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We consider the minimal energy problem on the unit sphere \({\mathbb {S}}^2\) in the Euclidean space \({\mathbb {R}}^3\) immersed in an external field Q, where the charges are assumed to interact via Newtonian potential 1/r, r being the Euclidean distance. The problem is solved by finding the support of the extremal measure, and obtaining an explicit expression for the equilibrium density. We then apply our results to an external field generated by a point charge, and to a quadratic external field.

In this article we obtain the Coulombian potential of spinor and scalar particle with the quantum field theory, there is no difference between the spinor and scalar particle. Similarly the Newtonian potential of spinor particle and scalar particle are also obtained with the quantum field theory, however it is found that the Newtonian potential of scalar particle is half of the Newtonian potential of spinor particle with same mass, it contradicts with the equality of gravitational and inertial mass. The Newtionian potential of the spinor particle can be used to obtain the Newtonian potentials of the sun,the earth and so on, therefore we suggest that it should be considered to be the experimental test of the gravitational theory of Einstein.

We analyze the existence of equilibrium points for a particle or dust grain in the framework of unperturbed and perturbed Robe’s motion. This particle is moving in a spherical nebula consisting of a homogeneous incompressible fluid, which is considered as the primary body. The second primary body creates the modified Newtonian potential. The perturbed mean motion and equations of motion are found. The equilibrium points (i.e. collinear, noncollinear and out–of–plane points), along with the required conditions of their existence are also analyzed. We emphasize that this analysis can be used to study the oscillations of the Earth’s core under the attraction of the Moon and it is also applicable to study the motion of underwater vehicles.

The generation and detection of entanglement between mesoscopic systems would have major fundamental and applicative implications. In this work, we demonstrate the utility of continuous variable tools to evaluate the Gaussian entanglement arising between two homogeneous levitated nanobeads interacting through a central potential. We compute the entanglement for the steady state and determine the measurement precision required to detect the entanglement in the laboratory.

The gravitational theory including spinor particle is formulated with the suggestion of tetrad fields [Formula: see text] by Utiyama, Kibble and Weinberg.[Formula: see text] We might take the metric [Formula: see text] as a constraint, that is used the method of Lagrange multiplier. In the weak-field approximation we calculate the metric of a static Dirac spinor particle proton and obtain that the metric is a diagonal [Formula: see text] which coincides with the anticipative result of Newtonian Gravitational potential.

We study the properties of the well-known Newtonian potential operator including its domain, codomain, compactness, relation with the Poisson’s equation and the spectral radii of the weight Newtonian potential operators. These new properties are then applied to study the existence of positive classical solutions of nonlinear Poisson’s equations in bounded open sets, which are required neither to be connected nor smooth on their boundaries. The connectedness and smoothness are common requirements for studying the existence of classical or weak solutions of linear or nonlinear elliptic boundary value problems in the literature. Some applications to population models are provided.

The growing availability of increasingly accurate data on transiting exoplanets suggests the possibility of using these systems as possible testbeds for modified models of gravity. In particular, we suggest that the post-Keplerian (pK) dynamical effects from the perturbations of the Newtonian potential falling off as the square or the cube of the distance from the mass of the host star break the degeneracy of the anomalistic, draconitic and sidereal periods. The latter are characteristic temporal intervals in the motion of a binary system, and all coincide in the purely Keplerian case. We work out their analytical expressions in presence of the aforementioned perturbations to yield preliminary insights on the potential of the method proposed for constraining the modified models of gravity considered. A comparison with other results existing in the literature is made., 6 pages, references added; revised to match the version accepted for publication in JCAP

We characterize functions $$V\le 0$$ for which the heat kernel of the Schrodinger operator $$\Delta +V$$ is comparable with the Gauss–Weierstrass kernel uniformly in space and time. In dimension 4 and higher the condition turns out to be more restrictive than the condition of the boundedness of the Newtonian potential of V. This resolves the question of V. Liskevich and Y. Semenov posed in 1998. We also give specialized sufficient conditions for the comparability, showing that local $$L^p$$ integrability of V for $$p>1$$ is not necessary for the comparability.

Ongoing and upcoming cosmological surveys will significantly improve our ability to probe the equation of state of dark energy, ${w}_{\mathrm{DE}}$, and the phenomenology of large scale structure. They will allow us to constrain deviations from the $\mathrm{\ensuremath{\Lambda}}$ cold dark matter predictions for the relations between the matter density contrast and the weak lensing and the Newtonian potential, described by the functions $\mathrm{\ensuremath{\Sigma}}$ and $\ensuremath{\mu}$, respectively. In this work, we derive the theoretical prior for the joint covariance of ${w}_{\mathrm{DE}}$, $\mathrm{\ensuremath{\Sigma}}$ and $\ensuremath{\mu}$, expected in general scalar-tensor theories with second order equations of motion (Horndeski gravity), focusing on their time-dependence at certain representative scales. We employ Monte Carlo methods to generate large ensembles of statistically independent Horndeski models, focusing on those that are physically viable and in broad agreement with local tests of gravity, the observed cosmic expansion history and the measurement of the speed of gravitational waves from a binary neutron star merger. We identify several interesting features and trends in the distribution functions of ${w}_{\mathrm{DE}}$, $\mathrm{\ensuremath{\Sigma}}$ and $\ensuremath{\mu}$, as well as in their covariances; we confirm the high degree of correlation between $\mathrm{\ensuremath{\Sigma}}$ and $\ensuremath{\mu}$ in scalar-tensor theories. The derived prior covariance matrices will allow us to reconstruct jointly ${w}_{\mathrm{DE}}$, $\mathrm{\ensuremath{\Sigma}}$ and $\ensuremath{\mu}$ in a nonparametric way.

In this paper, we verify the large scale structure consistency relations using N-body simulations, including modes in the highly non-linear regime. These relations (pointed out by Kehagias & Riotto and Peloso & Pietroni) follow from the symmetry of the dynamics under a shift of the Newtonian potential by a constant and a linear gradient, and predict the absence of certain poles in the ratio between the (equal time) squeezed bispectrum and power spectrum. The consistency relations, as symmetry statements, are exact, but have not been previously checked beyond the perturbative regime. Our test using N-body simulations not only offers a non-perturbative check, but also serves as a warm-up exercise for applications to observational data. A number of subtleties arise when taking the squeezed limit of the bispectrum--we show how to circumvent or address them. An interesting by-product of our investigation is an explicit demonstration that the linear-gradient symmetry is unaffected by the periodic boundary condition of the simulations. Lastly, we verify using simulations that the consistency relations are violated when the initial conditions are non-gaussian (of the local fNL type). The methodology developed here paves the way for constraining primordial non-gaussianity using large scale structure data, including (numerous) highly non-linear modes that are otherwise hard to interpret and utilize., 10 pages, 5 figures, 1 table

We investigate the nature of additional scalar degrees of freedom contained in extended hybrid metric-Palatini gravity, outlining the emergence of two coupled dynamical scalar modes. In particular, we discuss the weak field limit of the theory, both in the static case and from a gravitational waves perspective. In the first case, performing an analysis at the lowest order of the post parameterized Newtonian (PPN) structure of the model, we stress the settling of Yukawa corrections to the Newtonian potential. In this respect, we show that one scalar field can have long range interactions and used in the principle for mimicking dark matter effects. Concerning the gravitational waves propagation, instead, we demonstrate that is possible to have well-defined physical degrees of freedom, provided by suitable constraints on model parameters. Moreover, the study of the geodesic deviation points out the presence of breathing and longitudinal polarizations due to these novel scalar waves, which on peculiar assumptions can give rise to beating phenomena during their propagation., 14 pages, 1 figure. Comments are welcome

We investigate in detail the interaction between the spin-${1/2}$ fields endowed with mass dimension one and the graviton. We obtain an interaction vertex that combines the characteristics of scalar-graviton and Dirac's fermion-graviton vertices, due to the scalar-dynamic attribute and the fermionic structure of this field. It is shown that the vertex obtained obeys the Ward-Takahashi identity, ensuring the gauge invariance for this interaction. In the contribution of the mass dimension one fermion to the graviton propagator at one-loop, we found the conditions for the cancellation of the tadpole term by a cosmological counter-term. We calculate the scattering process for arbitrary momentum. For low energies, the result reveals that only the scalar sector present in the vertex contributes to the gravitational potential. Finally, we evaluate the non relativistic limit of the gravitational interaction and obtain an attractive Newtonian potential, as required for a dark matter candidate., Comment: 8 pages, 3 figures

We use the planetary ephemeris INPOP17b to constrain the mass of the graviton in the Newtonian limit. We also give an interpretation of this result for a specific case of fifth force framework. We find that the residuals for the Cassini spacecraft significantly (90\% C.L.) degrade for Compton wavelengths of the graviton smaller than $1.83\times 10^{13}$ km, corresponding to a graviton mass bigger than $6.76\times 10^{-23} eV/c^2$. This limit is comparable in magnitude to the one obtained by the LIGO-Virgo collaboration in the radiative regime. We also use this specific example to illustrate why constraints on alternative theories of gravity obtained from postfit residuals are generically overestimated., Comment: New version. Technical details moved into Supplemental Materials. Author Pierre Deram added. Interpretation in terms of fifth force added. Minor corrections. 8 pages. 4 figures. 2 tables

Context. Ground-based astro-geodetic observations and atmospheric radio occultations are two examples of observational techniques requiring a scrutiny analysis of atmospheric refraction. In both cases, the measured changes of the observables are geometrically related to changes in the photon path through the refractive profile of the crossed medium. Therefore, having a clear knowledge of how the refractivity governs the photon path evolution is of prime importance to clearly understand observational features. Aims. We analytically performed the integration of the photon path and the light time of rays traveling across a non-spherically symmetric planetary atmosphere. Methods. Assuming that the atmospheric refraction evolves linearly with the Newtonian potential, we derived an exact solution to the equations of geometrical optics. By varying the solution’s arbitrary constants of integration, we reformulated the equation of geometrical optics into a new set of osculating equations describing the constants’ evolution following any changes in the refractive profile. We have highlighted the capabilities of the formalism, carrying out five realistic applications in which we derived analytical expressions. Finally, we assessed the accuracy by comparing the solution to results from a numerical integration of the equations of geometrical optics in the presence of a quadrupolar moment (J2). Results. Analytical expressions for the light time and the refractive bending are given with relative errors at the level of one part in 108 and one part in 105, for typical values of the refractivity and J2 at levels of 10−4 and 10−2, respectively. Conclusions. The establishment of the osculating equations for the ray propagation has two main advantages. Firstly, it provides an easy and comprehensive geometrical picture for interpreting the photon path. Secondly, it allows the analytical solving of the ray propagation in the presence of non-radial dependencies in the refractive profile.

60 pags., 5 figs., 5 tabs., We investigate possible signatures of quantum gravity which could be tested with current and future gravitational-wave (GW) observations. In particular, we analyze how quantum gravity can influence the GW luminosity distance, the time dependence of the effective Planck mass and the instrumental strain noise of interferometers. Using both model-dependent and model-independent formulæ, we show that these quantities can encode a non-perturbative effect typical of all quantum-gravity theories, namely the way the dimension of spacetime changes with the probed scale. Effects associated with such dimensional flow might be tested with GW observations and constrained significantly in theories with a microscopically discrete spacetime geometry, more strongly than from propagation-speed constraints. Making use of public LIGO data as well as of a simulated higher-redshift LISA source, we impose the first, respectively, actual and mock constraints on quantum-gravity parameters affecting the GW luminosity distance and discuss specific theoretical examples. If also the Newtonian potential is modified but light geodesics are not, then solar-system bounds may be stronger than GW ones. Yet, for some theories GW astronomy can give unique information not available from solar-system tests., G.C. and S.K. are supported by the I+D grant FIS2017-86497-C2-2-P of the Spanish Ministry of Science, Innovation and Universities. S.K. is supported by JSPS KAKENHI No. 17K14282 and Career Development Project for Researchers of Allied Universities. M.S. is supported in part by the Science and Technology Facility Council (STFC), United Kingdom, under the research grant ST/P000258/1. G.T. is partially supported by STFC grant ST/P00055X/1.

It is shown here in the framework of standard General Relativity that the gravitational potential in static spacetime, equivalently the redshift factor, inside any kind of matter, can be derived from maximum entropy principle. It is used only the Hamiltonian constraint, without further invoking Einstein's equations or any new principle. The Newtonian potential arises from the same procedure., Comment: Accepted in CQG

We consider minimization problems of the Thomas–Fermi–Dirac–von Weizsacker (TFDW) type in which the Newtonian potential is perturbed by a background potential satisfying mild conditions, which ensures the existence of minimizers. We describe the structure of minimizing sequences for those variants and obtain a more precise characterization of patterns in minimizing sequences for the TFDW functionals regularized by long-range perturbations.We consider minimization problems of the Thomas–Fermi–Dirac–von Weizsacker (TFDW) type in which the Newtonian potential is perturbed by a background potential satisfying mild conditions, which ensures the existence of minimizers. We describe the structure of minimizing sequences for those variants and obtain a more precise characterization of patterns in minimizing sequences for the TFDW functionals regularized by long-range perturbations.

The continuity of the Laplacian operator of a function u : Ω ⊂ Rn −→ R does not guarantee that the function is of class C 2 (Ω). However, Schauder’s regularity theory will assure us that if Δu is α-H¨older continuous, then u will be locally C 2,α (Ω), this is, each second-order partial derivative ∂ 2 u ∂xixj , 1 ≤ i, j ≤ n, will be C 0,α loc. In this dissertation, we will establish Schauder’s Theory C 2,α by three methods: by the Maximum Principle, by the Method of Newton’s Compassion and by Potential. In each method, we work on its peculiarities to obtain the desired estimates. Some techniques that we will cover here can be extended to more general operators. On the other hand, all the central difficulties already appear when we consider the Laplacian operator (Poisson equation). For this reason, this is the operator we will consider throughout the work. A continuidade do operador Laplaciano aplicado em uma fun¸c˜ao u : Ω ⊂ Rn −→ R não garante que a função seja de classe C 2 (Ω). Todavia, a teoria de regularidade de Schauder nos garantirá que se Δu for α-Hölder contínuo, então u será localmente C 2,α (Ω), isto é, cada derivada parcial de segunda ordem ∂ 2 u ∂xixj , 1 ≤ i, j ≤ n, será C 0,α loc. Nesta dissertação estabeleceremos a Teoria C 2,α de Schauder por três métodos: via Princípio do Máximo, pelo Método da Compacidade e Potencial Newtoniano. Em cada método trabalhamos suas peculiaridades para obtermos as estimativas almejadas. Algumas técnicas que abordaremos aqui podem ser estendidas para operadores mais gerais. No entanto, todas as dificuldades centrais já aparecem quando consideramos o operador Laplaciano (Equação de Poisson). Por esta razão este é o operador que consideraremos em toda a extensão do trabalho.

The common nature of the dark sector - dark energy and dark matter - as shown in [1] follows readily from the consideration of generalized Newtonian potential as a weak-field General Relativity. That generalized potential satisfying the Newton's theorem on the equivalence of sphere's gravity and that of a point-mass located in its center, contains an additional constant which along with the gravitational constant is able to explain quantitatively both the dark energy (cosmological constant) and dark matter. So, gravity is defined not by one but two fundamental constants. We show that, the second constant is dimensional-independent and matter-uncoupled and hence is even more universal than the gravitational constant, thus affecting the strategy of observational studies of dark energy and of the search of dark matter., Comment: To appear in Eur Phys J C; 5 pages

PyUltraLight simulates the dynamics of ultralight dark matter in a non-expanding background. PyUltraLight can describe the evolution of several interacting ultralight dark matter halos or one or more halos orbiting a central, fixed Newtonian potential, the latter scenario corresponding to dwarf galaxies orbiting a massive central galaxy. We verify PyUltraLight by showing that it reproduces qualitative dynamical features of previously published simulations and demonstrate that it has excellent energy-conservation properties. PyUltraLight is implemented in a Python-based Jupyter notebook, solving the Schrodinger-Poisson equation governing ultralight scalar field dark matter dynamics in the non-relativistic regime using a symmetrised split-step pseudospectral algorithm. The notebook interface makes it simple to specify simulation parameters and visualise the resulting output but performance-critical routines are managed via calls to computationally efficient compiled libraries. PyUltraLight runs on standard desktop hardware with support for shared memory mutlithreading and is available on GitHub., available at: https://github.com/auckland-cosmo/PyUltraLight

The Newton-Raphson basins of convergence, corresponding to the coplanar libration points (which act as attractors), are unveiled in the Copenhagen problem, where instead of the Newtonian potential and forces, a quasi-homogeneous potential created by two primaries is considered. The multivariate version of the Newton-Raphson iterative scheme is used to reveal the attracting domain associated with the libration points on various type of two-dimensional configuration planes. The correlations between the basins of convergence and the corresponding required number of iterations are also presented and discussed in detail. The present numerical analysis reveals that the evolution of the attracting domains in this dynamical system is very complicated, however, it is a worth studying issue., Published in International Journal of Non-Linear Mechanics (IJNLM)

Alternative theories of gravity may serve to overcome several shortcomings of the standard cosmological model but, in their weak field limit, general relativity must be recovered so as to match the tight constraints at the Solar System scale. Therefore, testing such alternative models at scales of stellar systems could give a unique opportunity to confirm or rule them out. One of the most straightforward modifications is represented by analytical f(R)-gravity models that introduce a Yukawa-like modification to the Newtonian potential thus modifying the dynamics of particles. Using the geodesics equations, we have illustrated the amplitude of these modifications. First, we have integrated numerically the equations of motion showing the orbital precession of a particle around a massive object. Second, we have computed an analytic expression for the periastron advance of systems having their semimajor axis much shorter than the Yukawa-scale length. Finally, we have extended our results to the case of a binary system composed of two massive objects. Our analysis provides a powerful tool to obtain constraints on the underlying theory of gravity using current and forthcoming data sets.

The Newtonian potential, or the potential of volume, associated to the Laplace equation \(\Delta u=0\) is, by definition, the following improper integral: $$\begin{aligned} U(\xi )=-\frac{1}{(n-2)\omega _n}\int _{\Omega }\frac{f(x)}{r^{n-2}}\mathrm {d}x, \end{aligned}$$ where \(r=r_{\xi x}=\left| \overline{\xi x}\right| =\sqrt{ \sum \limits _{i=1}^n\left( x_i-\xi _i\right) ^2}\), and \(\omega _n\) is the area of the unit sphere from the n-dimensional space \(\mathrm{I}\!\mathrm{R}^n\).

In the standard approach to studying cosmological structure formation, the overall expansion of the Universe is assumed to be homogeneous, with the gravitational effect of inhomogeneities encoded entirely in a Newtonian potential. A topic of ongoing debate is to what degree this fully captures the dynamics dictated by general relativity, especially in the era of precision cosmology. To quantitatively assess this, we directly compare standard N-body Newtonian calculations to full numerical solutions of the Einstein equations, for cold matter with various magnitude initial inhomogeneities on scales comparable to the Hubble horizon. We analyze the differences in the evolution of density, luminosity distance, and other quantities defined with respect to fiducial observers. This is carried out by reconstructing the effective spacetime and matter fields dictated by the Newtonian quantities, and by taking care to distinguish effects of numerical resolution. We find that the fully general relativistic and Newtonian calculations show excellent agreement, even well into the nonlinear regime. They only notably differ in regions where the weak gravity assumption breaks down, which arise when considering extreme cases with perturbations exceeding standard values., Comment: 17 pages, 14 figures; revised to match PRD version

An algorithm is presented for numerical computation of choreographies in the plane in a Newtonian potential and on the sphere in a cotangent potential. It is based on stereographic projection, approximation by trigonometric polynomials, and quasi-Newton and Newton optimization methods with exact gradient and exact Hessian matrix. New choreographies on the sphere are presented.

We prove that, for generic (open and dense) values of the masses, the Newtonian potential function of the collinear N-body problem has critical values when restricted to a fixed inertia level. In particular, we prove that for generic values of the masses, there is only one global minimal Moulton configuration.

We investigate a potential with a radially symmetric and strictly decreasing kernel depending on a parameter. We regard the potential as a function defined on the upper half-space R m × ( 0 , + ∞ ) \mathbb {R}^m \times (0,+\infty ) and study some geometric properties of its spatial maximizer. To be precise, we give some sufficient conditions for the uniqueness of a maximizer of the potential and study the asymptotic behavior of the set of maximizers. Using these results, we imply geometric properties of some specific potentials. In particular, we consider applications for the solution of the Cauchy problem for the heat equation, the Poisson integral (including a solid angle) and r α − m r^{\alpha -m} -potentials.

A simple existence criterion of inflection points of the potential in nonuniform spherical bodies is formulated. The criterion states that geometric loci of inflection points of the potential appear not only at density discontinuities but also at locations where the density ρ(r) is two-thirds of the average matter density $$\overline {\rho (r)}$$ inside a sphere of a specified radius. The criterion is universal and is fulfilled as for bodies with a continuous density distribution, and for bodies consisting of layers of finite thickness, and also in mixed models. The method of separation of extreme points method of separation of inflection points is given. The criterion is tested on many models including isothermal, polytropic, and isochronous spheres. The family of models within which there are no inflection points is specified. Testing of the method on the Earth’s model confirms its adequacy. A lower limit for the axial (polar) moment of inertia of a spherical body $$\frac{{I_{cr} }} {{M R^2 }} \approx 0.360$$ that separates planets and moons of the solar system into two groups is also obtained using the 3/2 criterion. The first and the largest group includes celestial bodies that have internal extreme points of the attraction force. The second group consists of planets and satellites that have no internal extreme points. These are the Moon, Io, Phobos, and, significantly, Mars.

The existence and stability of circular orbits (CO) in static and spherically symmetric (SSS) spacetime are important because of their practical and potential usefulness. In this paper, using the fixed point method, we first prove a necessary and sufficient condition on the metric function for the existence of timelike COs in SSS spacetimes. After analyzing the asymptotic behavior of the metric, we then show that asymptotic flat SSS spacetime that corresponds to a negative Newtonian potential at large $r$ will always allow the existence of CO. The stability of the CO in a general SSS spacetime is then studied using the Lyapunov exponent method. Two sufficient conditions on the (in)stability of the COs are obtained. For null geodesics, a sufficient condition on the metric function for the (in)stability of null CO is also obtained. We then illustrate one powerful application of these results by showing that an SU(2) Yang-Mills-Einstein SSS spacetime whose metric function is not known, will allow the existence of timelike COs. We also used our results to assert the existence and (in)stabilities of a number of known SSS metrics., 17 pages, no figure; references adjusted; error corrected

Particle trajectories in different Spherically Symmetric static space-time metrics are studied employing Pseudo Newtonian potential (PNP).

We develop an approach to parametrize cosmological perturbations beyond linear order for general dark energy and modified gravity models characterized by a single scalar degree of freedom. We derive the full nonlinear action, focusing on Horndeski theories. In the quasi-static, non-relativistic limit, there are a total of six independent relevant operators, three of which start at nonlinear order. The new nonlinear couplings modify, beyond linear order, the generalized Poisson equation relating the Newtonian potential to the matter density contrast. We derive this equation up to cubic order in perturbations and, in a companion article, we apply it to compute the one-loop matter power spectrum. Within this approach, we also discuss the Vainshtein regime around spherical sources and the relation between the Vainshtein scale and the nonlinear scale for structure formation., v3: typo fixed

We focus on several models of modified gravity which share the characteristic of leading to perturbations of the Newtonian potential $\propto K_2~r^{-2}$ and $\propto K_3~r^{-3}$. In particular, by using existing long data records of the LAGEOS satellites, tracked on an almost continuous basis with the Satellite Laser Ranging (SLR) technique, we set preliminary constraints on the free parameters $K_2,~K_3$ in a model-independent, phenomenological way. We obtain $\left|K_2\right|\lesssim 2.1\times 10^6~\textrm{m}^4~\textrm{s}^{-2},~ -2.5\times 10^{12}~\textrm{m}^5~\textrm{s}^{-2}\lesssim K_3 \lesssim 4.1\times 10^{12}~\textrm{m}^5~\textrm{s}^{-2}.$ They are several orders of magnitude tighter than corresponding bounds existing in the literature inferred with different techniques and in other astronomical and astrophysical scenarios. Then, we specialize them to the different parameters characterizing the various models considered. The availability of SLR data records of increasing length and accuracy will allow to further refine and strengthen the present results., Comment: 20 pages, 4 figures; revised to match the version accepted for publication in JCAP

We give a computer-assisted proof of the full listing of central configuration for n-body problem for Newtonian potential on the plane for $$n=5,6,7$$ n = 5 , 6 , 7 with equal masses. We show all these central configurations have a reflective symmetry with respect to some line. For $$n=8,9,10$$ n = 8 , 9 , 10 , we establish the existence of central configurations without any reflectional symmetry.

A form of infinite derivative gravity is free from ghost-like instabilities with improved small scale behavior. In this theory, we calculate the tree-level scattering amplitude and the corresponding weak field potential energy between two localized covariantly conserved spinning point-like sources that also have velocities and orbital motion. We show that the spin-spin and spin-orbit interactions take the same form as in Einstein's gravity at large separations, whereas at small separations, the results are different. We find that not only the usual Newtonian potential energy but also the spin-spin and spin-orbit interaction terms in the potential energy are non-singular as one approaches r->0., 11 pages, references added, version accepted for publication in PRD

In this paper, using the boundary conditions of the Newton potential, the wave potential and surface wave potentials in the mixed region D ⊂ R2 the Bitsadze-Samarskii boundary condition for the Lavrent’ev-Bitsadze equation is studied. Theorem on existence and uniqueness of the solution of the considered problem is proved by method of the theory of potential.

We consider a 3D quantum system of $N$ identical bosons in a trapping potential $|x|^p$, with $p\geq0$, interacting via a Newton potential with an attractive interaction strength $a_{N}$. For a fixed large $N$ and the coupling constant $a_{N}$ smaller than a critical value $a_{*}$ (Chandrasekhar limit mass), in an appropriate sense, the many-body system admits a ground state. We investigate the blow-up behavior of the ground state energy as well as the ground states when $a_{N}$ approaches $a_{*}$ sufficiently slowly in the limit $N\to\infty$. The blow-up profile is given by the Gagliardo-Nirenberg solutions., Comment: To appear in Reviews in Mathematical Physics

The gravitational potential of (a part of) the Earth is assumed to be available, the inverse gravimetry problem is to determine the density contrast function inside (the specified part of) the Earth from known potential values. This paper deals with the characteristic ill-posed features of transferring input gravitational information in the form of Newtonian volume integral values to geological output characteristics of the density contrast function. Some properties of the Newton volume integral are recapitulated. Different methodologies of the resolution of the inverse gravimetry problem and their numerical implementations are examined dependent on the data source. Three cases of input information may be distinguished, namely internal (borehole), terrestrial (surface), and/or external (spaceborne) gravitational data sets. Singular integral theory based inversion of the Newtonian integral equation such as Haar-type solutions are proposed in a multiscale framework to decorrelate specific geological signal signatures with respect to inherently available features. Reproducing kernel Hilbert space regularization techniques are studied (together with their transition to mollified variants) to provide geological contrast density distributions by “downward continuation” from terrestrial and/or spaceborne data. Finally, reproducing kernel Hilbert space solutions are formulated for use of gravimeter data, independent of a specifically chosen input area, i.e., in whole Euclidean space \(\mathbb{R}^3\)

We consider a class of L 1 critical nonlocal aggregation equations with linear or nonlinear porous media-type diffusion which are characterized by a long-range interaction potential that decays faster than the Newtonian potential at infinity. The fast decay breaks the L 1 scaling symmetry and we prove that all ‘sufficiently spread out’ initial data, even with supercritical mass, results in global, decaying solutions. In particular, we produce decaying solutions with arbitrary mass in cases for which finite time blow up solutions or non-decaying solutions are also known to exist for sufficiently large mass. This is in contrast to the classical parabolic-elliptic PKS for which essentially all solutions with supercritical mass blow up in finite time. The results with linear diffusion are proved using properties of the Fokker-Planck semi-group whereas the results with nonlinear diffusion are proved using a more interesting bootstrap argument coupling the entropy-entropy dissipation methods of the porous media...

Gravitomagnetism is universally and formally recognised in contemporary physics as being the linear first-order approximation of Einstein’s field equations emerging from the General Theory of Relativity (GTR). Herein, we argue that, as has been done by others in the past, gravitomagnetism can be viewed as a fully-fledged independent theory of gravitomagnetism that can be divorced from Professor Einstein’s GTR. The gravitomagnetic theory whose exposition we give herein is exactly envisioned by Professor Maxwell and Dr. Heaviside. The once speculative Maxwell-Heaviside Gravitomagnetic theory now finds full justification as a fully fledged theory from Professor Jose Hera’s Existence Theorem which states that all that is needed for there to exist the four Max-well-type field equations is that a mass-current conservation law be obeyed. Our contribution in the present work, if any, is that we demonstrate conclusively that like electromagnetism, the gravitomagnetic phenomenon leads to the prediction of gravitomagnetic waves that travel at the speed of light. Further, we argue that for the gravitational phenomenon, apart from the Newtonian gravitational potential, there are four more potentials and these operate concurrently with the Newtonian potential. At the end of it, it is seen that the present work sets the stage for a very interesting investigation of several gravitational anomalies such as the ponderous Pioneer Anomaly, the vexing Flyby Anomalies, the mysterious Anomalous Rotation Curves of Spiral Galaxies and as well, the possibility of the generation of stellar magnetic fields by rotating gravitational masses.

The central part of the nuclear potential energy is shown to depend on the interacting masses of the nuclear matter. This mass dependent potential energy reduces to the usual Newtonian potential energy of the interacting masses when both the interacting masses are more than a certain limiting mass. This strong potential energy results when both the interacting masses are less than the limiting mass. The potential energy is applied to two more systems here and out of which one nucleus is in the middle of periodic table.

We consider a Yukawa-type gravitational potential combined with the Poynting-Robertson effect. Dust particles originating within the asteroid belt and moving on circular and elliptic trajectories are studied and expressions for the time rate of change of their orbital radii and semimajor axes, respectively, are obtained. These expressions are written in terms of basic particle parameters, namely their density and diameter. Then, they are applied to produce expressions for the time required by the dust particles to reach the orbit of Earth. For the Yukawa gravitational potential, dust particles of diameter $10^{ - 3}$ m in circular orbits require times of the order of $8.557 \times 10^{6}$ yr and for elliptic orbits of eccentricities $e =0.1, 0.5$ require times of $9.396 \times 10^{6}$ and $2.129 \times 10^{6}$ yr respectively to reach Earth’s orbit. Finally, various cases of the Yukawa potential are studied and the corresponding particle times to reach Earth’s are derived per case along with numerical results for circular and various elliptical orbits.

A potential theory approach to an algorithm of conceptual space partitioningThis paper proposes a new classification algorithm for the partitioning of a conceptual space. All the algorithms which have been used until now have mostly been based on the theory of Voronoi diagrams. This paper proposes an approach based on potential theory, with the criteria for measuring similarities between objects in the conceptual space being based on the Newtonian potential function. The notion of a fuzzy prototype, which generalizes the previous definition of a prototype, is introduced. Furthermore, the necessary conditions that a natural concept must meet are discussed. Instead of convexity, as proposed by Gärdenfors, the notion of geodesically convex sets is used. Thus, if a concept corresponds to a set which is geodesically convex, it is a natural concept. This definition applies, for example, if the conceptual space is an Euclidean space. As a by-product of the construction of the algorithm, an extension of the conceptual space to d-dimensional Riemannian manifolds is obtained. Algorytm podziału przestrzeni konceptualnych przy użyciu teorii potencjałuW niniejszej pracy zaproponowany został nowy algorytm podziału przestrzeni konceptualnej. Dotąd podział taki zazwyczaj wykorzystywał teorię diagramów Voronoi. Nasze podejście do problemu oparte jest na teorii potencjału Miara podobieństwa pomiędzy elementami przestrzeni konceptualnej bazuje na Newtonowskiej funkcji potencjału. Definiujemy pojęcie rozmytego prototypu, który uogólnia dotychczas stosowane definicje prototypu. Ponadto zajmujemy się warunkiem koniecznym, który musi spełniać naturalny koncept. Zamiast wypukłości zaproponowanej przez Gärdenforsa, rozważamy linie geodezyjne w obszarze odpowiadającym danemu konceptowi naturalnemu, otrzymując warunek mówiący, że koncept jest konceptem naturalnym, jeżeli zbiór odpowiadający temu konceptowi jest geodezyjnie wypukły. Ta definicja pokrywa się w przypadku, gdy przestrzenią konceptualną jest przestrzeń euklidesowa. Jako produkt uboczny konstrukcji naszego algorytmu rozważamy dość ogólne przestrzenie konceptualne będące d-wymiarowymi rozmaitościami Reimanna.

It was proposed that if a higher-derivative gravity is renormalizable it implies necessarily a finite Newtonian potential at the origin, but the reverse of this statement is not true. Here, we show that the reverse is true when taking into account the van Dam-Veltman-Zakharov discontinuity, which states that the theory obtained from the massive one by taking a zero mass limit is not equivalent to the theory obtained in the zero mass case. The surviving degree of freedom in the zero mass limit is an extra scalar that does not affect the light bending angle but affects the Newtonian potential. This asserts that in order to make the singularity cancellation the number of massive ghost and healthy tensors matches with that of massive ghost and healthy scalars.