Geodesic Trajectories Research Articles

Chaos in quadratic gravity

While recent gravitational wave observations by LIGO and Virgo allow for tests of general relativity in the extreme gravity regime, these observations are still blind to a large swath of phenomena outside these instruments' sensitivity curves. Future gravitational-wave detectors, such as LISA, will enable probes of longer-duration and lower-frequency events. In particular, LISA will enable the characterization of the non-linear dynamics of extreme mass-ratio inspirals, when a small compact object falls into a supermassive black hole. In this paper, we study the motion of test particles around spinning black holes in two quadratic gravity theories: scalar Gauss-Bonnet and dynamical Chern-Simons gravity. We show that geodesic trajectories around rotating black holes in these theories are likely to not have a fourth constant of the motion. In particular, we show that Poincar\'e sections of the orbital phase space present chaotic features that will affect the inspiral of small compact objects into supermassive black holes in these theories. Nevertheless, the characteristic size of these chaotic features is tiny and their location in parameter space is very close to the event horizon of the supermassive black hole. Therefore, the detection of such chaotic features with LISA is likely very challenging, at best.

Detecting intrinsic global geometry of an obstacle via layered scattering.

Given a closed k-dimensional submanifold K, encapsulated in a compact domain M ⊂ E, k ≤ n - 2, we consider the problem of determining the intrinsic geometry of the obstacle K (such as volume, integral curvature) from the scattering data, produced by the reflections of geodesic trajectories from the boundary of a tubular ϵ-neighborhood T ( K , ϵ ) of K in M. The geodesics that participate in this scattering emanate from the boundary ∂ M and terminate there after a few reflections from the boundary ∂ T ( K , ϵ ). However, the major problem in this setting is that a ray (a billiard trajectory) may get stuck in the vicinity of K by entering some trap there so that this ray will have infinitely many reflections from ∂ T ( K , ϵ ). To rule out such a possibility, we modify the geometry of a tube T ( K , ϵ ) by building it from spherical bubbles. We need to use ⌈ dim ⁡ ( K ) / 2 ⌉ many bubbling tubes { T ( K , ϵ ) } for detecting certain global invariants of K, invariants that reflect its intrinsic geometry. Thus, the words "layered scattering" are in the title. These invariants were studied by Hermann Weyl in his classical theory of tubes T ( K , ϵ ) and their volumes.

Precisely computing bound orbits of spinning bodies around black holes. II. Generic orbits

In this paper, we continue our study of the motion of spinning test bodies orbiting Kerr black holes. Non-spinning test bodies follow geodesics of the spacetime in which they move. A test body's spin couples to the curvature of that spacetime, introducing a "spin-curvature force" which pushes the body's worldline away from a geodesic trajectory. The spin-curvature force is an important example of a post-geodesic effect which must be modeled carefully in order to accurately characterize the motion of bodies orbiting black holes. One motivation for this work is to understand how to include such effects in models of gravitational waves produced from the inspiral of stellar mass bodies into massive black holes. In this paper's predecessor, we describe a technique for computing bound orbits of spinning bodies around black holes with a frequency-domain description which can be solved very precisely. In that paper, we present an overview of our methods, as well as present results for orbits which are eccentric and nearly equatorial (i.e., the orbit's motion is no more than $\mathcal{O}(S)$ out of the equatorial plane). In this paper, we apply this formulation to the fully generic case -- orbits which are inclined and eccentric, with the small body's spin arbitrarily oriented. We compute the trajectories which such orbits follow, and compute how the small body's spin affects important quantities such as the observable orbital frequencies $\Omega_r$, $\Omega_\theta$ and $\Omega_\phi$.

Null geodesics and shadow of 4D Einstein–Gauss–Bonnet black holes surrounded by quintessence

We study the properties of the null geodesics of four-dimensional Einstein–Gauss–Bonnet (4D EGB) black holes surrounded by quintessence matter, recently derived by Shah et al. [Phys. Lett. B 818 (2021) 136383]. Due to the quintessence correction, we discuss the radial and nonradial geodesics. For the case of nonradial geodesics we obtain the effective potential, photon sphere and impact parameter associated to the null geodesics in detail. We analyze the different possible orbits of massless particles (i.e. photons) such as unstable circular orbits and unbounded orbits and investigate the role of the Gauss–Bonnet coupling [Formula: see text] and the quintessence parameter q on the null geodesic trajectories. Moreover, we study the effects of the model parameters on the shadow cast by such black holes. The results are compared to that of Schwarzschild black hole with and without quintessence matter and 4D EGB black holes without quintessence. As a physical application of null geodesics, the deflection angle is calculated and the effect of the quintessence matter on it is investigated.

Abdominal motion tracking with free-breathing XD-GRASP acquisitions using spatio-temporal geodesic trajectories.

Free-breathing external beam radiotherapy remains challenging due to the complex elastic or irregular motion of abdominal organs, as imaging moving organs leads to the creation of motion blurring artifacts. In this paper, we propose a radial-based MRI reconstruction method from 3D free-breathing abdominal data using spatio-temporal geodesic trajectories, to quantify motion during radiotherapy. The prospective study was approved by the institutional review board and consent was obtained from all participants. A total of 25 healthy volunteers, 12 women and 13 men (38years ± 12 [standard deviation]), and 11 liver cancer patients underwent imaging using a 3.0T clinical MRI system. The radial acquisition based on golden-angle sparse sampling was performed using a 3D stack-of-stars gradient-echo sequence and reconstructed using a discretized piecewise spatio-temporal trajectory defined in a low-dimensional embedding, which tracks the inhale and exhale phases, allowing the separation between distinct motion phases. Liver displacement between phases as measured with the proposed radial approach based on the deformation vector fields was compared to a navigator-based approach. Images reconstructed with the proposed technique with 20 motion states and registered with the multiscale B-spline approach received on average the highest Likert scores for the overall image quality and visual SNR score 3.2 ± 0.3 (mean ± standard deviation), with liver displacement errors varying between 0.1 and 2.0mm (mean 0.8 ± 0.6mm). When compared to navigator-based approaches, the proposed method yields similar deformation vector field magnitudes and angle distributions, and with improved reconstruction accuracy based on mean squared errors. Schematic illustration of the proposed 4D-MRI reconstruction method based on radial golden-angle acquisitions and a respiration motion model from a manifold embedding used for motion tracking. First, data is extracted from the center of k-space using golden-angle sampling, which is then mapped onto a low-dimensional embedding, describing the relationship between neighboring samples in the breathing cycle. The trained model is then used to extract the respiratory motion signal for slice re-ordering. The process then improves the image quality through deformable image registration. Using a reference volume, the deformation vector field (DVF) of sequential motion states are extracted, followed by deformable registrations. The output is a 4DMRI which allows to visualize and quantify motion during free-breathing.

Widened R<sup><i>n</i></sup> General Relativity

An extension of General Relativity is presented based on a scalar Lagrangian density which is a general function of all the independent invariant scalars that one is able to build by the powers of the Ricci tensor. It is shown how the new terms arising in the generalized Einstein filed equations may be interpreted as dark matter and dark energy contributions. Metricity condition fulfilled by a new tensor different than the usual metric tensor is also obtained. Moreover, it is shown that Schwarzschild-De Sitter, Robertson-Walker-De Sitter and Kerr-De Sitter metrics are exact solutions to the new field equations. Remarkably, the form of the equation of the geodesic trajectories of particle motions across space-time remains the same as in Einstein General Relativity unless the cosmological constant Lambda is no longer a constant becoming a function of the space-time co-ordinates.

Temporal No-Linearity: An Alternative to Dark Energy

Mathematical compatibilities and constraints of a hypothetical 5D space-time with time referenced by two coordinates (3-2) have been revisited in detail in several recent papers. It has been prescribed from the GR the compatibility constraints of the FLRW metric in each temporal brane, to be restricted to Closed Universes, Smooth Initial Singularities, and “Open CTC”. In a first view, this leads to leaving these works considering mathematical games discarded by the Standard Candles data. However, if time would be referred by two coordinates, they would not be linearly related, and it will be mathematically stated that space-time may not be flat in any case because time-like branes geometry will never be. If so, the time scale “lived” over a two-time dimension geodesic necessarily is not constant over its linear projection on one of both coordinates. Consequently, the correlations between Redshift and Distance Modulus—Distance Ladder—may be corrected by a synchronization function (if a no-linear two-time geodesic trajectory over a “warped temporal geometry” is linearly divided into constant segments, then their projections are not linear in any case). We apply an example of time-trajectory over the time slices, matching the Standard Candle data for a Closed Universe dominated by matter (>90%) in a bulk 3-2 configuration, with open temporal branes and smooth singularity. If time can be referred by two coordinates, then there is no need of Darkness to explain astrophysical data and Universe can be closed.

Finiteness in polygonal billiards on hyperbolic plane

\textsc{J. Hadamard} studied the geometric properties of geodesic flows on surfaces of negative curvature, thus initiating "Symbolic Dynamics". In this article, we follow the same geometric approach to study the geodesic trajectories of billiards in "rational polygons" on the hyperbolic plane. We particularly show that the billiard dynamics resulting thus are just 'Subshifts of Finite Type' or their dense subsets. We further show that 'Subshifts of Finite Type' play a central role in subshift dynamics and while discussing the topological structure of the space of all subshifts, we demonstrate that they approximate any shift dynamics.

Some Characterizations on Geodesic, Asymptotic and Slant Helical Trajectories According to PAFORS

In this paper, we study the geodesic, asymptotic and slant helical trajectories according to PAFORS in three-dimensional Euclidean space and give some characterizations on them. Also, we explain how we determine the helix axis for slant helical trajectories (according to PAFORS). Moreover, we develop a method which enables us to find the slant helical trajectory (if exists) lying on a given implicit surface which accepts a given fixed unit direction as an axis and a given angle as the constant angle. This method also gives information when the desired slant helical trajectory does not exist. The results obtained here involve some differential and partial differential equations or they are based on these equations. The aforementioned results are new contributions to the field and they may be useful in some specific applications of particle kinematics and differential geometry.

Computational Geometry-Based 3D Yarn Path Modeling of Wound SiCf/SiC-Cladding Tubes and Its Application to Meso-Scale Finite Element Model

The filament winding process is a competitive performing technology for nuclear fuel cladding tubes due to its high automation. The study of the yarn path on the mandrel surface is vital to design and produce the cladding tube with the desired mechanical properties, reducing manufacturing time and costs. The geodesic and semi-geodesic trajectories are used to create a 3D yarn path in this paper. A 3D yarn path optimization method based on the principle of minimum potential energy is proposed to simulate the overlap effect in accord with the real winding process. The finite element (FE) mesh based on the 3D yarn path has been used for the mechanical analysis of the cladding tube. The embedded region constraint is applied to define the interaction between the matrix mesh and the yarn mesh to model the meso-structure of the cladding tube. Based on the meso-scale FE model, the mechanical behavior of the wound SiCf/SiC nuclear fuel cladding tube is studied in detail. The results show that due to the neglect of the overlap effect, the conventional laminate model overestimates the cladding tube strength. The proposed meso-scale FE model can accurately predict the failure of the cladding tube. The results also confirm that the creation of a 3D yarn path and the derived meso-scale FE model, representing an accurate wound structure, are of importance to the prediction of the performance of the cladding tube.

Equivalent continuum method for anisogrid composite lattice conical shells with equiangular, equidistant and geodesic spiral ribs

This paper is concerned with filament winding trajectories of helical ribs of anisogrid composite lattice conical shells. Filament winding can be designed to coincide with equiangular, equidistant and geodesic spiral trajectories. The unified geometrical formulas of the three types of conical spiral and circumferential curves are established. As a consequence, three types of lattice conical shells are built and the corresponding geometric parameters of lattice conical shells are obtained. Additionally, the smearing method, adopting straight-line and curved-line assumptions for helical ribs, respectively, is applied to calculate the elastic properties and the global stiffness properties along with the classical membrane theory. The finite element analysis verifies the analytical results and also shows a closer correspondence with numerical results for the curved-line assumption than the straight-line assumption. Further, the global stiffness properties of three types of lattice conical shells are compared using the single variable method. The results indicate that the flexural stiffness and the torsional stiffness of the lattice conical shell with equiangular helical ribs are the maximum and the axial stiffness of the shell with geodesic helical ribs is the maximum among them, while the values of all stiffness properties of the shell with equidistant helical ribs are the minimum among them.

A Brief Theory of Everything – the Redesign, Declination and the Prediction of the Electron, the Atom, with All its Implications on Thermonuclear Reactions, Lenr Reactions, Matter Creation, Chemistry, Quantumtheory vs. Einsteins Relativity, and All Particle Accelerators Based Science Ahead.

The new atomic model is based on the axiom of two different charges within protons andor neutrons, leading to a chargeimbalance spikedout in protons, commonly called electron. While this rips out a bit from the protoncharge along a geodesic minimal wineglass trajectory, the viscosity of the proton charge sticking on this spikedout chargeimbalance defines many traits of the wineglass, including chemical bonding traits, shape and position within the nuclei with adjacent protons, as deep down into matter creation itself, the core coat charge integration causing the neutron and why it bonds on protons at all. It is a timeless theory of everything based on the axiom of cumulative bondingflux with charge equilibrium as the basis leading to the unanswered questions after this paper of what energy really is.

Investigation of the in-flight anomalies of the LISA Pathfinder Test Mass release mechanism

The LISA Pathfinder mission was flown to test some key technologies for the in-flight detection of gravitational waves. The mechanism developed to initialise the science phase, releasing the test mass into a geodesic trajectory, produced unexpected test mass states, requiring dedicated in-flight and on-ground test campaigns. Due to the limited information available from the telemetry, the analysis of the results required a combination of experimental and analytical approaches. A dynamic model of the release mechanism, integrated by an impact model, is developed to explain the in-flight test results. The proposed interpretation of the mechanism anomalies may provide useful guidelines for the design and development of the flight units for the forthcoming LISA mission.

Stability of filament-wound hyperbolic flexible pipes under internal pressure based on non-geodesic winding

Filament-wound flexible pipes are widely used to transport fluid in pipeline systems, proved extremely useful in marine engineering. The hyperbolic flexible pipes have good vibration suppression performance, but they are easily deformed under internal pressure. This paper focuses on the stability of hyperbolic flexible pipes based on the composite Reissner shell theory and the transfer-matrix method. The nonlinear stretch of the reinforced filament and the fiber bridge effect are considered in the model. The calculation results show that a large winding angle reduces the deformation and the meridional stress. The available initial winding angle is limited by the geometry and the slippage coefficient of flexible pipe. The reinforced filament of high tensile modulus will reduce the deformation of the pipe. Compared with the geodesic winding trajectory, non-geodesic winding trajectories improves the stability of the pipe. The theoretical result is verified by the finite element analysis. The investigation method and results present in this paper will guide the design and optimization of more novel flexible pipes in the future.

Analytical modeling and numerical analysis of the effect of mosaic patterns on composite pressure vessels with dome

The aim of this work is to provide an analytical tool and numerical analysis for the optimum design of composite pressure vessels with the dome, incorporating triangular mosaic patterns. This article presents the analytical modeling involving kinematic constraints based on geodesic trajectory: the helical angle and dome thickness. The structural analysis is performed using a commercial finite element analysis tool. The results show that this new analytical method gives more accurate dome thickness than cubic spline function and Gramoll and Namiki’s methods. The incorporation of mosaic patterns based on winding kinematics provides more realistic modeling of the real stress distribution and the stress values compared to the vessel without mosaic patterns and vessels with mosaic patterns based on nongeodesic trajectories. The results have been validated and are quite promising with regard to better accuracy and safety.

DYNAMICS IN FRACTAL SPACES

We study the dynamics of a particle in a space that is non-differentiable. Non-smooth geometrical objects have an inherently probabilistic nature and, consequently, introduce stochasticity in the motion of a body that lives in their realm. We use the mathematical concept of fiber bundle to characterize the multivalued nature of geodesic trajectories going through a point that is non-differentiable. Then, we generalize our concepts to everywhere non-smooth structures. The resulting theoretical framework can be considered a hybridization of the theory of surfaces and the theory of stochastic processes. We keep the concepts as general as possible, in order to allow for the introduction of other fundamental processes capable of modeling the fractality or the fluctuations of any conceivable continuous, but non-differentiable space.

Merging the weak gravity and distance conjectures using BPS extremal black holes

We analyze the charge-to-mass structure of BPS states in general infinite-distance limits of mathcal{N} = 2 compactifications of Type IIB string theory on Calabi-Yau three-folds, and use the results to sharpen the formulation of the Swampland Conjectures in the presence of multiple gauge and scalar fields. We show that the BPS bound coincides with the black hole extremality bound in these infinite distance limits, and that the charge-to-mass vectors of the BPS states lie on degenerate ellipsoids with only two non-degenerate directions, regardless of the number of moduli or gauge fields. We provide the numerical value of the principal radii of the ellipsoid in terms of the classification of the singularity that is being approached. We use these findings to inform the Swampland Distance Conjecture, which states that a tower of states becomes exponentially light along geodesic trajectories towards infinite field distance. We place general bounds on the mass decay rate λ of this tower in terms of the black hole extremality bound, which in our setup implies lambda ge 1/sqrt{6} . We expect this framework to persist beyond mathcal{N} = 2 as long as a gauge coupling becomes small in the infinite field distance limit.

In-flight testing of the injection of the LISA Pathfinder test mass into a geodesic

LISA Pathfinder is a technology demonstrator space mission, aimed at testing key technologies for detecting gravitational waves in space. The mission is the precursor of LISA, the first space gravitational waves observatory, whose launch is scheduled for 2034. The LISA Pathfinder scientific payload includes two gravitational reference sensors (GRSs), each one containing a test mass (TM), which is the sensing body of the experiment. A mission critical task is to set each TM into a pure geodesic motion, i.e. guaranteeing an extremely low acceleration noise in the sub-Hertz frequency bandwidth. The grabbing positioning and release mechanism (GPRM), responsible for the injection of the TM into a geodesic trajectory, was widely tested on ground, with the limitations imposed by the 1-g environment. The experiments showed that the mechanism, working in its nominal conditions, is capable of releasing the TM into free-fall fulfilling the very strict constraint imposed on the TM residual velocity, in order to allow its capture on behalf of the electrostatic actuation.However, the first in-flight releases produced unexpected residual velocity components, for both the TMs. Moreover, all the residual velocity components were greater than maximum value set by the requirements. The main suspect is that unexpected contacts took place between the TM and the surroundings bodies. As a consequence, ad hoc manual release procedures had to be adopted for the few following injections performed during the nominal mission. These procedures still resulted in non compliant TM states which were captured only after impacts. However, such procedures seem not practicable for LISA, both for the limited repeatability of the system and for the unmanageable time lag of the telemetry/telecommand signals (about 4400 s). For this reason, at the end of the mission, the GPRM was deeply tested in-flight, performing a large number of releases, according to different strategies. The tests were carried out in order to understand the unexpected dynamics and limit its effects on the final injection. Some risk mitigation maneuvers have been tested aimed at minimizing the vibration of the system at the release and improving the alignment between the mechanism and the TM. However, no overall optimal release strategy to be implemented in LISA could be found, because the two GPRMs behaved differently.

Applications of metaplectic cohomology and global-local contact holonomy

The purpose of the paper is to elucidate by global-dual contact holonomy why one might expect the Foucault spherical pendulum’s spin echo-stabilized, symplectic swing-plane through the rotation axis of the spinning earth to follow a parallel vector field as time passes at a velocity depending on the latitude of the swivel’s location. The spinor geometric foundations of the paradigmatic Foucault spherical pendulum experiment of global-local contact holonomy exploit Hopf–Rinow type minimizing geodesic trajectories which appear as inertial traces designed by swing-planes on the terrestrial circular floor panel which is situated beneath the covariant motion of the oscillating pendulum device and where the linear traces are designing diameters of the spin structure induced Poincaré conformal hyperbolic open disc model. The quantum field theoretical evidence of the existence and uniqueness of categorizing parallel vector fields around smooth loops follows in terms of the open-book foliation with Hilbert space structures placed on the covariant pages.

Simulation of geodesic trajectory of charged BTZ black holes in massive gravity

In order to classify and understand structure of the spacetime, investigation of the geodesic motions of massive and massless particles is a key tool. So the geodesic equation is a central equation of gravitating systems and the subject of geodesics in the black hole dictionary attracted much attention. In this paper, we give a full description of geodesic motions in three-dimensional spacetime. We investigate the geodesics near charged BTZ black holes and then generalize our prescriptions to the case of massive gravity. We show that electric charge is a critical parameter for categorizing the geodesic motions of both lightlike and timelike particles. In addition, we classify the type of geodesics based on the particle properties and geometry of spacetime.