Geodesic Equations Research Articles

The shadow and gamma-ray bursts of a Schwarzschild black hole in asymptotic safety

The effects and rules of the dimensionless parameter ξ on neutrino annihilation ν+ν¯→e−+e+ dominated gamma-ray bursts are analysed and investigated within the context of black holes in asymptotic safety. We also computationally model photon orbits around black holes, as photons and neutrinos have the same geodesic equations near black holes. We show that the black hole shadow radius decreases with increasing ξ. Calculations are made to determine the temperature of the accretion disk surrounding the black hole and the ratio Q̇/Q̇Newt of energy deposition per unit time and compared to that of the Newtonian scenario. The accretion disk temperature peaks at a higher temperature due to quantum gravity corrections, which increases the probability of neutrino emission from the black hole. It is interesting to note that larger quantum gravity effects cause the ratio value to significantly decline. In the neutrino–antineutrino annihilation process, the energy deposition rate is sufficient even while the energy conversion is inhibited because of quantum corrections. Gamma-ray bursts might originate from the corrected annihilation process. Additionally, we examine the derivative dQ̇/dr about the star radius r. The findings demonstrate that the ratio is lowered by the black hole’s quantum influence. The neutrino pair annihilation grows weaker the more prominent the influence of quantum gravity.

The deflection angle and quasi-periodic oscillations of an extended gravitational decoupled black hole solution

In weak field limits, we compute the deflection angle of a gravitational decoupling extended black hole (BH) solution. We obtained the Gaussian optical curvature by examining the null geodesic equations with the help of Gauss–Bonnet theorem (GBT). We also looked into the deflection angle of light by a black hole in weak field limits with the use of the Gibbons–Werner method. We verify the graphical behavior of the black hole after determining the deflection angle of light. Additionally, in the presence of the plasma medium, we also determine the deflection angle of the light and examine its graphical behavior. Furthermore, we compute the Einstein ring via gravitational decoupling extended black hole solution. We also compute the quasi-periodic oscillations and discuss their graphical behavior.

Blind source separation algorithm for biomedical signal based on lie group manifold

Independent Component Analysis (ICA) is a powerful tool for solving blind source separation problem in biomedical engineering. The traditional ICA algorithm ignores the Lie group structure of constrained matrix manifold. In this paper, a gradient descent algorithm on Lie group manifold is proposed based on the geometric framework of optimization algorithm on Riemann manifold. Firstly, the orthogonal constraint separation matrices are regarded as a Lie group manifold, and the gradient of ICA objective function on the Lie group manifold is given by using Riemann metric; Secondly, the geodesic equation of the current iteration point along the gradient descent direction is calculated; Finally, a new iteration point is obtained by moving a certain step along the geodesic line, meanwhile, the step length can be adjusted adaptively. Simulation results show that the gradient algorithm on Lie group manifold is feasible for blind Source Separation, and its performance (convergence speed, stability and error) is better than other algorithms.

Information geometry of the Otto metric

AbstractWe introduce the dual of the mixture connection with respect to the Otto metric which represents a new kind of exponential connection. This provides a dual structure consisting of the mixture connection, the Otto metric as a Riemannian metric, and the new exponential connection. We derive the geodesic equation of this exponential connection, which coincides with the Kolmogorov forward equation of a gradient flow. We then derive the canonical contrast function of the introduced dual structure.

Observational signatures of strong gravitational lensing in GUP-modified Schwarzschild black holes

We investigate gravitational lensing in strong-field limit by a static spherically symmetric GUP-modified Schwarzschild black holes having additional parameter ϵ . We found that influences of negative and positive values of ϵ parameter differ from Schwarzschild black hole. We expand the deflection angle of the photon in the neighborhood of complete capture, defining a strong field limit in opposition to the standard weak field limit. This expansion is performed for a generic spherically symmetric spacetime without referencing the field equations and assuming that the light ray follows the geodesics equation. We prove that the deflection angle always diverges logarithmically when the minimum impact parameter is reached. We apply this general formalism to Schwarzschild GUP black holes. We then compare the coefficients characterizing these metrics and find that different strong field limits characterize different collapsed objects. The strong field limits coefficients, such as the relativistic images’ position and magnification, are directly connected to the observables. As a concrete example, we consider the black hole at the center of our galaxy and estimate the optical resolution needed to investigate its strong field behavior through its relativistic images.

Shadows and quasinormal modes of rotating black holes in Horndeski theory: Parameter constraints using EHT observations of M87* and Sgr A*

One way to test the presence of scalar & vector fields in the spacetime of black holes is to get constraints on the fields and black hole parameters using astrophysical observations such as black hole shadow, quasiperiodic oscillations, etc. In this work, we aimed to investigate spacetime properties around black holes in Horndeski gravity. First, we obtain a rotating black hole solution using the Newman–Janis Algorithm (NJA), and we study the event horizon, statistic limits, and ergoregion of the black hole spacetime. In addition, we calculate the effective gravitational mass of the black hole. We derive equations for null geodesics and show the combined effects of the black hole spin and Horndeski gravity parameter on the shadow shape of the black hole and its radius distortion. We find that black hole spin and Horndeski parameters cause an increase in the values of shadow radius and its deformation. Next, we obtain constraints for the upper and lower values for black hole spin and Horndeski parameters using event horizon telescope (EHT) data from Sgr A* and M87*shadow sizes. Finally, we investigated the shadow radius, equatorial, and polar quasinormal modes using the geometric-optic relationship between the parameters of the quasinormal mode and the conserved values along the geodesics in the spacetime of the rotating Horndeski black hole.

Covariant and manifestly projective invariant formulation of Thomas-Whitehead gravity

Thomas-Whitehead (TW) gravity is a recently formulated projectively invariant extension of Einstein-Hilbert gravity. Projective geometry was used long ago by Thomas to succinctly package equivalent paths encoded by the geodesic equation. Projective invariance in gravity has further origins in string theory through a geometric action constructed from the method of coadjoint orbits using the Virasoro algebra. A projectively invariant connection arises from this construction, a part of which is known as the diffeomorphism field. TW gravity exploits projective Gauss-Bonnet terms in the action functional to endow the diffeomorphism field with dynamics, while allowing the theory to collapse to general relativity in the limit that the diffeomorphism field vanishes and the connection becomes Levi-Civita. In the original formulation of TW gravity, the diffeomorphism field is projectively invariant but not tensorial and the connection is projectively invariant but not affine. In this paper we reformulate TW gravity in terms of projectively invariant tensor fields and a projectively invariant covariant derivative, derive field equations respecting these symmetries, and show that the field equations obtained are classically equivalent across formulations. This provides a “Rosetta Stone” between this newly constructed covariant and projective invariant formulation of TW gravity and the original formulation that was manifestly projective invariant, but not covariant. Published by the American Physical Society 2024

Fractional mimetic dark matter: A fractional action-like variational approach

In this paper, we propose a nonlocal extension of the mimetic dark matter model based on the FALVA implementation of fractional calculus. Our primary objective to explore how certain properties of dark matter can be modeled within the FALVA framework rather than formulate a phenomenological model. We begin by constructing the action functional of the cosmological extension of the mimetic dark model and deriving the nonlinear equations of motion in the general case. Next, we focus on a fractional-power homogeneous mimetic dark field with an exponential expansion factor. We derive the equation of motion for the lapse field and obtain its general solution. Analytical solutions are then obtained for small time intervals and arbitrary values of the fractionality parameter. These solutions enable us to establish the physical line element of spacetime, incorporating the mimetic dark matter field. From this line element, we derive the Ricci tensor, Ricci scalar, and geodesic equations.

A five-dimensional vacuum-defect wormhole space-time

In this study, we present a novel extension to the Klinkhamer vacuum-defect model by incorporating a fifth spatial dimension. This modification results in the formulation of a comprehensive five-dimensional wormhole space-time. Crucially, this extension preserves its classification as a vacuum solution to the field equations, thereby automatically satisfying all energy conditions. We demonstrate that this five-dimensional vacuum-defect space-time can be expressed in a pure canonical form, and thus, locally reduces to five-dimensional Minkowski space. Finally, we explore the geodesic equations in the vicinity of this five-dimensional vacuum-defect wormhole, offering insights into its intriguing properties.

Deflection of light by a compact object with electric charge and magnetic dipole in Einstein–Born–Infeld gravity

We consider the bending of light around a compact astrophysical object with both the electric field and the magnetic field in Einstein–Born–Infeld theory. From the null geodesic of a light ray passing a massive object with electric charge and magnetic dipole, the effective metric was obtained from the light cone condition reflecting the nonlinear electromagnetic effects. We found the asymptotic form of the effective metric up to the first order in gravitational constant G and Born–Infeld parameter 1/β2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1/\\beta ^2$$\\end{document} on the equatorial plane. Then we compute the bending angle of light from the geodesic equation. The result includes particular cases where only one type of field is present taking the appropriate limits.

The common solution space of general relativity

We review the solution space for the field equations of Einstein's General Relativity for various static, spherically symmetric spacetimes. We consider the vacuum case, represented by the Schwarzschild black hole; the de Sitter-Schwarzschild geometry, which includes a cosmological constant; the Reissner-Nordström geometry, which accounts for the presence of charge. Additionally we consider the homogenenous and anisotropic locally rotational Bianchi II spacetime in the vacuum. Our analysis reveals that the field equations for these scenarios share a common three-dimensional group of point transformations, with the generators being the elements of the D⊗sT2 Lie algebra, known as the semidirect product of dilations and translations in the plane. Due to this algebraic property the field equations for the aforementioned gravitational models can be expressed in the equivalent form of the null geodesic equations for conformally flat geometries. Consequently, the solution space for the field equations is common, and it is the solution space for the free particle in a flat space. This approach open new directions on the construction of analytic solutions in gravitational physics and cosmology.

A Riemannian geometric approach for timelike and null spacetime geodesics

The geodesic motion in a Lorentzian spacetime can be described by trajectories in a 3-dimensional Riemannian metric. In this article we present a generalized Jacobi metric obtained from projecting a Lorentzian metric over the directions of its Killing vectors. The resulting Riemannian metric inherits the geodesics for asymptotically flat spacetimes including the stationary and axisymmetric ones. The method allows us to find Riemannian metrics in three and two dimensions plus the radial geodesic equation when we project over three different directions. The 3-dimensional Riemannian metric reduces to the Jacobi metric when static, spherically symmetric and asymptotically flat spacetimes are considered. However, it can be calculated for a larger variety of metrics in any number of dimensions. We show that the geodesics of the original spacetime metrics are inherited by the projected Riemannian metric. We calculate the Gaussian and geodesic curvatures of the resulting 2-dimensional metric, we study its near horizon and asymptotic limit. We also show that this technique can be used for studying the violation of the strong cosmic censorship conjecture in the context of general relativity.

Revisiting the fastest way to circle a black hole

The shortest time for a null particle traveling between two arbitrary points outside a static spherically symmetric black hole is revisited. We introduce a functional for the time taken by a null particle in traveling on the path between the two points. By variating the time functional, we analyze the possible path with the shortest travel time for the null particle. It is analytically proven that the Euler–Lagrange equation corresponding to the time-functional for finding the path with the shortest traveling time is equivalent to the geodesics equation. This is in agreement with Hod’s conjecture on the fastest way to circle a black hole. We apply the formalism to the dirty black hole in Einstein-square-root nonlinear electrodynamics-dilaton theory. We calculate explicitly the time measured by an asymptotic observer which is needed for a null particle to circle the dirty black hole. Accordingly, a null particle circling the dirty black hole on an almost circular path of radius infinity achieves the shortest time.

Metrics and geodesics on fuzzy spaces

Abstract We study the fuzzy spaces (as special examples of noncommutative manifolds) with their quasicoherent states in order to find their pertinent metrics. We show that they are naturally endowed with two natural “quantum metrics” which are associated with quantum fluctuations of “paths”. The first one provides the length the mean path whereas the second one provides the average length of the fluctuated paths. Onto the classical manifold associated with the quasicoherent state (manifold of the mean values of the coordinate observables in the state minimising their quantum uncertainties) these two metrics provides two minimising geodesic equations. Moreover, fuzzy spaces being not torsion free, we have also two different autoparallel geodesic equations associated with two different adiabatic regimes in the move of a probe onto the fuzzy space. We apply these mathematical results to quantum gravity in BFSS matrix models, and to the quantum information theory of a controlled qubit submitted to noises of a large quantum environment.

Impact of chaplygin-like equation of state on Joule–Thomson expansion and tidal forces of AdS black holes

This study examines a recently hypothesized black hole. We study the Joule–Thomson coefficient, the inversion temperature and also the isenthalpic curves in the Ti−Pi plane. The Joule–Thomson coefficient, the inversion curves and the isenthalpic curves are discussed in AdS black holes surrounded by Chaplygin dark fluid. In T−P plane, the inversion temperature curves and isenthalpic curves are obtained with different parameters. Next, we explore the radial time-like geodesics that leads us to explore the tidal force effects for a radially in-falling particle in such black hole spacetime. We also numerically solve the geodesic deviation equation for two nearby radial geodesics for a freely falling particle. Our analysis shows that contrary to the Schwarzschild spacetime, the tidal forces do not become zero at spatial infinity due to the lack of asymptotic flatness because of the presence of a non-zero cosmological constant. The geodesic separation profile shows an oscillating trend and depends on the dynamic spacetime parameters q,B and Λ.

Black hole evaporation process and Tangherlini–Reissner–Nordström black holes shadow

In this article, we study the black hole evaporation process and shadow property of the Tangherlini-Reissner-Nordström (TRN) black holes. The TRN black holes are the higher-dimensional extension of the Reissner-Nordström (RN) black holes and are characterized by their mass M, charge q, and spacetime dimensions D. In higher-dimensional spacetime, the black hole evaporation occurs rapidly, causing the black hole’s horizon to shrink. We derive the rate of mass loss for the higher-dimensional charged black hole and investigate the effect of higher-dimensional spacetime on charged black hole shadow. We derive the complete geodesic equations of motion with the effect of spacetime dimensions D. We determine impact parameters by maximizing the black hole’s effective potential and estimate the critical radius of photon orbits. The photon orbits around the black hole shrink with the effect of the increasing number of spacetime dimensions. To visualize the shadows of the black hole, we derive the celestial coordinates in terms of the black hole parameters. We use the observed results of M87 and Sgr A∗ black hole from the Event Horizon Telescope and estimate the angular diameter of the charge black hole shadow in the higher-dimensional spacetime. We also estimate the energy emission rate of the black hole. Our finding shows that the angular diameter of the black hole shadow decreases with the increasing number of spacetime dimensions D.

Space-time structures, light trajectories and shadows for Kerr-Newman-AdS black hole with surrounding perfect fluid matter in Rastall gravity

We present the horizon structures and the shadows for Kerr-Newman-AdS black hole located in perfect fluid matter field in Rastall gravity. By considering the Hamilton-Jacobi equation and Carter constant separable method, we obtain the null geodesic equations of the black hole space-time as well as the unstable circular photon orbits in terms of the impact parameters. The horizon structures of the space-time are calculated numerically. The results show that there are four roots for a general case of the horizon equation (the Cauchy horizon r−h , the event horizon r+h , the cosmological horizon r q of Rastall gravity and the cosmological horizon r c of cosmological constant), which lead to a specific characteristic with the subregions of the space-time. The influences of the cosmological horizons r q and r c on the light trajectories around the black hole are also investigated. Furthermore, we study the effects of the parameters, including the perfect fluid matter intensity N s , the Rastall parameter k λ, the cosmological constant Λ, the black hole spin a and the black hole charge Q on black hole shadows in detail.

Mass fluctuations in non-rotating BTZ black holes

We investigate the impact of oscillations of a black hole's mass around its average value on the three-dimensional black hole geometry. Drawing on a classical framework that conceptualizes fluctuations near an event horizon as mass variations, we introduce a model where the metric of a black hole, formed from the collapse of a massive null shell, exhibits oscillatory behavior in spherical modes. This dynamic is encapsulated by a non-rotating BTZ-Vaidya solution, characterized by the black hole mass fluctuating at a resonant frequency ω and a small amplitude parameter μ0. Using a perturbative approach, solutions to the null geodesic equation are determined up to the second order in μ0. The temporal fluctuations of the event horizon's location induce alterations in the thermodynamic variables' values. Upon calculating the time-averaged values, the mean Hawking temperature experiences a slight decrease due to these fluctuations, while the mean entropy exhibits an increase. Furthermore, the study explores the influence of these fluctuations on the trajectories of null rays near the horizon, which ultimately reach the anti-de Sitter boundary at late times. The analytical computation of the general solution for the perturbed rays up to the second order underscores the novel approach of this study in examining the effects of mass oscillations on black hole thermodynamics and geometry, thereby contributing a unique perspective to the field.

Higher-dimensional MOG dark compact object: shadow behaviour in the light of EHT observations

Consideration of extra spatial dimensions is motivated by the unification of gravity with other interactions, the achievement of the ultimate framework of quantum gravity, and fundamental problems in particle physics and cosmology. Much attention has been focused on the effect of these extra dimensions on the modified theories of gravity. Analytically examining astrophysical phenomena like black hole shadows is one approach to understand how extra dimensions would affect the modified gravitational theories. The purpose of this study is to derive a higher-dimensional metric for a dark compact object in STVG theory and then examine the behaviour of the shadow shapes for this solution in STVG theory in higher dimensions. We apply the Carter method to formulate the geodesic equations and the Hamilton–Jacobi method to find photon orbits around this higher-dimensional MOG dark compact object. We investigate the effects of extra dimensions and the STVG parameter α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} on the black hole’s shadow size. Next, we compare the shadow radius of this higher-dimensional MOG dark compact object to the shadow size of the supermassive black hole M87*, which has been realized by the Event Horizon Telescope (EHT) collaborations, in order to restrict these parameters. We find that extra dimensions in the STVG theory typically lead to a reduction in the shadow size of the higher-dimensional MOG dark compact object, whereas the effect of parameter α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} on this black hole’s shadow is suppressible. Remarkably, given the constraints from EHT observations, we find that the shadow size of the four-dimensional MOG dark compact object lies in the confidence levels of the EHT data. Finally, we investigate the issue of acceleration bounds in higher-dimensional MOG dark compact object in confrontation with EHT data of M87*.

Displacement versus velocity memory effects from a gravitational plane wave

This article demonstrates that additionally to the well-known velocity memory effect, a vacuum gravitational plane wave can also induce a displacement memory on a couple of test particles. A complete classification of the conditions under which a velocity or a displacement memory effect occur is established. These conditions depend both the initial conditions of the relative motion and on the wave profile. The two cases where the wave admits a pulse or a step profile are treated. Our analytical expressions are then compared to numerical integrations to exhibit either a velocity or a displacement memory, in the case of these two families of profiles. Additionally to this classification, the existence of a new symmetry of polarized vacuum gravitational plane wave under Möbius reparametrization of the null time is demonstrated. Finally, we discuss the resolution of the geodesic deviation equation by means of the underlying symmetries of vacuum gravitational plane wave.