Light Deflection Research Articles

Schlieren Image Velocimetry of Swirl Sprays

Schlieren image velocimetry (SIV) is based on light deflection through flow heterogeneities and image cross-correlations. This is a low-cost and relatively low complexity technique that allows measurement of the droplet velocity field in a large region of a spray. A Z-type Toepler schlieren system with a high-speed camera was used to determine mean vertical and horizontal droplet velocities, as well as the cone angles of sprays produced by a pressure swirl injector with characteristic geometric constant K = 2. Different LEDs and digital filters were evaluated for edge detection and improvement of image contrast. Open software was adopted for digital image processing and velocimetry. Interrogation windows and overlaps of different sizes were tested to obtain an appropriate correlation for determination of the velocity field. The digital images were obtained with 5 × 103 fps and a resolution of 2.77 pixels/mm. Since the swirl sprays analysed presented instabilities, a number of 100 cross-correlations of images was required to reduce mean velocity fluctuations. Injection pressures varied from 0.05 to 7 bar and mass flow rates varied from 1.389 to 13.89 g/s, using water as test fluid. The wideband warm white LED with Laplacian or high-pass filters provided velocity data for a larger range of injection pressures. Mean axial velocities varied from 3.3 to 11.3 m/s, approximately, with mean horizontal velocities varying from around 0.17 to 3.3 m/s for pressures from 0.05 to 3.22 bar. The velocity data were compared to microscopic shadowgraphy results, showing a good agreement. Spray cone angles ranged from about 32.5o to 69.5°, for injection pressures from 0.05 to 7 bar, and results of triangulation with a blue LED were closer to semi-empirical data.

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

The Effect of Light Deflection by Solar System Objects on High-precision Square Kilometre Array Astrometry

We have computed the deflection angles caused by 195 objects in the solar system, including 177 satellites, and eight asteroids; 21 satellites and six asteroids can bend light from distant compact extragalactic sources by more than 0.1 μas, and 14 satellites, and the asteroid Ceres can deflect light by more than 1.0 μas. We calculated the zones and durations of perturbations posed by the gravitational fields of five planets (excluding Earth, Jupiter, and Saturn), Pluto, and Ceres, where the perturbations would affect astrometry measured with the Square Kilometre Array (SKA). Perturbed zones with deflection angles larger than 0.1 and 1.0 μas appear as ribbons. Their widths range from dozens of degrees for Uranus, Neptune, and Venus to several degrees or less for other objects at 0.1 μas, and from ∼16° for Venus to several degrees or less for other objects at 1.0 μas. From the calculated perturbation durations, the influence of the gravitational fields of selected objects can be divided into four levels: (i) hardly affects SKA astrometry, (ii) may have little effect, (iii) may have a great effect on single-epoch astrometry, and (iv) may greatly affect both single- and multiepoch astrometry. The objects corresponding to these levels are (i) Ceres, (ii) Pluto, (iii) Mercury and Mars, and (iv) other objects.

Novel features of Schwarzschild-like black hole of Lorentz violating bumblebee gravity

A possible avenue for observing quantum gravity (QG) effects at low energy scales is to introduce spontaneous Lorentz violation (LV) in new models of gravity theories. One such model in the literature is bumblebee gravity yielding Schwarzschild-like black hole and weak field Solar System observations involve LV corrections characterized by the parameter ℓ. Here we first show that these LV corrections have a novel genesis in the conical angle Δ = πb subtended at the origin of the spacetime of massless bumblebee gravity. Exploiting the resultant asymptotic light deflection angle πb −1 as a new input in the exact deflection formula, we next study the strong field lensing properties of the Schwarzschild-like black hole exploring how they differ from those of the Schwarzschild black hole of general relativity. It is shown that the angular image separation and ratio of fluxes could respectively be s ∼ e πℓ and r ∼ e−πℓ times those of the Schwarzschild black hole (ℓ = 0). However, the shadow of the Schwarzschild-like black hole is independent of ℓ suggesting that observations of shadow radii cannot reveal QG effects. Finally, we raise an interesting issue about the measurability of the LV corrections caused by strong field lensing. An appendix briefly outlines lensing by the spinning bumblebee black hole.

A new camera model combining an analytical model and a discrete correction to overcome refractive index variation challenges

When performing Tomo-PIV experiments, the calculation of accurate camera models is a key point for successful measurements. In severe configurations, where optical interfaces are involved and refractive index variations occur along the line of sight, analytical models can fail to accurately represent the projection and back-projection functions. This inability can lead to significant inaccuracies in particle location and volume reconstruction, which has a considerable impact on the calculation of velocity fields. In order to overcome these limitations, an innovative camera model based on the combination of an analytical model, such as a pinhole model or a polynomial model, with discrete corrections is proposed. In this method, the analytical projection and back-projection are adjusted with a discrete correction stored in two adaptive grids that save both memory and computation time. These correction grids require calibration which is performed with triangulation procedures similar to those used in misalignment corrections. The calculation and operation of the model are described in this paper. The performance of the camera model is evaluated on simulated and experimental setups based on a large depth-of-field calibration performed in a glass water tank. The presence of multiple optical interfaces and fluids, resulting in large light deviations, makes it difficult to compute a high accuracy camera model. In this configuration, the proposed technique successfully reduces the triangulation error from 1 pixel to less than 0.01 pixel. The usefulness of the model is demonstrated in a Tomo-PIV experiment where the deflection of light through the water tunnel walls prevents classical analytical functions from accurately modelling the projection and back-projection functions. This corrected model can also solve discontinuity problems in the projection functions and can be used when there are interfaces in the measurement volume. It opens new perspectives in the study of fluid-structure interaction when transparent solids are involved.

On the meaning of various mass definitions for asymptotically flat spacetimes

In general relativity, the energy density of the gravitational field is not well defined. However, for asymptotically flat spacetimes, various definitions for total mass of the system have been proposed. In this paper, eight masses are considered and it is shown that some of them correspond to the active gravitational mass while the others correspond to the inertial mass. For example, the ADM mass corresponds to the inertial mass, while the Møller mass corresponds to the active gravitational mass. In general, the inertial and active gravitational masses are not equal. If the spacetime is vacuum at large r, the Einstein equations force the inertial and active gravitational masses to be the same. The Einstein equations also force these masses to be the same if any matter that extends out to large r satisfies the weak, strong, or dominant energy condition. The contributions of the inertial and active gravitational masses to the gravitational redshift, the deflection of light, the Shapiro time delay, the precession of perihelia, and the motion of test bodies in the spacetime are also examined.

Magnetic and electric black holes in the vector-tensor Horndeski theory

We construct exact solutions of magnetically charged black holes in the vector-tensor Horndeski gravity and discuss their main features. Unlike the analogous electric case, the field equations are linear in a simple (quite standard) parametrization of the metric tensor, and they can be solved analytically even when a cosmological constant is added. The solutions are presented in terms of hypergeometric functions, which makes the analysis of the black hole properties relatively straightforward. Some of the aspects of these black holes are quite ordinary, like the existence of extremal configurations with maximal magnetic charge for a given mass or the existence of a mass with maximal temperature for a given charge, but others are somewhat unexpected, like the existence of black holes with a repulsive gravitational field. We perform our analysis for both signs of the nonminimal coupling constant and find black hole solutions in both cases but with significant differences between them. The most prominent difference is the fact that the black holes for the negative coupling constant have a spherical surface of curvature singularity rather than a single point. On the other hand, the gravitational field produced around this kind of black holes is always attractive. Also, for small enough magnetic charge and negative coupling constant, extremal black holes do not exist, and all magnetic black holes have a single horizon. In addition, we study the trajectories around these magnetic black holes for light as well as massive particles either neutral or electrically charged. Finally, we compare the main features of these black holes with their electric counterparts, adding some aspects that have not been discussed before, like temperature, particle trajectories, and light deflection by electrically charged Horndesky black holes.

Is the equivalence principle useful for understanding general relativity?

The equivalence principle (EP) is at the heart of general relativity (GR), tested in many aspects. In discussion, it is often referred to explaining qualitatively the influence of gravity on physical phenomena, but is this qualitative approach supported by the theory? We compare clock rates, frequency shifts, light deflection, and time delay in simple static spacetimes to the analogous phenomena seen by accelerated observers in Minkowski space. In contrast to previous studies, we do not assume that the gravitational field is weak and see, as we proceed, how the field is constrained by the EP. Special care is taken that results are only observer-, but not coordinate-dependent. By this, we clarify some of the issues raised in the literature and show which gravitational effects can and which cannot be simulated by acceleration. The paper may also serve as a starting point for discussions on the implications of the EP.

Post-Newtonian gravity and Gaia-like astrometry

Context. Relativistic models of light propagation adopted for high-precision astrometry are based on the parametrised post-Newtonian formalism, which provides a framework for examining the effects of a hypothetical violation of general relativity on astrometric data. Astrometric observations are strongly affected by the post-Newtonian (PPN) parameter γ describing the strength of gravitational light deflection. Aims. We study both analytically and numerically how a deviation in the PPN parameter γ from unity, which is the value predicted by general relativity, affects the parallax estimations in Gaia-like astrometry. Methods. Changes in the observable quantities produced by a small variation in PPN γ were calculated analytically. We then considered how such variations of the observables are reflected in the parallax estimations, and we performed numerical simulations to check the theoretical predictions. Results. A variation in the PPN γ results in a global shift of parallaxes and we present a formula describing the parallax bias in terms of the satellite barycentric distance, the angle between the spin axis and the direction to the Sun, and the PPN γ uncertainty. Numerical simulations of the astrometric solutions confirm the theoretical result. The up-to-date estimation of PPN γ suggests that a corresponding contribution to the Gaia parallax zero point unlikely exceeds 0.2 μas. The numerical simulations indicate that the parallax shift is strongly dependent on ecliptic latitude. It is argued that this effect is due to an asymmetry in the Gaia scanning law and this conclusion is fully validated by additional simulations with a reversed direction of the precession of the spin axis around the direction to the Sun.

Observables from spherically symmetric modified dispersion relations

In this work, we continue the systematic study of observable effects emerging from modified dispersion relations. We study the motion of test particles subject to a general first-order modification of the general relativistic dispersion relation as well as subject to the [Formula: see text]-Poincaré dispersion relation in spherical symmetry. We derive the corrections to the photon sphere, the black hole shadow, the and the light deflection and identify the additional dependence of these observables on the photons’ four momentum, which leads to measurable effects that can be compared to experimental data. The results presented here can be interpreted in two ways, depending on the origin of the modified dispersion relation: on the one hand as prediction for traces of quantum gravity, when the modified dispersion relation is induced by phenomenological approaches to quantum gravity, on the other hand as predictions of observables due to the presence of a medium, like a plasma, which modifies the dispersion relation of light on curved spacetimes.

Null geodesics and shadow of hairy black holes in Einstein-Maxwell-dilaton gravity

The timelike and null-like geodesics around compact objects are some of the best tools to classify and understand the structure of a space-time. We study the null geodesics around charged static dilaton black holes in Einstein-Maxwell-dilaton gravity. The physical parameters for nonradial geodesics, including the effective potential, effective force, radius of the photon sphere, and impact parameter, are obtained, and effects of the charge parameter and dilaton coupling constant on these quantities are studied. Possible photon motions for different values of the impact parameter are analyzed, and unstable circular orbits and unbounded orbits are plotted. These results are compared to that of Schwarzschild, Reissner-Nordstr\"om, and Gibbons-Maeda-Garfinkle-Horowitz-Strominger black holes. Also, we study the shadow cast by a dilaton black hole and investigate how the dilaton coupling affects the size of the black hole shadow. Finally, as an application of null geodesics, we calculate the deflection of light and investigate the effects of the model parameters on the bending angle.

Henry Cavendish on Gravitational Deflection of Light

AbstractA recent analysis of two handwritten sheets of paper by Henry Cavendish (1731–1810) reveals the details of his calculation of the deflection of light induced by the gravitational field of isolated celestial bodies. Except for a few lines from one of these pages, where Cavendish communicated the result, the two documents have never been published. This gave rise to speculations about how, if at all, Cavendish derived his formula for the deflection angle by his own calculations. Such doubts are dispelled and it is demonstrated how Cavendish, in these documents, applied Newtonian mechanics to the corpuscles of light.

Double-sided liquid crystal metasurfaces for electrically and mechanically controlled broadband visible anomalous refraction

Abstract Liquid crystals self-assemble on nanopatterned alignment layers into purely soft matter metasurfaces sensitive to external stimuli and imparting tailored spatial modulations to transmitted light wavefronts. Upon fine optimization, they are capable of efficient light deflection by virtue of anomalous refraction into a dominating transmission diffraction order. To expand the spectral range and acquire additional functionality, we put forward the double-sided metasurface design based on the liquid crystal alignment by a pair of complementing patterned substrates. We numerically optimize, fabricate, and experimentally characterize metasurfaces refracting red light with an efficiency of up to 70% and sustaining the efficiency above 50% in a broad range of visible wavelengths exceeding 500 nm. We verify that the refraction is reversibly switched in less than 10 ms by voltages of a few volts. We also report on a remarkable mechanical reconfigurability, as micrometer-scale relative substrate shift flips the refraction direction.

Light bending and gravitational lensing in energy-momentum-squared gravity

In the present work, we derive the motion of light in the weak-field limit of energy-momentum-squared gravity (EMSG). To do so, we introduce the post-Newtonian expansion of this modified theory of gravity. It is shown that, in addition to the Newtonian potential, a new EMSG potential affects the trajectory of photons. As a result, in this theory, photons do not behave as predicted by general relativity (GR). To evaluate the EMSG theory by the Solar System tests, we study light deflection and Shapiro time delay. Regarding the results obtained in Bertotti et al. [Nature (London) 425, 374 (2003)] and Shapiro et al. [Phys. Rev. Lett. 92, 121101 (2004)], we restrict the free parameter of the theory and show that it lies within the range $\ensuremath{-}4.0\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}27}\text{ }\text{ }\mathrm{m}\text{ }{\mathrm{s}}^{2}\text{ }{\mathrm{kg}}^{\ensuremath{-}1}<{f}_{0}^{\ensuremath{'}}<8.7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}26}\text{ }\text{ }\mathrm{m}\text{ }{\mathrm{s}}^{2}\text{ }{\mathrm{kg}}^{\ensuremath{-}1}$. This interval is in agreement with those derived in Nazari et al. [Phys. Rev. D 105, 044014 (2022)] and Akarsu et al. [Phys. Rev. D 97, 124017 (2018)]. This consistency manifests that this theory passes these Solar System tests with flying colors. Interestingly, it turns out that the magnitude of the EMSG correction strongly depends on the density of the deflector. So, we investigate the possible effects of EMSG on images of a light source microlensed by a compact dense object such as neutron stars. It is estimated that the EMSG correction to the position of lensed images could be as large as (1--0.1) microarcseconds, which may be detected by future high-resolution missions. Moreover, the total magnification and the shape of light curves are obtained in the EMSG theory. It is revealed that except for a small deviation, the overall behavior of the EMSG light curves is similar to that in GR. We also show that as long as the light source and the dense lens are aligned, the EMSG correction is effective, and the combined light of the lensed images is different from the GR case. This issue makes it possible to observe signatures of this theory in the microlensing regime.

Deflection of light by magnetars in the generalized Born–Infeld electrodynamics

We study the deflection of light by a magnetic dipole field in the generalized Born–Infeld electrodynamics. Using the effective index of refraction and the trajectory equation based on geometric optics, we compute the weak bending angle of light passing on the equator of the magnetic dipole. In the limit where the classical Born–Infeld parameter is infinite, the bending angle agree with the one computed from the Euler–Heisenberg Lagrangian. We also compute the bending angle using the geodesic equation of the effective metric induced by a massive object with magnetic dipole. In the massless limit the bending angle agrees with the computation using the trajectory equation. We apply the result to magnetars to estimate the order of magnitude for the bending angle.

Unified f(R) gravity at local scales

We explore the shifted f(R) (propto R^{1+delta }) model with {delta } as a distinguishing physical parameter for the study of constraints at local scales. The corresponding dynamics confronted with different geodesics (null and non-null) along with their conformal analog are investigated. For null geodesics, we discuss the light deflection angle, whereas, for non-null geodesics under the weak field limit, we investigate the perihelion advance of the Mercury orbit in f(R) Schwarzschild background, respectively. The extent of an additional force, appearing for non-null geodesics, depends on delta . Such phenomenological investigations allow us to strictly constrain delta to be approximately {mathcal {O}}(10^{-6}) with a difference of unity in orders at galactic and planetary scales and seem to provide a unique f(R) at local scales. Our results suggest that the present form of the model is suitable for the alternative explanation of dark matter-like effects at local scales.

Solar system constraints of a polymer black hole in loop quantum gravity

A new polymer black hole solution in loop quantum gravity was proposed recently. The difference between the polymer black hole and Schwarzschild black hole is captured by a quantum parameter $A$. In order to get the constraints on parameter $A$, we consider the observational constraints imposed on $A$ by using the Solar System experiments and calculate the deflection of light, Shapiro time delay, perihelion precession and obtain the effects associated with parameter $A$. Moreover, the parameterized post-Newtonian approach of this loop quantum gravity black hole was also carried out. It turns out the tightest constraint on $A$ can be improved to $0<A<4.0\times10^{-6}$.

Light deflection angle by superentropic black holes

Motivated by recent works on light deflection and shadow behaviors on AdS geometries [S. Haroon, M. Jamil, K. Jusufi, K. Lin and R. B. Mann, Phys. Rev. D 99 (2019) 044015, arXiv:1810.04103; A. Belhaj, H. Belmahi and M. Benali, Phys. Lett. B 821 (2021) 136619, arXiv:2110.06771], we investigate the deflection angle of light rays by superentropic black holes. Taking appropriate approximations, we first obtain the involved expression. For large values of the impact parameter, we get a specific value being zero for ordinary black holes without AdS backgrounds. Then, we examine and analyze such an optical quantity by providing graphical discussions in terms of a bounded region of the moduli space required by superentropic black hole conditions. Concretely, we study the deflection angle aspects by varying the black hole mass being fixed in the previous findings.

Nondivergent deflection of light around a photon sphere of a compact object

We demonstrate that the location of a stable photon sphere (PS) in a compact object is not always an edge such as the inner boundary of a black hole shadow, whereas the location of an unstable PS is known to be the shadow edge notably in the Schwarzschild black hole. If a static spherically symmetric (SSS) spacetime has the stable outermost PS, the spacetime cannot be asymptotically flat. A nondivergent deflection is caused for a photon traveling around a stable PS, though a logarithmic divergent behavior is known to appear in most of SSS compact objects with an unstable photon sphere. The reason for the nondivergence is that the closest approach of a photon is prohibited in the immediate vicinity of the stable PS when the photon is emitted from a source (or reaches a receiver) distant from a lens object. The finite gap size depends on the receiver and source distances from the lens as well as the lens parameters. The mild deflection angle of light can be approximated by an arcsine function. A class of SSS solutions in Weyl gravity exemplify the nondivergent deflection near the stable outer PS.

Optical gravity in a graviton spacetime

While the Hubble redshift is generally linked to expansion of spacetime, an open question concerns where the energy lost from redshifted photons, gravitons and gravitational waves goes. One possibility is that it gives rise to gravity. In the optical-mechanical analogy in general relativity, relativistic light deflection is treated as refraction in an optical medium of varying density gradient. We now model the optical medium analogue of spacetime as a real graviton conjugate interlinking all masses. Gravitons in local mass systems are assumed to be organized in coherent quantum states overlapping with those of particles within spacetime structures. With the Hubble redshift, however, gravitons acquire longer wavelengths as they are shifted to less coherent states. Assuming that the momentum and energy lost from gravitons is transferrable to particles, mutual screening by masses then gives rise to an attractive force with a magnitude consistent with gravity. Some possible cosmological implications of the model are discussed.