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General Relativity describes the trajectories of light-rays through curved spacetime near a massive object. In addition to gravitational lensing, we include an absorbing dielectric medium given by a complex refractive index known as the Drude model. When absorption is included the eikonal becomes complex, with the imaginary part related to the absorption along a ray between emission and observation points. We extend results from the literature to include dispersion in the index of refraction. The complex Hamiltonian splits into a real part that describes the equations of motion and a constraint equation that governs the momentum loss in the system. We work in coordinates which are fully real, with a real metric in physical spacetime. We assume the dust and plasma distributions of the Drude matter to coincide and vary as a power-law $1/r^h$. We find that transmission requires $h>1$, otherwise exponential absorption occurs along ray paths. We use ray-tracing through strongly absorbing matter near the surface of the compact star, as well as specializing to a point-lens in the weak-field limit with weakly absorbing matter to generate potentially observable light curves for distant observers. In the appropriate limits, our theory reproduces results from the literature., Comment: Comments Welcome!

Extreme scattering events (ESEs) are observed as dramatic ($>50\%$) drops in flux density that occur over an extended period of weeks to months. Discrete plasma lensing structures are theorized to scatter the radio waves produced by distant sources such as pulsars, causing the signature decrease in flux density and characteristic caustic spikes in ESE light curves. While plasma lens models in the extant literature have reproduced key features of ESE light curves, they have all faced the problem of being highly over-dense and over-pressured relative to the surrounding interstellar medium (ISM) by orders of magnitude. We model ESEs by numerically ray-tracing through analytic, volumetric plasma lens models by solving the eikonal equation. Delaunay triangulation connecting the rays approximates the wavefront, generating a mapping from the observer plane to the source plane to account for multiple-imaging. This eikonal method of ray-tracing is tested against known analytic solutions and is then applied to a three-dimensional Gaussian-distributed electron volume density lens, and a filament model inspired by Grafton et al. (2023). We find convergence of our numerical results with established analytic solutions validating our numerical method, and reproduce ESE-like light curves. Our numerical ray-tracing method lends itself well to exploring the lensing effects of volumetric turbulence as well as sheet-like lenses, which is currently in progress., Comment: 24 pages, 9 figures, accepted for publication in The Astrophysical Journal

The effect of ionized gas on the propagation of radio signals is known as plasma lensing. Unlike gravitational lensing, plasma lensing causes both magnification and strong de-magnification effects to background sources. We study the cross section of plasma lensing for two density profiles, the Gaussian and power-law models. In general, the cross section increases with the density gradient. Radio sources can be used to measure the free electron density along the line of sight. However, plasma lensing de-magnification causes an underestimate of the electron density. Such a bias increases with the electron density, and can be up to $\sim 15\%$ in the high density region. There is a large probability that high density clumps will be missed due to this bias. The magnification of plasma lensing can also change the luminosity function of the background sources. The number density of sources on both the high and low luminosity ends can be overestimated due to this biasing effect., Comment: MNRAS accepted, comments welcome

In this paper we investigate the relation between the potential and geometric time delays in gravitational lensing. In the original paper of Shapiro (1964), it is stated that there is a time delay in the radar signals between Earth and Venus that pass near a massive object (the Sun), compared to the path taken in the absence of any mass. The reason for this delay is connected with the influence of gravity on the coordinate velocity of a light ray in a gravitational potential. The contribution from the change of the path length, which happens to be of second order, is considered as negligible. Nevertheless, in the gravitational lens theory the geometrical delay, related to the change of path length, is routinely taken into account along with the potential term. In this work we explain this apparent discrepancy. We address the contribution of the geometric part of the time delay in different situations, and introduce a unified treatment with two limiting regimes of lensing. One of these limits corresponds to the time delay experiments near the Sun where the geometrical delay is shown to be negligible. The second corresponds to the typical gravitational lens scenario with multiple imaging where the geometrical delay is shown to be significant. We introduce a compact, analytical, and quantitative criteria based on relation between the angular position of source and the Einstein radius. This criterion allows one to find out easily when it is necessary to take the geometrical delay into account. In particular, it is shown that the geometrical delay is non-negligible in the case of good alignment between source, lens and observer, because in such a case it becomes a first order quantity (the same order as the potential term)., Comment: 21 pages, 5 figures

Spherical plasma lens models are known to suffer from a severe over-pressure problem, with some observations requiring lenses with central pressures up to millions of times in excess of the ambient ISM. There are two ways that lens models can solve the over-pressure problem: a confinement mechanism exists to counter the internal pressure of the lens, or the lens has a unique geometry, such that the projected column-density appears large to an observer. This occurs with highly asymmetric models, such as edge-on sheets or filaments, with potentially low volume-density. In the first part of this work we investigate the ability of non-magnetized plasma filaments to mimic the magnification of sources seen behind spherical lenses and we extend a theorem from gravitational lens studies regarding this model degeneracy. We find that for plasma lenses, the theorem produces unphysical charge density distributions. In the second part of the work, we consider the plasma lens over-pressure problem. Using magnetohydrodynamics, we develop a non self-gravitating model filament confined by a helical magnetic field. We use toy models in the force-free limit to illustrate novel lensing properties. Generally, magnetized filaments may act as lenses in any orientation with respect to the observer, with the most high density events produced from filaments with axes near the line of sight. We focus on filaments that are perpendicular to the line of sight that show the toroidal magnetic field component may be observed via the lens rotation measure., Comment: Accepted for publication in MNRAS. 19 pages, 10 figures. Comments welcome!

Radio signals are delayed when propagating through plasma. This type of delay is frequency-dependent and is usually used for estimating the projected number density of electrons along the line of sight, called the dispersion measure. The dense and clumpy distribution of plasma can cause refractive deflections of radio signals, analogous to lensing effects. Similar to gravitational lensing, there are two contributions to the time delay effect in plasma lensing: a geometric delay, due to increased path length of the signal, and a dispersive delay due to the change of speed of light in a plasma medium. We show the delay time for two models of the plasma distribution, and point out that the estimated dispersion measure can be biased. Since the contribution of the geometric effect can be comparable to that of the dispersive delay, the bias in the measured dispersion measure can be dramatically large if plasma lensing effects are not taken into account when signals propagate through a high-density gradient clump of plasma., Comment: comments welcome

Plasma lensing is the refraction of low-frequency electromagnetic rays due to free electrons in the interstellar medium. Although the phenomenon has a distinct similarity to gravitational lensing, particularly in its mathematical description, plasma lensing introduces other additional features, such as wavelength dependence, radial rather than tangential image distortions, and strong demagnification of background sources. Axisymmetrical models of plasma lenses have been well-studied in the literature, but density distributions with more complicated shapes can provide new and exotic image configurations and increase the richness of the magnification properties. As a first step towards non-axisymmetrical distributions, we study two families of elliptical plasma lens, softened power-law and exponential plasma distributions. We perform numerical studies on each lens model, and present them over a parameter space. In addition to deriving elliptical plasma lens formulae, we also investigate the number of critical curves that the lens can produce by studying the lens parameter space, in particular the dependence on the lensing ellipticity. We find that the introduction of ellipticity into the plasma distribution can enhance the lensing effects as well as the complexity of the magnification map., Comment: 14 pages, comments welcome

We present a Chandra and XMM-Newton study of the SNR CTB 37B, along with archival radio observations. In radio wavelengths, CTB 37B is an incomplete shell showing bright emission from the eastern side while the X-ray morphology shows diffuse emission from regions surrounding the magnetar CXOU J171405.7-381031. We used archival HI absorption measurements to constrain the distance to the remnant as 9.8+/-1.5 kpc. The X-ray spectrum of the remnant is described by a thermal model in the 1-5 keV energy range, with a temperature of 1.3+/-0.1 keV. The abundances from the spectral fits are consistent with being solar or sub-solar. A small region of diffuse emission is seen to the southern side of the remnant, best fitted by a nonthermal spectrum with an unusually hard photon index of 1.3+/-0.3. Assuming a distance of 9.8 kpc to the SNR, we infer a shock velocity of 915+/-70 km/s and explosion energy of (1.8+/-0.6)e50 ergs. The overall imaging and spectral properties of CTB 37B favor the interpretation of a young SNR (6200 yr) propagating in a low-density medium, under the assumption of a Sedov evolutionary phase., Comment: Accepted for publication in MNRAS, 10 pages, 9 figures, 4 tables

In contrast to the converging, achromatic behaviour of axisymmetric gravitational lenses, diverging frequency-dependent lensing occurs from refraction due to a distribution of over-dense axisymmetric plasma along an observer's line of sight. Such plasma lenses are particularly interesting from the point of view of astronomical observations because they can both magnify and dim the appearance of background sources as a function of frequency. Plasma lensing is believed to be involved in a number of separate phenomena involving the scintillation of radio pulsars, extreme scattering events of background radio sources and may also play a role in the generation of fast radio bursts. These lensing phenomena are believed to occur in dense environments, in which there may be many density perturbations between an observer and background source. In this work we generalize individual plasma lens models to produce dual component lenses using families of plasma lens models previously studied in the literature, namely the exponential and softened power-law lenses. Similar to binary gravitational lens models, these dual component plasma lenses feature a rich and complex critical and caustic morphology, as well as generate more complicated light curves. We map the number of criticals formed for a given component separation and angular size, and highlight a relevant degeneracy between two particular models. This work provides an argument in favor of close monitoring of extreme scattering events in progress in order to break such model degeneracies., Comment: 19 pages, 20 figures, accepted for publication in MNRAS. Comments welcome!

The recent discovery of the neutron star merger and multi-messenger event GW170817 (also known as kilonova GRB170817A) provides an unprecedented laboratory in which to study these mysterious objects, as well as an opportunity to test cutting-edge theories of gravity in the strong field regime. Before this event, such tests of our understanding of the nature of gravity were not possible. In this study, we analyze the X-ray observations of GW170817 obtained with NASA's Chandra X-ray Observatory following the 17 August 2017 detection of the event by the Laser Interferometer Gravitational Wave Observatory (LIGO). Motivated by understanding the emission mechanism for X-ray light and the outcome of the merger event, we fit the Chandra data with both non-thermal (as done previously in the literature) and thermal models. We specifically explore thermal plasma models that would be expected from a kilonova remnant (KNR). We reproduced the non-thermal results which were recently published by Nynka et al. (2018). We also find that thermal bremsstrahlung emission from hot plasma can account for the X-ray emission from this source. Furthermore, we consider models allowing for an intrinsic absorption from the merger event, yielding a softer power-law model photon index than previously published, or a lower plasma temperature. We also report on evidence for line emission, or excess above the continuum model, near 1.3 keV and 2.2 keV which sheds new light on the interpretation of the KNR and its nucleosynthesis products. We discuss the feasibility for the KNR as the origin for thermal X-ray emission at this stage of the kilonova evolution., Comment: 14 pages, 5 figures; research article to appear in JRASC (Feb 2019 issue)

The aim of the present work is twofold: first, we present general remarks about the application of recent procedures to compute the deflection angle in spherically symmetric and asymptotically flat spacetimes, taking into account finite distance corrections based on the Gauss-Bonnet theorem. Second, and as the main part of our work, we apply this powerful technique to compute corrections to the deflection angle produced by astrophysical configurations in the weak gravitational regime when a plasma medium is taken into account. For applications, we use these methods to introduce new general formulae for the bending angle of light rays in plasma environments in different astrophysical scenarios, generalizing previously known results. We also present new and useful formulae for the separation angle between the images of two sources when they are lensed by an astrophysical object surrounded by plasma. In particular, for the case of an homogeneous plasma we study these corrections for the case of light rays propagating near astrophysical objects described in the weak gravitational regime by a Parametrized-Post-Newtonian (PPN) metric which takes into account the mass of the objects and a possible quadrupole moment. Even when our work concentrates on finite distances corrections to the deflection angle, we also obtain as particular cases of our expressions new formulae which are valid for the more common assumption of infinite distance between receiver, lens and source. We also consider the presence of an inhomogeneous plasma media introducing as particular cases of our general results explicit expressions for particular charge number density profiles., Comment: 31 pages. Substantial additions. New section added, new theorem proved and a broader discussion on the deflection of light in the extended solar corona plasma model

Bell inequality experiments measure the correlation coefficients of two spatially separated systems. In an EPR setup, at one location Alice has $N_a\geq 2$ observables $A =\{\A_j\}_1^{N_a}$ while at a second remote location Bob has $N_b \geq2 $ observables $B= \{\B_k\}_1^{N_b}$. Within this bipartite environment each real $N_a \times N_b$ weight matrix $W$ constructs a Bell operator $\widehat{S}_W$ defined by the $jk$ sum of $W_{jk}\, \A_j \otimes \B_k$. Operator $\widehat{S}_W$ has the Bell non-locality boundary given by a hidden variable norm of $W$. As the $(A,B)$ composition varies, quantum extremes arise when the $\widehat{S}_W$ operator norm has the greatest possible Bell violation. A genetic algorithm (GA) search over all $(A,B)$ is used to find examples of the Alice and Bob operators that realize quantum extremes. A class $\XX_N$ of weights of special interest is given by the square $N_a=N_b=N$ matrices having two $\pm 1$ entries in each row and column with an odd number of minus signs. The class $\XX_N$ is a natural extension of the $2 \times 2 $ CHSH family. For dimensions $N=2\sim10$ the GA search finds that both the EPR correlation matrices and the Bell operator extremes do saturate their respective quantum bounds. Maximum Bell operator expectations fall between two benchmarks: the Bell inequality threshold and the quantum bound. The difference between these benchmarks is the quantum gap. Weight matrices $W$ that have zero quantum gap are determined by a row, column sum criteria., Comment: 25 pages, 2 figures. Comments welcome!

In the standard gravitational lensing scenario, rays from a background source are bent in the direction of a foreground lensing mass distribution. Diverging lens behaviour produces deflections in the opposite sense to gravitational lensing, and is also of astrophysical interest. In fact, diverging lensing due to compact distributions of plasma has been proposed as an explanation for the extreme scattering events (ESEs) that produce frequency-dependent dimming of extra-galactic radio sources, and may also be related to the refractive radio-wave phenomena observed to affect the flux density of pulsars. In this work we study the behaviour of two families of astrophysical diverging lenses in the geometric optics limit, the power-law and the exponential plasma lenses. Generally, the members of these model families show distinct behaviour in terms of image formation and magnification, however the inclusion of a finite core for certain power-law lenses can produce a caustic and critical curve morphology that is similar to the well-studied Gaussian plasma lens. Both model families can produce dual radial critical curves, a novel distinction from the tangential distortion usually produced by gravitational (converging) lenses. The deflection angle and magnification of a plasma lens varies with the observational frequency, producing wavelength-dependent magnifications that alter the amplitudes and the shape of the light curves. Thus, multi-wavelength observations can be used to physically constrain the distribution of the electron density in such lenses., Comment: 13 pages, 8 figures, accepted by MNRAS, comments still welcome

When doubly-affine matrices such as Latin and magic squares with a single non-zero eigenvalue are powered up they become constant matrices after a few steps. The process of compounding squares of orders m and n can then be used to generate an infinite series of such squares of orders mn. The Cayley-Hamilton theorem is used to understand this property, where their characteristic polynomials have just two terms., Comment: 24 pages, 10 tables

Weighted sums of left and right hand Kronecker products of Integer Sequence Doubly Affine (ISDA) as well as Generalized Arithmetic Progression Doubly Affine (GAPDA) arrays are used to generate larger ISDA arrays of multiplicative order (compound squares) from pairs of smaller ones. In two dimensions we find general expressions for the eigenvalues (EVs) and singular values (SVs) of the larger arrays in terms of the EVs and SVs of their constituent matrices, leading to a simple result for the rank of these highly singular compound matrices. Since the critical property of the smaller constituent matrices involves only identical row and column sums (often called semi-magic), the eigenvalue and singular value results can be applied to both magic squares and Latin squares. Additionally, the compounding process works in arbitrary dimensions due to the generality of the Kronecker product, providing a simple method to generate large order ISDA cubes and hypercubes. The first examples of compound magic squares are found in manuscripts that date back to the 10th century CE, and other representative applications are outlined through judicious examples., Comment: 23 pages, 3 tables. Comments welcome!

Electromagnetic rays travel on curved paths under the influence of gravity. When a dispersive optical medium is included, these trajectories are frequency-dependent. In this work we consider the behaviour of rays when a spherically symmetric, luminous compact object described by the Schwarzschild metric is surrounded by an optically thin shell of plasma supported by radiation pressure. Such levitating atmospheres occupy a position of stable radial equilibrium, where radiative flux and gravitational effects are balanced. Using general relativity and an inhomogeneous plasma we find the existence of a stable circular orbit within the atmospheric shell for low-frequency rays. We explore families of bound orbits that exist between the shell and the compact object, and identify sets of novel periodic orbits. Finally, we examine conditions necessary for the trapping and escape of low-frequency radiation., Comment: Invited contribution to Universe, special issue "Gravitational Lensing and Astrometry", academic eds. Valerio Bozza, Francesco De Paolis and Achille A. Nucita. 16 pages, 7 figures. Comments welcome: rogers@physics.umanitoba.ca

We consider the gravitational lensing of rays emitted by a compact object (CO) within a distribution of plasma with power-law density $\propto r^{-h}$. For the simplest case of a cloud of spherically symmetric cold non-magnetized plasma, the diverging effect of the plasma and the converging effect of gravitational lensing compete with one another. When $h<2$, the plasma effect dominates over the vacuum Schwarzschild curvature, potentially shifting the radius of the unstable circular photon orbit outside the surface of the CO. When this occurs, we define two relatively narrow radio-frequency bands in which plasma effects are particularly significant. Rays in the escape window have $\omega_{0} < \omega \leq \omega_{+}$ and are free to propagate to infinity from the CO surface. To a distant observer, the visible portion of the CO surface appears to shrink as the observed frequency is reduced, and vanishes entirely at $\omega_{0}$, in excess of the plasma frequency at the CO surface. We define the anomalous propagation window for frequencies $\omega_{-}< \omega \leq \omega_{0}$. Rays emitted from the CO surface within this frequency range are dominated by optical effects from the plasma and curve back to the surface of the CO, effectively cloaking the star from distant observers. We conclude with a study of neutron star (NS) compactness ratios for a variety of nuclear matter equations of state (EoS). For $h=1$, NSs generated from stiff EoS should display significant frequency dependence in the EW, and lower values of $h$ with softer EoS can also show these effects., Comment: 10 pages, 4 figures. Updated to match published version

Frequency-dependent gravitational lens effects are found for trajectories of electromagnetic rays passing through a distribution of plasma near a massive object. Ray propagation through plasma adds extra terms to the equations of motion that depend on the plasma refractive index. For low-frequency rays these refractive effects can dominate, turning the gravitational lens into a mirror. While light rays behave like particles with an effective mass given by the plasma frequency in a medium with constant density, an inhomogeneous plasma introduces more complicated behavior even for the spherically symmetric case. As a physical example, the pulse profile of a compact object sheathed in a dense plasma is examined, which introduces dramatic frequency-dependent shifts from the behavior in vacuum., Comment: 8 pages, 2 figures. Summary of an invited parallel session talk given in the GL3 session at the 14th Marcel Grossmann meeting, University of Rome "La Sapienza", Rome, July 12-18, 2015. For more details, see A. Rogers, 2015, MNRAS, 451, 1, 17-25

Energy losses from isolated neutron stars are commonly attributed to the emission of electromagnetic radiation from a rotating point-like magnetic dipole in vacuum. This emission mechanism predicts a braking index $n=3$, which is not observed in highly magnetized neutron stars. Despite this fact, the assumptions of a dipole field and rapid early rotation are often assumed a priori, typically causing a discrepancy between the characteristic age and the associated supernova remnant (SNR) age. We focus on neutron stars with `anomalous' magnetic fields that have established SNR associations and known ages. Anomalous X-ray pulsars (AXPs) and soft gamma repeaters (SGRs) are usually described in terms of the magnetar model, which posits a large magnetic field established by dynamo action. The high magnetic field pulsars (HBPs) have extremely large magnetic fields just above QED scale (but below that of the AXPs and SGRs), and central compact objects (CCOs) may have buried fields that will emerge in the future as nascent magnetars. In the first part of this series we examined magnetic field growth as a method of uniting the CCOs with HBPs and X-ray dim isolated neutron stars (XDINSs) through evolution. In this work we constrain the characteristic age of these neutron stars using the related SNR age for a variety of energy loss mechanisms and allowing for arbitrary initial spin periods. In addition to the SNR age, we also make use of the observed braking indices and X-ray luminosities to constrain the models., Comment: 14 pages, 7 figures, 2 tables. Accepted for publication in MNRAS

A wealth of X-ray and radio observations has revealed in the past decade a growing diversity of neutron stars (NSs) with properties spanning orders of magnitude in magnetic field strength and ages, and with emission processes explained by a range of mechanisms dictating their radiation properties. However, serious difficulties exist with the magneto-dipole model of isolated neutron star fields and their inferred ages, such as a large range of observed braking indices ($n$, with values often $<$3) and a mismatch between the neutron star and associated supernova remnant (SNR) ages. This problem arises primarily from the assumptions of a constant magnetic field with $n$=3, and an initial spin period that is much smaller than the observed current period. It has been suggested that a solution to this problem involves magnetic field evolution, with some NSs having magnetic fields buried within the crust by accretion of fall-back supernova material following their birth. In this work we explore a parametric phenomenological model for magnetic field growth that generalizes previous suggested field evolution functions, and apply it to a variety of NSs with both secure SNR associations and known ages. We explore the flexibility of the model by recovering the results of previous work on buried magnetic fields in young neutron stars. Our model fits suggest that apparently disparate classes of NSs may be related to one another through the time-evolution of the magnetic field., Comment: Updated to match published version: MNRAS 457, 1180, 2016. 12 pages, 4 figures, 3 tables