Spherical Metric Research Articles

Astrophysical Tests of General Relativity

At the initial stage of its development, general relativity (GR) was verified and confirmed in a weak gravitational field limit. However, with the development of astronomical observation technologies, GR predictions in a strong gravitational field began to be discussed and confirmed, such as the profile of the X-ray iron \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K\\alpha $$\\end{document} line (in the case if the emission region is very close to the event horizon), the trajectories of stars near black holes and the shapes and sizes of shadows of supermassive black holes in M87* and Sgr A*. In 2005 it was predicted that a shadow formed near a supermassive black hole at the Galactic Center could be reconstructed from observations of ground based global VLBI system or ground—space interferometer acting in mm or sub-mm bands. In 2022 this prediction was confirmed since the Event Horizon Telescope (EHT) collaboration reported about a shadow reconstructions for Sgr A*. In 2019 the EHT collaboration presented the first image reconstruction around the shadow for the supermassive black hole in M87. In 2021 the EHT collaboration constrained parameters (“charges”) of spherical symmetrical metrics of black holes from an allowed interval for shadow radius. In 2022 the EHT collaboration constrained charges of metrics for the supermassive black hole at the Galactic Center. Earlier, we obtained analytical expressions for the shadow radius as a function of charge (including a tidal one) in the case of Reissner–Nordström metric. Based on results of the shadow size evaluation for M87* done by the EHT collaboration we constrained a tidal charge. We discussed opportunities to use shadows to test alternative theories of gravity and alternative models for galactic centers.

Data-driven Bayesian analysis of marine accidents in the English Channel

The English Channel, renowned for its heavy traffic and challenging conditions, frequently witnesses marine accidents. To understand the causal mechanisms and identify risk factors, this study analyzed 833 accidents from 2017 to 2022, employing a Bayesian network predictive model. Validation using log loss, quadratic loss, and spherical gain metrics confirmed the model’s accuracy. The research elucidates the primary influences on accident occurrence: ship type, tonnage, accident location, and flag state. Ship-related factors surpass environmental ones in significance. Older ships face heightened risks of mechanical failures and collisions, especially in adverse weather with limited visibility. Yachts and fishing vessels are particularly susceptible to fire, explosion, collision, and grounding incidents. The study offers theoretical guidance for management departments to devise prevention strategies and enhance navigation safety in the English Channel.

Spherical warp-based bubble with non–trivial lapse function and its consequences on matter content

In the present work, we study the consequences of including the lapse function as an additional degree of freedom for a general spherical warp-based geometry. By allowing a non-uniform lapse function to evolve, we find that it is possible to accommodate a fluid that includes heat flow. This broadens the range of fluid types that have been studied in these systems and is consistent with the spherical warp metric. Having added the lapse function, we solved the system of equations using an anisotropic fluid with heat flow. In this way, we can examine the different characteristics of the variables of the system. Next, we study the energy conditions and establish how these are modified by including heat flux for an appropriate generic observer in a locally flat spacetime. Finally, we explore all energy conditions using the numerical solutions and verify the regions where they are satisfied.

On circle patterns and spherical conical metrics

The Koebe-Andreev-Thurston circle packing theorem, as well as its generalization to circle patterns due to Bobenko and Springborn, holds for Euclidean and hyperbolic metrics possibly with conical singularities, but fails for spherical metrics because of the nonuniqueness coming from Möbius transformations. In this paper, we show that a unique existence result for circle pattern with spherical conical metric holds if one prescribes the total geodesic curvature of each circle instead of the cone angles.

Physical analysis of anisotropic compact stars in f(𝒬) gravity

The main objective of this study is to investigate the stability and viability of anisotropic compact stellar objects using the Krori–Burua solutions in [Formula: see text] gravity, where [Formula: see text] is a nonmetricity scalar that explains the gravitational effects. We use a static spherical metric in the inner region and Schwarzschild spacetime in the outer region of the Her X-I, SAX J 1808.4-3658 and 4U1820-30 compact stars to investigate their physical properties. By using observed values of the radius and mass of the considered compact stars, the unknown parameters are determined. We use a particular model of this theory to examine the behavior of energy density, pressure components, anisotropy, equation of state parameters and energy constraints in the interior of the proposed stellar objects. The Tolman–Oppenheimer–Volkoff equation is used to analyze the equilibrium state of these stars and causality condition, Herrera cracking method, adiabatic index methods are used to determine their stability. It is found that revolutionary technique sheds new light on the development of observationally accurate gravity model.

Anisotropic compact stars admitting karmarkar condition in f(𝒬) theory

The major goal of this study is to examine the viability and stability of anisotropic compact stellar objects using the Karmarkar condition in [Formula: see text] gravity, where [Formula: see text] is a scalar of nonmetricity that explains gravitational effects. We consider a static spherical metric in inner and Schwarzschild spacetime in outer regions of the star to analyze the physical attribute of Vela [Formula: see text], SAX J 1808.4−3658, 4U 1608−52, PSR J0348+0432 and 4U 1820−30 compact stars. The unknown parameters are determined through observational values of the radius and mass of the considered compact stars. We take a specific model of this theory to evaluate the energy density, pressure elements, anisotropy, equation of state parameters and energy bounds in inner region of suggested stellar objects. The equilibrium position of these stars is examined through Tolman–Oppenheimer–Volkoff equation and their stability checked by Herrera cracking method and adiabatic index. We find more viable as well as stable compact stars in this modified theory.

Dihedral monodromy of cone spherical metrics

Dihedral monodromy of cone spherical metrics

Topology Design and Optimization of Modular Soft Robots Capable of Homogenous and Heterogenous Reconfiguration

Abstract The deformability of soft material robots provides them with the ability to transform between complex shapes and forms. This unique ability facilitates Modular Soft Robots (MSoRos) to assemble and reconfigure into different configurations, e.g., planar and spherical. These topologies display widely different locomotion modes that are desirable to navigate different environments, e.g., crawling or rolling for these cases. This research presents topology design and optimization methodology of MSoRos capable of both homogeneous and heterogeneous reconfiguration in spherical and planar configurations. Homogeneous reconfiguration refers to the scenario when all the modules are identical, while the heterogeneous contains nonidentical modules. The sequential design approach uses a polyhedron (Archimedean or Platonic) as the base solid to define module characteristics. As the design processes involve nonlinear projections, the base polyhedron also dictates the type of reconfiguration—heterogeneous (Archimedean) or homogeneous (Platonic). Thereafter, it applies the polyhedron vertex alignment principle to ensure geometric alignment of the modules during reconfiguration. Planar and spherical distortion metrics are defined to quantify distortions due to reconfiguration. Subsequently, the optimal topology is obtained by minimizing a cost function that is a weighted sum of the two distortion metrics. The result is a set of MSoRos capable of distinct 1D and 2D planar configurations (both heterogeneous and homogeneous) and multiple 3D spherical configurations of varying radii (both heterogeneous and homogeneous). The methodology is validated on a MSoRo system based on the combination of a cuboctahedron (Archimedean solid) and a cube and an octahedron (Platonic solids).

Holonomy of complex projective structures on surfaces with prescribed branch data

AbstractWe characterize the representations of the fundamental group of a closed surface to that arise as the holonomy of a branched complex projective structure with fixed branch divisor. In particular, we compute the holonomies of the spherical metrics with prescribed integral conical angles and the holonomies of affine structures with fixed conical angles on closed surfaces.

Higher-Dimensional Generalizations of Some Theorems on Normality of Meromorphic Functions

In [5], Grahl and Nevo obtained a significant improvement for the well-known normality criterion of Montel. They proved that for a family of meromorphic functions F in a domain D⊂C, and for a positive constant ε, if for each f∈F there exist meromorphic functions af,bf,cf such that f omits af,bf,cf in D and min{ρ(af(z),bf(z)),ρ(bf(z),cf(z)),ρ(cf(z),af(z))}≥ε for all z∈D, then F is normal in D. Here, ρ is the spherical metric in Cˆ. In this paper, we establish the high-dimensional versions for the above result and for the following well-known result of Lappan: A meromorphic function f in the unit disc △:={z∈C:|z|<1} is normal if there are five distinct values a1,…,a5 such that sup{(1−|z|2)|f′(z)| 1+|f(z)|2:z∈f−1{a1,…,a5}}<∞.

Parabolic bundles and spherical metrics

We use the Kobayashi-Hitchin correspondence for parabolic bundles to reprove the results of Troyanov [Trans. Amer. Math. Soc. 324 (1991), pp. 793-821] and Luo-Tian [Proc. Amer. Math. Soc. 116 (1992), pp. 1119-1129] regarding existence and uniqueness of conformal spherical metrics on the Riemann sphere with prescribed cone angles in the interval ( 0 , 2 π ) (0, 2\pi ) at a given configuration of three or more points.

Spherical doubly warped spacetimes for radiating stars and cosmology

Spherically symmetric spacetimes are ambient spaces for models of stellar collapse and inhomogeneous cosmology. We obtain results for the Weyl tensor and the covariant form of the Ricci tensor on general doubly warped (DW) spacetimes. In a spherically symmetric metric, the Ricci and electric tensors become rank-2, built with the metric tensor, a velocity vector field and its acceleration. Their structure dictates the general form of the energy-momentum tensor in the Einstein equations in DW spherical metrics. The anisotropic pressure and the heat current of an imperfect fluid descend from the gradient of the acceleration and the electric part of the Weyl tensor. For radiating stellar collapse with heat flow, the junction conditions of the doubly warped metric with the Vaidya metric are reviewed, with the boundary condition for the radial pressure. The conditions for isotropy simply accomodate various models in the literature. The anisotropy of the Ricci tensor in the special case of spherical GRW space-times (geodesic velocity), gives Friedmann equations deviating from standard FRW cosmology by terms due to the electric tensor. We introduce “perfect 2-scalars” to discuss f(R) gravity with anisotropic fluid source in a doubly warped spacetime, and show that the new geometric terms in the field equations do not change the tensor structure of the fluid energy-momentum tensor.

Non-existence of Yamabe Minimizers on Singular Spheres

We prove that a minimizer of the Yamabe functional does not exist for a sphere \(\mathbb S^n\) of dimension \(n\ge 3\), endowed with a standard edge-cone spherical metric of cone angle greater than or equal to \(4\pi \), along a great circle of codimension two. When the cone angle along the singularity is smaller than \(2\pi \), the corresponding metric is known to be a Yamabe metric, and we show that all Yamabe metrics in its conformal class are obtained from it by constant multiples and conformal diffeomorphisms preserving the singular set.

Constraints on a Tidal Charge of the Supermassive Black Hole in M87* with the EHT Observations in April 2017

More than two years ago the Event Horizon Telescope collaboration presented the first image reconstruction around the shadow for the supermassive black hole in M87*. It gives an opportunity to evaluate the shadow size. Recently, the Event Horizon Telescope collaboration constrained parameters (“charges”) of spherical symmetrical metrics of black holes from an estimated allowed interval for shadow radius from observations of M87* in 2017. Earlier, analytical expressions for the shadow radius as a function of charge (including a tidal one) in the case of Reissner–Nordström metric have been obtained. Some time ago, Bin-Nun proposed to apply a Reissner–Nordström metric with a tidal charge as an alternative to the Schwarzschild metric in Sgr A*. If we assume that a Reissner–Nordström black hole with a tidal charge exists in M87*, therefore, based on results of the shadow size evaluation for M87* done by the Event Horizon Telescope collaboration we constrain a tidal charge. Similarly, we evaluate a tidal charge from shadow size estimates for Sgr A*.

Spherical conical metrics and harmonic maps to spheres

A spherical conical metric g g on a surface Σ \Sigma is a metric of constant curvature 1 1 with finitely many isolated conical singularities. The uniformization problem for such metrics remains largely open when at least one of the cone angles exceeds 2 π 2\pi . The eigenfunctions of the Friedrichs Laplacian Δ g \Delta _g with eigenvalue λ = 2 \lambda =2 play a special role in this problem, as they represent local obstructions to deformations of the metric g g in the class of spherical conical metrics. In the present paper we apply the theory of multivalued harmonic maps to spheres to the question of existence of such eigenfunctions. In the first part we establish a new criterion for the existence of 2 2 -eigenfunctions, given in terms of a certain meromorphic data on Σ \Sigma . As an application we give a description of all 2 2 -eigenfunctions for metrics on the sphere with at most three conical singularities. The second part is an algebraic construction of metrics with large number of 2 2 -eigenfunctions via the deformation of multivalued harmonic maps. We provide new explicit examples of metrics with many 2 2 -eigenfunctions via both approaches, and describe the general algorithm to find metrics with arbitrarily large number of 2 2 -eigenfunctions.

Soliton-Like Spherical Symmetric Solutions to the Electromagnetic and Scalar Nonlinear Induction Field Equations in the General Relativity Theory

In this paper, we have used the static spherical symmetric metric. The parameter of the nonlinearity fields is included in the arbitrary function characterizing the interaction between the electromagnetic and scalar fields. Taking into account the own gravitational field of elementary particles, we have obtained exact static spherical symmetric solutions to the electromagnetic and scalar field equations of nonlinear induction. Considering all forms of the solution of Liouville equation, we proved that the metric functions are regular with localized energy density. Moreover, the total energy of the nonlinear induction fields is bounded and the total charge of the elementary particles has a finite value (soliton-like). In the flat space-time, soliton-like solutions exist.

Image-Based Angular Distortion Metric of Map Projections by Using Surface Fitting for Noise Reduction

Measuring, analyzing, reducing, and optimizing distortions in map projections is important in cartography. In this study, we introduced a novel image-based angular distortion metric based on the previous spherical great circle arcs-based metric. Images with predefined patterns were used to generate distorted images using mapping software. The generated distorted images with known patterns were then exploited to calculate the proposed angular distortion metric. The mapping software performed the underlying transformation of map projections. Therefore, there was no direct explicit dependence on the forward equations of the map projections in our proposed metric. However, there were fairly large computation errors in the ordinary image-based approach without special correction. To reduce the error, we introduced surface-fitting-based noise reduction in our approach. We established and solved systems of linear equations based on bivariate polynomial functions in the process of noise reduction. Sufficient experiments were made to validate the proposed image-based metric and the accompanying noise reduction approach. In the experiment, the NASA G.Projector was employed as the mapping software for evaluating more than 200 map projections. Experimental results demonstrated that the proposed image-based approach and surface fitting-based noise reduction are feasible and practical for the evaluation of the angular distortion of map projections.

Rigidity through a Projective Lens

In this paper, we offer an overview of a number of results on the static rigidity and infinitesimal rigidity of discrete structures which are embedded in projective geometric reasoning, representations, and transformations. Part I considers the fundamental case of a bar–joint framework in projective d-space and places particular emphasis on the projective invariance of infinitesimal rigidity, coning between dimensions, transfer to the spherical metric, slide joints and pure conditions for singular configurations. Part II extends the results, tools and concepts from Part I to additional types of rigid structures including body-bar, body–hinge and rod-bar frameworks, all drawing on projective representations, transformations and insights. Part III widens the lens to include the closely related cofactor matroids arising from multivariate splines, which also exhibit the projective invariance. These are another fundamental example of abstract rigidity matroids with deep analogies to rigidity. We conclude in Part IV with commentary on some nearby areas.

Geometric Decompositions of Surfaces with Spherical Metric and Conical Singularities

Abstract We prove that any compact surface with constant positive curvature and conical singularities can be decomposed into irreducible components of standard shape, glued along geodesic arcs connecting conical singularities. This is a spherical analog of the geometric triangulations for flat surfaces with conical singularities. The irreducible components include not only spherical triangles but also other interesting spherical polygons. In particular, we present the class of half-spherical concave polygons that are spherical polygons without diagonals and that can be arbitrarily complicated. Finally, we introduce the notion of core as a geometric invariant in the settings of spherical surfaces. We use it to prove a reducibily result for spherical surfaces with a total conical angle at least $(10g-10+5n)2\pi$.

Irreducible cone spherical metrics and stable extensions of two line bundles

A cone spherical metric is called irreducible if any developing map of the metric does not have monodromy in U(1). By using the theory of indigenous bundles, we construct on a compact Riemann surface X of genus gX≥1 a canonical surjective map from the moduli space of stable extensions of two line bundles to that of irreducible metrics with cone angles in 2πZ>1, which is generically injective in the algebro-geometric sense as gX≥2. As an application, we prove the following two results about irreducible metrics:• as gX≥2 and d is even and greater than 12gX−7, the effective divisors of degree d which could be represented by irreducible metrics form an arcwise connected Borel subset of Hausdorff dimension ≥2(d+3−3gX) in Symd(X);• as gX≥1, for almost every effective divisor D of degree odd and greater than 2gX−2 on X, there exist finitely many cone spherical metrics representing D.