International Journal of Geometric Methods in Modern Physics

We analyze the conservation laws in the gauge gravity theory which are derived for the general class of gravitational models with the action invariant under the local Poincaré and the diffeomorphism group. The consistent Noether-Lagrange formalism is developed, revealing the important role of the auxiliary Goldstone and Stueckelberg fields, with the help of which we construct the composite gauge fields that have a clear geometrical and physical meaning.

The more precise definition and the more fundamental understanding of the concepts of time, energy, entropy and information are building upon the new, relativistic foundation of gravity. This lecture is an attempt to explain the basic principles that underpin this progress, by focusing on the simple but subtle universal definition of energy. The principles are unearthed from Einstein’s theory and Noether’s theorems, beneath a century of misconceptions.

Local Lorentz transformations play an important role in teleparallel gravity theories, in which a tetrad is conventionally employed as a fundamental field variable describing the gravitational field. It is commonly understood that modifications of general relativity in the teleparallel framework break a certain notion of local Lorentz invariance, which is present in the pure tetrad formulation of such theories, while another notion present in the covariant formulation is preserved. We illuminate these different notions from a geometric perspective, and distinguish them from what is commonly understood as breaking of local Lorentz invariance in the context of gravity phenomenology. Based on physical arguments, we present a geometric interpretation of the dynamical fields in teleparallel gravity, which unifies and refines the conventional approaches.

We review the possible violation of the Equivalence Principle at finite temperature $T$ in the framework of curvature-based Extended Theories of Gravity. Specifically, we first show how it is possible to derive Equivalence Principle violation from Quantum Field Theory at $T\ne 0$. Subsequently, we exhibit how this result can be precisely recovered by following an alternative path that envisages the employment of generalized Einstein equations with a temperature-dependent energy-momentum tensor. Finally, we adopt the latter formalism in the context of some Extended Gravity models to quantify the amount of Equivalence Principle violation. Specifically, Brans–Dicke Theory, Standard Model Extension and Conformal Gravity are considered in detail.

In this review, we discuss the emergence of unimodular gravity (or, more precisely, Weyl transverse gravity) from the thermodynamics of spacetime. By analyzing three different ways to obtain gravitational equations of motion by thermodynamic arguments, we show that the results point to unimodular rather than fully diffeomorphism invariant theories and that this is true even for modified gravity. The unimodular character of dynamics is especially evident from the status of cosmological constant and energy-momentum conservation.

Astrophysical objects and the surrounding geometry must be described by suitable states in the complete quantum theory of matter and gravity. We attempt at reconstructing such a state for a Schwarzschild black hole from the analysis of a collapsing ball of dust and describing the outer geometry by means of a coherent state.

In this paper, we consider the quadratic Metric-Affine Gauge gravity Lagrangian, which contains all the algebraic invariants up to quadratic order in torsion, nonmetricity and curvature. The goal is to collect the known exact solutions for this theory that describe the propagation of a gravitational wave and some useful properties. We concentrate on solutions whose connections are different from the Levi-Civita one and provide also the extension of already known axial torsion solutions to more general Lagrangians. In the end, we will briefly discuss colliding waves and Riemannian solutions.

In this paper, we explore the possibility to find exact solutions for Teleparallel Gravity (TG) of the type of spherically symmetric Lemaître–Tolman–Bondi (LTB) dust models. We apply to the LTB metric the formalism of Teleparallel Gravity in its extension to $f(T,B)$ models, which can be seen as the analagous from the Schwarzschild solution in General Relativity. An exact LTB solution is obtained which is compatible with a specific $f(T,B)$ model whose observational constraints are cosmologically viable in a standard spatially flat Robertson–Walker geometry.

The improved description of the planets’ interior is provided. We examine the modified gravity effects on the Earth-like planets composed of the iron core and silicate mantle. We confirm that the mass–radius relations as well as density profiles differ with respect to the commonly adopted Newtonian models.

This paper considers the relationship between geometry, symmetry and fundamental interactions — gravity and those mediated by gauge fields. We explore product spacetimes which (a) have the necessary symmetries for gauge interactions and four-dimensional gravity and (b) reduce to an $N$-dimensional isotropic universe in their flat space limit. The key technique is looking at orbits of the operator form of symmetric rank-two tensors under changes of coordinate system. Orbits containing diagonal matrices are seen to correspond to product manifolds. The $GL(N,ℝ)$ symmetry of the decompactified universe acts nonlinearly on such a product spacetime. We explore the resulting Kaluza–Klein theories, in which the internal symmetries act indirectly on space of the extra dimensions, and give two examples: a six-dimensional model in which the gauge symmetry is $U(1)$ and a seven-dimensional model in which it is $SU(2)$. We identify constraints that can be placed on any rank-two symmetric tensor to obtain such spacetimes: relationships between polynomial invariants. The multiplicities of its eigenvalues determine the dimensionalities of the factor spaces and hence the gauge symmetries. If the tensor in question is the Ricci tensor, other than two-dimensional factor spaces all the factor spaces are Einstein manifolds. This situation represents the classical vacuum of the Kaluza–Klein theory.