Geodesic Structure Research Articles

Rotating black holes in Einstein-dilaton-Gauss-Bonnet gravity with finite coupling

Among various strong-curvature extensions to General Relativity, Einstein-Dilaton-Gauss-Bonnet gravity stands out as the only nontrivial theory containing quadratic curvature corrections while being free from the Ostrogradsky instability to any order in the coupling parameter. We derive an approximate stationary and axisymmetric black-hole solution of this gravitational theory in closed form, which is quadratic in the black-hole spin angular momentum and of seventh order in the coupling parameter of the theory. This extends previous work that obtained the corrections to the metric only at the leading order in the coupling parameter, and allows us to consider values of the coupling parameter close to the maximum permitted by theoretical constraints. We compute some geometrical properties of this solution, such as the dilaton charge, the moment of inertia and the quadrupole moment, and its geodesic structure, including the innermost-stable circular orbit and the epicyclic frequencies for massive particles. The latter represent a valuable tool to test General Relativity against strong-curvature corrections through observations of the electromagnetic spectrum of accreting black holes.

Time-like geodesic structure in massive gravity

In the present paper we analyze the geodesic structure of black hole spacetime in massive gravity with the scalar charge Q representing the modification to Einstein’s general relativity. By solving the geodesic equation and analyzing the behavior of effective potential, we investigate the time-like geodesic types of the test particle around a black hole in massive gravity. At the same time, all kinds of orbits, which are allowed according to the energy levels of the effective potential, are numerically simulated in detail.

Electrogeodesics in the di-hole Majumdar-Papapetrou spacetime

We investigate the (electro-)geodesic structure of the Majumdar-Papapetrou solution representing static charged black holes in equilibrium. We assume only two point sources, thus imparting spacetime axial symmetry. We study electrogeodesics both on and off the equatorial plane and explore the stability of circular trajectories via the geodesic deviation equation. In contrast to the classical Newtonian situation, we find regions of spacetime admitting two different angular frequencies for a given radius of the circular electrogeodesic. We look both at the weak- and near-field limits of the solution. We use analytic as well as numerical methods in our approach.

Holographic mutual information in global Vaidya-BTZ spacetime

We investigate the evolution of the mutual information between two spatial subsystems in a compact 1+1 dimensional CFT after a quantum quench. To this end, we use the dual holographic process, given by the 2+1 dimensional Vaidya-BTZ spacetime in global coordinates, which describes the collapse of a spherically symmetric null shell. So, we first discuss the spacelike geodesic structure of this geometry and then we present the various behaviors of the holographic mutual information observed in this case. We also consider the analogous process in the adiabatic limit and compare these two cases from a geometrical point of view.

Geodesic structure of five-dimensional nonasymptotically flat 2-branes

We study the geodesics of a five dimensional non-asymptotically flat dilatonic 2-brane. Although the metric's warp function diverges logarithmically in the far field region of the transverse space, the curvature does not. The brane has two naked singularities, the usual central singularity and a circular singularity at the radius where the warp function vanishes. This creates two causally disconnected regions of the transverse space. Using the methods of energy conservation and effective potentials we study the null and timelike orbital and radial geodesics in both regions and show that they exhibit opposing energy requirements.

Geometric aspects of charged black holes in Palatini theories

Charged black holes in gravity theories in the Palatini formalism present a number of unique properties. Their innermost structure is topologically nontrivial, representing a wormhole supported by a sourceless electric flux. For certain values of their effective mass and charge curvature divergences may be absent, and their event horizon may also disappear yielding a remnant. We give an overview of the mathematical derivation of these solutions and discuss their geodesic structure and other geometric properties.

Heat kernel asymptotic expansions for the Heisenberg sub-Laplacian and the Grushin operator.

The sub-Laplacian on the Heisenberg group and the Grushin operator are typical examples of sub-elliptic operators. Their heat kernels are both given in the form of Laplace-type integrals. By using Laplace's method, the method of stationary phase and the method of steepest descent, we derive the small-time asymptotic expansions for these heat kernels, which are related to the geodesic structure of the induced geometries.

Circular geodesic of Bardeen and Ayon–Beato–Garcia regular black-hole and no-horizon spacetimes

In this paper, we study circular geodesic motion of test particles and photons in the Bardeen and Ayon–Beato–Garcia (ABG) geometry describing spherically symmetric regular black-hole or no-horizon spacetimes. While the Bardeen geometry is not exact solution of Einstein's equations, the ABG spacetime is related to self-gravitating charged sources governed by Einstein's gravity and nonlinear electrodynamics. They both are characterized by the mass parameter m and the charge parameter g. We demonstrate that in similarity to the Reissner–Nordstrom (RN) naked singularity spacetimes an antigravity static sphere should exist in all the no-horizon Bardeen and ABG solutions that can be surrounded by a Keplerian accretion disc. However, contrary to the RN naked singularity spacetimes, the ABG no-horizon spacetimes with parameter g/m > 2 can contain also an additional inner Keplerian disc hidden under the static antigravity sphere. Properties of the geodesic structure are reflected by simple observationally relevant optical phenomena. We give silhouette of the regular black-hole and no-horizon spacetimes, and profiled spectral lines generated by Keplerian rings radiating at a fixed frequency and located in strong gravity region at or nearby the marginally stable circular geodesics. We demonstrate that the profiled spectral lines related to the regular black-holes are qualitatively similar to those of the Schwarzschild black-holes, giving only small quantitative differences. On the other hand, the regular no-horizon spacetimes give clear qualitative signatures of their presence while compared to the Schwarschild spacetimes. Moreover, it is possible to distinguish the Bardeen and ABG no-horizon spacetimes, if the inclination angle to the observer is known.

Geodesic Structure of Janis-Newman-Winicour Space-time

In the present paper we study the geodesic structure of the Janis-Newman-Winicour(JNW) space-time which contains a strong curvature naked singularity. This metric is an extension of the Schwarzschild geometry included a massless scalar field. We find that the strength parameter μ of the scalar field takes affection on the geodesic structure of the JNW space-time. By solving the geodesic equation and analyzing the behavior of effective potential, we investigate all geodesic types of the test particle and the photon in the JNW space-time. At the same time we simulate all the geodesic orbits corresponding to the energy levels of the effective potential in the JNW space-time.

Jacobi Fields and Conjugate Points on Timelike Geodesics in Special Spacetimes

Several physical problems such as the `twin paradox' in curved spacetimes have purely geometrical nature and may be reduced to studying properties of bundles of timelike geodesics. The paper is a general introduction to systematic investigations of the geodesic structure of physically relevant spacetimes. The investigations are focussed on the search of locally and globally maximal timelike geodesics. The method of dealing with the local problem is in a sense algorithmic and is based on the geodesic deviation equation. Yet the search for globally maximal geodesics is non-algorithmic and cannot be treated analytically by solving a differential equation. Here one must apply a mixture of methods: spacetime symmetries (we have effectively employed the spherical symmetry), the use of the comoving coordinates adapted to the given congruence of timelike geodesics and the conjugate points on these geodesics. All these methods have been effectively applied in both the local and global problems in a number of simple and important spacetimes and their outcomes have already been published in three papers. Our approach shows that even in Schwarzschild spacetime (as well as in other static spherically symetric ones) one can find a new unexpected geometrical feature: instead of one there are three different infinite sets of conjugate points on each stable circular timelike geodesic curve. Due to problems with solving differential equations we are dealing solely with radial and circular geodesics.

Geodesic completeness and homogeneity condition for cosmic inflation

There are two disjointed problems in cosmology within General Relativity (GR), which can be addressed simultaneously by studying the nature of geodesics around $t\rightarrow 0$, where $t$ is the physical time. One is related to the past geodesic completeness of the inflationary trajectory due to the presence of a cosmological singularity, and the other one is related to the homogeneity condition required to inflate a local space-time patch of the universe. We will show that both the problems have a common origin, arising from how the causal structure of null and timelike geodesics are structured within GR. In particular, we will show how a non-local extension of GR can address both problems, while satisfying the null energy condition for the matter sources.

A syndecan-4 binding peptide derived from laminin 5 uses a novel PKCε pathway to induce cross-linked actin network (CLAN) formation in human trabecular meshwork (HTM) cells

In this study, we examined the role(s) of syndecan-4 in regulating the formation of an actin geodesic dome structure called a cross-linked actin network (CLAN) in which syndecan-4 has previously been localized. CLANs have been described in several different cell types, but they have been most widely studied in human trabecular meshwork (HTM) cells where they may play a key role in controlling intraocular pressure by regulating aqueous humor outflow from the eye. In this study we show that a loss of cell surface synedcan-4 significantly reduces CLAN formation in HTM cells. Analysis of HTM cultures treated with or without dexamethasone shows that laminin 5 deposition within the extracellular matrix is increased by glucocorticoid treatment and that a laminin 5-derived, syndecan-4-binding peptide (PEP75), induces CLAN formation in TM cells. This PEP75-induced CLAN formation was inhibited by heparin and the broad spectrum PKC inhibitor Ro-31–7549. In contrast, the more specific PKCα inhibitor Gö 6976 had no effect, thus excluding PKCα as a downstream effector of syndecan-4 signaling. Analysis of PKC isozyme expression showed that HTM cells also expressed both PKCγ and PKCε. Cells treated with a PKCε agonist formed CLANs while a PKCα⧸γ agonist had no effect. These data suggest that syndecan-4 is essential for CLAN formation in HTM cells and that a novel PKCε-mediated signaling pathway can regulate formation of this unique actin structure.

Operators which are the difference of two projections

We study the set D of differencesD={A=P−Q:P,Q∈P}, where P denotes the set of orthogonal projections in H. We describe models and factorizations for elements in D, which are related to the geometry of P. The study of D throws new light on the geodesic structure of P (we show that two projections in generic position are joined by a unique minimal geodesic). The topology of D is examined, particularly its connected components are studied. Also we study the subsets Dc⊂DF, where Dc are the compact elements in D, and DF are the differences A=P−Q such that the pair (P,Q) is a Fredholm pair ((P,Q) is a Fredholm pair if QP|R(P):R(P)→R(Q) is a Fredholm operator).

Geodesic study of regular Hayward black hole

This paper is devoted to study the geodesic structure of regular Hayward black hole. The timelike and null geodesic have been studied explicitly for radial and non-radial motion. For timelike and null geodesic in radial motion there exists analytical solution, while for non-radial motion the effective potential has been plotted, which investigates the position and turning points of the particle. It has been found that massive particle moving along timelike geodesics path are dragged towards the BH and continues move around BH in particular orbits.

Motion of particles on a $$z=2$$ z = 2 Lifshitz black hole background in 3 $$+$$ + 1 dimensions

The object of study is the geodesic structure of a \(z=2\) Lifshitz black hole in 3+1 space–time dimensions, which is an exact solution to the Einstein-scalar-Maxwell theory. The motion of massless and massive particles in this background is researched using the standard Lagrangian procedure. Analytical expressions are obtained for radial and angular motions of the test particles, where the polar trajectories are given in terms of the \(\wp \)-Weierstraß elliptic function. It will be demonstrated that an external observer can see that photons with radial motion arrive at spatial infinity in a finite coordinate time. For particles with non-vanished angular momentum, the motion is studied on the invariant plane \(\phi = \pi /2\) and, it is shown that bounded orbits are not allowed for this space–time on this plane. These results are consistent with other recent studies on Lifshitz black holes.

An effective biomedical image retrieval framework in a fuzzy feature space employing Phase Congruency and GeoSOM

This paper presents a detailed study about a biomedical image retrieval framework by extracting Phase Congruency (PC) features from L*a*b* triplets of images (query, target) and representing them in fuzzy feature space. These features correspond to an edge-corner map of the given image. The resulting map is then processed by Scale Invariant Feature Transform (SIFT) to derive keypoints, that are invariant to affine transformations. The ensuing features were vector quantized to build a codebook of keypoints. The codebook was produced using a Spherical Self-Organizing Map (SOM) built with a geodesic data structure termed as GeoSOM. Then keypoints of the query image are mapped with the codebook and their occurrences are counted to formulate a histogram termed as Phase Congruency-based Bag of Keypoints (PC-BoK). This histogram is generated offline for target images and a similarity measure was performed with the query image to yield the nearest match based on a global fuzzy membership function. Exhaustive experiments of the proposed framework named as BIRS (Biomedical Image Retrieval System) were performed on a diverse medical image collection (NBIA, MESSIDOR, DRIVE). Finally, performance of BIRS demonstrates the advantage of the proposed image representation approach in terms of Precision (P)–Recall (R) parameters. Furthermore relative comparison of the proposed scheme with existing feature descriptors depicts improved P–R values. The proposed feature extraction and representation scheme was also robust against quantization errors.

Development of Geodesic Composite Fuselage Structure

The paper is concerned with development of geodesic composite fuselage structure whose stiffness and strength are provided by a system of ribs made of unidirectional carbon-epoxy composite materials by continuous wet filament winding. Design and weight efficiency of the geodesic structures are discussed along with the efficiency of traditional composite stringer structures. Results of design, analysis and fabrication of a full-scale 4m diameter fuselage section are presented.

The Bubble Hall: Bamboo Reticular Geodesic Structure with the Shape of a Soap Bubble

This article aims to disclose some aspects of the research and constructive methods on lightweight structures made of tied-up bamboos developed by the Laboratory for Investigation in Living Design, LILD, from PUC-Rio. In this paper, we demonstrate the way of obtaining a shape similar to the one of a soap bubble when blown and manipulated by the researcher, according to previously established parameters. The approximation of such a geometry is achieved through a variety of interactive experiments between the states of a model electronic, manufactured /miniature, and in use that follow the logics of geodesic lines, obtained by means of a grid when inflated. Finally, we present results of initial observations of the assemblage of the bamboo reticular structure in the in use state, that we call The Bubble Hall.

Fourth class of convex equilateral polyhedron with polyhedral symmetry related to fullerenes and viruses

The three known classes of convex polyhedron with equal edge lengths and polyhedral symmetry--tetrahedral, octahedral, and icosahedral--are the 5 Platonic polyhedra, the 13 Archimedean polyhedra--including the truncated icosahedron or soccer ball--and the 2 rhombic polyhedra reported by Johannes Kepler in 1611. (Some carbon fullerenes, inorganic cages, icosahedral viruses, geodesic structures, and protein complexes resemble these fundamental shapes.) Here we add a fourth class, "Goldberg polyhedra," which are also convex and equilateral. We begin by decorating each of the triangular facets of a tetrahedron, an octahedron, or an icosahedron with the T vertices and connecting edges of a "Goldberg triangle." We obtain the unique set of internal angles in each planar face of each polyhedron by solving a system of n equations and n variables, where the equations set the dihedral angle discrepancy about different types of edge to zero, and the variables are a subset of the internal angles in 6gons. Like the faces in Kepler's rhombic polyhedra, the 6gon faces in Goldberg polyhedra are equilateral and planar but not equiangular. We show that there is just a single tetrahedral Goldberg polyhedron, a single octahedral one, and a systematic, countable infinity of icosahedral ones, one for each Goldberg triangle. Unlike carbon fullerenes and faceted viruses, the icosahedral Goldberg polyhedra are nearly spherical. The reasoning and techniques presented here will enable discovery of still more classes of convex equilateral polyhedra with polyhedral symmetry.

Methodology for Definition of the Three-Dimensional Geodesic Structures in the Olinda’s Historical Site

Methodology for Definition of the Three-Dimensional Geodesic Structures in the Olinda’s Historical Site