitable principal bundle. Moreover, we present a direct way to characterize Lie group fiber bundle connections and generalized principal connections in terms of horizontal lifts and of local conditions. Finally, we recover in our setting some already known results, including that generalized principal connections are associated only to Lie group fiber bundle connections and that they reduce to usual principal connections on standard principal bundles. Our results are needed in order to understand how generalized principal connections might fit in the fiber bundle treatment of classical field theories, aiming towards a notion of generalized gauge theory.

math-ph↗
arXiv2026

Lecture Notes in Loop Quantum Gravity. LN4: Hamiltonian framework

We discuss a covariant setting for Hamiltonian formalism in a relativistic field theory and we use this to obtain again the properties of Hamilton principal functional in Newtonian mechanics, relativistic mechanics, Klein-Gordon, electromagnetism, and Ashtekar-Barbero-Immirzi gravitational theory.

gr-qc↗
arXiv2020

Einstein, Planck and Vera Rubin: relevant encounters between the Cosmological and the Quantum Worlds

In Cosmology and in Fundamental Physics there is a crucial question like: where the elusive substance that we call Dark Matter is hidden in the Universe and what is it made of?, that, even after 40 years from the Vera Rubin seminal discovery does not have a proper answer. Actually, the more we have investigated, the more this issue has become strongly entangled with aspects that go beyond the established Quantum Physics, the Standard Model of Elementary particles and the General Relativity and related to processes like the Inflation, the accelerated expansion of the Universe and High Energy Phenomena around compact objects. Even Quantum Gravity and very exotic DM particle candidates may play a role in framing the Dark Matter mystery that seems to be accomplice of new unknown Physics. Observations and experiments have clearly indicated that the above phenomenon cannot be considered as already theoretically framed, as hoped for decades. The Special Topic to which this review belongs wants to penetrate this newly realized mystery from different angles, including that of a contamination of different fields of Physics apparently unrelated. We show with the works of this ST that this contamination is able to guide us into the required new Physics. This review wants to provide a good number of these "paths or contamination" beyond/among the three worlds above; in most of the cases, the results presented here open a direct link with the multi-scale dark matter phenomenon, enlightening some of its important aspects. Also in the remaining cases, possible interesting contacts emerges.

gr-qc↗
arXiv2020

The generally covariant meaning of space distances

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.

gr-qc↗
arXiv2020

Solar System Tests in Brans-Dicke and Palatini f(R)-theories

We compare Mercury's precession test in standard General Relativity (GR), Brans-Dicke theories (BD), and Palatini f(R)-theories. We avoid post Newtonian (PN) approximation and compute exact precession in these theories. We show that the well-known mathematical equivalence between Palatini 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 different incompatible precession for Mercury depending on the solution one chooses. As a result one cannot use BD equivalence to rule out Palatini f(R)-theories. On the contrary, we directly discuss that Palatini f(R)-theories can (and specific models do) easily pass Solar System tests as Mercury's precession.

gr-qc↗
arXiv2020

Barbero-Immirzi connections and how to build them

We introduce a covariant formulation of Barbero-Immirzi connections, which are used in Loop Quantum Gravity to describe gravity. We show that Barbero-Immirzi connections can be uniquely defined out of a given spin connection for any $(n+1)$-dimensional lorentzian manifold which is spin. A remarkable result is that the presence of a real Barbero-Immirzi parameter is a feature unique to the $4$-dimensional case.

math.DG↗
arXiv2020

Discrete Relativistic Positioning Systems

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.

gr-qc↗
arXiv2020

Extended Cosmology in Palatini f(R)-theories

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.

gr-qc↗
arXiv2016

Emergent Gravity from an Augmented Variational Principle

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.

gr-qc↗
arXiv2016

Principal Symbol of Euler-Lagrange Operators

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.

gr-qc↗
arXiv2016

Extended Cosmologies

We shall discuss cosmological models in extended theories of gravitation. We shall define a surface, called the model surface, in the space of observable parameters which characterises families of theories. We also show how this surface can be used to compare with observations. The model surface can potentially be used to falsify whole families of models instead reasoning on a single model basis as it is usually done by best fit arguments with observations.

gr-qc↗
arXiv2016

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.

gr-qc↗
arXiv2015

A further study on Palatini f(R)-theories for polytropic stars

After briefly reviewing the results about polytropic stars in Palatini f(R)-theories, we first show how these results rely on the assumption of a regular function f(R). In particular, singular models allow to extend the parameter interval in which no singularity is formed. Furthermore, we present how the conformal metric can be matched smoothly in the cases where the original metric generates a singularity. In fact, the singularity comes from a singular conformal factor which is continuous though not differentiable at the stellar surface. This suggests that the correct metric to be considered as physical is the conformal metric. This is relevant because, even also when matching the original metric is possible, the use of the conformal metric generates different stellar models.

gr-qc↗
arXiv2014

Symmetry operators and separation of variables for Dirac's equation on two-dimensional spin manifolds with external fields

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.

math-ph↗