Geodesic Equations Research Articles

Exploring tidal force effects and shadow constraints for Schwarzschild-like black hole in Starobinsky–Bel-Robinson gravity

The current manuscript deals with the tidal force effects, geodesic deviation, and shadow constraints of the Schwarzschild-like black hole theorised in Starobinsky–Bel-Robinson gravity exhibiting M-theory compactification. In the current analysis, we explore the radial and angular tidal force effects on a radially in-falling particle by the central black hole, which is located in this spacetime. We also numerically solve the geodesic deviation equation and study the variation of the geodesic separation vector with the radial coordinate for two nearby geodesics using suitable initial conditions. All the obtained results are tested for Sag A* and M87* by constraining the value of the stringy gravity parameter β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document} using the shadow data from the event horizon telescope observations. All the results are compared with Schwarzschild black hole spacetime. In our study, we found that both the radial and angular tidal forces experienced by a particle switch their initial behaviour and turn compressive and stretching, respectively, before reaching the event horizon. The geodesic deviation shows an oscillating trend as well for the chosen initial condition. For the constrained value of β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}, we see that the spacetime geometry generated by Sag A* and M87* is effectively same for both Schwarzschild and Starobinsky–Bel-Robinson black hole. Furthermore, we also calculated the angular diameter of the shadow in Starobinsky–Bel-Robinson black hole and compared with the Schwarzschild black hole. It is observed that the angular diameter of shadow for M87* and Sgr A* in Starobinsky–Bel-Robinson black hole is smaller than the Schwarzschild black hole. The calculated results satisfy the event horizon telescope observational constraints.

Action Principle for Scale Invariance and Applications (Part I)

On the basis of a general action principle, we revisit the scale invariant field equation using the cotensor relations by Dirac (1973). This action principle also leads to an expression for the scale factor λ, which corresponds to the one derived from the gauging condition, which assumes that a macroscopic empty space is scale-invariant, homogeneous, and isotropic. These results strengthen the basis of the scale-invariant vacuum (SIV) paradigm. From the field and geodesic equations, we derive, in current time units (years, seconds), the Newton-like equation, the equations of the two-body problem, and its secular variations. In a two-body system, orbits very slightly expand, while the orbital velocity keeps constant during expansion. Interestingly enough, Kepler’s third law is a remarkable scale-invariant property.

Hidden symmetry in the double copy

We show that the Killing tensor of the Kerr spacetime has an analogue in the Kerr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{\ extrm{Kerr}} $$\\end{document} gauge theory solution related to it by the classical double copy. This hidden symmetry of Kerr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{\ extrm{Kerr}} $$\\end{document} leads to an additional constant of motion for color-charged point particles moving in it, implying integrability of the equation of motion. These are the gauge theory counterparts to the Carter constant and the integrability of the geodesic equation in a Kerr background.

Geodesic equations for guided wave helical path separation for a pipe bend

The sections of pipe bends are susceptible to wall thinning by flow-accelerated corrosion due to the sudden change in the fluid flow direction and velocity. Compared to existing screening approaches, Guided Wave Tomography (GWT) has been demonstrated to provide more accurate information about the defect. In GWT an inversion problem is solved, in which a forward model is used to predict a synthetic dataset for a given physical model. Two-dimensional (2D) forward models are preferred for the inversion scheme due to the feasibility of combining these with tomographic algorithms. Since 2D models resemble the GW propagation of a plate, they do not consider the cyclic nature of the pipe. To overcome this limitation, a robust helical path separation was introduced in Huthwaite and Seher (2015) for a straight pipe. However, it relies on time-reversing the measured wave field ϕ(t) as a function of the propagation distance xs,r from the source to the receiver. Although ray-tracing and grid-based methods are capable of computing the distance xs,r for complex geometries such as the pipe bend, they are complex and time-consuming. Hence, in this paper a straight forward approach is proposed to compute xs,r based on the pipe geometry and the geodesic equations of a torus. Then the computed distance xs,r is used for removing the helical path trajectories. Compared to the distance obtained by using the Hilbert envelope, the results show that the geodesic distances preserve the information of the wavefront of interest, including the focusing effect and the scattered wave field from a Hann-shaped defect. The technique is applied to experimental data and demonstrated to separate the wavefront of interest.

4D Einstein-Gauss-Bonnet black hole in Power-Yang-Mills field: a shadow study

We consider a static black hole immersed in the Power-Yang-Mills field in four-dimensional Einstein-Gauss-Bonnet gravity and investigate the effect of various parameters on the radius of the photon sphere. The modified form of the Newman-Janis algorithm is used for obtaining a rotating black hole solution in this gravity. Further, we try to explore the influence of the Yang-Mills magnetic charge Q with power q, Gauss-Bonnet parameter α, and spin a on the horizon radius. The geodesic equations are constructed by incorporating the Hamilton-Jacobi formalism. The radial component of the geodesic equations gives the effective potential which is further used in deriving the mathematical structure for the shadows by using Bardeen's procedure for a fixed observer at infinity. The shadows are calculated and plotted in terms of two celestial coordinates for an equatorial observer. It is observed that all the parameters have a very significant effect on the shadow and related physical observables. We also obtain the constraint values for the spin, magnetic charge, and Gauss-Bonnet parameters, using the shadow size of supermassive black holes Sagittarius A* and M87* from the EHT observations for the cases of q = 0.6 and q = 0.9. It is shown that there are upper and lower bounds for the charge and spin of M87* at q = 0.6, while only the upper bounds for the charge and spin of Sagittarius A*. Finally, we investigate the energy emission rate in the Hawking radiation around the 4D Einstein-Gauss-Bonnet black hole in the Power-Yang-Mills field.

Thick branes via higher order field theory models with exponential and power-law tails

In this work, we obtain exact thick brane models in 4+1 dimensions generated by higher order field theory kinks, inspired by specific potentials for phi ^{10} and phi ^{18} models. We verify that the geodesic equation along the fifth dimension confirms the confining effects of the scalar field on the brane for all of these models. These models provide new solutions with exponential and power-law tails which live in different topological sectors. We show that the resulting branes of specific exponential law models do not possess Z_2-symmetry. Furthermore, we examine the stability of the thick branes, by determining the sign of the w^2 term in the expansion of the potential for the resulting Schrödinger-like equation. It turns out that two of the three models of the phi ^{10} brane are stable, while another contains unstable modes for certain ranges of the model parameters. We also show that the brane solution from the specific phi ^{18} models are stable, while the others involve neutral equilibrium. The asymptotic behaviour of the brane solutions are also discussed.

Photon region and shadow of a rotating 5D black string

To explore the possible clues for the extra dimension from the Event Horizon Telescope (EHT) observations, we study the shadow of the rotating 5D black string in General Relativity (GR). Instead of investigating the shadow in the effective 4D theory, we concern the motion of photons along the extra dimension z with a conserved momentum P_z, which appears as an effective mass in the geodesic equations of photons. The existence of P_z enlarges the photon regions and the shadow of the rotating 5D black string while it has slight impact on the distortion. The EHT observations of M87* and SgrA* can rule out the black string model with an infinite length along the extra dimension, and support the hypothesis that the extra dimension is compact to avoid the Gregory-Laflamme (GL) instability, where the length of the black string/the compact extra dimension can be constrained as 2.03125~textrm{mm} lesssim ell lesssim 2.6~textrm{mm} and 2.28070~textrm{mm} lesssim ell lesssim 2.6~textrm{mm} respectively.

Derivation of MOND from Hossenfelder–Verlinde gravity

Verlinde proposed emergent gravity, which naturally explains the Tully–Fisher relation, an empirical relation in galaxy rotation curves. Inspired by this theory, Hossenfelder constructed a covariant formulation of Verlinde’s emergent gravity. In this work, we show that the equation of motion gains an extra acceleration in addition to the usual geodesic equation, according to Hossenfelder’s theory. Moreover, we show that the extra acceleration is precisely the square root of the Newtonian gravitational acceleration if the mass of the imposter field is negligible, thus completing the proof that Hossenfelder’s theory reduces to modified Newtonian dynamics (MOND) and determining which version of MOND it reduces to. We also obtain the value of L in Hossenfelder–Verlinde gravity theory, which is a constant, contrary to what Hossenfelder claimed. Finally, we suggest how the Newtonian limit that suitably describes our observations in Solar System is recovered in Hossenfelder’s theory, by considering the mass of the imposter field.

Quasi-Hyperbolically Symmetric γ-Metric.

We carry out a systematic study on the motion of test particles in the region inner to the naked singularity of a quasi-hyperbolically symmetric γ-metric. The geodesic equations are written and analyzed in detail. The obtained results are contrasted with the corresponding results obtained for the axially symmetric γ-metric and the hyperbolically symmetric black hole. As in this latter case, it is found that test particles experience a repulsive force within the horizon (naked singularity), which prevents them from reaching the center. However, in the present case, this behavior is affected by the parameter γ which measures the departure from the hyperbolical symmetry. These results are obtained for radially moving particles as well as for particles moving in the θ-r subspace. The possible relevance of these results in the explanation of extragalactic jets is revealed.

Complete separation of variables in the geodesic Hamilton-Jacobi equation in four dimensions

We list all metrics of arbitrary signature in four dimensions which admit complete separation of variables in the Hamilton–Jacobi equation for geodesic Hamiltonians. There are only ten classes of separable metrics admitting Killing vector fields, indecomposable quadratic conservation laws, and coisotropic coordinates. Canonical separable metrics parameterized by several (up to twelve) arbitrary functions of single coordinates are written explicitly. The full set of independent conservation laws in involution for each canonical metrics is also found.

Classification of Killing magnetic curves in ℍ3

In this paper, we study classification of magnetic curves corresponding to Killing vector fields of [Formula: see text]. First, we solve the geodesic equation analytically. Then we calculate the trajectories generated by all the six Killing vector fields, which are considered as magnetic field vectors, by using perturbation method up to first-order with respect to the strength of the magnetic field. We present a comparison of our solution with the numerical solution for one case. We also prove that 3-dimensional [Formula: see text]-Kenmotsu manifolds cannot have any magnetic vector field in the direction of their Reeb vector fields.

Metric fluctuations in higher-dimensional black holes

We investigated the impact of metric fluctuations on the higher-dimensional black hole geometry. We generalized the four-dimensional model to higher dimensions to treat quantum vacuum fluctuations by the classical approach. A fluctuating black hole is portrayed by a higher-dimensional Vaidya metric with a spherically oscillating mass. Assuming a small fluctuation amplitude, we employed a perturbation method to obtain a radially outgoing null geodesic equation up to the second order in the fluctuation. Furthermore, the fluctuation of the event horizon up to the second order depends on the number of spacetime dimensions. Therefore, the time-averaged values of the thermodynamic variables defined at the horizon also feature dimension-dependent correction terms. A general solution was obtained for rays propagating near the horizon within a fluctuating geometry. Upon examining this in a large D limit, we found that a complete solution can be obtained in a compact form.

Analog of a Laplace–Runge–Lenz vector for particle orbits (time-like geodesics) in Schwarzschild spacetime

In Schwarzschild spacetime, time-like geodesic equations, which define particle orbits, have a well-known formulation as a dynamical system in coordinates adapted to the time-like hypersurface containing the geodesic. For equatorial geodesics, the resulting dynamical system is shown to possess a conserved angular quantity and two conserved temporal quantities, whose properties and physical meaning are analogs of the conserved Laplace–Runge–Lenz vector, and its variant known as Hamilton’s vector, in Newtonian gravity. When a particle orbit is projected into the spatial equatorial plane, the angular quantity yields the coordinate angle at which the orbit has either a turning point (where the radial velocity is zero) or a centripetal point (where the radial acceleration is zero). This is the same property as the angle of the respective Laplace–Runge–Lenz and Hamilton vectors in the plane of motion in Newtonian gravity. The temporal quantities yield the coordinate time and the proper time at which those points are reached on the orbit. In general, for orbits that have a single turning point, the three quantities are globally constant, and for orbits that possess more than one turning point, the temporal quantities are just locally constant as they jump at every successive turning point, while the angular quantity similarly jumps only if an orbit is precessing. This is analogous to the properties of a generalized Laplace–Runge–Lenz vector and generalized Hamilton vector which are known to exist for precessing orbits in post-Newtonian gravity. The angular conserved quantity is used to define a direct analog of these vectors at spatial infinity.

Geodesics as Equations of Motion

The empirical description of the evolution of a physical system is the account of observed changes of state of the system in time, the measure of time being that of the Systeme International. To describe the Sun-Mercury system Einstein proposed a model based on the assumption that the evolution of the system, i.e. its relative space-time trajectory describes a geodesic on space-time endowed with the Schwarzschild metric. The evolution parameter of a geodesic is, however, an affine parameter or equivalently the proper time. However, the geodesic equation can be expressed in coordinate time. It has been shown that the solution of this equation reproduces the empirical evolution. A possiblee interptetation of the proper time is presented.

Asymptotically locally flat and AdS higher-dimensional black holes of Einstein–Horndeski–Maxwell gravity in the light of EHT observations: shadow behavior and deflection angle

Unification of gravity with other interactions, achieving the ultimate framework of quantum gravity, and fundamental problems in particle physics and cosmology motivate to consider extra spatial dimensions. The impact of these extra dimensions on the modified theories of gravity has attracted a lot of attention. One way to examine how extra dimensions affect the modified gravitational theories is to analytically investigate astrophysical phenomena, such as black hole shadows. In this study, we aim to investigate the behavior of the shadow shapes of higher-dimensional charged black hole solutions including asymptotically locally flat (ALF) and asymptotically locally AdS (ALAdS) in Einstein–Horndeski–Maxwell (EHM) gravitational theory. We utilize the Hamilton–Jacobi method to find photon orbits around these black holes as well as the Carter approach to formulate the geodesic equations. We examine how extra dimensions, negative cosmological constant, electric charge, and coupling constants of the EHM gravity affect the shadow size of the black hole. Then, we constrain these parameters by comparing the shadow radius of these black holes with the shadow size of M87* supermassive black hole captured by the Event Horizon Telescope (EHT) collaborations. We discover that generally the presence of extra dimensions within the EHM gravity results in reducing the shadow size of higher-dimensional ALF and ALAdS charged black holes, whereas the impact of electric charge on the shadow of these black holes is suppressible. Interestingly, we observe that decreasing the negative cosmological constant, i.e., increasing its absolute value, leads to increase the shadow size of the ALAdS charged higher-dimensional black hole in the EHM gravity. Surprisingly, based on the constraints from EHT observations, we discover that only the shadow size of the four dimensional ALF charged black hole lies in the confidence levels of EHT data, whereas owing to the presence of the negative cosmological constant, the shadow radius of the four, five, and seven dimensional ALAdS charged black holes lie within the EHT data confidence levels.

Geodesics on metrics of Eguchi–Hanson type

Geodesic equations are solved when at least two of theta , phi and psi are constant, or r is constant, on scalar flat metrics of Eguchi–Hanson type. They can also be solved also on Eguchi–Hanson metrics which are Ricci flat if only phi is constant. However, the explicit solution of the geodesic equations is not available yet if only psi is constant.

Shadows of quintessential dark energy black holes in the domain of outer communication

The rotating black holes in the quintessential dark energy correspond to three horizons: inner, outer, and quintessential horizon. The domain of outer communication is the region between outer and quintessential horizon. Here in this work, we study the photon region and shadows of the quintessential dark energy black holes when the observer stays statically in the domain of outer communication. The quintessential dark energy black holes shadow characterizes by its mass (M), spin parameter (a), quintessential dark energy parameter (ωq), and normalization factor (γ). The dark energy parameter ωq can take values between −1.1<ωq<−1/3 and follows the equation of state ωq = pressure(p)/energy density(ρq). This state parameter significantly affects the shape and the size of the black hole shadow. We generalize all the geodesic equations of motion for ωq and obtain relation to visualizing the black hole shadow by a static observer at any arbitrary distance in the domain of outer communication. We estimate the black hole shadow observables: radius Rs, distortion parameter δs, and the shadow area A. Using the numerical values of shadow radius Rs and area A, we obtain the angular diameter (θd) of the black hole shadow. The angular diameter of the M87 and Sgr A∗ black holes shadow are 42±3μas and 48.7±7μas respectively as observe by Event Horizon Telescope (EHT). In this case, the angular diameter of the black hole shadow varies from θd≈21o±4 with γ from 0.08/M to 0.12/M for the state parameter ωq=−2/3 when the static observer at ro=5M in the domain of outer communication.

The geodesic flow of the BGPP metric is Liouville integrable

We prove that the geodesics equations corresponding to the BGPP metric are integrable in the Liouville sense. The symmetry of the model allows to reduce the system from four to two degrees of freedom. Moreover, solutions of the reduced system and its degenerations can be given explicitly or reduced to a certain quadrature. In degenerated cases BGPP metric coincides with the Eguchi–Hanson metric and for this case the mentioned quadrature can be calculated explicitly in terms of elliptic integrals.

COMPUTER ALGEBRA TOOLS FOR GEOMETRIZATION OF MAXWELL'S EUQATIONS

Calculations of optical devices in the geometrized Maxwell’s theory use well-known formalisms of general theory of relativity and differential geometry. In particular, for such calculations it is required to know the analytical form of the geodesic equations, which leads to the need to calculate a large number of monotonous mathematical expressions. One of the purposes of computer algebra is to facilitate the researcher’s work by automating cumbersome symbolic computations. Thus, the use of computer algebra systems seems to be quite an obvious way. Several free implementations of symbolic computations for the apparatus of general relativity are considered. A practical example of symbolic computations for the geometrized Maxwell’s theory is given.

Motion equations in a Kerr–Newman–de Sitter spacetime: some methods of integration and application to black holes shadowing in Scilab

In this paper, we recall some basic facts about the Kerr–Newman–(anti) de Sitter (KNdS) spacetime and review several formulations and integration methods for the geodesic equation of a test particle in such a spacetime. In particular, we introduce some basic general symplectic integrators in the Hamiltonian formalism and we re-derive the separated motion equations using Carter’s method. After this theoretical background, we explain how to ray-trace a KNdS black hole, equipped with a thin accretion disk, using Scilab. We compare the accuracy and execution time of the previous methods, concluding that the Carter equations is the best one. Then, inspired by Hagihara, we apply Weierstrass’ elliptic functions to the non-rotating case, yielding a fairly fast shadowing program for such a spacetime. We provide some illustrations of the code, including a depiction of the effects of the cosmological constant on shadows and accretion disk, as well as a simulation of M87*.