Geodesic Solutions Research Articles

A relativistic scalar model for fractional interaction between dark matter and gravity

Abstract In a series of recent papers we put forward a “fractional gravity”
framework striking an intermediate course between a modified gravity theory and an
exotic dark matter (DM) scenario, which envisages the DM component in virialized
halos to feel a non-local interaction mediated by gravity. The remarkable success
of this model in reproducing several aspects of DM phenomenology motivates us to
look for a general relativistic extension. Specifically, we propose a theory, dubbed
Relativistic Scalar Fractional Gravity or RSFG, in which the trace of the DM stressenergy tensor couples to the scalar curvature via a non-local operator constructed with
a fractional power of the d’Alembertian. We derive the field equations starting from
an action principle, and then we investigate their weak field limit, demonstrating that
in the Newtonian approximation the fractional gravity setup of our previous works is
recovered. We compute the first-order post-Newtonian parameter γ and its relation
with weak lensing, showing that although in RSFG the former deviates from its GR
values of unity, the latter is unaffected. We also perform a standard scalar-vectortensor-decomposition of RSFG in the weak field limit, to highlight that gravitational
waves propagate at the speed of light, though also an additional scalar mode becomes
dynamical. Finally, we derive the modified conservation laws of the DM stress energy
tensor in RSFG, showing that a new non-local force emerges, and hence that the DM
fluid deviates from the geodesic solutions of the field equations.

Energy Benefits of Tourist Accommodation Using Geodesic Domes

Over the last decade there has been a proliferation of glamping architecture. This study analyses the energy performance of geodesic domes for use in tourist glamping compared to more conventional prismatic architectural solutions. The energy analysis of geodesic domes applied to this type of singular construction project currently lacks detailed studies that provide conclusions about their relevance and suitability with respect to other types of architecture. The main objective of this research is to demonstrate the energy benefits of tourist accommodations that use geodesic structures compared to those with a simple geometry. A comparative study of a traditional and a geodesic geometry accommodation is carried out, considering that they share the same characteristics and they are built with the same construction solution. An energy simulation of both architectures is carried out by using DesignBuilder software. The most influential strategies, such as Direct Passive Solar Gain, Heating, Natural Ventilation Cooling, Fan-Forced Ventilation Cooling and Window Solar Shading are considered. After demonstrating the greater efficiency of geodesic domes, this study analyses the relevance of subdividing the accommodations into several geodesic dome spaces. The results quantify an energy benefit of 52% for cooling consumption using the geodesic dome solution compared to a traditional prismatic solution.

Analytical solutions of equatorial geodesic motion in Kerr spacetime* *Y. L. is financially supported by the Natural Science Foundation of Shandong Province (ZR2023QA133) and Yantai University (WL22B218). B.S. is Supported by the National Natural Science Foundation of China (12375046) and Beijing University of Agriculture (QJKC-2023032).

The study of Kerr geodesics has a long history, particularly for those occurring within the equatorial plane, which are generally well-understood. However, when compared with the classification introduced by one of the authors [Phys. Rev. D 105, 024075 (2022)], it becomes apparent that certain classes of geodesics, such as trapped orbits, still lack analytical solutions. Thus, in this study, we provide explicit analytical solutions for equatorial timelike geodesics in Kerr spacetime, including solutions of trapped orbits, which capture the characteristics of special geodesics, such as the positions and conserved quantities of circular, bound, and deflecting orbits. Specifically, we determine the precise location at which retrograde orbits undergo a transition from counter-rotating to prograde motion due to the strong gravitational effects near a rotating black hole. Interestingly, the trajectory remains prograde for orbits with negative energy despite the negative angular momentum. Furthermore, we investigate the intriguing phenomenon of deflecting orbits exhibiting an increased number of revolutions around the black hole as the turning point approaches the turning point of the trapped orbit. Additionally, we find that only prograde marginal deflecting geodesics are capable of traversing through the ergoregion. In summary, our findings present explicit solutions for equatorial timelike geodesics and offer insights into the dynamics of particle motion in the vicinity of a rotating black hole.

Circular geodesics in the field of double-charged dilatonic black holes

A non-extreme dilatonic charged (by two “color electric” charges) black hole solution is examined within a four-dimensional gravity model that incorporates two scalar (dilaton) fields and two Abelian vector fields. The scalar and vector fields interact through exponential terms containing two dilatonic coupling vectors. The solution is characterized by a dimensionless parameter a(0<a<2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(0< a < 2)$$\\end{document}, which is a specific function of dilatonic coupling vectors. The paper presents solutions for timelike and null circular geodesics that may play a crucial role in different astrophysical scenarios, including quasinormal modes of various test fields in the eikonal approximation. For a=1/2,1,3/2,2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a = 1/2,1, 3/2,2 $$\\end{document}, the radii of the innermost stable circular orbit are presented and analyzed.

Complete characterization of the orbital shapes of the noncircular Kerr geodesic solutions with circular orbit constants of motion

Complete characterization of the orbital shapes of the noncircular Kerr geodesic solutions with circular orbit constants of motion

Explicit Solutions for Geodetic Problems on the Deformed Sphere as Reference Model for the Geoid

In this article, we consider deformed spheres as a new reference model for the geoid, alternatively to the classical ellipsoidal one. The parametrization of deformed spheres is furnished through the incomplete elliptic integrals. From the other side, the solutions for geodesics on those surfaces are given entirely via elementary analytical functions, contrary to the case of ellipsoids of revolution. We explicitly described algorithms (all necessary computational steps) for the solution of the direct and inverse geodetic problems on the deformed spheres. Finally, we presented a few illustrative numerical solutions of the inverse geodetic problems for two conceptual cases of near and far points. It had turned out that even in the non-optimized case we obtained the good agreement with the predictions of the World Geodetic System 1984's ellipsoidal reference model.

Using the Theory of Functional Connections to Solve Boundary Value Geodesic Problems

This study provides a least-squares-based numerical approach to estimate the boundary value geodesic trajectory and associated parametric velocity on curved surfaces. The approach is based on the Theory of Functional Connections, an analytical framework to perform functional interpolation. Numerical examples are provided for a set of two-dimensional quadrics, including ellipsoid, elliptic hyperboloid, elliptic paraboloid, hyperbolic paraboloid, torus, one-sheeted hyperboloid, Moëbius strips, as well as on a generic surface. The estimated geodesic solutions for the tested surfaces are obtained with residuals at the machine-error level. In principle, the proposed approach can be applied to solve boundary value problems in more complex scenarios, such as on Riemannian manifolds.

Information Geometry, Fluctuations, Non-Equilibrium Thermodynamics, and Geodesics in Complex Systems.

Information theory provides an interdisciplinary method to understand important phenomena in many research fields ranging from astrophysical and laboratory fluids/plasmas to biological systems. In particular, information geometric theory enables us to envision the evolution of non-equilibrium processes in terms of a (dimensionless) distance by quantifying how information unfolds over time as a probability density function (PDF) evolves in time. Here, we discuss some recent developments in information geometric theory focusing on time-dependent dynamic aspects of non-equilibrium processes (e.g., time-varying mean value, time-varying variance, or temperature, etc.) and their thermodynamic and physical/biological implications. We compare different distances between two given PDFs and highlight the importance of a path-dependent distance for a time-dependent PDF. We then discuss the role of the information rate and relative entropy in non-equilibrium thermodynamic relations (entropy production rate, heat flux, dissipated work, non-equilibrium free energy, etc.), and various inequalities among them. Here, is the information length representing the total number of statistically distinguishable states a PDF evolves through over time. We explore the implications of a geodesic solution in information geometry for self-organization and control.

Dynamics of particle near time conformal slowly rotating Kerr black hole

In this work, we have developed the time conformal slowly rotating Kerr black hole using the approximate Noether symmetry approach. It is found that the slowly rotating Kerr black hole admits time conformal factor eεf(t), where ε is very small perturbation and f(t) is an arbitrary function of time. We have obtained the numerical solutions of geodesics for lowly rotating Kerr black hole and time conformal slowly rotating Kerr black hole. We have discussed the neutral and charged particle dynamics moving around time conformal slowly rotating Kerr black hole in the absence/presence of magnetic field. Moreover, we have analyzed graphically the behavior of effective force, effective potential and escape velocity of particles in the absence/presence of magnetic field near the event horizon. We have examined the conditions under which particle can escape from the surrounding of black hole after colliding with another particle. We have found the unstable orbits of particles near event horizon due to the effect of magnetic field and angular momentum.

Deformed contour segment matching for multi-source images

Robust and accurate multi-source matching is a difficult task due to significant nonlinear radiometric differences, background clutter, and geometric deformation in corresponding regions. Motivated by these existing problems, a discriminating yet robust combined descriptor for multi-source image matching, called deformed contour segment similarity (DCSS), is proposed in this work. First, the proposed DCSS, which is constructed by histogram of the combined contour features rather than the commonly used corner point and gradient, presents the accurate correspondence between image pairs and improves the descriptive ability to radiometric differences. Second, the deformed curve is presented via a finite-dimensional matrix Lie group to determine the similarity metric with an explicit geodesic solution. The geodesic distance, which indicates the nearest distance between curves in fluid space, is defined as the weight coefficient of the constructed histogram to enhance the robustness of the descriptor. The proposed algorithm utilizes the holistic contour information for the scoring and ranking of the shape similarity hypothesis, which can effectively reduce the influence of partially missing contours. Finally, a precise bilateral matching rule is used to perform the matching between the corresponding contour segments. Some experiments are carried out on various infrared-visible image data sets. The results demonstrate that the proposed DCSS achieves more robust and accurate matching performance than many popular multi-source image matching methods.

Studying the Global Spatial Randomness of Impact Craters on Mercury, Venus, and the Moon With Geodesic Neighborhood Relationships

AbstractImpact crater records on planetary surfaces are often analyzed for their spatial randomness. Generalized approaches such as the mean second closest neighbor distance (M2CND) and standard deviation of adjacent area (SDAA) are available via a software tool but do not take the influence of the planetary curvature into account in the current implementation. As a result, the measurements are affected by map distortion effects and can lead to wrong interpretations. This is particularly critical for investigations of global data sets as the level of distortion typically increases with increasing distance from the map projection center. Therefore, we present geodesic solutions to the M2CND and SDAA statistics that can be implemented in future software tools. We apply the improved methods to conduct spatial randomness analyses on global crater data sets on Mercury, Venus, and the Moon and compare the results to known crater population variations and surface evolution scenarios. On Mercury, we find that the emplacement of smooth plain deposits strongly contributed to a global clustering of craters and that a random distribution of Mercury's basins is not rejected. On Venus, the randomness analyses show that craters are largely randomly distributed across all sizes but where local nonrandom distributions due to lower crater densities in regions of recent volcanic activity may appear. On the Moon, the global clustering of craters is more pronounced than on Mercury due to mare volcanism and the Orientale impact event. Furthermore, a random distribution of lunar basins is not rejected.

Gravitational redshift/blueshift of light emitted by geodesic test particles, frame-dragging and pericentre-shift effects, in the Kerr\u2013Newman\u2013de Sitter and Kerr\u2013Newman black hole geometries

We investigate the redshift and blueshift of light emitted by timelike geodesic particles in orbits around a Kerr–Newman–(anti) de Sitter (KN(a)dS) black hole. Specifically we compute the redshift and blueshift of photons that are emitted by geodesic massive particles and travel along null geodesics towards a distant observer-located at a finite distance from the KN(a)dS black hole. For this purpose we use the killing-vector formalism and the associated first integrals-constants of motion. We consider in detail stable timelike equatorial circular orbits of stars and express their corresponding redshift/blueshift in terms of the metric physical black hole parameters (angular momentum per unit mass, mass, electric charge and the cosmological constant) and the orbital radii of both the emitter star and the distant observer. These radii are linked through the constants of motion along the null geodesics followed by the photons since their emission until their detection and as a result we get closed form analytic expressions for the orbital radius of the observer in terms of the emitter radius, and the black hole parameters. In addition, we compute exact analytic expressions for the frame dragging of timelike spherical orbits in the KN(a)dS spacetime in terms of multivariable generalised hypergeometric functions of Lauricella and Appell. We apply our exact solutions of timelike non-spherical polar KN geodesics for the computation of frame-dragging, pericentre-shift, orbital period for the orbits of S2 and S14 stars within the 1^{prime prime } of SgrA*. We solve the conditions for timelike spherical orbits in KN(a)dS and KN spacetimes. We present new, elegant compact forms for the parameters of these orbits. Last but not least we derive a very elegant and novel exact formula for the periapsis advance for a test particle in a non-spherical polar orbit in KNdS black hole spacetime in terms of Jacobi’s elliptic function sn and Lauricella’s hypergeometric function F_D.

Mechanics of incompressible test bodies moving in Riemannian spaces

In the present paper, we have discussed the mechanics of incompressible test bodies moving in Riemannian spaces with nontrivial curvature tensors. For Hamilton's equations of motion, the solutions have been obtained in the parametrical form and the special case of the purely gyroscopic motion on the sphere has been discussed. For the geodetic case when the potential is equal to zero, the comparison between the geodetic and geodesic solutions has been done and illustrated in details for the case of a particular choice of the constants of motion of the problem. The obtained results could be applied, among others, in geophysical problems, for example, for description of the movement of continental plates or the motion of a drop of fat or a spot of oil on the surface of the ocean (e.g., produced during some “ecological disaster”), or generally in biomechanical problems, for example, for description of the motion of objects with internal structure on different curved two‐dimensional surfaces (e.g., transport of proteins along the curved biological membranes).

Efficiency for variational control problems on Riemann manifolds with geodesic quasiinvex curvilinear integral functionals

In this paper, we formulate and prove necessary and sufficient conditions of geodesic efficiency associated with a new class of multiobjective fractional variational control problems governed by geodesic quasiinvex path-independent curvilinear integral functionals and mixed constraints involving first order PDE of m-flow type. Under $$ \displaystyle (\rho , b) $$ -geodesic quasiinvexity assumptions, by using the new notion of (normal) geodesic efficient solution, we set sufficient conditions of geodesic efficiency for a feasible solution in the considered variational control problems.

Global existence of smooth solutions for wave maps in de Sitter spacetime

In this paper, we study the problem of a harmonic map from de Sitter spacetime into a geodesic on an n-dimensional complete Riemannian manifold. We show that the global smooth solution exists for such wave map when the solution is a small perturbation along the geodesic solution by the method of weighted energy estimates.

On a New Class of Vector Variational Control Problems

In this paper, we introduce geodesic (strongly) b-V-KT-pseudoinvex multidimensional control problems. This new class of multiobjective variational control problems, involving multiple integral cost functionals, is described such that every geodesic Kuhn–Tucker point is geodesic efficient solution. In addition, to illustrate the effectiveness of our main result, the paper is completed with an application.

Intersection and point-to-line solutions for geodesics on the ellipsoid

The paper presents two algorithms for the computation of intersection of geodesics and minimum distance from a point to a geodesic on the ellipsoid, respectively. They are based on the iterative use of direct and inverse problems of geodesy by means of their implementations with machine-precision accuracy in GeographicLib. The algorithms yield the same results as those obtained by Karney’s approach based on the use of auxiliary ellipsoidal gnomonic projections, with the advantage on our side that the algorithms are not limited to distances below 10000 km. This results in our algorithm being the only general solution for the problem of minimum distance from a point to a geodesic on the ellipsoid.

A Time–Space Symmetry Based Cylindrical Model for Quantum Mechanical Interpretations

Following a bi-cylindrical model of geometrical dynamics, our study shows that a 6D-gravitational equation leads to geodesic description in an extended symmetrical time–space, which fits Hubble-like expansion on a microscopic scale. As a duality, the geodesic solution is mathematically equivalent to the basic Klein–Gordon–Fock equations of free massive elementary particles, in particular, the squared Dirac equations of leptons. The quantum indeterminism is proved to have originated from space–time curvatures. Interpretation of some important issues of quantum mechanical reality is carried out in comparison with the 5D space–time–matter theory. A solution of lepton mass hierarchy is proposed by extending to higher dimensional curvatures of time-like hyper-spherical surfaces than one of the cylindrical dynamical geometry. In a result, the reasonable charged lepton mass ratios have been calculated, which would be tested experimentally.

Geometric structure and information change in phase transitions.

We propose a toy model for a cyclic order-disorder transition and introduce a geometric methodology to understand stochastic processes involved in transitions. Specifically, our model consists of a pair of forward and backward processes (FPs and BPs) for the emergence and disappearance of a structure in a stochastic environment. We calculate time-dependent probability density functions (PDFs) and the information length L, which is the total number of different states that a system undergoes during the transition. Time-dependent PDFs during transient relaxation exhibit strikingly different behavior in FPs and BPs. In particular, FPs driven by instability undergo the broadening of the PDF with a large increase in fluctuations before the transition to the ordered state accompanied by narrowing the PDF width. During this stage, we identify an interesting geodesic solution accompanied by the self-regulation between the growth and nonlinear damping where the time scale τ of information change is constant in time, independent of the strength of the stochastic noise. In comparison, BPs are mainly driven by the macroscopic motion due to the movement of the PDF peak. The total information length L between initial and final states is much larger in BPs than in FPs, increasing linearly with the deviation γ of a control parameter from the critical state in BPs while increasing logarithmically with γ in FPs. L scales as |lnD| and D^{-1/2} in FPs and BPs, respectively, where D measures the strength of the stochastic forcing. These differing scalings with γ and D suggest a great utility of L in capturing different underlying processes, specifically, diffusion vs advection in phase transition by geometry. We discuss physical origins of these scalings and comment on implications of our results for bistable systems undergoing repeated order-disorder transitions (e.g., fitness).

Deformation Based Curved Shape Representation.

In this paper, we introduce a deformation based representation space for curved shapes in . Given an ordered set of points sampled from a curved shape, the proposed method represents the set as an element of a finite dimensional matrix Lie group. Variation due to scale and location are filtered in a preprocessing stage, while shapes that vary only in rotation are identified by an equivalence relationship. The use of a finite dimensional matrix Lie group leads to a similarity metric with an explicit geodesic solution. Subsequently, we discuss some of the properties of the metric and its relationship with a deformation by least action. Furthermore, invariance to reparametrization or estimation of point correspondence between shapes is formulated as an estimation of sampling function. Thereafter, two possible approaches are presented to solve the point correspondence estimation problem. Finally, we propose an adaptation of k-means clustering for shape analysis in the proposed representation space. Experimental results show that the proposed representation is robust to uninformative cues, e.g., local shape perturbation and displacement. In comparison to state of the art methods, it achieves a high precision on the Swedish and the Flavia leaf datasets and a comparable result on MPEG-7, Kimia99 and Kimia216 datasets.