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We study for the first time the propagation of the packets of plane waves of the Maxwell free field in the de Sitter expanding universe as detected by an observer staying at rest in his proper frame with physical de Sitter–Painlevé coordinates. This observes an accelerate propagation of the wave packet along to a null geodesic, laying out a severe exponential decay and a moderate dispersion, increasing exponentially in time during propagation. The example we give is the usual anisotropic Gaussian packet for which we present a short graphical analysis pointing out the accelerated propagation, decay and dispersion. Moreover, we show that the observer perceives his horizon as a mirror stopping the wave packets prepared on it and reflecting those prepared beyond it.

We establish a new lower bound for the null injectivity radius of a null cone. The idea is to use a function closely related to the null second fundamental form which codifies the directional expansion of the null cone along any null geodesic in it. This approach uses first derivatives of the metric instead of curvature bounds. The technique is applied to a family of null cones with the null conjugate radius less or equal to the null crossing radius, or folded due to the curvature in a precise way that we call geometric fold. This condition is necessary because there are trivial examples with bounded null injectivity radius due to global identifications. We show two examples where we compute the real null injectivity radius and the lower bound provided in this paper, in order to compare both quantities. We also give an analogous result for Riemannian manifolds.

In Minkowski space-time, (locally) space-filling families of null geodesics i.e., null geodesic congruences, appear in a variety different physical contexts. We here describe what appears to be the general explicit generic expression for these congruences. From each congruence one can construct a (virtually) unique form of the flat Lorentzian metric that is associated with the congruence. In addition, the “so-called” optical parameters for each congruence are calculated. The very important special cases of the shear-free congruences (twisting and non-twisting) are described in detail. These results are closely related to the asymptotically shear-free null geodesic congruences of asymptotically flat space-times and their rather surprising physical implications.

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.

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.

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?

In the 1950’s Herman Bondi observed that a very effective way to study gravitational radiation was to use null surfaces as part of the coordinate system for analyzing the Einstein (Einstein–Maxwell) equations. A particular class of such surfaces, (referred to as Bondi null surfaces) with their associated null tetrad, has now been the main tool for this analysis for many years; their use—until recently—has been almost ubiquitous. Several years ago we realized that there was an attractive alternative to the use of Bondi coordinates, namely to use coordinates (in the asymptotic null future space–time region) that were as close to ordinary flat-space light-cones as possible—very different from Bondi surfaces. There were initially serious impediments to this program: these new null surfaces (referred to as asymptotically shear-free surfaces, ASF) were determined by solving a non-linear differential equation (the “good-cut” equation) whose solutions were most often complex. Eventually these problems were overcome and the program was implemented. In a series of papers, using the ASF null surfaces to study the asymptotically flat Einstein (or Einstein–Maxwell) equations, a variety of surprising (strange) results were obtained. Using only the Einstein and Maxwell equations, we found a large number of the basic relations of classical mechanics. They included very detailed conservation laws, well know kinematic relations and dynamic equations and even the Abraham–Lorentz–Dirac radiation reaction force and the rocket force. As interesting as these were, they came with a serious enigma. These relations from classical mechanics had no relationship with the physical space–time. The space for the action of these relations was the parameter space of solutions of the good-cut equation—a complex space, known as H-space. The enigma—what possible relationship did these standard appearing classical relations have with physical space–time? It is the purpose of this work to establish such a relationship—objects in H-space are related to structures in physical space–time. For example, a complex world-line in H-space becomes in physical space–time an asymptotically shear-free null geodesic congruence where its twist describes its intrinsic spin and if charged, its magnetic dipole.

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.

Position determination of photon emitters and associated strong field parallax effects are investigated using relativistic optics when the photon orbits are confined to the equatorial plane of the Schwarzschild spacetime. We assume the emitter is at a fixed space position and the receiver moves along a circular geodesic orbit. This study requires solving the inverse problem of determining the (spatial) intersection point of two null geodesic initial data problems, serving as a simplified model for applications in relativistic astrometry as well as in radar and satellite communications., 21 pages, 5 figures, GRG macros

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

We point out an unusual relationship among a variety of null geodesic congruences; (a) the generators of ordinary light-cones and (b) certain (related) shear-free but twisting congruences in Minkowski Space-time as well as (c) asymptotically shear-free null geodesic congruences that exist in the neighborhood of Penrose's Scri in Einstein or Einstein-Maxwell asymptotically flat-space-times. We refer to these geodesic congruences respectively as: Lignt-Cones (LCs), as Almost-Complex- Light-Cones, (ACLCs), [though they are real they resemble complex light-cones in complex Minkowski space] and finally to a family of congruences in asymptotically flat-spaces as `Almost Light-Cones', (ALC). The two essential points of resemblance among the three families are: (1) they are all either shear-free or asymptotically shear-free and (2) in each family the individual members of the family can be labeled by the points in a real or complex four-dimensional manifold. As an example, the Minkowski space LCs are labeled by the (real) coordinate value of their apex. In the case of (ACLCs) (complex coordinate values), the congruences will have non-vanishing twist whose magnitude is determined by the imaginary part of the complex coordinate values. In studies of gravitational radiation, Bondi-type of null surfaces and their associated Bondi coordinates have been almost exclusively used for calculations. Some surprising relations arise if, instead of the Bondi coordinates, one uses ALCs and their associated coordinate systems in the analysis of the Einstein-Maxwell equations near Scri. More explicitly and surprisingly, the asymptotic Bianchi Identities expressed in the coordinates of the ALCs, turn directly into many of the standard definitions and relations of classical mechanics.

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.

The geometrical symmetries corresponding to the continuous groups of collineations and motions generated by a null vector l are considered. These symmetries have been translated into the language of Newman-Penrose formalism for pure radiation (PR) type D fields. It is seen that for such fields, conformal, special conformal and homothetic motions degenerate to motion. The concept of free curvature, matter curvature and matter affine collineations have been introduced and the conditions under which PR type D fields admit such collineations have been obtained. Moreover, it is shown that the projective collineation degenerate to matter affine, special projective, conformal, special conformal, null geodesic and special null geodesic collineations. It is also seen that type D pure radiation fields admit Maxwell collineation along the propagation vector l.

We present a new, fully anisotropic, criterion for formation of trapped surfaces in vacuum. More precisely we provide local conditions on null data, concentrated in a neighborhood of a short null geodesic segment (possibly flat in all other directions) whose future development contains a trapped surface. This extends considerably the previous result of Christodoulou (Monographs in Mathematics. European Mathematical Society, Switzerland, 2009) which required instead a uniform condition along all null geodesic generators. To obtain our result we combine Christodoulou’s mechanism for the formation of a trapped surface with a new deformation process which takes place along incoming null hypersurfaces., National Science Foundation (U.S.) (Grant DMS-1001500), National Science Foundation (U.S.). Focused Research Group (Grant DMS-1065710)

Based on the principle of general relativity, geometrization of interaction, the interaction of the inhomogeneous isotropic medium to light can be equated to a non-Euclidean geometry field just like the situation of gravity, i.e., light travels in the null geodesic in the non-Euclidean curved space-time, which is equivalent to the Fermat principle for the inhomogeneous media. In this paper, the propagation of light in an inhomogeneous media is studied by means of the effective metric theory. The modification to the classical ray equation of photons is derived from the geodesic equation of photon by considering the spin effect of photons, which is induced via the spin-orbit coupling of photons, and the corresponding Hamiltonian of photon is proposed. Based on the spin-orbit coupling of photon, a light splitting phenomenon emerges in the inhomogeneous media, which is the spin hall effect of photon.

The deflection of a ray of light passing close to a gravitational mass is generally calculated from the null geodesic which the light ray (photon) follows. However, there is an alternate approach, where the effect of gravitation on the ray of light is estimated by considering the ray to be passing through a material medium. Calculations have been done in this paper, following the latter approach, to estimate the amount of deflection due to a static non-rotating mass. The refractive index of such a material medium, has been calculated in a more rigorous manner in the present work, and the final expression for the amount of deflection calculated here is claimed to be more exact than all other expressions derived so far, using material medium approach. Based on this expression, the amount of deflection for a sun grazing ray has also been calculated. The exact amount of deflection can be performed in a number of ways, without the material medium approach. However, the method presented here using the material medium approach and without any weak field approximation is believed to be original.

Let $(X^{m+1}, g)$ be a globally hyperbolic spacetime with Cauchy surface diffeomorphic to an open subset of $\mathbb R^m$. The Legendrian Low conjecture formulated by Nat\'ario and Tod says that two events $x,y\in\ss$ are causally related if and only if the Legendrian link of spheres $\mathfrak S_x, \mathfrak S_y$ whose points are light geodesics passing through $x$ and $y$ is non-trivial in the contact manifold of all light geodesics in $X$. The Low conjecture says that for $m=2$ the events $x,y$ are causally related if and only if $\mathfrak S_x, \mathfrak S_y$ is non-trivial as a topological link. We prove the Low and the Legendrian Low conjectures. We also show that similar statements hold for any globally hyperbolic $(X^{m+1}, g)$ such that a cover of its Cauchy surface is diffeomorphic to an open domain in $\mathbb R^m.$, Comment: Version 3 - minor improvements, references added 11 pages, 1 figure

Assuming a modification of general relativity which implies a direct coupling between the Ricci curvature scalar and the matter Lagrangian, a non-geodesic motion of test particles is obtained, which, performing a repulsive force, could, in principle, be connected with Dark Matter and Pioneer anomaly problems.

A key result in the proof of black hole uniqueness in 4-dimensions is that a stationary black hole that is ``rotating''--i.e., is such that the stationary Killing field is not everywhere normal to the horizon--must be axisymmetric. The proof of this result in 4-dimensions relies on the fact that the orbits of the stationary Killing field on the horizon have the property that they must return to the same null geodesic generator of the horizon after a certain period, $P$. This latter property follows, in turn, from the fact that the cross-sections of the horizon are two-dimensional spheres. However, in spacetimes of dimension greater than 4, it is no longer true that the orbits of the stationary Killing field on the horizon must return to the same null geodesic generator. In this paper, we prove that, nevertheless, a higher dimensional stationary black hole that is rotating must be axisymmetric. No assumptions are made concerning the topology of the horizon cross-sections other than that they are compact. However, we assume that the horizon is non-degenerate and, as in the 4-dimensional proof, that the spacetime is analytic., Comment: 24 pages, no figures, v2: footnotes and references added, v3: numerous minor revisions

The properties of an asymptotically flat spacetime with a mass and a magnetic dipole are studied. We focus on the breaking of spherical symmetry by the magnetic dipole. We investigate the geometry of the spacetime through the trajectories of null geodesics and the embedding diagrams of various hypersurfaces, included constant t and r surfaces and the infinite red shift surface. We find that the effects of the magnetic dipole on the spacetime are larger at the poles than on the equator.

In this article, we first investigate the bending of light rays in 4D line elements representing spherically symmetric, static on–brane geometries. The amount of bending in these four dimensional, strong and weak field solutions is derived. Signatures, which appear in the bending formulae, due to the presence of extra dimensions are discussed. Subsequently, as a separate exercise, we calculate the bending of null geodesic trajectories in a bulk five dimensional spacetime with a Schwarzschild line element on the 3-brane section. We interpret the deviation of null trajectories as that of gravitational signals or five dimensional ‘photons,’ which are, unlike light rays confined to four dimensions, allowed to propagate along the fifth (extra) dimension. Features of the presence of extra dimensions in the effective potentials and the bending formula are analysed in this context.

We have developed a new tool for numerical work in General Relativity: GRworkbench. We discuss how GRworkbench's implementation of a numerically-amenable analogue to Differential Geometry facilitates the development of robust and chart-independent numerical algorithms. We consider, as an example, geodesic tracing on two charts covering the exterior Schwarzschild space-time.

This paper explores conformal Killing vectors in teleparallel theory for the plane-symmetric static spacetime. Ten non-linear coupled teleparallel conformal Killing equations are solved to get the general form of teleparallel conformal Killing vectors and the conformal factor along with the integrability conditions. Three different possibilities are considered where the spacetime may exhibit a teleparallel conformal Killing vector. In two cases the integrability conditions are solved completely and teleparallel conformal Killing vectors are obtained along with the conformal factor for the special choice of metric functions. Another possibility reveals that no teleparallel proper conformal Killing vector exists and the spacetime admits only a teleparallel homothetic vector.

We investigate the geodesics’ kinematics and dynamics in the Linet–Tian metric with $$\Lambda

We analyze a class of 5D non-compact warped-product spaces characterized by metrics that depend on the extra coordinate via a conformal factor. Our model is closely related to the so-called canonical coordinate gauge of Mashhoon et al. We confirm that if the 5D manifold in our model is Ricci-flat, then there is an induced cosmological constant in the 4D sub-manifold. We derive the general form of the 5D Killing vectors and relate them to the 4D Killing vectors of the embedded spacetime. We then study the 5D null geodesic paths and show that the 4D part of the motion can be timelike -- that is, massless particles in 5D can be massive in 4D. We find that if the null trajectories are affinely parameterized in 5D, then the particle is subject to an anomalous acceleration or fifth force. However, this force may be removed by reparameterization, which brings the correct definition of the proper time into question. Physical properties of the geodesics -- such as rest mass variations induced by a variable cosmological ``constant'', constants of the motion and 5D time-dilation effects -- are discussed and are shown to be open to experimental or observational investigation., Comment: 19 pages, REVTeX, in press in Gen. Rel. Grav

In this work we integrate the null geodesic equations in three-dimensional Minkowski space-time in order to obtain the light-cone cut function; that is, the function that describes the intersection, Cxa, of the light cone from each space-time point, xa, with future null infinity I+. Furthermore, using this result, we locate the singularities of the null surface obtained as the envelope of the past light cones from points on a deformed light-cone cut of I+.

Cylindrical spacetimes with rotation are studied using the Newmann-Penrose formulas. By studying null geodesic deviations the physical meaning of each component of the Riemann tensor is given. These spacetimes are further extended to include rotating dynamic shells, and the general expression of the surface energy-momentum tensor of the shells is given in terms of the discontinuation of the first derivatives of the metric coefficients. As an application of the developed formulas, a stationary shell that generates the Lewis solutions, which represent the most general vacuum cylindrical solutions of the Einstein field equations with rotation, is studied by assuming that the spacetime inside the shell is flat. It is shown that the shell can satisfy all the energy conditions by properly choosing the parameters appearing in the model, provided that $ 0 \le \sigma \le 1$, where $\sigma$ is related to the mass per unit length of the shell., Comment: Typed in Revtex, including three figures. To appear in General Relativity and Gravity

Einstein-Maxwell spacetimes endowed with twocommuting spacelike Killing vector fields areconsidered. Subject to the hypotheses that one of thetwo null geodesic congruence orthogonal to thetwo-surface generated by the two commuting spacelikeKilling vector fields is shearfree and theelectromagnetic field is non null, it is shown that,with a specific choice of null tetrad, theNewman-Penrose equations together with the Maxwell equations for theclass of spacetime considered may be reduced to asecond-order ode of Sturm-Liouville type, from whichexact solutions of the class of spacetimes consideredmay be constructed. Examples of exact solutions arethen given. Exact solutions with distribution-valuedWeyl curvature describing the scattering ofelectromagnetic shock wave with gravitational impulsiveor shock wave of variable polarisation are also constructed.

We consider smooth electrovac spacetimes which represent either (A) an asymptotically flat, stationary black hole or (B) a cosmological spacetime with a compact Cauchy horizon ruled by closed null geodesics. The black hole event horizon or, respectively, the compact Cauchy horizon of these spacetimes is assumed to be a smooth null hypersurface which is non-degenerate in the sense that its null geodesic generators are geodesically incomplete in one direction. In both cases, it is shown that there exists a Killing vector field in a one-sided neighborhood of the horizon which is normal to the horizon. We thereby generalize theorems of Hawking (for case (A)) and Isenberg and Moncrief (for case (B)) to the non-analytic case., 16 pages, no figures

The field-to-particle method of H. P. Robertson as applied by Noonan, in order to obtain the general relativistic equations describing the trajectory of a photon in a refractive medium, is compared with Synge’s general relativistic Hamiltonian theory of waves and rays. For a photon in vacuum it is known that both approaches yield the same equation for the trajectory, i.e., a null geodesic. However for a photon in a medium, in contradistinction to the Hamiltonian theory, the field-to-particle method (a) yields equations of the photon trajectory valid only in a nondispersive medium, (b) the time component u0of the tangent to the ray remains an undetermined quantity, (c) agreement with the Hamiltonian theory is achieved by substituting into Noonan’s equations the Hamiltonian expression for u0.

In the framework of general relativity theory, the Einstein equations are obtained using a modified procedure, i.e., by introducing a congruence of timelike curves in the Riemann space and taking into account the relation between the variations. The aim is to satisfy the least action principle with respect to any frame of reference

We review the main mathematical concepts of cosmic topology and the main observational methods to study the topology of the universe. We then show how topology can play a crucial role in the early universe.

The light trajectory in an inhomogeneous medium is studied by the variation of LagrangiansL andℒ which correspond to Fermat’s principle in the geometrical optics and the null geodesic in the metric optics, respectively. The relation between the metric coefficients of the three-dimensional space and of the four-dimensional space-time is established. The physical meaning of the equivalence and difference of both the descriptions is revealed. It is shown that Fermat’s principle is a direct result of the null geodesic.

An exact cylindrically symmetric solution to Einstein equations for scalar waves is presented. The asymptotic behaviour of the null geodesics near the singularity is discussed.

The center of traceX CT of the energy-momentum tensor of a spinning particle in General Relativity is defined. It is shown to be an invariant point of the particle, and its path is shown to be what is specified by the original side condition of Mathisson,S αβ u β=0.

A quantized automorphic scalar field in two-dimensional spacetime with closed null geodesics (Cauchy horizon) is considered. The renormalized energy—momentum tensor 〈Tμv〉ren is obtained. It is shown that for specific values of the automorphic parameter 〈Tμv〉ren remains regular on the Cauchy horizon.

We study the properties of the solution space of local surface-forming null “sub-congruences” in the neighbourhood of a given null geodesic in a pseudo-Riemannian space-time. This solution space is a three-dimensional manifold, naturally endowed with a conformai Minkowski metric.

The problem of a rigorous theory of singularities in space-times with torsion is addressed. We define geodesics as curves whose tangent vector moves by parallel transport. This is different from what other authors have done, because their definition of geodesics only involves the Christoffel connection, though studying theories with torsion. We propose a preliminary definition of singularities which is based on timelike or null geodesic incompleteness, even though for theories with torsion the paths of particles are not geodesics. The study of the geodesic equation for cosmological models with torsion shows that the definition has a physical relevance. It can also be motivated, as done in the literature, remarking that the causal structure of a space-time with torsion does not get changed with respect to general relativity. We then prove how to extend Hawking's singularity theorem without causality assumptions to the space-time of the ECSK theory. This is achieved studying the generalized Raychaudhuri equation in the ECSK theory, the conditions for the existence of conjugate points and properties of maximal timelike geodesics. Hawking's theorem can be generalized, provided the torsion tensor obeys some conditions. Thus our result can also be interpreted as a no-singularity theorem if these additional conditions are not satisfied. In other words, it turns out that the occurrence of singularities in closed cosmological models based on the ECSK theory is less generic than in general relativity. Our work is to be compared with previous papers in the literature. There are some relevant differences, because we rely on a different definition of geodesics, we keep the field equations of the ECSK theory in their original form rather than casting them in a form similar to general relativity with a modified energy momentum tensor, Comment: 17 pages, plain-tex, published in Nuovo Cimento B, volume 105, pages 75-90, year 1990

We investigate a class of semi-Riemannian submersions satisfying a Lorentzian analogue of the classical Clairaut's relation for surfaces of revolution. We show that a Lorentzian submersion with one-dimensional fibers is Clairaut if and only if the fibers are totally umbilic with a gradient field as the normal curvature vector field. We also investigate the behavior of timelike and null geodesics in Lorentzian Clairaut submersions. In particular, every null geodesic of a Lorentzian Clairaut submersion with one-dimensional fibers projects to a pregeodesic in the base space with respect to a conformally related metric on the base space if and only if the integrability tensor of the submersion vanishes.

The solar gravitational deflection angle of a graviton is calculated through the scattering cross section of the graviton by the sun and shown to be equal to the light deflection angle as calculated from the null geodesic equation of general relativity.

We use geodesic probes to recover the entire bulk metric in certain asymptotically AdS spacetimes. Given a spectrum of null geodesic endpoints on the boundary, we describe two remarkably simple methods for recovering the bulk information. After examining the issues which affect their application in practice, we highlight a significant advantage one has over the other from a computational point of view, and give some illustrative examples. We go on to consider spacetimes where the methods cannot be used to recover the complete bulk metric, and demonstrate how much information can be recovered in these cases., Comment: 33 pages, 11 figures; v2 references added

We study the Penrose limit about a null geodesic with 3 equal angular momenta in the recently obtained type IIB solution dual to an exactly marginal $\gamma$-deformation of N=4 SYM. The resulting background has non-trivial NS 3-form flux as well as RR 5- and 3-form fluxes. We quantise the light-cone Green-Schwarz action and show that it exhibits a continuum spectrum. We show that this is related to the dynamics of a charged particle moving in a Landau plane with an extra interaction induced by the deformation. We interpret the results in the dual N=1 SCFT.

We obtain the Penrose limit of NCYM theories in dimensions $3 \leq d \leq 6$ which originate from (D$(p-2)$, D$p$) supergravity bound state configurations for $2 \leq p \leq 5$ in the so-called NCYM limit. In most cases the Penrose limit does not lead to solvable string theories except for six-dimensional NCYM theory. We obtain the masses of various bosonic coordinates and observe that they are light-cone time dependent and their squares can be negative as has also been observed in other cases in the literature. When the non-commutative effect is turned off we recover the results of Penrose limit of ordinary D$p$-branes in the usual YM limit. We point out that for $d = 6$ NCYM theory, there exists another null geodesic in the neighborhood of which the Penrose limit leads to a solvable string theory. We briefly discuss the quantization of this theory and show that this pp-wave background is half supersymmetric., Comment: latex, 22 pages, physical meaning of the choice of the parameter $l=1$ is given, comments on the negative mass^2 added and the background is shown to be half supersymmetric, references added, published version

The Weyl neutrino equation is integrated in flat space-time assuming that the energy-momentum tensor of the neutrino field vanishes. It is shown that the flux vector of the neutrino field is tangent to a twist-free and shear-free congruence of null geodesics, which is a special Robinson congruence and constitutes a geometrical representation of a null twistor. It is also shown that, conversely, given such a congruence, a ghost neutrino field can be constructed.

The concept of conformai dynamical symmetry (CDS) is introduced as a basic tool for the analysis of null geodesic motion. CDSs are characterized as generators of Jacobi fields along null geodesies. Methods for the construction of CDS's based on the connections between CDSs and conformai Killing tensors or on the behavior of CDSs under conformai transformations are then discussed. Finally, a comprehensive analysis of conserved quantities canonically related to CDSs is given.

A number of recent theorems by Krolak and Newman purport to prove cosmic censorship by showing that “strong curvature” singularities must be hidden behind horizons. I prove that Newman's “null, strong curvature” condition, which he imposes on certain classes of null geodesics to restrict curvature growth in the space-time, does not hold in many physically realistic space-times: it is not satisfied by any null geodesic in the relevant class in any open Friedmann cosmological model, nor does it hold for any null geodesic in the relevant class in maximal Schwarzschild space. More generally, I argue that the singularity predicted by the Penrose singularity theorem is unlikely to be of the type eliminated by Newman. Thus the Newman theorems are probably without physical significance. The Krolak theorems, although based on a physically significant definition of strong curvature singularity, are mathematically invalid, and his approach cannot be used to obtain a cosmic censorship theorem.

It is shown that the nonlocally rotationally symmetric Bianchi type-V vacuum solution recently discussed by Nayak is identical, after trivial transformations, with the solution found some years ago by Joseph. In addition some calculations and remarks concerning the integration of the null geodesic equations are given.

It is shown that if the past light cone of an observer is isotropic and has a caustic (i.e., contains points which are conjugate to the observer along a null geodesic) then the spacetime admits a compact (S3) spacelike hypersurface. The result is valid for space-times with dimension ⩾ 4.

The null cut locus of a spacelike submanifold of codimension 2 in a space-time is defined. In globally hyperbolic space-times, it is shown that the future (past) null cut locusc n + (H) [c n - (H)] of a compact, acausal, spacelike submanifoldH of codimension 2 is a closed subset of the space-time, and each pointx∈ c n + (H) is either a focal point ofH along some future-directed null geodesic meetingH orthogonally or there exist at least two null geodesics fromH tox, realizing the distance betweenH andx or both. Also, it can be shown that the assumptions of the Penrose's singularity theorem for open globally hyperbolic space-times may be weakened to the space-times which are conformal to an open subset of an open globally hyperbolic space-time.