Null Geodesic Equations Research Articles

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.

Deflection angle and quasi-periodic oscillations of extended gravitational decoupled black hole solution

Abstract In weak field limits, we compute the deflection angle of gravitational decoupling extended black hole Solution black hole solution. We obtained the Gaussian optical curvature by examining the null geodesic equations with the help of Gauss–Bonnet theorem. We also looked into the deflection angle of light by black hole in weak field limits with the use of the Gibbons Werner method. We verify the graphical behaviour 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 the graphical behaviour. Further, we compute the Einstein ring via gravitational decoupling extended black hole solution. We also compute the quasi-periodic oscillations and discuss their graphical behaviour.

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.

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.

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*.

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.

Shadow behaviors of rotating Ayón–Beato–García black holes in four-dimensional Einstein Gauss–Bonnet gravity

By applying the Newman–Janis Algorithm, we investigate optical behaviors of rotating Ayón–Beato–García black holes in four-dimensional Einstein Gauss–Bonnet Gravity. Exploiting the Hamilton–Jacobi mechanism, we first obtain the needed null geodesic equations of motion. We then provide analytic and numerical methods to study the associated optical aspect for certain regions of the involved moduli space. Concretely, we analyze and examine the shadow behaviors in terms of one-dimensional real closed curves. Particularly, we find various shapes including the D-ones. After that, we study graphically the corresponding astronomical observables. We compute and discuss the energy emission rate. We observe that the rotating parameter and the Gauss–Bonnet coupling have a relevant influence on the shadow geometric configurations and the energy emission rate. Moreover, we find that the nonlinear electrodynamic charge does not affect such optical behaviors. Finally, we provide a possible link with the event horizon telescope activities by imposing certain conditions on the involved black hole parameters in the [Formula: see text] imaging aspect.

Light deflection angle in General Relativity via Daftardar-Jafari method

In this paper, the iterative method suggested by Daftardar and Jafari hereafter called Daftardar-Jafari method (DJM) is applied for studying the deflection of light in General Relativity. For this purpose, a brief review of the nonlinear geodesic equations in the spherical symmetry spacetime and the main ideas of DJM are given. As an illustrative example, the simple case of the Schwarzschild metric is considered for which the approximate solution to the null-geodesic equation and the deflection angle of light are obtained. We also compare the obtained result with some similar results presented earlier in the literature. Â

Superentropic black hole shadows in arbitrary dimensions

We investigate the shadow behaviors of the superentropic black holes in arbitrary dimensions. Using the Hamilton–Jacobi mechanism, we first obtain the associated null geodesic equations of motion. By help of a spheric stereographic projection, we discuss the shadows in terms of one-dimensional real curves. Fixing the mass parameter m, we obtain certain shapes being remarkably different than four dimensional geometric configurations. We then study theirs behaviors by varying the black hole mass parameter. We show that the shadows undergo certain geometric transitions depending on the spacetime dimension. In terms of a critical value m_c, we find that the four dimensional shadows exhibit three configurations being the D-shape, the cardioid and the naked singularity associated with m>m_c, m=m_c and m<m_c, respectively. We reveal that the D-shape passes to the naked singularity via a critical curve called cardioid. In higher dimensions, however, we show that such transitional behaviors are removed.

Bilocal geodesic operators in static spherically-symmetric spacetimes

We present a method to compute exact expressions for optical observables for static spherically symmetric spacetimes in the framework of the bilocal geodesic operator formalism. The expressions are obtained by solving the linear geodesic deviation equations for null geodesics, using the spacetime symmetries and the associated conserved quantities. We solve the equations in two different ways: by varying the geodesics with respect to their initial data and by directly integrating the equation for the geodesic deviation. The results are very general and can be applied to a variety of spacetime models and configurations of the emitter and the observer. We illustrate some of the aspects with an example of Schwarzschild spacetime, focusing on the behaviour of the angular diameter distance, the parallax distance, and the distance slip between the observer and the emitter outside the photon sphere.

Higher-dimensional charged black holes in f(R)-gravity's rainbow surrounded by cloud of strings: Exact solution, shadow, and effective potential barrier

We construct a new class of spherically symmetric black hole solutions in f(R)-gravity's rainbow1 framework which is surrounded by string cloud configuration. In the thermodynamic analysis, we show that the black hole's temperature grows up for such small values of b parameter and also big values of it, graphically. In addition to the mass and charge of the black holes, we show that the presence of the rainbow function gr(ε) and cloud's strength b can also affect the size of the shadow. For the black hole solution, it is shown that the shadow size increases with the parameter b, but decreases with the parameter gr(ε). In addition, we have shown that the energy emission rate decreases with increase in gr(ε) and decrease in b parameter. We have analyzed the concept of effective potential barrier by transforming the radial equation of motion into standard Schrodinger form. The most important result derived from this study is that the height of this potential increases with decrease in gr(ε) parameter and increases in the coupling constant λ and the b parameter. We also investigate the impact of these parameters on the other thermodynamical quantities like temperature and heat capacity. There is a minimum on that between for one special value of b. From the temperature diagram versus the b parameter (T−b) and analogy with T−r+ diagrams, we can say that the b parameter behaves as the event horizon radius r+. Regarding the cosmological constant as a thermodynamic pressure and its conjugate quantity as a thermodynamic volume, we investigate the critical behavior of our black hole solutions. In d=5, the P-V diagrams are more complex than that of the standard Van der Waals. These diagrams behave like the Born–Infeld-AdS P-V diagrams. When d=6, the P-V diagrams are also more complex than that of the standard Van der Waals and we decompose them into two parts. One part behaves like the standard Van der Waals system. The other part is similar to the Schwarzschild-AdS black hole P-V diagrams where the isotherms turn and enter a region with negative pressures. The null geodesic equations are computed in d=5 spacetime dimensions by using the concept of symmetries and Hamilton-Jacobi equation and Carter separable method. With the null geodesics in hand, we then evaluate the celestial coordinates (x, y) and the radius Rs of the black hole shadow and represent it graphically.

Exact solutions to the null-geodesics in Ellis–Bronnikov wormhole spacetime via (G′/G)-expansion method

In this paper, the [Formula: see text]-expansion method is used to obtain two certain classes of one-parameter exact solutions to the null-geodesics in Ellis–Bronnikov wormhole spacetime. This method has been developed as an effective technique to construct exact analytical solutions for some kind of nonlinear evolution equations and nonlinear partial derivative equations (PDE). At the first stage of this method, a nonlinear PDE is transformed into nonlinear ordinary derivative equation (ODE) of a polynomial form. Therefore, if we initially have a nonlinear ODE of a polynomial form, say, the geodesic equation, then sometimes its solutions can be obtained following to the procedure of [Formula: see text]-expansion method. Here, this method made it possible to obtain some classes of exact analytical solutions of null-geodesic equations in the metric of Ellis–Bronnikov wormholes.

Orbits of light rays in (1＋2)-dimensional Einstein–power–Maxwell gravity: Exact analytical solution to the null geodesic equations

We study photon orbits within (1＋2)-dimensional Einstein–Power–Maxwell non-linear electrodynamics assuming a static and circularly symmetric background. An exact analytical solution to the null geodesic equations for light rays is obtained in terms of the Weierstraß function. We investigate in detail the impact of the photon energy, the electric charge of the black hole, and the integration constant (initial condition) on the shape of the orbits.

Magrathea-Pathfinder: a 3D adaptive-mesh code for geodesic ray tracing in N-body simulations

We introduce MAGRATHEA-PATHFINDER, a relativistic ray-tracing framework that can reconstruct the past light cone of observers in cosmological simulations. The code directly computes the 3D trajectory of light rays through the null geodesic equations, with the weak-field limit as its only approximation. This approach offers high levels of versatility while removing the need for many of the standard ray-tracing approximations such as plane-parallel, Born, or multiple-lens. Moreover, the use of adaptive integration steps and interpolation strategies based on adaptive-mesh refinement grids allows MAGRATHEA-PATHFINDER to accurately account for the nonlinear regime of structure formation and fully take advantage of the small-scale gravitational clustering. To handle very large N-body simulations, the framework has been designed as a high-performance computing post-processing tool relying on a hybrid paralleliza-tion that combines MPI tasks with C++11 std::threads. In this paper, we describe how realistic cosmological observables can be computed from numerical simulation using ray-tracing techniques. We discuss in particular the production of simulated catalogues and sky maps that account for all the observational effects considering first-order metric perturbations (such as peculiar velocities, gravitational potential, integrated Sachs-Wolfe, time-delay, and gravitational lensing). We perform convergence tests of our gravitational lensing algorithms and conduct performance benchmarks of the null geodesic integration procedures. MAGRATHEA-PATHFINDER introduces sophisticated ray-tracing tools to make the link between the space of N-body simulations and light-cone observables. This should provide new ways of exploring existing cosmological probes and building new ones beyond standard assumptions in order to prepare for the next generation of large-scale structure surveys.

Shadow and deflection angle of charged rotating black hole surrounded by perfect fluid dark matter

We analysed the shadow cast by charged rotating black hole (BH) in presence of perfect fluid dark matter (PFDM). We studied the null geodesic equations and obtained the shadow of the charged rotating BH to see the effects of PFDM parameter γ, charge Q and rotation parameter a, and it is noticed that the size as well as the shape of BH shadow is affected due to PFDM parameter, charge and rotation parameter. Thus, it is seen that the presence of dark matter around a BH affects its spacetime. We also investigated the influence of all the parameters (PFDM parameter γ, BHs charge Q and rotational parameter a) on effective potential, energy emission by graphical representation, and compare all the results with the non rotating case in usual general relativity. To this end, we have also explored the effect of PFDM on the deflection angle and the size of Einstein rings.

The density distributions of cosmic structures: impact of the local environment on weak-lensing convergence

ABSTRACT Whilst the underlying assumption of the Friedman-Lemaître-Robertson-Walker (FLRW) cosmological model is that matter is homogeneously distributed throughout the universe, gravitational influences over the life of the universe have resulted in mass clustered on a range of scales. Hence we expect that, in our inhomogeneous Universe, the view of an observer will be influenced by the location and local environment. Here, we analyse the one-point probability distribution functions and angular power spectra of weak-lensing (WL) convergence and magnification numerically to investigate the influence of our local environment on WL statistics in relativistic N-body simulations. To achieve this, we numerically solve the null geodesic equations which describe the propagation of light bundles backwards in time from today, and develop a ray-tracing algorithm, and from these calculate various WL properties. Our findings demonstrate how cosmological observations of large-scale structure through WL can be impacted by the locality of the observer. We also calculate the constraints on the cosmological parameters as a function of redshift from the theoretical and numerical study of the angular power spectrum of WL convergence. This study concludes the minimal redshift for the constraint on the parameter Ωm (H0) is $z$ ∼ 0.2 ($z$ ∼ 0.6) beyond which the local environment’s effect is negligible and the data from WL surveys are more meaningful above that redshift. The outcomes of this study will have direct consequences for future surveys, where per cent-level-precision is necessary.

Progress toward an acoustic point to point ray tracing method in a stratified atmosphere with wind

NASA is investigating acoustic propagation capabilities to accommodate a point to point propagation prediction scheme defining a ray between a specific source–observer pair in an inhomogeneous stratified atmosphere with wind. A 4D space-time ray method based on the null geodesic equations yields two main benefits over classical 3D Euclidian reduced ray equations: (1) it poses a valid two point boundary value problem for the rays connecting two locations; and (2) the solution method is valid throughout the whole domain including at ray turning points without the need for piecewise definitions. The resulting two point boundary value problem and the constant horizontal slowness vectors, which are a result of the stratified medium assumption, allow the desired (t, x, y, z) ray trace. The presentation highlights the equations and inputs required to solve the 4D ray trace. The solution to this two point boundary value problem will create the groundwork for efficient future NASA propagation codes.

Strong gravitational lensing by Kerr–Newman-NUT-Quintessence black hole

We investigate the strong gravitational lensing on equatorial plane as well as quasi-equatorial plane by the Kerr–Newman-NUT-Quintessence (KNNQ) black hole (BH) with the equation of state (EoS) parameter of the quintessence [Formula: see text] and the quintessence density [Formula: see text]. Our results show that the strong gravitational lensing in the KNNQ black hole space–time has some distinct behaviors from those in the backgrounds of the four dimension Kerr black hole. Also, we investigate the strong gravitational lensing on equatorial plane as well as quasi-equatorial plane by the KNNQ BH with the effects of Nut charge, spin parameter and quintessence parameter. First, we calculate the null geodesic equations using the Hamilton–Jacobi separation method. Then we investigate the equatorial lensing by KNNQ black hole. We obtain the deflection angle and deflection coefficients in the equatorial plane, which is affected by EoS parameter of the quintessence [Formula: see text], quintessence density [Formula: see text], Nut parameter [Formula: see text], spin parameter [Formula: see text] and quintessence parameter [Formula: see text] [Formula: see text]. Next, we discuss the lens equation and the observables in the equatorial plane. Finally, we investigate gravitational lensing by the KNNQ black hole in the quasi-equatorial plane. In this work, the quintessence density [Formula: see text], the EoS parameter of the quintessence [Formula: see text], Nut parameter [Formula: see text], spin parameter [Formula: see text] and quintessence parameter [Formula: see text] [Formula: see text] have significant effects on the strong gravitational lensing both in equatorial plane as well as quasi-equatorial plane.

Superentropic AdS black hole shadows

We study shadow aspects of superentropic black holes in four dimensions. Using Hamilton-Jacobi formalism, we first get the null geodesic equations. In the celestial coordinate framework relying on with fixed positions of observers, we investigate the shadow behaviors in terms of the mass and the cosmological scale variation parameters. Among others, we obtain ellipse shaped geometries contrary to usual black hole solutions. Modifying the ordinary relations describing geometrical observables, we discuss the size and the shape deformation parameters of these non-trivial geometric forms. Due to horizonless limits associated whit certain mass values, we explore the shadow of the naked singularity of such black holes.