Gravitational Light Bending Research Articles

An accurate equation for the gravitational bending of light by a static massive object

An accurate equation for the gravitational bending of light by a static massive object

Orbital Precession in Janis–Newman–Winicour Spacetime

We have investigated the Janis–Newman–Winicour spacetime through three fundamental tests of theories of gravity, namely, gravitational lensing, perihelion shift, and redshift due to gravitational force. Focusing initially on the circular motion of a massive particle within the equatorial plane, the analysis disregards external scalar field interactions. The Janis–Newman–Winicour (JNW) spacetime’s unique parameters, mass (M) and the scalar parameter (n), are examined, revealing an intriguing relationship between the innermost stable circular orbit position of the test particle and the scalar field parameter. The study also explores photon motion around a gravitational object in JNW spacetime, revealing the expansion of the photon sphere alongside a diminishing shadow, influenced by the external scalar field. Despite these complexities, gravitational bending of light remains consistent with general relativity predictions. The investigation extends to perihelion precession, where the trajectory of a massive particle in JNW spacetime exhibits eccentricity-dependent shifts, distinguishing it from Schwarzschild spacetime. Finally, oscillatory motion of massive particles in JNW spacetime is explored, providing analytical expressions for epicyclic frequencies using perturbation methods. The study concludes with the application of MCMC analyses to constrain the JNW spacetime parameters based on observational data.

Shadow and deflection angle of asymptotic, magnetically-charged, non-singular black hole

In this paper, we investigate a solution for an asymptotic, magnetically-charged, non-singular (AMCNS) black hole. By utilizing the Gauss–Bonnet theorems, we aim to unravel the intricate astrophysics associated with this unique black hole. The study explored various aspects including the black hole’s gravitational field, intrinsic properties, light bending, the shadow and greybody bounding of the black hole. Through rigorous calculations and simulations, we derive the weak deflection angle of the optical metric of AMCNS black hole. Additionally, we investigate the impact of the dark matter medium on the deflection angle, examined the distinctive features of the black hole’s shadow, and bound its greybody factors. Our findings not only deepen our understanding of gravitational lensing but also pave the way for future improvements in black hole theories by minimizing restrictive assumptions and incorporating a more realistic representation of these cosmic phenomena.

A polarimetrically oriented X-ray stare at the accreting pulsar EXO 2030+375

Accreting X-ray pulsars (XRPs) are presumed to be ideal targets for polarization measurements, as their high magnetic field strength is expected to polarize the emission up to a polarization degree of ∼80%. However, such expectations are being challenged by recent observations of XRPs with the Imaging X-ray Polarimeter Explorer (IXPE). Here, we report on the results of yet another XRP, namely, EXO 2030+375, observed with IXPE and contemporarily monitored with Insight-HXMT and SRG/ART-XC. In line with recent results obtained with IXPE for similar sources, an analysis of the EXO 2030+375 data returns a low polarization degree of 0%–3% in the phase-averaged study and a variation in the range of 2%–7% in the phase-resolved study. Using the rotating vector model, we constrained the geometry of the system and obtained a value of ∼60° for the magnetic obliquity. When considering the estimated pulsar inclination of ∼130°, this also indicates that the magnetic axis swings close to the observer’s line of sight. Our joint polarimetric, spectral, and timing analyses hint toward a complex accreting geometry, whereby magnetic multipoles with an asymmetric topology and gravitational light bending significantly affect the behavior of the observed source.

Correlated variability of the reflection fraction with the X-ray flux and spectral index for Mrk 478

ABSTRACT The X-ray spectrum of Mrk 478 is known to be dominated by a strong soft excess that can be described using relativistic blurred reflection. Using observations from XMM–Newton, AstroSat, and Swift, we show that for the long-term (∼years) and intermediate-term (days to months) variability, the reflection fraction is anticorrelated with the flux and spectral index, which implies that the variability is due to the hard X-ray producing corona moving closer to and further from the black hole. Using flux-resolved spectroscopy of the XMM–Newton data, we show that the reflection fraction has the same behaviour with flux and index on short time-scales of hours. The results indicate that both the long- and short-term variability of the source is determined by the same physical mechanism of strong gravitational light bending causing enhanced reflection and low flux as the corona moves closer to the black hole.

A new Monte Carlo radiative transfer simulation of cyclotron resonant scattering features

ABSTRACT We present a new Monte Carlo radiative transfer code, which we have used to model the cyclotron line features in the environment of a variable magnetic field and plasma density. The code accepts an input continuum and performs only the line transfer by including the three cyclotron resonant processes (cyclotron absorption, cyclotron emission, cyclotron scattering). Subsequently, the effects of gravitational redshift and light bending on the emergent spectra are computed. We have applied our code to predict the observable spectra from three different emission geometries; (1) an optically thin slab near the stellar surface, (2) an accretion mound formed by the accumulation of the accreted matter, (3) an accretion column representing the zone of a settling flow on to the star. Our results show that the locally emergent spectra from the emission volume are significantly anisotropic. However, in the presence of strong light bending the anisotropy reduces considerably. This averaging also drastically reduces the strength of harmonics higher than second in the observable cyclotron spectra. We find that uniform field slabs produce line features that are too narrow, and mounds with large magnetic distortions produce features that are too wide compared to the average widths of the spectral features observed from various sources. The column with a gently varying (dipole) field produces widths in the intermediate range, similar to those observed.

Measuring the Nonaxially Symmetric Surface Temperature Distribution of the Central Compact Object in Puppis A

Abstract The surface temperature distributions of central compact objects (CCOs) are powerful probes of their crustal magnetic field strengths and geometries. Here we model the surface temperature distribution of RX J0822−4300, the CCO in the Puppis A supernova remnant, using 471 ks of XMM-Newton data. We compute the energy-dependent pulse profiles in 16 energy bands, fully including the general relativistic effects of gravitational redshift and light bending, to accurately model the two heated surface regions of different temperatures and areas, in addition to constraining the viewing geometry. This results in precise measurements of the two temperatures: kT warm = ( 1 + z ) × 0.222 − 0.019 + 0.018 keV and kT hot = (1 + z) × 0.411 ± 0.011 keV. The two heated surface regions are likely located very close to the rotational poles, with the most probable position of the hotter component ≈ 6° from the rotational pole. For the first time, we are able to measure a deviation from a pure antipodal hot-spot geometry, with a longitudinal offset δ γ = 11 .° 7 − 2 .° 5 + 2 .° 6 . The discovery of this asymmetry, along with the factor of ≈2 temperature difference between the two emitting regions, may indicate that RX J0822−4300 was born with a strong, tangled crustal magnetic field.

Light bending by the cosmological constant

We revisit the question of whether the cosmological constant Λ affects the cosmological gravitational bending of light, by numerical integration of the geodesic equations for a Swiss cheese model consisting of a point mass and a compensated vacuole, in a Friedmann-Robertson-Walker background. We find that there is virtually no dependence of the light bending on the cosmological constant that is not already accounted for in the angular diameter distances of the standard lensing equations, plus small modifications that arise because the bending is restricted to a finite region covered by the hole. The residual Λ dependence for a 1013 M ☉ lens is at the level of 1 part in 107, and even this might be accounted for by small changes in the hole size evolution as the photon crosses. We therefore conclude that there is no need for modification of the standard cosmological lensing equations in the presence of a cosmological constant.

Ultralight dark matter: constraints from gravitational waves and other astrophysical observations

The orbital period loss of the compact binary systems is the first indirect evidence of gravitational waves which agrees well with Einstein’s general theory of relativity to a very good accuracy. However, there is less than one percent uncertainty in the measurement of orbital period loss from the general reltivistic prediction. Perihelion precession of planets, Gravitational light bending and Shapiro delay are three other successful tests of general relativity theory. Though there are uncertainties in the measurements of those observations from the general reltivistic predictions as well. To resolve these uncertainties, we assume radiation of ultralight axions and light gauge boson particles of L i – Lj type from those systems which can be a possible candidate of fuzzy dark matter. In this article, we obtain bounds on new physics parameters from those astrophysical observations.

Null geodesics and QNMs in the field of regular black holes

In this paper, we analyze the null geodesics of regular black holes (BHs). A detailed analysis of geodesic structure, both null geodesics and timelike geodesics, has been investigated for the said BH. As an application of null geodesics, we calculate the radius of photon sphere and gravitational bending of light. We also study the shadow of the BH spacetime. Moreover, we determine the relation between radius of photon sphere [Formula: see text] and the shadow observed by a distance observer. Furthermore, we discuss the effect of various parameters on the radius of shadow [Formula: see text]. Also, we compute the angle of deflection for the photons as a physical application of null-circular geodesics. We find the relation between null geodesics and quasinormal mode (QNM) frequency in the eikonal approximation by computing the Lyapunov exponent. It is also shown that (in the eikonal limit) the QNMs of BHs are governed by the parameter of null-circular geodesics. The real part of QNMs frequency determines the angular frequency, whereas the imaginary part determines the instability timescale of the circular orbit. Next, we study the massless scalar perturbations and analyze the effective potential graphically. Massive scalar perturbations are also discussed. As an application of timelike geodesics, we compute the innermost stable circular orbit (ISCO) and marginally bound circular orbit (MBCO) of the regular BHs which are closely related to the BH accretion disk theory. In the appendix, we calculate the relation between angular frequency and Lyapunov exponent for null-circular geodesics.

Extreme relativistic reflection in the active galaxy ESO 033-G002

ABSTRACTWe present the first high signal-to-noise broad-band X-ray spectrum of the radio-quiet type-2 Seyfert ESO 033-G002, combining data from XMM–Newton and NuSTAR. The nuclear X-ray spectrum is complex, showing evidence for both neutral and ionized absorption, as well as reflection from both the accretion disc and more distant material, but our broad-band coverage allows us to disentangle all of these different components. The total neutral column during this epoch is NH ∼ (5−6) × 1022 $\rm \, cm^{-2}$, consistent with the optical classification of ESO 033-G002 as a type-2 Seyfert but not so large as to prevent us from robustly determining the properties of the innermost accretion flow. The ionized absorption – dominated by lines from Fe xxv and Fe xxvi – reveals a moderately rapid outflow (vout ∼ 5400 $\rm \, km~s^{-1}$) which has a column comparable to the neutral absorption. We find the disc reflection from the innermost regions to be extreme, with a reflection fraction of Rfrac ∼ 5. This requires strong gravitational lightbending and, in turn, both an extremely compact corona (within ∼2 RG of the black hole) and a rapidly rotating black hole (a* > 0.96). Despite this tight size constraint, with a temperature of kTe = 40–70 keV the X-ray corona in ESO 033-G002 appears similar to other active galactic nucle in terms of its placement in the compactness–temperature plane, consistent with sitting close to the limit determined by runaway pair production. Finally, combining X-ray spectroscopy, timing, and updated optical spectroscopy, we also estimate the mass of the black hole to be log[MBH/M⊙] ∼ 7.0–7.5.

Relativistic rotating vector model for X-ray millisecond pulsars

The X-ray radiation produced on the surface of accreting magnetised neutron stars is expected to be strongly polarised. A swing of the polarisation vector with the pulsar phase gives a direct measure of the source inclination and magnetic obliquity. In the case of rapidly rotating millisecond pulsars, the relativistic motion of the emission region causes additional rotation of the polarisation plane. Here, we develop a relativistic rotating vector model, where we derive analytical expression for the polarisation angle as a function of the pulsar phase accounting for relativistic aberration and gravitational light bending in the Schwarzschild metric. We show that in the case of fast pulsars the rotation of the polarisation plane can reach tens of degrees, strongly influencing the observed shape of the polarisation angle’s phase dependence. The rotation angle grows nearly linearly with the spin rate but it is less sensitive to the neutron star radius. Overall, this angle is large even for large spots. Our results have implications with regard to the modelling of X-ray polarisation from accreting millisecond pulsars that are to be observed with the upcoming Imaging X-ray Polarimeter Explorer and the enhanced X-ray Timing and Polarimetry mission. The X-ray polarisation may improve constraints on the neutron star mass and radius coming from the pulse profile modelling.

Returning radiation in strong gravity around black holes: reverberation from the accretion disc

ABSTRACT We study reflected X-ray emission that returns to the accretion disc in the strong gravitational fields around black holes using General Relativistic ray-tracing and radiative transfer calculations. Reflected X-rays that are produced when the inner regions of the disc are illuminated by the corona are subject to strong gravitational light bending, causing up to 47 per cent of the reflected emission to be returned to the disc around a rapidly spinning black hole, depending upon the scale height of the corona. The iron Kα line is enhanced relative to the continuum by 25 per cent, and the Compton hump by up to a factor of 3. Additional light traveltime between primary and secondary reflections increases the reverberation time lag measured in the iron K band by 49 per cent, while the soft X-ray lag is increased by 25 per cent and the Compton hump response time is increased by 60 per cent. Measured samples of X-ray reverberation lags are shown to be consistent with X-rays returning to the accretion disc in strong gravity. Understanding the effects of returning radiation is important in interpreting reverberation observations to probe black holes. Reflected X-rays returning to the disc can be uniquely identified by blueshifted returning iron K line photons that are Compton scattered from the inner disc, producing excess, delayed emission in the 3.5–4.5 keV energy range that will be detectable with forthcoming X-ray observatories, representing a unique test of General Relativity in the strong field limit.

Gravitational light deflection in Earth-based laser cavity experiments

As known from Einstein's theory of general relativity, the propagation of light in the presence of a massive object is affected by gravity. In this work, we discuss whether the effect of gravitational light bending can be observed in Earth-based experiments, using high-finesse optical cavities. In order to do this, we theoretically investigate the dynamics of electromagnetic waves in the spacetime of a homogeneous gravitational field and give an analytical expression for the resulting modifications to Gaussian beam propagation. This theoretical framework is used to calculate the intensity profile at the output of a Fabry-P\'erot cavity and to estimate the imprints of Earth's gravity on the cavity output signal. In particular, we found that gravity causes an asymmetry of the output intensity profile. Based on that, we discuss a measurement scheme, that could be realized in facilities like the GEO600 gravitational wave detector and the AEI 10 m detector prototype.

The Legacy of Einstein’s Eclipse, Gravitational Lensing

A hundred years ago, two British expeditions measured the deflection of starlight by the Sun’s gravitational field, confirming the prediction made by Einstein’s General Theory of Relativity. One hundred years later many physicists around the world are involved in studying the consequences and use as a research tool, of the deflection of light by gravitational fields, a discipline that today receives the generic name of Gravitational Lensing. The present review aims to commemorate the centenary of Einstein’s Eclipse expeditions by presenting a historical perspective of the development and milestones on gravitational light bending, covering from early XIX century speculations, to its current use as an important research tool in astronomy and cosmology.

First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole

Abstract When surrounded by a transparent emission region, black holes are expected to reveal a dark shadow caused by gravitational light bending and photon capture at the event horizon. To image and study this phenomenon, we have assembled the Event Horizon Telescope, a global very long baseline interferometry array observing at a wavelength of 1.3 mm. This allows us to reconstruct event-horizon-scale images of the supermassive black hole candidate in the center of the giant elliptical galaxy M87. We have resolved the central compact radio source as an asymmetric bright emission ring with a diameter of 42 ± 3 μas, which is circular and encompasses a central depression in brightness with a flux ratio ≳10:1. The emission ring is recovered using different calibration and imaging schemes, with its diameter and width remaining stable over four different observations carried out in different days. Overall, the observed image is consistent with expectations for the shadow of a Kerr black hole as predicted by general relativity. The asymmetry in brightness in the ring can be explained in terms of relativistic beaming of the emission from a plasma rotating close to the speed of light around a black hole. We compare our images to an extensive library of ray-traced general-relativistic magnetohydrodynamic simulations of black holes and derive a central mass of M = (6.5 ± 0.7) × 109 M ⊙. Our radio-wave observations thus provide powerful evidence for the presence of supermassive black holes in centers of galaxies and as the central engines of active galactic nuclei. They also present a new tool to explore gravity in its most extreme limit and on a mass scale that was so far not accessible.

Effect of power-law Maxwell field to the gravitational lensing

In this paper, we extend the gravitational bending of light studies in Kottler metrics to comprise nonlinear electrodynamics within the framework of Einstein - power - Maxwell theory. We show that the closest approach distance and the gravitational bending of light are affected from the presence of charge for particular values of the power parameter $k$, which is defined by means of energy conditions. It is shown that the bending angle of light is stronger in the case of a strong electric field, which is the case for $k=1.2$.

Pulsar timing in extreme mass ratio binaries: a general relativistic approach

The detection of a pulsar (PSR) in a tight, relativistic orbit around a supermassive or intermediate mass black hole - such as those in the Galactic centre or in the centre of Globular clusters - would allow for precision tests of general relativity (GR) in the strong-field, non-linear regime. We present a framework for calculating the theoretical time-frequency signal from a PSR in such an Extreme Mass Ratio Binary (EMRB). This framework is entirely relativistic with no weak-field approximations and so able to account for all higher-order strong-field gravitational effects, relativistic spin dynamics, the convolution with astrophysical effects and the combined impact on the PSR timing signal. Specifically we calculate both the spacetime path of the pulsar radio signal and the complex orbital and spin dynamics of a spinning pulsar around a Kerr black hole, accounting for spacetime curvature and frame dragging, relativistic and gravitational time delay, gravitational light bending, temporal and spatial dispersion induced by the presence of plasma along the line of sight and relativistic aberration. This then allows for a consistent time-frequency solution to be generated. Such a framework is key for assessing the use of PSR as probes of strong field GR, helping to inform the detection of an EMRB system hosting a PSR and, most essentially, for providing an accurate theoretical basis to then compare with observations to test fundamental physics.

Influence of bulk motion of an infalling plasma in line-forming region on cyclotron line in accreting X-ray pulsars

Abstract We examine the effect of the bulk motion of an infalling plasma on the properties (energy, width and depth) of a cyclotron line, including the effect of gravitational light bending as well as variation of the magnetic field. The cyclotron line energy decreases with increasing bulk velocity. Moreover, the dependence of the line energy on viewing angle becomes stronger as bulk velocity increases. As a result, the amount of a variation of the line energy with angle increases with increasing bulk velocity. On considering the two-dimensional structure of an accretion column, the bulk velocity in the line-forming region that is expected to be located around the walls of an accretion column can be considerably different from that in the continuum-formation region because of the decrease in the radiation energy density near the walls of the accretion column. In addition, the bulk velocity in the line-forming region can considerably depend on the beam pattern generated in the continuum-forming region because the bulk velocity is decelerated by photons propagating not across the magnetic field but along it. In the present paper, we demonstrate that the observed amount of a variation of the line energy with pulse phase can be explained by a variation of the bulk velocity with beam pattern, i.e., luminosity, using a cylindrical geometry with an accretion mound where a continuum spectrum is formed.

Gravitational lensing in conformal and emergent gravity

The gravitational bending of light in the framework of conformal gravity is considered where an exact solution for null geodesics in the Mannheim-Kazanas is obtained. The linear coefficient γ characteristic to conformal gravity is shown to contribute enhanced deflection compared to the angle predicted by General Relativity for small γ. We also briefly consider gravitational lensing in covariant emergent gravity.