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We study real-time propagation of a massive scalar field on the extremal BTZ black hole spacetime, focusing on the Aretakis instability of the event horizon. We obtain a simple time-domain expression for the $\textrm{AdS}_3$ retarded Green function with Dirichlet boundary conditions and construct the corresponding time-domain BTZ retarded Green function using the method of images. The field decays at different rates on and off the horizon, indicating that transverse derivatives grow with time on the horizon (Aretakis instability). We solve the null geodesic equation in full generality and show that the instability is associated with a class of null geodesics that orbit near the event horizon arbitrarily many times before falling in. In an appendix we also treat the problem in the frequency domain, finding consistency between the methods., Comment: 23 pages, 6 figures

In this paper, we investigate the effect of higher curvature corrections from Gauss-Bonnet gravity on the shadow of charged black holes in both $AdS$ and Minkowski spacetimes. The null geodesic equations are computed in $d=5$ spacetime dimensions by using the directions of symmetries and Hamilton-Jacobi equation. With the null geodesics in hand, we then proceed to evaluate the celestial coordinates ($\alpha, \beta$) and the radius $R_s$ of the black hole shadow and represent it graphically. The effects of charge $Q$ of the black hole and the Gauss-Bonnet parameter $\gamma$ on the radius of the shadow $R_s$ is studied in detail. It is observed that the Gauss-Bonnet parameter $\gamma$ affects the radius of the black hole shadow $R_s$ differently for the $AdS$ black hole spacetime in comparison to the black hole spacetime which is asymptotically flat. In particular the radius of the black hole shadow increases with increase in the Gauss-Bonnet parameter in case of the $AdS$ black hole spacetime and decreases in case of the asymptotically flat black hole spacetime. We then introduce a plasma background in order to observe the change in the silhouette of the black hole shadow due to a change in the refractive index of the plasma medium. Finally, we study the effect of the Gauss-Bonnet parameter $\gamma$ on the energy emission rate of the black hole which depends on the black hole shadow radius and represent the results graphically., Comment: 18 pages Latex, comments are welcome

Roger Penrose's 2020 Nobel Prize in Physics recognises that his identification of the concepts of "gravitational singularity" and an "incomplete, inextendible, null geodesic" is physically very important. The existence of an incomplete, inextendible, null geodesic doesn't say much, however, if anything, about curvature divergence, nor is it a helpful definition for performing actual calculations. Physicists have long sought for a coordinate independent method of defining where a singularity is located, given an incomplete, inextendible, null geodesic, that also allows for standard analytic techniques to be implemented. In this essay we present a solution to this issue. It is now possible to give a concrete relationship between an incomplete, inextendible, null geodesic and a gravitational singularity, and to study any possible curvature divergence using standard techniques., This essay received an Honorable Mention in the 2021 Essay Competition of the Gravity Research Foundation (see https://static1.squarespace.com/static/5852e579be659442a01f27b8/t/609d66c823a9a352bc3b24c3/1620928201758/2021-GRF-Abstracts.pdf). Keywords: Singularity, completion, boundary, endpoint theorem, coordinates, singularity theorem, curvature, Kerr, Boyer-Lindquist, black hole

We study string theory on the pp-wave geometry obtained by taking the Penrose limit around a certain null geodesic of the non-supersymmetric Schrodinger background. We solve for the spectrum of bosonic excitations and find compelling agreement with the dispersion relation of the giant magnons in the Schrodinger background obtained previously in arXiv:1712.03091. Inspired by the pp-wave spectrum we conjecture an exact in the t'Hooft coupling dispersion relation for the magnons in the original Schrodinger background. We show that the pp-wave background admits exactly 16 Killing spinors. We use the explicit form of the latter in order to derive the supersymmetry algebra of the background which explicitly depends on the deformation parameter. Its bosonic subalgebra is of the Newton-Hooke type., Comment: 35 pages; v2: Minor changes, JHEP version

We consider spherically symmetric spacetimes embedded in a 5-dimensional static Lorentzian manifold. Studying the null geodesic equations in warped spacetimes bounded by static 3-branes, we discuss possible relations of particles following the geodesics with the gravitational waves detected recently.

The null curvature condition (NCC) is the requirement that the Ricci curvature of a Lorentzian manifold be nonnegative along null directions, which ensures the focusing of null geodesic congruences. In this note, we show that the NCC together with the causal structure significantly constrain the metric. In particular, we prove that any conformal rescaling of a vacuum spacetime introduces either geodesic incompleteness or negative null curvature, provided the conformal factor is non-constant on at least one complete null geodesic. In the context of bulk reconstruction in AdS/CFT, our results combined with the technique of light-cone cuts can be used in vacuum spacetimes to reconstruct the full metric in regions probed by complete null geodesics reaching the boundary. For non-vacuum spacetimes, our results constrain the conformal factor, giving an approximate reconstruction of the metric., Comment: 12 pages

In this contribution we study the birefringence and the quasinormal modes (QNM) in the eikonal approximation, of the Einstein-Euler-Heisenberg black hole (EEH-BH) . The EEH-BH is an exact solution of the Einstein equations coupled to the Euler-Heisenberg nonlinear electrodynamics. It is well known that in the presence of strong electromagnetic fields light rays modify their trajectories and do follow the null geodesics of an effective metric as a result of the nonlinear interaction between light and the intense magnetic or electric background. In the Euler-Heisenberg theory the phenomenon of birefringence arises and then exist two possible light trajectories for polarized light in the vicinity of the EEH-BH. On the other hand, using the correspondence between the parameters of the null unstable geodesics and the quasinormal modes in the eikonal approximation. We determine these QNM for both, the electric and the magnetic EEH-BH and compare them with its linear counterpart, the Reissner-Nordstr\"{o}m black hole, since there are two effective metrics, to each one corresponds one null geodesic that renders two possible QNM from the same compact object.

We show that the recipe for computing the expansions $\theta_\ell$ and $\theta_n$ of outgoing and ingoing null geodesics normal to a surface admits a covariance group with nonconstant scalar $\kappa(x)$, corresponding to the mapping $\theta_\ell \to \kappa \theta_\ell$, $\theta_n \to \kappa^{-1} \theta_n$. Under this mapping, the product $\theta_\ell \theta_n$ is invariant, and thus the marginal surface computed from the vanishing of $\theta_\ell$, which is used to define the apparent horizon, is invariant. This covariance group naturally appears in comparing the expansions computed with different choices of coordinate system., Comment: Latex, 6 pages

Since, in Einstein gravity, a massless scalar field with lightlike gradient behaves as a null dust, one could expect that it can act as the matter source of Vaidya geometries. We show that this is impossible because the Klein–Gordon equation forces the null geodesic congruence tangent to the scalar field gradient to have zero expansion, contradicting a basic property of Vaidya solutions. By contrast, exact plane waves travelling at light speed and sourced by a scalar field acting as a null dust are possible.

We investigate photon surfaces and their stability in a less symmetric spacetime, a general static warped product with a warping function acting on a Riemannian submanifold of codimension two. We find a one-dimensional pseudopotential that gives photon surfaces as its extrema regardless of the spatial symmetry of the submanifold. The maxima and minima correspond to unstable and stable photon surfaces, respectively. It is analogous to the potential giving null circular orbits in a spherically symmetric spacetime. We also see that photon surfaces indeed exist for the spacetimes which are solutions to the Einstein equation. The parameter values for which the photon surfaces exist are specified. As we show finally, the pseudopotential arises due to the separability of the null geodesic equation, and the separability comes from the existence of a Killing tensor in the spacetime. The result leads to the conclusion that photon surfaces may exist even in a less symmetric spacetime if the spacetime admits a Killing tensor., 21 pages, no figure

We discuss dynamics of massive Klein-Gordon fields in two-dimensional Anti-de Sitter spacetimes ($AdS_2$), in particular conserved quantities and non-modal instability on the future Poincar\'e horizon called, respectively, the Aretakis constants and the Aretakis instability. We find out the geometrical meaning of the Aretakis constants and instability in a parallel-transported frame along a null geodesic, i.e., some components of the higher-order covariant derivatives of the field in the parallel-transported frame are constant or unbounded at the late time, respectively. Because $AdS_2$ is maximally symmetric, any null hypersurfaces have the same geometrical properties. Thus, if we prepare parallel-transported frames along any null hypersurfaces, we can show that the same instability emerges not only on the future Poincar\'e horizon but also on any null hypersurfaces. This implies that the Aretakis instability in $AdS_2$ is the result of singular behaviors of the higher-order covariant derivatives of the fields on the whole $AdS$ infinity, rather than a blow-up on a specific null hypersurface. It is also discussed that the Aretakis constants and instability are related to the conformal Killing tensors. We further explicitly demonstrate that the Aretakis constants can be derived from ladder operators constructed from the spacetime conformal symmetry., Comment: 24 pages, 1 figure, v2: minor revisions, v3: minor revisions, accepted for publication in Phys. Rev. D

Material particles and photons are of paramount importance in constructing and analysing models of gravitational fields in general relativity. Their histories in space-time are congruences of time-like and light-like (or null) world lines respectively. Material particles and photons therefore play a central role in this book and we begin by describing the standard theory of congruences of time-like world lines and of null congruences. In the light-like case we shall only require a knowledge of null geodesic congruences. For use later we describe geodesic deviation equations with the help of Synge’s world function and parallel propagator.

An optical equation for null strings is derived. The equation is similar to Sachs' optical equations for null geodesic congruences. The string optical equation is given in terms of a single complex scalar function $Z$, which is a combination of spin coefficients at the string trajectory. Real and imaginary parts of $Z$ determine expansion and rotation of strings. Trajectories of strings can be represented by diagrams in a complex $Z$-plane. Such diagrams allow one to draw some universal features of null strings in different backgrounds. For example, in asymptotically flat space-times $Z$ vanishes as future null infinity is approached, that is, gradually shapes of strings are "freezing out". Outgoing gravitational radiation and flows of matter leave ripples on the strings. These effects are encoded in subleading terms of $Z$. String diagrams are demonstrated for rotating and expanding strings in a flat space-time and in cosmological models., Comment: 21 pages, 3 figures

We investigate a particular regularization of big bang singularity, which remains within the domain of 4-dimensional general relativity but allowing for degenerate metrics. We study the geodesics and geodesic congruences in the modified Friedmann-Lema\^itre-Robertson-Walker universe. In particular, we calculate the expansion of timelike and null geodesic congruences. Based on these results, we also briefly discuss the cosmological singularity theorems., Comment: 16 pages, 1 figure, published version

We obtain a high-dimensional Schwarzschild black hole solution in the scalar-tensor-vector gravity (STVG), and then analyze the influence of parameter $\alpha$ associated with a deviation of the STVG theory from General Relativity on event horizons and Hawking temperature. We calculate the quasinormal mode frequencies of massless scalar field perturbations for the high-dimensional Schwarzschild STVG black hole by using the sixth-order WKB approximation method and the unstable null geodesic method in the eikonal limit. The results show that the increase of parameter $\alpha$ makes the scalar waves decay slowly, while the increase of the spacetime dimension makes the scalar waves decay fast. In addition, we study the influence of parameter $\alpha$ on the shadow radius of this high-dimensional Schwarzschild STVG black hole and find that the increase of parameter $\alpha$ makes the black hole shadow radius increase, but the increase of the spacetime dimension makes the black hole shadow radius decrease. Finally, we investigate the energy emission rate of the high-dimensional Schwarzschild STVG black hole, and find that the increase of parameter $\alpha$ makes the evaporation process slow, while the increase of the spacetime dimension makes the process fast., Comment: v1: 18 pages, 6 figures, 4 tables; v2: clarifications and references added; v3: 19 pages, final version to appear in European Physical Journal C

We obtain the solution of a generalized Ay\'on-Beato--Garc\'{\i}a (ABG) black hole with the nonlinear tensor field ${B}_{\ensuremath{\mu}\ensuremath{\nu}}$ in scalar-tensor-vector gravity (STVG), called the generalized ABG STVG black hole. This black hole is endowed with four parameters, the black hole mass $M$, the parameter $\ensuremath{\alpha}$ associated with STVG theory, and the two parameters $\ensuremath{\lambda}$ and $\ensuremath{\beta}$ that are related to the dipole and quadrupole moments of the nonlinear tensor field, respectively. By analyzing the characteristics of the black hole, we find that the generalized ABG STVG black hole is regular when $\ensuremath{\lambda}\ensuremath{\ge}3$ and $\ensuremath{\beta}\ensuremath{\ge}4$, and we study the effects of the parameters $\ensuremath{\alpha}$, $\ensuremath{\lambda}$, and $\ensuremath{\beta}$ on the black hole horizon. We calculate the quasinormal mode frequencies of the odd parity gravitational perturbation for the generalized ABG STVG black hole by using the sixth order WKB approximation method and simultaneously using the null geodesic method at the eikonal limit. The results show that the increase of the parameters $\ensuremath{\alpha}$ and $\ensuremath{\lambda}$ makes the gravitational waves decay slowly, while the increase of the parameter $\ensuremath{\beta}$ makes the gravitational waves decay fast at first, then slowly. In addition, we verify that the improved correspondence between the real part of quasinormal frequencies at the eikonal limit and the shadow radius is valid for the generalized ABG STVG black hole.

In this paper we consider a massive vector field in the background of a space-time obtained by certain Z2 quotient of the BTZ black hole. We analyse the back reaction of the matter field on the space-time geometry up to first order in metric perturbation. The expectation value of the stress-energy tensor can be computed exactly by considering its pull-back onto the covering space. Upon a suitable choice of the boundary condition on the vector field around a non-contractible cycle of the quotient manifold it is possible to obtain the average energy on a null geodesic to be negative there by resulting a traversable wormhole., 20 pages, 3 figures

We study horizon shells and soldering freedom for extreme black holes and how supertranslation-like Bondi-Metzner-Sachs (BMS) symmetries appear as soldering transformations. Further, for a null shell placed infinitesimally close to the horizon of an extreme Reissner-Nordstr$\ddot{o}$m (RN) black hole, we show superrotation-like symmetries also arise as soldering freedom. Next, considering the interaction of impulsive gravitational waves supported at the horizon shell with test particles, we study how the "memory" (or the imprints) of BMS-like symmetries gets encoded in the geodesics (test particles) crossing the shell. Our study shows, timelike test particles get displaced from their initial plane when they cross the horizon shell. For a null geodesic congruence crossing the horizon shell, the optical tensors corresponding to the congruence suffer jumps. In both the cases, the changes are induced by BMS parameters that constitute the gravity wave and matter degrees of freedom of the shell., 23 pages, no figure. Text substantially modified for better presentation. Version published in Phys. Rev D

The null geodesic equation in the Kerr spacetime can be expressed as a set of integral equations involving certain potentials. We classify the roots of these potentials and express the integrals in manifestly real Legendre elliptic form. We then solve the equations using Jacobi elliptic functions, providing the complete set of null geodesics of the Kerr exterior as explicit parametrized curves.

We provide an exhaustive and illustrated classification of timelike and null geodesics in the near-horizon region of near-extremal Kerr black holes. The classification of polar motion extends to Kerr black holes of arbitrary spin. The classification of radial motion leads to a simple parametrization of the separatrix between bound and unbound motion. Furthermore, we prove that each timelike or null geodesic is related via conformal transformations and discrete symmetries to spherical orbits, and we provide the explicit mappings. We detail the high-spin behavior of both the innermost stable and the innermost bound spherical orbits., SCOPUS: ar.j, info:eu-repo/semantics/published

We consider the high spin expansion for the null geodesics in the Kerr spacetime. We expand the null geodesic equation successively to higher orders in deviation from extremity. Via the method of matched asymptotic expansion, the radial integrals are obtained analytically. It turns out that the analytic expressions are very sensitive to the value of the shifted Carter constant $q$. We show that for a large $q$, the analytic expressions can be used to study observational electromagnetic signatures for astrophysical black holes like M87*. However, for a small $q$, the high spin expansion method can only be applied to (near-) extreme black holes., Comment: 25pages, 6 figures, nb source is available, typos fixed, accepted in CQG

We derive, in a manifestly covariant and electromagnetic gauge independent way, the evolution law of the electric field $E^\alpha = Ee^\alpha\ (e^\alpha e_\alpha=1)$, relative to an arbitrary set of instantaneous observers along a null geodesic ray, for an arbitrary Lorentzian spacetime, in the geometrical optics limit of Maxwell's equations in vacuum. We show that, in general, neither the magnitude $E$ nor the direction $e^\alpha$ of the electric field (here interpreted as the observed polarization of light) are parallel transported along the ray. For an extended reference frame around the given light ray, we express the evolution of the direction $e^\alpha$ in terms of the frame's kinematics, proving thereby that its expansion never spoils parallel transport, which bears on the unbiased inference of intrinsic properties of cosmological sources, such as, for instance, the polarization field of the cosmic microwave background (CMB). As an application of the newly derived laws, we consider a simple setup for a gravitational wave (GW) interferometer, showing that, despite the (kinematic) shear induced by the GW, the change in the final interference pattern is negligible since it turns out to be of the order of the ratio of the GW and laser frequencies., Comment: 6 pages, 3 figures

We propose a new bound on the average null energy along a finite portion of a null geodesic. We believe our bound is valid on scales small compared to the radius of curvature in any quantum field theory that is consistently coupled to gravity. If correct, our bound implies that regions of negative energy density are never strongly gravitating, and that isolated regions of negative energy are forbidden., Comment: 28 pages, 3 figures

The massless Klein–Gordon equation on arbitrary curved backgrounds allows for solutions which develop “tails” inside the light cone and, therefore, do not strictly follow null geodesics as discovered by DeWitt and Brehme almost 60 years ago. A modification of the massless Klein–Gordon equation is presented, which always exhibits null geodesic propagation of waves on arbitrary curved spacetimes. This new equation is derived from a Lagrangian which exhibits current–current interaction. Its non-linearity is due to a self-coupling term which is related to the quantum mechanical Bohm potential.

It can be seen from some theorems proved by Penrose that when Strong Causality or Distinguishing fail, they fail along at least a segment of a null geodesic. In this paper, we investigate the behavior of the other causality conditions. In particular, we prove that when causal continuity and stable causality fail at some point, there is a segment of a null geodesic through that point along which the conditions fail.

In the literature, the Newman-Janis algorithm (NJA) has been widely used to construct stationary and axisymmetric spacetimes to describe rotating black holes. In addition, it has been recently shown that the general stationary and axisymmetric spacetime generated through NJA allows the complete separability of the null geodesic equations. In fact, the Hamilton-Jacobi equation in this spacetime is also separable if one of the metric functions is additively separable. In this work, we further study the conditions for a separable Klein-Gordon equation in such a general spacetime. The relations between the NJA spacetime and other parameterized axially symmetric spacetimes in the literature are also discussed., Comment: 8 pages, 1 figure

The retarded Green function for linear field perturbations of black hole spacetimes is notoriously difficult to calculate. One of the difficulties is due to a Dirac-$\delta$ divergence that the Green function possesses when the two spacetime points are connected by a "direct" null geodesic. We present a procedure which notably aids its calculation in the case of Schwarzschild spacetime by separating this direct $\delta$-divergence from the remainder of the retarded Green function. More precisely, the method consists of calculating the multipolar $\ell$-modes of the direct $\delta$-divergence and subtracting them from the corresponding modes of the retarded Green function. We illustrate the usefulness of the method with some specific calculations in the case of the scalar Green function and self-field for a point scalar charge in Schwarzschild spacetime., Comment: 13 pages, 4 figures; 2 Mathematica notebooks as ancillary files

Causal concept for the general black hole shadow is investigated, instead of the photon sphere. We define several `wandering null geodesics' as complete null geodesics accompanied by repetitive conjugate points, which would correspond to null geodesics on the photon sphere in Schwarzschild spacetime. We also define a `wandering set', that is, a set of totally wandering null geodesics as a counterpart of the photon sphere, and moreover, a truncated wandering null geodesic to symbolically discuss its formation. Then we examine the existence of a wandering null geodesic in general black hole spacetimes mainly in terms of Weyl focusing. We will see the essence of the black hole shadow is not the stationary cycling of the photon orbits which is the concept only available in a stationary spacetime, but their accumulation. A wandering null geodesic implies that this accumulation will be occur somewhere in an asymptotically flat spacetime., The title and some statements are corrected. This is the version which will be published in CQG

A nontrivial solution of the field equations of the null-surface formulation in $$3+1$$ dimensions has not, as yet, been found and, even in $$2+1$$ dimensions, the field equations are extremely difficult to solve. Only two nontrivial solutions in $$2+1$$ dimensions have previously been found and this work presents a third (exact) solution. It differs markedly from the two earlier solutions not only in its functional nature (parametric, as opposed to explicit or implicit) but also in its Petrov type and the physical nature of the source term. The solution is expressed in terms of elementary functions and has positive non-constant scalar curvature. The spacetime has a curvature singularity and the source of the gravitational field can be interpreted as a minimally coupled scalar field with a Liouville potential. The strong energy condition is violated but all of the other energy conditions hold. There is a congruence of null geodesic Killing vectors with the expansion, shear, and vorticity scalars all being zero.

We present a complete study of the geodesics around naked singularities in AdS3, the three-dimensional anti–de Sitter spacetime. These stationary spacetimes, characterized by two conserved charges—mass and angular momentum—are obtained through identifications along spacelike Killing vectors with a fixed point. They are interpreted as massive spinning point particles and can be viewed as three-dimensional analogues of cosmic strings in four spacetime dimensions. The geodesic equations are completely integrated, and the solutions are expressed in terms of elementary functions. We classify different geodesics in terms of their radial bounds, which depend on the constants of motion. Null and spacelike geodesics approach the naked singularity from infinity and either fall into the singularity or wind around and go back to infinity, depending on the values of these constants, except for the extremal and massless cases, for which a null geodesic could have a circular orbit. Timelike geodesics never escape to infinity and do not always fall into the singularity; namely, they can be permanently bounded between two radii. The spatial projections of the geodesics (orbits) exhibit self-intersections, whose number is particularly simple for null geodesics. As a particular application, we also compute the lengths of fixed-time spacelike geodesics of the static naked singularity using two different regularizations.

We study the null geodesics in a static circularly symmetric (SCS) black hole spacetime, which is a solution in the $(2+1)$D massive gravity proposed by Bergshoeff, Hohm, and Townsend (BHT massive gravity). We obtain analytic solutions for the null geodesic equation in the SCS black hole background and find the explicit form of deflection angles. We see that, for various values of the impact parameter, the deflection angle can be positive, negative, or even zero in this black hole spacetime. The negative deflection angle indicates the repulsive behavior of the gravity that comes from the gravitational hair parameter that is the most characteristic quantity of the BHT massive gravity.

We study the detector's response when moving along an ingoing null geodesic. The backgrounds are chosen to be black hole spacetimes ($(1+1)$ dimensional Schwarzschild metric and near horizon effective metric for any stationary black hole in arbitrary dimensions) as well as Friedmann-Lemaitre-Robertson-Walker (FLRW) Universe. For black holes the trajectories are defined in Schwarzschild coordinates and the field modes are corresponding to {\it Boulware vacuum}. Whereas for FLRW case, the detector is moving along the path defined in original {\it cosmic time} and the field modes are related to {\it conformal vacuum}. The analysis is done for three stages (de-Sitter, radiation dominated and matter dominated) of the Universe. We find that, although the detector is freely falling, it registered particles in the above mentioned respective vacuums. We confirm this by different approaches. The detection probability distributions, in all situations, are thermal in nature., Expanded version, to appear in Phys. Rev. D

Approximately 60 years ago, Herman Bondi took the major step in the study of gravitational radiation of introducing the use of null surfaces as coordinates for the study and integration of the Einstein equations. This led to the well-known Bondi mass loss theorem, the basis of the recent observation by the Ligo team of gravitational radiation. In Bondi’s approach to the integration procedure, he used special null surfaces (now referred to as Bondi null surfaces) where the null generators possessed (in general) non-vanishing asymptotic shear—the free radiation data. The use of this Bondi strategy over the years has become almost sacrosanct—being the central approach in almost all discussions of gravitational radiation issues. It led to the idea of an asymptotic symmetry—the BMS group. Eventually Bondi’s description of null infinity became elegantly formalized via Penrose’s future null infinity and associated structures. However recently an alternative picture of null infinity has been developed—now based on the similarity of the null surfaces to those of flat-space near null infinity. The new null surfaces, that are now asymptotically shear-free, are very different from the Bondi surfaces. These surfaces are as similar as possible to flat-space light cones near infinity. Totally new structures—e.g., the geometric asymptotically shear-free null geodesic congruences and even the physical classical equations of motion, now appear in this new picture. The Bondi–Sachs energy–momentum conservations laws remain but are augmented by angular momentum (orbital and spin) conservation laws. The BMS group again reappears, not as an asymptotic symmetry group, but as a transformation group acting on these new structures. An unanswered question arises: with this new point of view, have we lost the asymptotic symmetries of the BMS group?

The ambition of the present work is to highlight the phenomena of strong gravitational lensing and deflection angle for the photons coupling with Weyl tensor in a Kiselev black hole. Here, we have extended the prior work of Chen and Jing \cite{1} for Schwarzschild black hole to Kiselev black hole. For this purpose, the equation of motion for the photons coupled to Weyl tensor, null geodesic and equation of photon sphere in a Kiselev black hole spacetime have been formulated. It is found that the equation of motion of the photons depends not only on the coupling between photon and Weyl tensor, but also on the polarization direction of the photons. There is a critical value of the coupling parameter $\alpha$ for existence of the marginally circular photon orbit outside the event horizon, which depends on the parameters of black hole and the polarization direction of photons. Further, the polarization directions of coupled photon and the coupling parameter $\alpha$, both modify the features of the photon sphere, the angle of deflection and the functions $(\bar{a}$ and $\bar{b})$ for the strong gravitational lensing in Kiselev black hole spacetime. In addition to this, the observable gravitational lensing quantities and the shadows of the Kiselev black hole spacetime are presented in detail., Comment: 26 pages, 10 figures, major revision included and accepted for publications in Chinese Physics C. arXiv admin note: text overlap with arXiv:1611.08783, arXiv:1502.01088 by other authors

The memory effect at null infinity, $\mathcal{I}^+$, can be defined in terms of the permanent relative displacement of test particles (at leading order in $1/r$) resulting from the passage of a burst of gravitational radiation. In $D=4$ spacetime dimensions, the memory effect can be characterized by the supertranslation relating the "good cuts" of $\mathcal{I}^+$ in the stationary eras at early and late retarded times. It also can be characterized in terms of charges and fluxes associated with supertranslations. Black hole event horizons are in many ways analogous to $\mathcal{I}^+$. We consider here analogous definitions of memory for a black hole, assuming that the black hole is approximately stationary at early and late advanced times, so that its event horizon is described by a Killing horizon (assumed nonextremal) at early and late times. We give prescriptions for defining preferred foliations of nonextremal Killing horizons. We give a definition of the memory tensor for a black hole in terms of the "permanent relative displacement" of the null geodesic generators of the event horizon between the early and late time stationary eras. We show that preferred foliations of the event horizon in the early and late time eras are related by a Chandrasekaran-Flanagan-Prabhu (CFP) supertranslation. However, we find that the memory tensor for a black hole horizon does not appear to be related to the CFP symmetries or their charges and fluxes in a manner similar to that occurring at $\mathcal{I}^+$., Comment: 23 pages. v2: references and minor clarifications added; version accepted for publication in PRD

We examine the achronal averaged null energy condition (ANEC) for a class of conformal field theories (CFT) at strong coupling in curved spacetime. By applying the AdS/CFT duality, we find holographic models which violate the achronal ANEC for 3+1 and 4+1-dimensional boundary theories. In our model, the bulk spacetime is an asymptotically AdS vacuum bubble solution with neither causality violation nor singularities. The conformal boundary of our bubble solution is asymptotically flat and is causally proper in the sense that a “fastest null geodesic” connecting any two points on the boundary must lie entirely on the boundary. We show that conversely, if the spacetime fails to have this causally proper nature, then there must be a naked singularity in the bulk.

E. Cartan's method of moving frames is applied to 3-dimensional manifolds $M$ which are CR-embedded in 5-dimensional real hyperquadrics $Q$ in order to classify $M$ up to CR symmetries of $Q$ given by the action of one of the Lie groups $SU(3,1)$ or $SU(2,2)$. In the latter case, the CR structure of $M$ derives from a shear-free null geodesic congruence on Minkowski spacetime, and the relationship to relativity is discussed. In both cases, we compute which homogeneous CR 3-folds appear in $Q$., Substantial revision of Levi-nondegenerate classification, both intrinsic and extrinsic, for clarity; Levi-flat case now included; 32 pages

We study null geodesic congruences (NGCs) in the presence of spacetime torsion, recovering and extending results in the literature. Only the highest spin irreducible component of torsion gives a proper acceleration with respect to metric NGCs, but at the same time obstructs abreastness of the geodesics. This means that it is necessary to follow the evolution of the drift term in the optical equations, and not just shear, twist and expansion. We show how the optical equations depend on the non-Riemannian components of the curvature, and how they reduce to the metric ones when the highest spin component of torsion vanishes., v2: improved text, a few equations and a subsection added. v3: typo in one equation corrected, improved conclusions and other minor amendments

The generic null geodesic of the Schwarzschild–Kruskal–Szekeres geometry has a natural complexification, an elliptic curve with a cusp at the singularity. To realize that complexification as a Riemann surface without a cusp, and also to ensure conservation of energy at the singularity, requires a branched cover of the space-time over the singularity, with the geodesic being doubled as well to obtain a genus two hyperelliptic curve with an extra involution. Furthermore, the resulting space-time obtained from this branch cover has a Hamiltonian that is null geodesically complete. The full complex null geodesic can be realized in a natural complexification of the Kruskal–Szekeres metric.

We consider analytic, vacuum spacetimes that admit compact, non-degenerate Cauchy horizons. Many years ago we proved that, if the null geodesic generators of such a horizon were all \textit{closed} curves, then the enveloping spacetime would necessarily admit a non-trivial, horizon-generating Killing vector field. Using a slightly extended version of the Cauchy-Kowaleski theorem one could establish the existence of infinite dimensional, analytic families of such `generalized Taub-NUT' spacetimes and show that, generically, they admitted \textit{only} the single (horizon-generating) Killing field alluded to above. In this article we relax the closure assumption and analyze vacuum spacetimes in which the generic horizon generating null geodesic densely fills a 2-torus lying in the horizon. In particular we show that, aside from some highly exceptional cases that we refer to as `ergodic', the non-closed generators always have this (densely 2-torus-filling) geometrical property in the analytic setting. By extending arguments we gave previously for the characterization of the Killing symmetries of higher dimensional, stationary black holes we prove that analytic, 4-dimensional, vacuum spacetimes with such (non-ergodic) compact Cauchy horizons always admit (at least) two independent, commuting Killing vector fields of which a special linear combination is horizon generating. We also discuss the \textit{conjectures} that every such spacetime with an \textit{ergodic} horizon is trivially constructable from the flat Kasner solution by making certain `irrational' toroidal compactifications and that degenerate compact Cauchy horizons do not exist in the analytic case., Comment: 55 pages. arXiv admin note: substantial text overlap with arXiv:0805.1451

We present a definition of null G-structures on Lorentzian manifolds and investigate their geometric properties. This definition includes the Robinson structure on 4-dimensional black holes as well as the null structures that appear in all supersymmetric solutions of supergravity theories. We also identify the induced geometry on some null hypersurfaces and that on the orbit spaces of null geodesic congruences in such Lorentzian manifolds. We give the algebra of diffeomorphisms that preserves a null G-structure and demonstrate that in some cases it interpolates between the BMS algebra of an asymptotically flat spacetime and the Lorentz symmetry algebra of a Killing horizon., Comment: 26 pages, section 5 has been re-worked

It is known that using the Gaussian beam approximation one can show that there exist solutions of the wave equation on a general globally hyperbolic Lorentzian manifold whose energy is localised along a given null geodesic for a finite, but arbitrarily long time. In this paper, we show that the energy of such a localised solution is determined by the energy of the underlying null geodesic. This result opens the door to various applications of Gaussian beams on Lorentzian manifolds that do not admit a globally timelike Killing vector field. In particular we show that trapping in the exterior of Kerr or at the horizon of an extremal Reissner-Nordstr\"om black hole necessarily leads to a `loss of derivative' in a local energy decay statement. We also demonstrate the obstruction formed by the red-shift effect at the event horizon of a Schwarzschild black hole to scattering constructions from the future (where the red-shift turns into a blue-shift): we construct solutions to the backwards problem whose energies grow exponentially for a finite, but arbitrarily long time. Finally, we give a simple mathematical realisation of the heuristics for the blue-shift effect near the Cauchy horizon of sub-extremal and extremal black holes: we construct a sequence of solutions to the wave equation whose initial energies are uniformly bounded, whereas the energy near the Cauchy horizon goes to infinity., Comment: 46 pages, v2 extends the version accepted for publication

We examine some recently-constructed families of asymptotically-AdS$_3 \times$S$^3$ supergravity solutions that have the same charges and mass as supersymmetric D1-D5-P black holes, but that cap off smoothly with no horizon. These solutions, known as superstrata, are quite complicated, however we show that, for an infinite family of solutions, the null geodesic problem is completely integrable, due to the existence of a non-trivial conformal Killing tensor that provides a quadratic conservation law for null geodesics. This implies that the massless scalar wave equation is separable. For another infinite family of solutions, we find that there is a non-trivial conformal Killing tensor only when the left-moving angular momentum of the massless scalar is zero. We also show that, for both these families, the metric degrees of freedom have the form they would take if they arose from a consistent truncation on S$^3$ down to a (2+1)-dimensional space-time. We discuss some of the broader consequences of these special properties for the physics of these black-hole microstate geometries., Comment: 24 pages, 1 figure

The near horizon geometry of general black holes in equilibrium can be conveniently characterized in the formalism of weakly isolated horizons in the form of the Bondi-like expansions (Krishnan B, Class.\ Quantum Grav.\ 29, 205006, 2012). While the intrinsic geometry of the Kerr-Newman black hole has been extensively investigated in the weakly isolated horizon framework, the off-horizon description in the Bondi-like system employed by Krishnan has not been studied. We extend Krishnan's work by explicit, non-perturbative construction of the Bondi-like tetrad in the full Kerr-Newman spacetime. Namely, we construct the Bondi-like tetrad which is parallelly propagated along a nontwisting null geodesic congruence transversal to the horizon and provide all Newman-Penrose scalars associated with this tetrad. This work completes the description of the Kerr-Newman spacetime in the formalism of weakly isolated horizons and is a starting point for the investigation of deformed black holes., 16 pages, 6 figures

We study the gravitational lensing scenario where the lens is a spherically symmetric charged black hole (BH) surrounded by quintessence matter. The null geodesic equations in the curved background of the black hole are derived. The resulting trajectory equation is solved analytically via perturbation and series methods for special choice of parameters and the distance of the closest approach to black hole is calculated. We also derive the lens equation giving the bending angle of light in the curved background. In the strong field approximation, the solution of the lens equation is also obtained for all values of the quintessence parameter $w_q$. For all $w_q$, we show that there are no stable closed null orbits and that corrections to the deflection angle for the Reissner-Nordstr\"om black hole when the observer and the source are at large, but finite, distances from the lens do not depend on the charge up to the inverse of the distances squared. A part of the present work, analyzed however with a different approach, is the extension of {\it Phys. Rev. D \textbf{92}, 084042 (2015)} where the uncharged case has been treated, Comment: 14 pages, 5 figures, accepted for publication in Eur. Phys. J. C

In $$(2+1)$$ -dimensional AdS spacetime, we obtain new exact black hole solutions, including two different models (power parameter $$k=1$$ and $$k\ne 1$$ ), in the Einstein–Power–Maxwell (EPM) theory with nonminimally coupled scalar field. For the charged hairy black hole with $$k\ne 1$$ , we find that the solution contains a curvature singularity at the origin and is nonconformally flat. The horizon structures are identified, which indicates the physically acceptable lower bound of mass in according to the existence of black hole solutions. Later, the null geodesic equations for photon around this charged hairy black hole are also discussed in detail.

In this paper, we re-examine the light deflection in the Schwarzschild and the Schwarzschild-de Sitter spacetime. First, supposing a static and spherically symmetric spacetime, we propose the definition of the total deflection angle $\alpha$ of the light ray by constructing a quadrilateral $\Sigma^4$ on the optical reference geometry ${\cal M}^{\rm opt}$ determined by the optical metric $\bar{g}_{ij}$. On the basis of the definition of the total deflection angle $\alpha$ and the Gauss-Bonnet theorem, we derive two formulas to calculate the total deflection angle $\alpha$; (i) the angular formula that uses four angles determined on the optical reference geometry ${\cal M}^{\rm opt}$ or the curved $(r, \phi)$ subspace ${\cal M}^{\rm sub}$ being a slice of constant time $t$ and (ii) the integral formula on the optical reference geometry ${\cal M}^{\rm opt}$ which is the areal integral of the Gaussian curvature $K$ in the area of a quadrilateral $\Sigma^4$ and the line integral of the geodesic curvature $\kappa_g$ along the curve $C_{\Gamma}$. As the curve $C_{\Gamma}$, we introduce the unperturbed reference line that is the null geodesic $\Gamma$ on the background spacetime such as the Minkowski or the de Sitter spacetime, and is obtained by projecting $\Gamma$ vertically onto the curved $(r, \phi)$ subspace ${\cal M}^{\rm sub}$. We demonstrate that the two formulas give the same total deflection angle $\alpha$ for the Schwarzschild and the Schwarzschild--de Sitter spacetime. In particular, in the Schwarzschild case, the result coincides with Epstein--Shapiro's formula when the source $S$ and the receiver $R$ of the light ray are located at infinity. In addition, in the Schwarzschild--de Sitter case, there appear order ${\cal O}(\Lambda m)$ terms in addition to the Schwarzschild-like part, while order ${\cal O}(\Lambda)$ terms disappear., Comment: 23 pages, 7 figures

This note describes the behavior of null-geodesics near nondegenerate Killing horizons in language amenable to the application of a general framework, due to Vasy and Hintz, for the analysis of both linear and nonlinear wave equations. Throughout, the viewpoint of Melrose's b-geometry on a suitable compactification of spacetime at future infinity is adopted., Comment: 27 pages; minor changes. To appear in Trans. Amer. Math. Soc

In calculations of gravitational collapse to form black holes, trapping horizons (foliated by marginally trapped surfaces) make their first appearance either within the collapsing matter or where it joins on to a vacuum exterior. Those which then move outwards with respect to the matter have been proposed for use in defining black holes, replacing the global concept of an "event horizon" which has some serious drawbacks for practical applications. We here present results from a study of the properties of both outgoing and ingoing trapping horizons, assuming strict spherical symmetry throughout. We have investigated their causal nature (i.e. whether they are spacelike, timelike or null), making contact with the Misner-Sharp- Hernandez formalism, which has often been used for numerical calculations of spherical collapse. We follow two different approaches, one using a geometrical quantity related to expansions of null geodesic congruences, and the other using the horizon velocity measured with respect to the collapsing matter. After an introduction to these concepts, we then implement them within numerical simulations of stellar collapse, revisiting pioneering calculations from the 1960s where some features of the emergence and subsequent behaviour of trapping horizons could already be seen. Our presentation here is aimed firmly at "real world" applications of interest to astrophysicists and includes the effects of pressure, which may be important for the asymptotic behaviour of the ingoing horizon., Comment: 33 pages, 11 figures

We discuss the Hamiltonian dynamics of general relativity with real connection variables on a null foliation, and use the Newman-Penrose formalism to shed light on the geometric meaning of the various constraints. We identify the equivalent of Sachs' constraint-free initial data as projections of connection components related to null rotations, i.e. the translational part of the ISO(2) group stabilising the internal null direction soldered to the hypersurface. A pair of second-class constraints reduces these connection components to the shear of a null geodesic congruence, thus establishing equivalence with the second-order formalism, which we show in details at the level of symplectic potentials. A special feature of the first-order formulation is that Sachs' propagating equations for the shear, away from the initial hypersurface, are turned into tertiary constraints; their role is to preserve the relation between connection and shear under retarded time evolution. The conversion of wave-like propagating equations into constraints is possible thanks to an algebraic Bianchi identity; the same one that allows one to describe the radiative data at future null infinity in terms of a shear of a (non-geodesic) asymptotic null vector field in the physical spacetime. Finally, we compute the modification to the spin coefficients and the null congruence in the presence of torsion., Comment: 23 pages + Appendix, 2 figures. v2: Improved text and some amendments throughout, added more details on the relation between 2+2 foliations and null tetrads, updated references. Version submitted for peer reviewing. v3: Few minor amendments, footnote added on a null congruence in the presence of torsion; matches published version