Narrow Results

In this brief review, we discuss the viability of a multi-dimensional geometrical theory with one compactified dimension. We discuss the case of a Kaluza–Klein (KK) fifth-dimensional theory, addressing the problem by an overview of the astrophysical phenomenology associated with this five-dimensional (5D) theory. By comparing the predictions of our model with the features of the ordinary (four-dimensional (4D)) Relativistic Astrophysics, we highlight some small but finite discrepancies, expectably detectible from the observations. We consider a class of static, vacuum solutions of free electromagnetic KK equations with three-dimensional (3D) spherical symmetry. We explore the stability of the particle dynamics in these spacetimes, the construction of self-gravitating stellar models and the emission spectrum generated by a charged particle falling on this stellar object. The matter dynamics in these geometries has been treated by a multipole approach adapted to the geometric theory with a compactified dimension.

We propose a consistent approach to the definition of electric, magnetic and toroidal multipole moments. Electric and magnetic fields are split into potential, vortex and radiative terms, with the latter ones dropped off in the quasistatic approximation. The potential part of the electric field, the vortex parts of the magnetic field and vector potential contain gradients of scalar functions. Formally introducing magnetic and toroidal analogs of the electric charge, we apply multipole expansions for those scalars. Closed-form expressions are derived in an arbitrary order for electric, magnetic and toroidal multipoles, which constitute a full system for expansions of the electromagnetic field.

We developed a highly accurate and fully Maxwellian conformal mapping method for calculation of main fields of electrostatic particle optical elements. A remarkable advantage of this method is the possibility of rapid recalculations with geometric asymmetries and mispowered plates. We used this conformal mapping method to calculate the multipole terms of the high voltage quadrupoles in the storage ring of the Muon $g-2$ Experiment (FNAL-E-0989). Next, we demonstrate that an effect where the observed tunes correspond to a voltage that is about $4\%$ higher compared to the voltage to which the Muon $g-2$ quadrupoles are set is explained by the conceptual and quantitative differences between the beam optics quadrupole voltage and the quadrupole voltage at the plates. Completing the methodological framework for field computations, we present a method for extracting multipole strength falloffs of a particle optical element from a set of Fourier mode falloffs. We calculated the quadrupole strength falloff and its effective field boundary (EFB) for the Muon $g-2$ quadrupole, which has explained the experimentally measured tunes, while simple estimates based on a linear model exhibited discrepancies up to $2\%$.

In this paper, we study some classical aspects of Podolsky electrodynamics in the static regime. We develop the multipole expansion for the theory in both the electrostatic and the magnetostatic cases. We also address the problem of consistently truncating the infinite series associated with the several kinds of multipoles, yielding approximations for the static Podolskian electromagnetic field to any degree of precision required. Moreover, we apply the general theory of multipole expansion to some specific physical problems. In those problems, we identify the first terms of the series with the monopole, dipole, and quadrupole terms in the generalized theory. We also propose a situation in which Podolsky theory can be experimentally tested.

The linearized field equations of general relativity in harmonic coordinates are given by an inhomogeneous wave equation. In the region exterior to the matter field, the retarded solution of this wave equation can be expanded in terms of 10 Cartesian symmetric and tracefree (STF) multipoles in post-Minkowskian approximation. For such a multipole decomposition only three and rather weak assumptions are required:

(1) No-incoming-radiation condition.

(2) The matter source is spatially compact.

(3) A spherical expansion for the metric outside the matter source is possible.

During the last decades, the STF multipole expansion has been established as a powerful tool in several fields of gravitational physics: celestial mechanics, theory of gravitational waves and in the theory of light propagation and astrometry. But despite its formidable importance, an explicit proof of the fundamental theorem of STF multipole expansion has not been presented so far, while only some parts of it are distributed into several publications. In a technical but more didactical form, an explicit and detailed mathematical proof of each individual step of this important theorem of STF multipole expansion is represented.

The accuracy of multipole expansion of density distribution for deformed nuclei is tested. The interaction potential for a deformed-spherical pair of nuclei was calculated using the folding model derived from zero-range nucleon–nucleon (NN) interaction. We considered two spherical projectiles Ca⁴⁰ and Pb²⁰⁸ scattered on U²³⁸ deformed target nucleus. The error in the heavy ion (HI) potential resulting from using a truncated multipole density expansion is evaluated for each case in the presence of octupole deformation δ₃ besides quadrupole δ₂. We are interested in the value of error for R ≥ R_T (touching distance). We found that for values of |δ₃|≤0.1 the error at R = R_T reaches reasonable values when six terms expansion is used. For |δ₃| = 0.2, we calculated the Coulomb barrier parameters using realistic NN force and found that the large error present in six terms for zero range force decreases strongly to less than 1% when the zero range is added to finite range forces and Coulomb interaction to form the Coulomb barrier. It is noted that the negative value of octupole deformation parameters δ₃ = -0.1 produce error at orientation angle θ equal in value to that produced at angle (180°-θ) for the positive values δ₃ = 0.1. We also found that the error decreases as the mass number of the projectile nucleus increases.

The deformed relativistic Hartree–Bogoliubov theory in continuum (DRHBc) has been proved as one of the best models to probe the exotic structures in deformed nuclei. In DRHBc, the potentials and densities are expressed in terms of the multipole expansion with Legendre polynomials, the dependence on which has only been touched for light nuclei so far. In this paper, taking a light nucleus ${}^{20}$Ne and a heavy nucleus ${}^{242}$U as examples, we investigated the dependence on the multipole expansion of the potentials and densities in DRHBc. It is shown that the total energy converges well with the expansion truncation both in the absence of and presence of the pairing correlation, either in the ground state or at a constrained quadrupole deformation. It is found that to reach the same accuracy of the total energy, even to the same relative accuracy by percent, a larger truncation is required by a heavy nucleus than a light one. The dependence of the total energy on the truncation increases with deformation. By decompositions of the neutron density distribution, it is shown that a higher-order component has a smaller contribution. With the increase of deformation, the high-order components get larger, while at the same deformation, the high-order components of a heavy nucleus play a more important role than that of a light one.

In this paper, a hydroelastic model is considered to examine the trapped modes supported by a horizontal submerged cylinder placed in either of the layers of a two-layer fluid flowing over an elastic bottom at a finite depth. The elastic bottom is modeled as a thin elastic plate and is based on the Euler–Bernoulli beam equation. Using multipole expansion method, an infinite system of homogenous linear equations is obtained. For a fixed geometrical configuration and a specific arrangement of a set of other parameters, the frequencies for which the value of the truncated determinant is zero are numerically computed and the trapped wavenumbers corresponding to those frequencies are obtained by using the dispersion relation. These trapped modes are compared with those for which the lower layer is of infinite depth. We also look into the effect of the variation of the elastic plate parameters on the existence of trapped modes. Significant difference is observed with respect to the existence and also in the pattern of the trapped modes between the present case and the one when the cylinder is placed in an infinite depth lower layer of a two-layer fluid.

A semi-analytical approach is presented to solve three-dimensional dynamic Green’s function for an infinitely extended acoustic field with multiple spheres subjected to the Robin boundary conditions. The multipole expansions of the acoustic field induced by a time-harmonic point source are expanded with spherical wave functions. As an alternative to the complex addition theorem, the multipole expansion is computed in a straightforward way. By taking the finite number of terms, an algebraic system is constructed and is used to obtain Green’s function. This result of one sphere agrees with the available analytical solution. For the case of more than one sphere, the proposed results are verified by the numerical method such as the boundary element method (BEM). It indicates that the present solution is more accurate than that of the BEM and shows a fast convergence. Finally, the parameter study is performed to explore the influences of the exciting frequency of the point source, the surface admittance, the number and the separation of spheres on the dynamic Green’s functions. The proposed results can be applied to solve the acoustic scattering problems and to increase the application of boundary integral equation method in the way of numerical Green’s functions.