International Journal of Geometric Methods in Modern Physics

We review thermodynamic properties of modified gravity theories, such as $F(R)$ gravity and $f(T)$ gravity, where $R$ is the scalar curvature and $T$ is the torsion scalar in teleparallelism. In particular, we explore the equivalence between the equations of motion for modified gravity theories and the Clausius relation in thermodynamics. In addition, thermodynamics of the cosmological apparent horizon is investigated in $f(T)$ gravity. We show both equilibrium and nonequilibrium descriptions of thermodynamics. It is demonstrated that the second law of thermodynamics in the universe can be met, when the temperature of the outside of the apparent horizon is equivalent to that of the inside of it.

In this work, we study a particle creation mechanism at the expense of the energy stored in an extra dimension with the respective process of compactification, and hence the emergence of 4D space. The spacetime resulting can be the Friedmann–Robertson–Walker (FRW) type, but also can be the cosmological term plus cold dark matter (ΛCDM) type, depending on the model’s parameters. Concurrent with the compactification process we have also the homogenization process, since initially the spacetime is anisotropic.

We consider the simplest extension of the standard model, where torsion couples to spinor as well as the scalar fields, and in which the cosmological constant problem is solved.

Classical field theory is considered as a theory of unparametrized surfaces embedded in a configuration space, which accommodates, in a symmetric way, spacetime positions and field values. Dynamics is defined by a (Hamiltonian) constraint between multivector-valued generalized momenta, and points in the configuration space. Starting from a variational principle, we derive local equations of motion, that is, differential equations that determine classical surfaces and momenta. A local Hamilton–Jacobi equation applicable in the field theory then follows readily. The general method is illustrated with three examples: non-relativistic Hamiltonian mechanics, De Donder–Weyl scalar field theory, and string theory.

We propose a detail study of integral curves or flow lines of a linear vector field in Lorentz $3$-space. We construct the matrix $A$ depending on the causal characters of the vector $x$ by analyzing the non-zero solutions of the equation $A(x)=0$, $x\in {𝔼}_1^3,$ in such a space, where $A$ is the skew-symmetric matrix corresponding to the linear map $A$. Considering the structure of a linear vector field, we obtain the linear first-order system of differential equations. The solutions of this system of equations give rise to integral curves of linear vector fields from which we provide a classification of such curves.

We give a generalized Weierstrass formula for a Lorentz surface conformally immersed in the four-dimensional space ${ℝ}^{2,2}$ using spinors and Lorentz numbers. We also study the immersions of a Lorentzian surface in the Anti-de Sitter space (a pseudo-sphere in ${ℝ}^{2,2}$): we give a new spinor representation formula and deduce the conformal description of a flat Lorentzian surface in that space.

Using M-theory compactification, we develop a three-factor separation for the scalar submanifold of $N=2$ seven-dimensional supergravity associated with 2-cycles of the K3 surface. Concretely, we give an interplay between the three-scalar submanifold factors and the extremal black holes obtained from M2-branes wrapping such 2-cycles. Then, we show that the corresponding black hole charges are linked to one, two and four qubit systems.

The paper aims to extend major equations in the electromagnetic and gravitational theories from the flat space into the complex octonion curved space. Maxwell applied simultaneously the quaternion analysis and vector terminology to describe the electromagnetic theory. It inspires subsequent scholars to study the electromagnetic and gravitational theories with the complex quaternions/octonions. Furthermore Einstein was the first to depict the gravitational theory by means of tensor analysis and curved four-space–time. Nowadays some scholars investigate the electromagnetic and gravitational properties making use of the complex quaternion/octonion curved space. From the orthogonality of two complex quaternions, it is possible to define the covariant derivative of the complex quaternion curved space, describing the gravitational properties in the complex quaternion curved space. Further it is possible to define the covariant derivative of the complex octonion curved space by means of the orthogonality of two complex octonions, depicting simultaneously the electromagnetic and gravitational properties in the complex octonion curved space. The result reveals that the connection coefficient and curvature of the complex octonion curved space will exert an influence on the field strength and field source of the electromagnetic and gravitational fields, impacting the linear momentum, angular momentum, torque, energy, and force and so forth.

In this paper, we introduce Bäcklund transformation of a pseudo null curve in Minkowski 3-space as a transformation mapping a pseudo null helix to another pseudo null helix congruent to the given one. We also give the sufficient conditions for a transformation between two pseudo null curves in the Minkowski 3-space such that these curves have equal constant torsions. By using the Da Rios vortex filament equation, based on localized induction approximation (LIA), we derive the vortex filament equation for a pseudo null curve and prove that the evolution equation for the torsion is the viscous Burger’s equation. As an application, we show that pseudo null curves and their Frenet frames generate solutions of the Da Rios vortex filament equation.

We consider generally relativistic gauge transformations for the spinorial fields finding two mutually exclusive but together exhaustive classes in which fermions are placed adding supplementary information to the results obtained by Lounesto, and identifying quantities analogous to the momentum vector and the Pauli–Lubanski axial vector. We discuss how our results are similar to those obtained by Wigner by taking into account the system of Dirac field equations. We will investigate the consequences for the dynamics and in particular we shall address the problem of getting the nonrelativistic approximation in a consistent way. We are going to comment on extensions.

A generalized Robertson–Walker (GRW) space-time is the generalization of the classical Robertson–Walker space-time. In the present paper, we show that a Ricci simple manifold with vanishing divergence of the conformal curvature tensor admits a proper concircular vector field and it is necessarily a GRW space-time. Further, we show that a stiff matter perfect fluid space-time or a mass-less scalar field with time-like gradient and with divergence-free Weyl tensor are GRW space-times.

In this paper, we study anti-self dual manifolds endowed with metrics of neutral signature. Since the metrics depend on solutions of, in some cases, well-known partial differential equations (PDEs), we determine exact solutions using Lie group methods. This concludes specific forms of the metrics. We then determine the isometries and the variational symmetries of the underlying metrics and corresponding Euler–Lagrange (geodesic) equations, respectively, and establish relationships between the resultant Lie algebras. In some cases, we construct conservation laws via these symmetries or the “multiplier approach”.

The field equations of the recent nonlocal generalization of Einstein’s theory of gravitation are presented in a form that is reminiscent of general relativity. The implications of the nonlocal field equations are studied in the case of conformally flat spacetimes. Even in this simple case, the field equations are intractable. Therefore, to gain insight into the nature of these equations, we investigate the structure of nonlocal gravity (NLG) in 2D spacetimes. While any smooth 2D spacetime is conformally flat and satisfies Einstein’s field equations, only a subset containing either a Killing vector or a homothetic Killing vector can satisfy the field equations of NLG.

We discuss general aspects of non-relativistic quantum chaos theory of scattering of a quantum particle on a system of a large number of naked singularities. We define such a system space-temporal Sinai billiard. We discuss the problem in semiclassical approach. We show that in semiclassical regime the formation of trapped periodic semiclassical orbits inside the system is unavoidable. This leads to general expression of survival probabilities and scattering time delays, expanded to the chaotic Pollicott–Ruelle resonances. Finally, we comment on possible generalizations of these aspects to relativistic quantum field theory.

We study in detail the underlying graded geometric structure of abelian $N=2$ supersymmetric Chern–Simons theory in (2 + 1)-dimensions. This structure is an attribute of the hidden unbroken one-dimensional $N=2$ supersymmetries that the system also possesses. We establish the result that the geometric structures corresponding to the bosonic and to the fermionic sectors are equivalent fiber bundles over the (2 + 1)-dimensional manifold. Moreover, we find a geometrical answer to the question why some and not all of the fermionic sections are related to a $N=2$ supersymmetric algebra. Our findings are useful for the quantum theory of Chern–Simons vortices.

After an overview of variational principles for discrete gravity, and on the basis of the approach to conformal transformations in a simplicial PL setting proposed by Luo and Glickenstein, we present at a heuristic level an improved scheme for addressing the gravitational (Euclidean) path integral and geometrodynamics.

In this paper, we study a class of Finsler metrics, which are defined by a Riemannian metric $\alpha$ and a one-form $\beta$. They are called general $(\alpha ,\beta )$-metrics. We have proven that, every Landsberg general $(\alpha ,\beta )$-metric is a Berwald metric, under a certain condition. This shows that the hunting for an unicorn, one of the longest standing open problem in Finsler geometry, cannot be successful in the class of general $(\alpha ,\beta )$-metrics.

Gravitation theory is formulated as gauge theory on natural bundles with spontaneous symmetry breaking, where gauge symmetries are general covariant transformations, gauge fields are general linear connections, and Higgs fields are pseudo-Riemannian metrics.