International Journal of Geometric Methods in Modern Physics

There is a review of the main mathematical properties of system described by singular Lagrangians and requiring Dirac–Bergmann theory of constraints at the Hamiltonian level. The following aspects are discussed:

• ⁽ⁱ⁾the connection of the rank and eigenvalues of the Hessian matrix in the Euler–Lagrange equations with the chains of first- and second-class constraints;
• ⁽ⁱⁱ⁾the connection of the Noether identities of the second Noether theorem with the Hamiltonian constraints;
• ⁽ⁱⁱⁱ⁾the Shanmugadhasan canonical transformation for the identification of the gauge variables and for the search of the Dirac observables, i.e. the quantities invariant under Hamiltonian gauge transformations.

The classical and the quantum expressions of the energy of the electromagnetic radiation are compared. This comparison shows that, contrary to the commonly expected, the dependency of the energy $𝜀$ of the photon with respect to the frequency $\nu$ on the Einstein–Planck relation $𝜀=h\nu$ is obtainable from classical electrodynamics, a fact that may provide a new historic perspective of the birth of quantum mechanics. It is suggested that the photon is a composite particle, constituted by quanta of electric and magnetic field fluxes maybe in an entangled state — being these two quanta either real or only as pseudo particles. It is also suggested a modification on the de Broglie’s fusion principle for the photon for the specific case in consideration.

We study the relativistic quantum dynamics of a Klein–Gordon scalar field subject to a Cornell potential in spinning cosmic-string space-time, in order to better understand the effects of gravitational fields produced by topological defects on the scalar field. We solve the Klein–Gordon equation in the presence of scalar and vector interactions by utilizing the Nikiforov–Uvarov formalism and two ansätze, one of which leads to a biconfluent Heun differential equation. We obtain the wave-functions and the energy levels of the relativistic field in that space-time. We discuss the effect of various physical parameters and quantum numbers on the wave-functions.

In the work reported in the present paper, we have considered the inflationary dynamics inspired by holographic Ricci dark energy (HRDE) with the energy density ${\rho }_{\mathrm{\Lambda }}=3{\beta }^2(Ḣ+2H^2)$ in the framework of $f(T)$ gravity. We have considered a $f(T)$-gravity model in linear form, which is sourced by a canonical scalar field due to the HRDE. We have investigated an inflationary Universe in this framework through slow-roll formalism. We also studied the scalar spectral index $(n_s)$ and tensor-to-scalar ratio $(r)$, which have come out to be in agreement with WMAP$+$BAO$+$$H_0$ and Planck 2015 data.

In this paper, spatially homogeneous and anisotropic Bianchi type-${\mathrm{\text{VI}}}_0$ dark energy (DE) cosmological transit models with string fluid source in $f(R,T)$ gravity [T. Harko et al., Phys. Rev. D 84 (2011) 024020], where $R$ is the Ricci scalar and $T$ the trace of the stress energy–momentum tensor, have been studied in the context of early time decelerating and late-time accelerating expansion of the Universe as suggested by the recent observations. The exact solutions of the field equations are obtained first by using generalized hybrid expansion law (HEL) $a={(t^{\alpha }{e}^{\beta t})}^{\frac{1}{n}}$ which yields a time-dependent deceleration parameter $q=-1+\frac{\alpha n}{{(\alpha +\beta t)}^2}$ and second by considering the metric coefficient $A=a^k$. By using recent constraints from supernovae type-Ia union data [Cunha, arXiv:0811.2379[astro-ph]], we obtain $\alpha =0.8$ and $\beta =0.1184$ for transit model $n=2$. The Universe has an initial singularity and is anisotropic closed and it tends to be flat at the late time, i.e. our Universe is in accelerating expansion. Our model shows a phase transition property from decelerating to accelerating. It is remarkable to mention here that our Universe is homogeneous and anisotropic in the early phase whereas it becomes homogeneous and isotropic for $k=1$. We have also discussed the stability of the background solution with respect to perturbations of the metric along with the properties of future singularities in the Universe dominated by DE including the phantom-type fluid. Various physical and dynamical parameters are also calculated and investigated in terms of time and redshift both.

Polymer quantum mechanics, as a non-standard representation of quantum mechanics, is based on a symmetric sector of loop quantum gravity known as loop quantum cosmology. In this work, by analyzing the Hamiltonian and Friedmann equations in the standard Hilbert space and polymer Hilbert space, we show that polymer quantization is a successful formalism for a non-Abelian gauge field driving the cosmological inflation, the so-called gauge-flation, in order to remove initial singularity and also keeping the inflationary trajectories in this model as attractors of dynamics after the bounce.

We consider the interacting scenario of pilgrim dark energy (with future event horizon) with cold dark matter in loop quantum cosmology. We explore various cosmological parameters (Hubble, equation of state and deceleration) and planes ($r-s$ and ${\omega }_{\mathrm{\Lambda }}$–${\omega }_{\mathrm{\Lambda }}^{\prime }$) in this framework. We also discuss the validity of generalized second law of thermodynamics and find that it remains valid in the early epoch as well as in late future epoch. Finally, it is remarked that all the above constraints of cosmological parameters show consistency with different observational data like Planck, WP, BAO, $H_0$, SNLS and nine-year WMAP.

In this work, we study the phase transition of the charged-AdS black hole surrounded by quintessence via an alternative extended phase space defined by the charge square $Q^2$ and her conjugate $\mathrm{\Psi }$, a quantity proportional to the inverse of horizon radius, while the cosmological constant is kept fixed. The equation of state is derived under the form $Q^2=Q^2(T,\mathrm{\Psi })$ and the critical behavior of such black hole analyzed. In addition, we examine the role of the quintessence parameter and its effects on phase transitions. Besides that, we explore the connection between the microscopic structure and Ruppeiner geothermodynamics. We also find that, at certain points of the phase space, the Ruppeiner curvature is characterised by the presence of singularities that are interpreted as a signal of the occurrence of the phase transitions.

In this study, the first-order radiative correction to the Casimir energy for massive and massless scalar fields confined with mixed boundary conditions (BCs) (Dirichlet–Neumann) between two points in ${\varphi }^4$ theory was computed. Two issues in performing the calculations in this work are essential: to renormalize the bare parameters of the problem, a systematic method was employed, allowing all influences from the BCs to be imported in all elements of the renormalization program. This idea yields our counterterms appeared in the renormalization program to be position-dependent. Using the Box Subtraction Scheme (BSS) as a regularization technique is the other noteworthy point in the calculation. In this scheme, by subtracting the vacuum energies of two similar configurations from each other, regularizing divergent expressions and their removal process were significantly facilitated. All the obtained answers for the Casimir energy with the mixed BC were consistent with well-known physical grounds. We also compared the Casimir energy for massive scalar field confined with four types of BCs (Dirichlet, Neumann, mixed of them and Periodic) in $1+1$ dimensions with each other, and the sign and magnitude of their values were discussed.

This paper discusses the stability concerning closed linear Weingarten spacelike hypersurfaces $M^n$ immersed in foliated spacetimes. Firstly, we show that a totally umbilic $M^n$ is strongly stable. Secondly, given a generalized Robertson–Walker spacetime ${\tilde{M}}^{n+1}(c)=I{\times }_{\varphi }{F}^n$ whose warping function $\varphi$ is under a suitable restriction, we prove that if $M^n$ is strongly stable, then it is totally umbilic.

The paper studies a spacetime endowed with two stationary metrics. The first one is a Riemannian one, called the R-Schwarzschild metric. It satisfies Einstein vacuum field equations, describes correctly the slowdown of clocks in the gravitational field, the orbits of the planets and the perihelion drift. The R-Schwarzschild metric can be seen as the basic texture of the spacetime. All objects having mass are ruled by this Riemannian metric. The second metric, the light-adapted one, is deduced both taking into account the Rosen-type bi-metric compatibility condition and by the preservation of the axiom of the speed of the light limit. This second metric offers the texture of the “light-like” objects. The main “normal” surprise is that this metric can be only the classical Schwarzschild metric. So, a Rosen-type bi-metric universe exists and its properties are in accordance with the experimental physical evidences.

We attempt to provide a mesoscopic treatment of the origin of black hole entropy in $(3+1)$-dimensional spacetimes. We treat the case of horizons having space-like sections $\mathrm{\Sigma }$ which are topological spheres, following Hawking’s and the Topological Censorship theorems. We use the injectivity radius of the induced metric on $\mathrm{\Sigma }$ to encode the linear dimensions of the elementary cells giving rise to such entropy. We use the topological entropy of $\mathrm{\Sigma }$ as the fundamental quantity expressing the complexity of $\mathrm{\Sigma }$ on which its entropy depends. We point out the significance, in this context, of the Berger and Croke isoembolic inequalities.

In this paper, we study the propagation of photons in a spacetime of a scalar–tensor screwed cosmic string. We determine the Stokes parameters and compute the rotation of the polarization angle. The latter turns out to depend on the wavelength of the propagating photons and on the cosmological redshift. We then compute the circular polarization of photons showing that, to leading order in the coupling constant, it is zero for the model under consideration.

We investigate quantum vacuum effects for a massive scalar field, induced by two planar boundaries in background of a linearly expanding spatially flat Friedmann–Robertson–Walker spacetime for an arbitrary number of spatial dimensions. For the Robin boundary conditions and for general curvature coupling parameter, a complete set of mode functions is presented and the related Hadamard function is evaluated. The results are specified for the most important special cases of the adiabatic and conformal vacuum states. The vacuum expectation values of the field squared and of the energy–momentum tensor are investigated for a massive conformally coupled field. The vacuum energy–momentum tensor, in addition to the diagonal components, has nonzero off-diagonal component describing energy flux along the direction perpendicular to the plates. The influence of the gravitational field on the local characteristics of the vacuum state is essential at distances from the boundaries larger than the curvature radius of the background spacetime. In contrast to the Minkowskian bulk, at large distances the boundary-induced expectation values follow as power law for both massless and massive fields. Another difference is that the Casimir forces acting on the separate plates do not coincide if the corresponding Robin coefficients are different. At large separations between the plates the decay of the forces is power law. We show that during the cosmological expansion the forces may change the sign.

As gravitation and electromagnetism are closely analogous long-range interactions, and the current formulation of gravitation is given in terms of geometry, we expect the latter also to appear through the geometry. This unification has however, remained an unfulfilled goal.

The goal is achieved here in a new theory, which results from the principles of equivalence and Mach supplemented with a novel insight that the field tensors in a geometric theory of gravitation and electromagnetism must be traceless, since these long-range interactions are mediated by virtual exchange of massless particles whose mass is expected to be related to the trace of the field tensors. Hence, the Riemann tensor, like the analogous electromagnetic field tensor, must be traceless.

Thence emerges a scale-invariant, Machian theory of gravitation and electrodynamics unified, wherein the vanishing of the Ricci tensor appears as a boundary condition. While the field equations of the theory are given by the vanishing divergence of the respective field tensors and their duals, the matter and charge emerge from the spacetime. A quantitative formulation thereof, embodied in “energy-momentum super tensors”, follows from the respective Bianchi identities for the two fields. The resulting theory is valid at all scales and explains the observations without invoking the non-baryonic dark matter, dark energy or inflation. Moreover, it answers the questions that the general relativity-based standard paradigm could not address.

This paper is devoted to possibilities of a semi-analytical approximation of nonlinear wave equations with nonlinearities depending on the absolute value of the unknown function, which arise in different areas of physics and mechanics. The main difficulty of analysis of such equations is that the derivation of their rigorous solution is highly sophisticated, while their numerical solution requires burdensome computational costs. Using the traveling wave ansatz, we first reduce the wave equation to a nonlinear ordinary differential equation. Then, applying Frasca’s method, we construct its general solution in terms of the nonlinear Green’s function. For particular nonlinearities, it is shown that the first-order approximation of the Green’s function solution is numerically comparable with the solution obtained by the well-known numerical method of lines. The contribution of the higher order terms is studied for a particular nonlinearity.

We investigate the relation between two types of space curves, the Mannheim curves and constant-pitch curves and primarily explicate a method of deriving Mannheim curves and constant-pitch curves from each other by means of a suitable deformation of a space curve. We define a “radius” function and a “pitch” function for any arbitrary regular space curve and use these to characterize the two classes of curves. A few non-trivial examples of both Mannheim and constant pitch curves are discussed. The geometric nature of Mannheim curves is established by using the notion of osculating helices. The Frenet–Serret motion of a rigid body in theoretical kinematics is studied for the special case of a Mannheim curve and the axodes in this case are deduced. In particular, we show that the fixed axode is developable if and only if the motion trajectory is a Mannheim curve.