Light Deflection Research Articles

Study of light deflection and shadow in the Rindler-like Schwarzschild solution

In this paper, we calculate the deflection angle in the non-plasma medium by expanding the Keeton–Petters approach. We use analysis to determine how non-magnetic plasma affects the shadow of a Schwarzschild–Rindler black hole. We only consider the spherically symmetric case in which the shadow is circular. It is assumed that the plasma is independent of pressure and magnetization. From a specified spherically symmetric spacetime, we compute the results for a spherically symmetric plasma density. Our main purpose is to derive analytically a formula for the orbital radius of the shadow. Due to the diffusive nature of plasma, the shadow’s radius is influenced by the frequency of the photons. The impacts of the non-magnetic plasma are important in radio establishment. We determine that the non-magnetic plasma incorporates a decreasing impact on the shadow size for a distant observer from the Schwarzschild–Rindler black hole. Moreover, we graphically analyze the impacts of different parameters on the shadow of the corresponding black hole.

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.

Deflection of light by wormholes and its shadow due to dark matter within modified symmetric teleparallel gravity formalism

Abstract We explore the possibility of traversable wormhole formation in the dark matter halos in the context of $f(Q)$ gravity. We obtain the exact wormhole solutions with anisotropic matter source based on the Bose-Einstein condensate, Navarro-Frenk-White, and pseudo-isothermal matter density profiles. Notably, we present a novel wormhole solution supported by these dark matters using the expressions for the density profile and rotational velocity along with the modified field equations to calculate the redshift and shape functions of the wormholes. With a particular set of parameters, we demonstrate that our proposed wormhole solutions fulfill the flare-out condition against an asymptotic background. Additionally, we examine the energy conditions, focusing on the null energy conditions at the wormhole's throat, providing a graphical representation of the feasible and negative regions. Our study also examines the wormhole's shadow in the presence of various dark matter models, revealing that higher central densities result in a shadow closer to the throat, whereas lower values have the opposite effect. Moreover, we explore the deflection of light when it encounters these wormholes, particularly noting that light deflection approaches infinity at the throat, where the gravitational field is extremely strong.

Gravitational lensing by an ellipsoidal Navarro–Frenk–White dark-matter halo: An analytic solution and its properties

Context. The analysis of gravitational lensing by galaxies and galaxy clusters typically relies on ellipsoidal lens models to describe the deflection of light by the involved dark-matter halos. These models are most often based on the isothermal density profile – not an optimal description of the halo, but easy to use because it leads to an analytic deflection-angle formula. Aims. Dark-matter halos are better described by the Navarro–Frenk–White (hereafter NFW) density profile. We set out to study lensing by a general triaxial ellipsoidal NFW halo, with the aim of providing an analytic model that would be more consistent with the current understanding of dark-matter halos. Methods. We computed the conversion between the properties of a triaxial ellipsoidal lens model and its elliptical surface-density profile. In the case of the NFW lens model, its angular scale is defined by the projected scale semi-major axis of the halo, while its lensing regime depends on two parameters: the projected eccentricity e and the convergence parameter κs. We employed the Bourassa & Kantowski formalism to compute the complex scattering function of the model, which yields the deflection-angle components when separated into its real and imaginary parts. Results. We present the obtained closed-form expressions for the deflection-angle components, valid for an arbitrary eccentricity of the surface-density profile. We use them to compute and describe the lensing properties of the model, including: the shear, its components, and the phase; the critical curves, caustics, and the parameter-space mapping of their different geometries; the deformations and orientations of images. Conclusions. The analytically solved ellipsoidal NFW lens model is available for implementation in gravitational lensing software. The techniques introduced here such as the image-plane analysis can prove to be useful for understanding the properties of other lens models as well.

Gravitational lensing and shadow by a Schwarzschild-like black hole in metric-affine bumblebee gravity

In this paper, we investigate the gravitational lensing effect and the shadow around a Schwarzschild-like black hole in metric-affine bumblebee gravity, which leads to the Lorentz symmetry breaking. We first present a generalized formalism for calculating higher-order corrections to light weak bending angle in a static, spherically symmetric and not asymptotically flat spacetime, and then applying this general formalism to the metric-affine bumblebee gravity. Moreover, we derive the light deflection angle and the size of the Einstein ring within the weak field in this scenario. In addition, we analyze the black hole shadow in this theory framework. By using observational data from the Einstein’s ring of the galaxy ESO325-G004 and the black hole shadow of the M87\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{M}87$$\\end{document} galaxy, we estimate the upper bounds of the Lorentz symmetry breaking coefficient ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document}, respectively.

Schwarzschild spacetime in fractal dimensions: Deflection of light, supermassive black holes and temperature effects

In this paper, we construct a Schwarzschild metric in the fractal dimension based on fractal calculus, and we study the deflection of light near the sun. It was observed that the new fractal spherically symmetric spacetime metric may describe supermassive black holes, which have been detected through various astronomical observations such as SDSS. We have discussed the deflection of light by the gravitational field of the sun. It was observed that the total deflection of light is affected by the fractal dimension. To estimate the numerical value of the fractal dimension, we have used the 2004[Formula: see text]analysis of almost 2 million VLBI observations of 541 radio sources made by 87 VLBI sites, which estimates the factor [Formula: see text] based on PNN formalism. It was observed that the fractal dimension is roughly less than unity. However, our analysis showed that large bending gravity gives fractal dimensions lower than unity to some extent. We also gave a naïve idea about the emission thermal Unruh process in fractal dimension. It was observed that both the temperature and the entropy of black holes are affected by the fractal dimension, and that for low numerical values, the entropy of the black hole is enhanced.

The corrections to the metric of Reissner-Nordström black hole from the generalized uncertainty principle

The emergence of the minimal observable length is commonly accepted in most of the quantum gravitational candidates. The existence of such a minimal length in high energy physics necessitates some revisions to the standard uncertainty principle. The generalized uncertainty principle is particularly suitable for incorporating such a finite resolution of the space-time and may provide a useful phenomenological approach to study the physics of quantum gravity. It is possible to use the generalized uncertainty principle to modify the black hole thermodynamics straightforwardly. However, it is also possible to use the generalized uncertainty principle to modify the black hole metric itself. In this paper, we are going to modify the Reissner-Nordström metric in the presence of the quantum gravitational effects via the generalized uncertainty principle. Then, we use the modified charged black hole metric to study the black hole thermodynamics. The modified metric is also applied to study the light deflection angle.

Geodesics structure and deflection angle of electrically charged black holes in gravity with a background Kalb–Ramond field

This article investigates the geodesic structure and deflection angle of charged black holes in the presence of a nonzero vacuum expectation value background of the Kalb–Ramond field. Topics explored include null and timelike geodesics, energy extraction by collisions, and the motion of charged particles. The photon sphere radius is calculated and plotted to examine the effects of both the black hole charge (Q) and the Lorentz-violating parameter (b) on null geodesics. The effective potential for timelike geodesics is analyzed, and second-order analytical orbits are derived. We further show that the combined effects of Lorentz-violating parameter and electric charge can mimic a Kerr black hole spin parameter up to its maximum values, i.e., a/M∼1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a/M \\sim 1$$\\end{document} thus suggesting that the current precision of measurements of highly spinning black hole candidates may not rule out the effect of Lorentz-violating parameter. The center of mass energy of colliding particles is also considered, demonstrating a decrease with increasing Lorentz-violating parameter. Circular orbiting particles of charged particles are discovered, with the minimum radius for a stable circular orbit decreasing as both b and Q increase. Results show that this circular orbit is particularly sensitive to changes in the Lorentz-violating parameter. Additionally, a timelike particle trajectory is demonstrated as a consequence of the combined effects of parameters b and Q. Finally, the light deflection angle is analyzed using the weak field limit approach to determine the Lorentz-breaking effect, employing the Gauss–Bonnet theorem for computation. Findings are visualized with appropriate plots and thoroughly discussed.

GUP corrected black holes with cloud of string

We investigate shadows, deflection angle, quasinormal modes (QNMs), and sparsity of Hawking radiation of the Schwarzschild string cloud black hole’s solution after applying quantum corrections required by the Generalised Uncertainty Principle (GUP). First, we explore the shadow’s behaviour in the presence of a string cloud using three alternative GUP frameworks: linear quadratic GUP (LQGUP), quadratic GUP (QGUP), and linear GUP. We then used the weak field limit approach to determine the effect of the string cloud and GUP parameters on the light deflection angle, with computation based on the Gauss–Bonnet theorem. Next, to compute the quasinormal modes of Schwarzschild string clouds incorporating quantum correction with GUP, we determine the effective potentials generated by perturbing scalar, electromagnetic and fermionic fields, using the sixth-order WKB approach in conjunction with the appropriate numerical analysis. Our investigation indicates that string and linear GUP parameters have distinct and different effects on QNMs. We find that the greybody factor increases due to the presence of string cloud while the linear GUP parameter shows the opposite. We then examine the radiation spectrum and sparsity in the GUP corrected black hole with the cloud of string framework, which provides additional information about the thermal radiation released by black holes. Finally, our inquiries reveal that the influence of the string parameter and the quadratic GUP parameter on various astrophysical observables is comparable, however the impact of the linear GUP parameter is opposite.

Evolution of light deflection and shadow from a gauge-potential-like AdS black hole under the influence of a non-magnetic plasma medium

We investigate the light deflection in the weak field approximation from the accelerating charged AdS black hole. For this purpose, we apply the Gauss–Bonnet theorem to calculate the light deflection in the weak field area and use the Gibbons–Werner approach to analyze the optical geometry of the accelerating charged AdS black hole in the non-magnetic plasma absence/presence of a non-magnetic medium. We also represent the graphical behavior of the light deflection angle w.r.t. the impact parameter. We also compute the light deflection angle using Keeton and Petters approximations under the impact of accelerating charged AdS black hole geometry. Furthermore, by using the ray-tracing approach, we determine the shadow in the non-magnetic plasma presence and also demonstrate that graphical shadow has an impact on the gauge potential, non-magnetic plasma frequencies and charge.

The gravitational lensing by rotating black holes in loop quantum gravity

In this paper, we explore gravitational lensing effects associated with rotating black holes within the framework of loop quantum gravity. Utilizing the Gauss-Bonnet theorem as extended by Ono et al., we compute the light deflection angle in the weak field limit for a lens that is finitely distanced from both the source and the observer. Our findings indicate that the weak deflection angle for rotating black holes in LQG is smaller than that observed for the classical Kerr black holes, albeit with minimal deviations. In the strong field limit, we determine the photon sphere radius, the light deflection angle, and lensing observables, including the image position θ∞, angular separation s, magnification rmag, and temporal delays among various relativistic images. By considering supermassive black holes, such as Sgr A* and M87*, within the LQG framework, we calculate these observables and investigate the influence of the quantum parameter Aλ on them, compared with the Kerr black hole outcomes. Our comparative analysis reveals that the image position θ∞ and separation s for Sgr A* consistently exceed those for M87*, whereas M87* exhibits considerably greater time delays than Sgr A*. These distinctions could be important in differentiating between rotating black holes in LQG and classical Kerr black holes in future astronomical observations.

On the (un)testability of the general free scalar–tensor gravity for the Solar System tests

Recently, Zhang et al. [Eur. Phys. J. C 84, 381, https://doi.org/10.1140/epjc/s10052-024-12723-8 (2024)] constrained the Yukawa correction in the general free scalar–tensor gravity by borrowing the measured values of the parametrized post-Newtonian (PPN) parameters γ-1=(2.1±2.3)×10-5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma -1=(2.1\\pm 2.3)\ imes 10^{-5}$$\\end{document} and β-1=(-4.1±7.8)×10-5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta -1=(-\\,4.1\\pm 7.8)\ imes 10^{-5}$$\\end{document} in the Solar System tests. They firstly defined two PPN parameters γ(r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma (r)$$\\end{document} and β(r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta (r)$$\\end{document} in the gravity, which depend on the radial distance r and include the Yukawa-type parameters. Then, by comparing between the experiments’ results of two PPN parameters and their γ(r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma (r)$$\\end{document} and β(r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta (r)$$\\end{document}, the authors claimed that they gave the tightest bound on the gravity by the Cassini tracking experiment. However, we find their approach is not rigorous. In this paper, corresponding astronomical experiments have been physically modelled by considering the lightlike and the timelike geodesics in the general free scalar–tensor gravity. Contrary to the wrong results in Zhang et al.’s work, it is shown that the Cassini tracking experiment is insensitive to the general free scaler–tensor gravity. Furthermore, we also find that the time delay and the light deflection are all independent of the gravity. Due to an additional Yukawa-type advance in the periastron shift, we derive very much improved bounds on the Yukawa-type parameters of this gravity.

Deflection of light by a Reissner–Nordström black hole and Painlevé VI equation

We consider the bending angle of the trajectory of a photon incident from and deflected to infinity around a Reissner–Nordström black hole. We treat the bending angle as a function of the squared reciprocal of the impact parameter and the squared electric charge of the background normalized by the mass of the black hole. It is shown that the bending angle satisfies a system of two inhomogeneous linear partial differential equations with polynomial coefficients. This system can be understood as an isomonodromic deformation of the inhomogeneous Picard–Fuchs equation satisfied by the bending angle in the Schwarzschild spacetime, where the deformation parameter is identified as the background electric charge. Furthermore, the integrability condition for these equations is found to be a specific type of the Painlevé VI equation that allows an algebraic solution. We solve the differential equations both at the weak and strong deflection limits. In the weak deflection limit, the bending angle is expressed as a power series expansion in terms of the squared reciprocal of the impact parameter and we obtain the explicit full-order expression for the coefficients. In the strong deflection limit, we obtain the asymptotic form of the bending angle that consists of the divergent logarithmic term and the finite O(1) term supplemented by linear recurrence relations which enable us to straightforwardly derive higher order coefficients. In deriving these results, the isomonodromic property of the differential equations plays an important role. Lastly, we briefly discuss the applicability of our method to other types of spacetimes such as a spinning black hole.

Two-dimensional electro-optical multiphoton microscopy.

The development of genetically encoded fluorescent indicators of neural activity with millisecond dynamics has generated demand for ever faster two-photon (2P) imaging systems, but acoustic and mechanical beam scanning technologies are approaching fundamental limits. We demonstrate that potassium tantalate niobate (KTN) electro-optical deflectors (EODs), which are not subject to the same fundamental limits, are capable of ultrafast two-dimensional (2D) 2P imaging in vivo. To determine if KTN-EODs are suitable for 2P imaging, compatible with 2D scanning, and capable of ultrafast in vivo imaging of genetically encoded indicators with millisecond dynamics. The performance of a commercially available KTN-EOD was characterized across a range of drive frequencies and laser parameters relevant to in vivo 2P microscopy. A second KTN-EOD was incorporated into a dual-axis scan module, and the system was validated by imaging signals in vivo from ASAP3, a genetically encoded voltage indicator. Optimal KTN-EOD deflection of laser light with a central wavelength of 960nm was obtained up to the highest average powers and pulse intensities tested (power: 350mW; pulse duration: 118fs). Up to 32 resolvable spots per line at a 560kHz line scan rate could be obtained with single-axis deflection. The complete dual-axis EO 2P microscope was capable of imaging a by field-of-view at over 10kHz frame rate with lateral resolution. We demonstrate in vivo imaging of neurons expressing ASAP3 with high temporal resolution. We demonstrate the suitability of KTN-EODs for ultrafast 2P cellular imaging in vivo, providing a foundation for future high-performance microscopes to incorporate emerging advances in KTN-based scanning technology.

Determining the difference between local acceleration and local gravity: Applications of the equivalence principle to relativistic trajectories

We show by direct calculation that the common equivalence principle explanation for why gravity must deflect light is quantitatively incorrect by a factor of three in Schwarzschild geometry. It is, therefore, possible, at least as a matter of principle, to tell the difference between local acceleration and a true gravitational field by measuring the local deflection of light. We calculate as well the deflection of test particles of arbitrary energy and construct a leading-order coordinate transformation from Schwarzschild to local inertial coordinates, which shows explicitly how the effects of spatial curvature manifest locally for relativistic trajectories of both finite and vanishing rest mass particles.

About Newton’s Law of Gravitation and the Energy Momentum Relations

Newtonian dynamics was based on the assumption of instantaneous propagation of interaction. Roamer’s studies showed that the velocity of interaction cannot exceed the velocity of light in free space. (c=3x108 mps). Lorentz’ electron theory and electrodynamics take this fact into account. It is possible to rectify the defects of Newtonian theory by deriving Maxwell-Lorentz equations and the Lorentz Force equation for the gravitational field and hence the modified Newtonian dynamics can explain (i) The Perihelion Shift of the orbit of the Planet Mercury (ii) Deflection of light by the Sun (iii) The Gravitational Redshift [1]. It is shown that the use of centre of mass coordinate system also implies a non-elliptic orbit in general, so that the inverse square law of Newton/Coulomb is always valid. Next, the energy and momentum of a dynamical system are shown to satisfy Finally, the Schrodinger’s wave function is interpreted as a quantum mechanical energy density [2].

Proposal for a New Differential High-Sensitivity Refractometer for the Simultaneous Measurement of Two Refractive Indices and Their Differences.

The refractive index of a liquid serves as a fundamental parameter reflecting its composition, thereby enabling the determination of component concentrations in various fields such as chemical research, the food industry, and environmental monitoring. Traditional methods for refractive index (RI) measurement rely on light deflection angles at interfaces between the liquid and a material with a known refractive index. In this paper, the authors present a new differential refractometer for the highly sensitive measurement of RI differences between two liquid samples. Using a configuration with two cells equipped with flat parallel plates as measuring elements, the instrument facilitates accurate analysis. Namely, the sensor signals from both the solution and the solvent cuvette are generated simultaneously with one laser pulse, reducing the possible fluctuations in the laser radiation intensity. Our evaluation shows the high sensitivity of RI measurements <7×10-6), so this differential refractometer can be proposed not only as a high-sensitivity sensing tool that can be used for mobile detection of nanoparticles in solution samples but also to determine the level of environmental nano-pollution using water (including rain, snow) samples from various natural as well as industrial sources, thus helping to solve some important environmental problems.

Interferometric measurement of the deflection of light by light in air

Interferometric measurement of the deflection of light by light in air

Gravitational lensing by a stable rotating regular black hole

Recent observational data from the Event Horizon Telescope (EHT) collaboration provide convincing realistic evidence for the existence of black hole rotation. From a phenomenological perspective, a recently proposed stable rotating regular (SRR) black hole circumvents the theoretical flaws of the Kerr solution. For the purpose of obtaining observational signatures of this black hole, we study its gravitational lensing effect. In the strong deflection limit, we calculate the deflection angle of light on the equatorial plane, the radius of the photon sphere, and other observables. The observables include the relativistic image position, separation, magnification, and time delays between different images. Then, by modeling M87* and Sgr A* as the SRR black hole, we compute their observables and evaluate the deviation of the observables from the Kerr case. In the weak deflection limit, we calculate the light deflection angle on the equatorial plane of M87* and Sgr A* via the Gauss-Bonnet theorem (GBT). With the growth of deviation parameter e, the gravitational lensing effect in the weak deflection limit intensifies monotonically, and the gravitational lensing effect in the strong deflection limit changes dramatically only at high spins. Our research may contribute to distinguish between SRR black holes from Kerr black holes under higher-precision astronomical observations.

Three-dimensional diagnosis of lean premixed turbulent swirl flames using tomographic background oriented Schlieren

The necessity of minimizing NOx emissions drives the pursuit of ultra-lean premixed combustion in aeroengines and gas turbines, characterized by susceptibility to combustion instabilities. To tackle this issue, swirling flow design is widely incorporated into lean premixed combustor design, enhancing flame stability, and shortening flame length. This study utilizes the tomographic background-oriented Schlieren (TBOS) to reconstruct the spatial distribution of the refractive index gradient of lean premixed turbulent swirl flames with an aeroengine combustor configuration. A parametric study of the TBOS reconstruction quality is conducted, and the results reveal that view sparseness primarily degrades the reconstruction quality compared to the specific iterative algorithm used. The classic visual hull approach is explored to address this challenge, highlighting the significance of visual hull size. Furthermore, to improve the reconstruction quality, a posterior support constraint method is proposed, involving the removal of voxels of nearly constant refractive index within the central volume surrounded by flames. Results demonstrate that implementing this posterior support constraint further improves the reconstruction quality of lean premixed turbulent swirl flames. Finally, the robustness of this posterior support constraint method is validated by introducing high-level noise to the light deflection data, showcasing the potential of combining the dedicated designed visual hull and proposed posterior support constraint in addressing the view sparseness challenge for TBOS measurements.