International Journal of Geometric Methods in Modern Physics

The space of perceived colors, before acquiring an industrial interest, has received a systematic theoretical attention from philosophers, physicists and mathematicians. The research about this topic is still active nowadays. In this paper, it will be presented a critical overview of a model based on differential geometry proposed by H. L. Resnikoff in 1974. It will be shown that, while some fundamental and elegant ideas behind this model can be still used as a guiding principle, some other parts of the model must be updated to comply with the modern findings about color perception.

We outline how Drinfeld twist deformation techniques can be applied to the deformation quantization of principal bundles into noncommutative principal bundles and, more in general, to the deformation of Hopf–Galois extensions. First, we twist deform the structure group in a quantum group, and this leads to a deformation of the fibers of the principal bundle. Next, we twist deform a subgroup of the group of automorphisms of the principal bundle, and this leads to a noncommutative base space. Considering both deformations, we obtain noncommutative principal bundles with noncommutative fiber and base space as well.

Obviously, in Galilean physics, the universality of free fall implies an inertial frame, which in turns implies that the mass $m$ of the falling body is omitted (because it is a test mass; put otherwise, the center of mass of the system coincides with the center of the main, and fixed, mass $M$; or else, we consider only a homogeneous gravitational field). Conversely, an additional (in the opposite or same direction) acceleration proportional to $m/M$ would rise either for an observer at the center of mass of the system, or for an observer at a fixed distance from the center of mass of $M$. These elementary, but overlooked, considerations fully respect the equivalence principle (EP) and the (local) identity of an inertial or a gravitational pull for an observer in the Einstein cabin. They value as fore-runners of the self-force and gauge dependency in general relativity. Because of its importance in teaching and in the history of physics, coupled to the introductory role to Einstein’s EP, the approximate nature of Galilei’s law of free fall is explored herein. When stepping into general relativity, we report how the geodesic free fall into a black hole was the subject of an intense debate again centered on coordinate choice. Later, we describe how the infalling mass and the emitted gravitational radiation affect the free fall motion of a body. The general relativistic self-force might be dealt with to perfectly fit into a geodesic conception of motion. Then, embracing quantum mechanics, real black holes are not classical static objects any longer. Free fall has to handle the Hawking radiation, and leads us to new perspectives on the varying mass of the evaporating black hole and on the varying energy of the falling mass. Along the paper, we also estimate our findings for ordinary masses being dropped from a Galilean or Einsteinian Pisa-like tower with respect to the current state of the art drawn from precise measurements in ground and space laboratories, and to the constraints posed by quantum measurements. Appendix A describes how education physics and high impact factor journals discuss the free fall. Finally, case studies conducted on undergraduate students and teachers are reviewed.

The paper presents a general geometric approach to energy–momentum tensors in Lagrangian field theories, based on a global Hilbert-type definition. The approach is consistent with the ones defining energy–momentum tensors in terms of hypermomentum maps given by the diffeomorphism invariance of the Lagrangian — and, in a sense, complementary to these, with the advantage of an increased simplicity of proofs and also, opening up new insights on the topic. A special attention is paid to the particular cases of metric and metric-affine theories.

A covariant Hamiltonian formalism for the dynamics of compact spinning bodies in curved space-time in the test-particle limit is described. The construction allows a large class of Hamiltonians accounting for specific properties and interactions of spinning bodies. The dynamics for a minimal and a specific non-minimal Hamiltonian is discussed. An independent derivation of the equations of motion from an appropriate energy–momentum tensor is provided. It is shown how to derive constants of motion, both background-independent and background-dependent ones.

In this paper, we show that, using a procedure of classical irreversible thermodynamics (CIT) with internal variables, it is possible to describe the relaxation of thermal phenomena, obtaining some well known results of extended irreversible thermodynamics (EIT). In particular, we introduce as internal variables a vector and a second rank tensor, that influence the thermal transport phenomena, and we derive in the anisotropic and isotropic case, the phenomenological equations for these variables. In the case, in which the medium is isotropic, it is obtained that the total heat flux can be split in two parts: a first contribution $J^{(0)}$, governed by Fourier law, and a second contribution $J^{(1)}$, obeying Maxwell–Cattaneo–Vernotte (MCV) equation, in which a relaxation time is present. The obtained results may have applications in describing the thermal behavior in nanosystems (semiconductors, nanotubes$,\dots$), where the phenomena are fast and there are high-frequency thermal waves.

We start with a short introduction of the role that constraints and Lagrange multiplers play in variational calculus. After recalling briefly the properties of the nonlinear sigma model, we show how the Hamilton–Jacobi method can be applied to find its solutions. We discuss the importance of the Hamiltonian constraint in the standard cosmological model, and finally, apply the Hamilton–Jacobi method to the solution of coupled gravitational and sigma-field equations.

In this paper, we perform a parallel analysis to the model proposed in [E. J. Beggs and S. Majid, Gravity induced from quantum spacetime, Class. Quantum Grav. 31 (2014) 035020, arXiv: 1305.2403 [gr-qc]]. By considering the central co-tetrad (instead of the central metric), we investigate the modifications in the gravitational metrics coming from the noncommutative spacetime of the $\kappa$-Minkowski type in four dimensions. The differential calculus corresponding to a class of Jordanian $\kappa$-deformations provides metrics, which lead either to cosmological constant or spatial curvature type solutions of nonvacuum Einstein equations. Among vacuum solutions, we find pp-wave type.

The Legendre transformation method, applied in 1987 to deal with purely metric gravitational Lagrangians with nonlinear dependence on the Ricci tensor, is extended to metric-affine models and is shown to provide a concise and insightful comparison of the dynamical content of the two variational frameworks.

Fundamental issues underlying gravitational physics and some of the shortcomings of Einstein’s general relativity (GR) are discussed. In particular, after taking into account the role of the two main objects of relativistic theories of gravity, i.e. the metric and the connection fields, we consider the possibility that they are not trivially related so that the geodesic structure and the causal structure of the spacetime could be disentangled, as supposed in the Palatini formulation of gravity. In this perspective, the equivalence principle (EP), in its weak and strong formulations, can play a fundamental role in discriminating among competing theories. The possibility of its violation at quantum level could open new perspectives in gravitational physics and in unification with other interactions. We shortly debate the possibility of EP measurements by ground-based and space experiments.

We show that generalizations of general relativity theory, which consist in replacing the Hilbert Lagrangian $L_{\mathrm{\text{Hilbert}}}=\frac{1}{16\pi }\sqrt{\left|g\right|}R$ by a generic scalar density $L=L(g_{\mu \nu },R_{\mu \nu \kappa }^{\lambda })$ depending upon the metric $g_{\mu \nu }$ and the curvature tensor $R_{\mu \nu \kappa }^{\lambda }$, are equivalent to the conventional Einstein theory for a (possibly) different metric tensor ${\tilde{g}}_{\mu \nu }$ and (possibly) a different set of matter fields. The simple proof of this theorem relies on a new approach to variational problems containing metric and connection.

We shall review conformal gravity as a gauge natural theory and discuss the consequences of Weyl covariance on the definition of physical states.

The variational Lie derivative of classes of forms in the Krupka’s variational sequence is defined as a variational Cartan formula at any degree, in particular for degrees lesser than the dimension of the basis manifold. As an example of application, we determine the condition for a Noether–Bessel-Hagen current, associated with a generalized symmetry, to be variationally equivalent to a Noether current for an invariant Lagrangian. We show that, if it exists, this Noether current is exact on-shell and generates a canonical conserved quantity.

The invariant metrizability problem for affine connections on a manifold, formulated by Tanaka and Krupka for connected Lie groups actions, is considered in the particular cases of Lorentz and Poincaré (inhomogeneous Lorentz) groups. Conditions under which an affine connection on the open submanifold $ℝ\times ({ℝ}^3\backslash \{(0,0,0)\})$ of the Euclidean space ${ℝ}^4$ coincides with the Levi-Civita connection of some $\mathrm{\text{SO}}(3,1)$, respectively $({ℝ}^4{\times }_s\mathrm{\text{SO}}(3,1))$-invariant metric field are studied. We give complete description of metrizable Lorentz-invariant connections. Explicit solutions (metric fields) of the invariant metrizability equations are found and their properties are discussed.