Geodesic Equations Research Articles

Eulerian and Lagrangian Stability in Zeitlin’s Model of Hydrodynamics

The two-dimensional (2-D) Euler equations of a perfect fluid possess a beautiful geometric description: they are reduced geodesic equations on the infinite-dimensional Lie group of symplectomorphims with respect to a right-invariant Riemannian metric. This structure enables insights to Eulerian and Lagrangian stability via sectional curvature and Jacobi equations. The Zeitlin model is a finite-dimensional analogue of the 2-D Euler equations; the only known discretization that preserves the rich geometric structure. Theoretical and numerical studies indicate that Zeitlin’s model provides consistent long-time behaviour on large scales, but to which extent it truly reflects the Euler equations is mainly open. Towards progress, we give here two results. First, convergence of the sectional curvature in the Euler–Zeitlin equations on the Lie algebra su(N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {su}(N)$$\\end{document} to that of the Euler equations on the sphere. Second, L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-convergence of the corresponding Jacobi equations for Lagrangian and Eulerian stability. The results allow geometric conclusions about Zeitlin’s model to be transferred to Euler’s equations and vice versa, which could expedite the ultimate aim: to characterize the generic long-time behaviour of perfect 2-D fluids.

Analysis of high-frequency limit in absorption cross-section for Kerr–Newman–de Sitter black hole

In this paper, we study the critical impact parameter and Lyapunov exponent for a massive charged particle and a photon in the Kerr–Newman–de Sitter (KNdS) spacetime. From the geodesic equations obtained for the particle traveling parallel to the rotation axis (on-axis) of the KNdS spacetime and using the complex angular momentum (CAM) technique, it is found that the total absorption cross-section at a high-frequency limit oscillates with decreasing amplitude. We show that the black hole charge and spin affect the total absorption cross-section in the case of both the massive charged particle and photon.

Underluminous tidal disruption events

Abstract We have evidence of X-ray flares in several galaxies consistent with a a star being tidally disrupted by a supermassive black hole (MBH). If the star starts on a nearly parabolic orbit relative to the MBH, one can derive that the fallback rate follows a t−5/3 decay. Depending on the penetration factor, β, a star will be torn apart differently, and relativistic effects play a role. We have modified the standard version of the smoothed-particle hydrodynamics (SPH) code Gadget to include a relativistic treatment of the gravitational forces between the gas particles of a main-sequence (MS) star and a MBH. We include non-spinning post-Newtonian corrections to incorpore the periapsis shift and the spin-orbit coupling up to next-to-lowest order. We find that tidal disruptions around MBHs in the relativistic cases are underluminous for values starting at β≧2.25; i.e. the fallback curves produced in the relativistic cases are progressively lower compared to the Newtonian simulations as the penetration parameter increases. While the Newtonian cases display a total disruption, we find that all relativistic counterparts feature a survival core for penetration factors going to values as high as 12.05. We perform a additional dynamical numerical study which shows that the geodesics of the elements in the star converge at periapsis. We confirm these findings with an analytical study of the geodesic separation equation. The luminosity of TDEs must be lower than predicted theoretically due to the fact that the star will partially survive when relativistic effects are taken into account. A survival core should consistently emerge from any TDE with β≧2.25.

Tidal force effects and bound orbits in null naked singularity spacetime

Naked singularities form during the gravitational collapse of inhomogeneous matter clouds. The final nature of the singularity depends on the initial conditions of the matter properties and types of matter profiles. These naked singularities can also be divided into two types: null-like and timelike singularities. The spacelike singularity of the Schwarzschild black hole can be distinguished from the null and timelike naked singularity spacetimes. In light of this, we investigate the precession of timelike bound orbits in the null naked singularity spacetime, as well as tidal force effects and geodesic deviation features. As a result, we find that the orbital precession of the timelike bound orbits in null naked singularity spacetime could be distinguished from the Schwarzschild precession case. The radial component of the tidal force has an intriguing profile, whereas the angular component has a profile that is comparable to that of a Schwarzschild black hole scenario. The geodesic deviation equation is then solved numerically, yielding results that resemble Schwarzschild black holes. These characteristic features can then be used to discern among these singularities.

Particle and light motion in Lyra–Schwarzschild spacetime

Abstract In this paper we recall the static and spherically symmetric solution derived within LyST gravity. LyST stands for “Lyra Scalar-Tensor” and is a scalar-tensor proposal for the gravitational interaction modifying general relativity by adopting Lyra manifold as the spacetime substrate. Lyra manifold is characterized both by the metric tensor gµν(x) and the scale φ(x); the latter being an integral part of the definition of reference frame. In a previous work, we have launched the formal geometrical basis of LyST, we have proposed an associated action integral for the gravitational interaction, we have derived the field equations generalizing Einstein field equation, and we have solved these equation to obtain the generalized version of Schwarzschild line element in Lyra manifold. This lead us to Lyra-Schwarzschild spacetime, which is further explored in the present paper. Herein, the trajectories of test particles is thoroughly studied. This is done for both massive particles and massless particles. The geodesic equations are built and solved. The effective potential associated to the motion of these particles around the source is determined and characterized. The possible trajectories include gravitational capture, scattering, bounded orbits, stable and unstable circular orbits, near-horizon motion. The innermost circular orbit allowed for massive particles is determined and compared to the one predicted by GR. The last photon orbit is also calculated for Lyra-Schwarzschild metric. The equation for the periastron shift in the context of Lyra-Schwarzschild solution is constructed; its application to observational data could be useful to constrain the parameters typical of LyST, telling it apart from GR.

Tidal effects based on a GUP-induced effective metric

In this paper, we study tidal forces in the Schwarzschild black hole, whose metric explicitly includes a generalized uncertainty principle (GUP) effect. We also investigate interesting features of the geodesic equations and tidal effects that are dependent on the GUP parameter α related to a minimum length. Then, by solving the geodesic deviation equations explicitly with appropriate boundary conditions, we show that α in the effective metric affects both the radial and angular components of the geodesic equation, particularly near the singularities.

Shadow and quasinormal modes of rotating charged black holes in Kalb–Ramond gravity and implications from EHT data

The recent observations of the Event Horizon Telescope for the supermassive black holes M87* and Sgr A* can potentially test gravity theories and acquire further knowledge on black holes and gravity parameters. An essential field of research focuses on the shadow and quasinormal modes of black holes. Therefore, we used the Newman–Janis algorithm to get a rotating charged black hole solution in Kalb–Ramond gravity. Then, we looked at the event horizon radius, the static limit, and the ergoregion of spacetime. We derive the null geodesic equations and the effective potential for photon radial motion. We also investigate how the black hole’s Kalb–Ramond, spin, and charge parameters affect the shadow size and distortion. It is obtained that an increase in the Kalb–Ramond parameter causes a decrease in the radius and increases the distortion of the shadow size. We obtain constraints on black hole charge and spin using the data obtained from Event Horizon Telescope observations of M87* and Sgr A* by testing the effects of the Kalb–Ramond gravity parameter. We examine the energy emission rate using Hawking radiation and demonstrate that a rise in the Kalb–Ramond parameter diminishes this rate. Finally, we calculated and analyzed the typical shadow radius, equatorial, and polar quasinormal modes using the geometric–optic connection between quasinormal mode parameters and conserved quantities along geodesics.

Maximal acceleration in Rainbow gravity

In this paper, we derive maximal acceleration of a massive particle in Rainbow gravity. Using eight-dimensional phase-space metric compatible with Rainbow gravity, we obtain the maximal acceleration, valid up to first order in the Rainbow gravity parameter η. Using the positivity condition on maximal acceleration, we find that the upper bound on the Rainbow gravity parameter is of the order of for positron and for a black hole. After obtaining the expression for maximal acceleration for different choices of Rainbow functions, we derive corresponding modifications to Unruh temperature. Comparing with the observational value of the Unruh temperature, we find the upper bound on η as for positron radiation. We then derive geodesic equations for different choices of Rainbow functions and also obtain the Newtonian limit of these geodesic equations. We find that the changes in the value of maximum acceleration, maximum temperature and Newtonian force equation are dependent on the choices of Rainbow functions.

Memory effect of gravitational wave pulses in PP-wave spacetimes

In this paper, we study the gravitational memory effect in pp-wave spacetimes due to the passage of a pulse having the form of a ramp profile through this spacetime. We have analyzed the effect of this pulse on the evolution of nearby geodesics, and have determined analytical solutions of the geodesic equations in the Brinkmann coordinates. We have also examined the changes in the separation between a pair of geodesics and their velocity profiles. The separation (along x or y-direction) increases monotonically from an initial constant value. In contrast, the relative velocity grows from zero and settles to a final non-zero constant value. These resulting changes are retained as memory after the pulse dies out. The nature of this memory is similar to that determined by earlier workers using Gaussian, square, and other pulse profiles, thereby validating the universality of gravitational wave memory.

Self-similar collapse in Painlevé–Gullstrand coordinates

We report a family of self-similar exact solutions in General Relativity. The solutions are found in a Painleve-Gullstrand coordinate system but can also be transformed smoothly into a diagonal form. The solutions represent a gravitational collapse leading to three possible outcomes, depending on the parameter space: (i) a collapse followed by a bounce and dispersal of the clustered matter distribution, (ii) a rapid collapse followed by a bounce and an eventual re-collapse, and (iii) a standard collapse leading to zero proper volume. Profiles of the energy conditions are studied for all of the scenarios, and it is noted that a bounce is usually associated with a violation of the Null Energy Condition. It is found that more than one null surfaces (apparent horizons) can develop during the collapse. We also discuss that for a general metric tensor having a conformal symmetry, some regions of the parameter space allows a formation of null throat, much like a wormhole. Matching the metric with a Schwarzschild metric in Painleve–Gullstrand form leads to the geodesic equation for a zero energy falling particle in the exterior.

Kinetic equation for stochastic vector bundles

The kinetic equation is crucial for understanding the statistical properties of stochastic processes, yet current equations, such as the classical Fokker–Planck, are limited to local analysis. This paper derives a new kinetic equation for stochastic systems on vector bundles, addressing global scale randomness. The kinetic equation was derived by cumulant expansion of the ensemble-averaged local probability density function, which is a functional of state transition trajectories. The kinetic equation is the geodesic equation for the probability space. It captures global and historical influences, accounts for non-Markovianity, and can be reduced to the classical Fokker–Planck equation for Markovian processes. This paper also discusses relative issues concerning the kinetic equation, including non-Markovianity, Markov approximation, macroscopic conservation equations, gauge transformation, and truncation of the infinite-order kinetic equation, as well as limitations that require further attention.

Observational signatures of charged rotating traversable wormhole: shadows and light rings with different accretions

In this work, we investigate the optical appearance of qualitatively new observational features of accretion disk images around the charged rotating traversable wormhole (TWH) space-time for different spin, throat, and charge values. To accomplish this, we first consider the Hamilton–Jacobi method to derive the geodesic equations for the motion of photons and study the effects of parameters on the photon orbit in the observer’s sky. We found that each parameter affects the size and shape of the wormhole (WH) shadow and flatness is observed in the shadow because of spin and other parameters. To produce shadow images of sufficient visual quality but within manageable computational times, we adopt the ray-trace procedure and characterize the significant features of light trajectories on the observer’s screen, depending on the interaction between the space-time structure and the accretion disk. In addition, we consider the static spherically symmetric accretion flow model to observe the specific intensity around the traversable WH space-time geometry. It is found that the intensity and positions of the photon ring vary with respect to the involved parameters. In future observation, this type of study may provide a fertile playground to test the nature of compact objects, specifically the WH in the strong-field regime.

Imaging thick accretion disks and jets surrounding black holes

Based on the horizon-scale magnetofluid model developed in [1], we investigate the millimeter-wave images of a geometrically thick accretion disk or a funnel wall, i.e., the magnetofluid that encloses the base of the jet region, around a Kerr black hole. By employing the numerical method to solve the null geodesic and radiative transfer equations, we obtain the optical appearances at various observational angles and frequencies, generated by the thermal synchrotron radiation within the magnetofluid. For the thick disk, we specifically examine the impact of emission anisotropy on images, concluding that anisotropic synchrotron radiation could play an important role in the observability of the photon ring. For the funnel wall, we find that both the outflow and inflow funnel walls exhibit annular structures on the imaging plane. The outflow funnel wall yields a brighter primary image than the photon ring, whereas the inflow one does not. Based on our investigation, the inflow funnel wall model can not be ruled out by current observations of M87*.

Geometric analysis of the generalized surface quasi-geostrophic equations

We investigate the geometry of a family of equations in two dimensions which interpolate between the Euler equations of ideal hydrodynamics and the inviscid surface quasi-geostrophic equation. This family can be realised as geodesic equations on groups of diffeomorphisms. We show precisely when the corresponding Riemannian exponential map is non-linear Fredholm of index 0. We further illustrate this by examining the distribution of conjugate points in these settings via a Morse theoretic approach

Weak cosmic censorship conjecture by overspinning the Kerr–Newman–Kasuya black hole

We study the weak cosmic censorship conjecture by overspinning the Kerr–Newman–Kasuya black hole with test particle. We consider the geodesic equation of motion and find the certain conditions on the angular momentum of the black hole. We find the expression of [Formula: see text] and [Formula: see text] and then check whether the inequality satisfies or not. The results suggest that the Kerr–Newman–Kasuya black hole can be overspun showing the violation of cosmic censorship conjecture.

A Study of Geodesic Equation From Variational Principle

This paper employs variational principle in studying geodesic. In mathematical studies, a variational principle enables a problem to be solved employing calculus of variations that concerns seeking functions that increase the values of quantities that rely on those functions. For instance, the problem of ascertaining the shape of a hanging chain suspended at both ends can be solved using variational calculus. Hence, the variational principle is a function that lessens the gravitational potential energy of the chain. Geodesicis a procedure used in mathematics, specifically in Riemannian geometry that results in obtaining geodesics. Actually, these represent the paths of particles with no proper acceleration, their motion pleasing the geodesic equations. As the particles are subject to no appropriate acceleration, the geodesics generally signify the straightest path between two points in a bent spacetime. The article has investigated into the relation between the variation principle and geodesic equations that are particularly used in general relativity in physics as well.

Equivalence principle for quantum mechanics in the Heisenberg picture

We present an exact quantum observable analog of the weak equivalence principle for a ‘relativistic’ quantum particle. The quantum geodesic equations are obtained from Heisenberg equations of motion as an exact analog of a fully covariant classical Hamiltonian evolution picture, with the proper identification of the canonical momentum variables as p µ , rather than p µ . We discuss the meaning of the equations in relation to projective measurements as well as equations with solution curves as ones in the noncommutative geometric picture of spacetime, and a plausible approach to quantum gravity as a theory about quantum observables as physical quantities including the notion of quantum coordinate transformation.

The space of closed G2-structures. I. Connections

Abstract In this article, we develop foundational theory for geometries of the space of closed G2-structures in a given cohomology class as an infinite-dimensional manifold. We construct Levi-Civita connections for Sobolev-type metrics, formulate geodesic equations and analyze the variational structures of torsion-free G2-structures under these Sobolev-type metrics.

Classical and quantum solutions in scalar field cosmology via the Eisenhart lift and linearization

This study introduces a novel approach for solving the cosmological field equations within scalar field theory by employing the Eisenhart lift. The field equations are reformulated as a system of geodesic equations for the Eisenhart metric. In the case of an exponential potential, the Eisenhart metric is shown to be conformally flat. By applying basic geometric principles, a new set of dynamical variables is identified, allowing for the linearization of the field equations and the derivation of classical cosmological solutions. However, the quantization of the Eisenhart system reveals a distinct set of solutions for the wavefunction, particularly in the presence of symmetry breaking at the quantum level.

To geometric aspects of in nite-dimensional dynamical systems

The main goal of the work is to construct analogues of Christoffel symbols for infinitedimensional systems and on this basis to obtain geodesic equations for such systems. These analogies are of particular interest in terms of identifying the relationship between the dynamics of systems with an infinite number of degrees of freedom and Riemannian geometry, as well as geometry defined by the pseudo-Riemannian metric.