The theory of gravitational lensing is reviewed from a spacetime perspective, without quasi-Newtonian approximations More precisely, the review covers all aspects of gravitational lensing where light propagation is described in terms of lightlike geodesics of a metric of Lorentzian signature It includes the basic equations and the relevant techniques for calculating the position, the shape, and the brightness of images in an arbitrary general-relativistic spacetime It also includes general theorems on the classification of caustics, on criteria for multiple imaging, and on the possible number of images The general results are illustrated with examples of spacetimes where the lensing features can be explicitly calculated, including the Schwarzschild spacetime, the Kerr spacetime, the spacetime of a straight string, plane gravitational waves, and others

https://link.springer.com/content/pdf/10.12942%2Flrr-2004-9.pdf

Gravitational Lensing from a Spacetime Perspective

The theory of gravitational lensing is reviewed from a spacetime perspective, without quasi-Newtonian approximations. More precisely, the review covers all aspects of gravitational lensing where light propagation is described in terms of lightlike geodesics of a metric of Lorentzian signature. It includes the basic equations and the relevant techniques for calculating the position, the shape, and the brightness of images in an arbitrary general-relativistic spacetime. It also includes general theorems on the classification of caustics, on criteria for multiple imaging, and on the possible number of images. The general results are illustrated with examples of spacetimes where the lensing features can be explicitly calculated, including the Schwarzschild spacetime, the Kerr spacetime, the spacetime of a straight string, plane gravitational waves, and others.

This article is divided in five parts, the first one having nothing to do with tachyons. In fact, to prepare the ground, in part I (sect.2) we shall merely show that special relativity - even without tachyons - can be given a form such to describe both particles and antiparticles. Part II is the largest one: initially, after some historical remarks and having revisited the postulates of special relativity, we presnt a review of the elegant ''model theory'' of tachyons in two dimensions; passing then to four dimensions, we review the main results of the classical theory of tachyons that do not depend on the existence of Superluminal reference frames (or that are at least independent of the explicit form of the Superluminal Lorentz ''transformations''). In particular, we discuss how tachyons would look like, i.e. their apparent ''shape''. Last but not least, all the common causality problems are thoroughly solved, on the basis of the previously reviewed tachyon kinematics. Part III deals with tachyons in general relativity; in particular the question of the apparent Superluminal expansions in astrophysics is reviewed. Part IV shows the interesting, possible role of tachyons in elementary-particle physcis and in quantum theory. In part V, the last one, we face the (still open) problem of the Superluminal Lorentz ''transformations'' in four dimensions, by introducing, for instance, an auxiliary six-dimensional space-time, and finally present the electromagnetic theory of tachyons: a theory that can be relevant also from the ''experimental'' point of view.

Classical tachyons and possible applications

We analytically investigate the influence of a cosmic expansion on the shadow of the Schwarzschild black hole. We suppose that the expansion is driven by a cosmological constant only and use the Kottler (or Schwarzschild--de Sitter) spacetime as a model for a Schwarzschild black hole embedded in a de Sitter universe. We calculate the angular radius of the shadow for an observer who is comoving with the cosmic expansion. It is found that the angular radius of the shadow shrinks to a nonzero finite value if the comoving observer approaches infinity.

Black hole shadow in an expanding universe with a cosmological constant

We find all higher dimensional solutions of Einstein-Maxwell theory that are the topological product of two manifolds of constant curvature. These solutions include the higher dimensional Nariai, Bertotti-Robinson and anti-Nariai solutions and the anti--de Sitter Bertotti-Robinson solutions with toroidal and hyperbolic topology (Pleba\ifmmode \acute{n}\else \'{n}\fi{}ski-Hacyan solutions). We give explicit results for any dimension $Dg~4.$ These solutions are generated from the appropriate extremal limits of the higher dimensional near-extreme black holes in de Sitter and anti--de Sitter backgrounds. Thus, we also find the mass and charge parameters of higher dimensional extreme black holes as a function of the radius of the degenerate horizon.

Nariai, Bertotti-Robinson and anti-Nariai solutions in higher dimensions

The conformal structure of timelike 2-surfaces of the vacuum manifold is examined in detail in the cases when the cosmological constant, $\ensuremath{\lambda}$, is not assumed to be zero. For $\ensuremath{\lambda}g0$ with $3m\ensuremath{\le}{\ensuremath{\Lambda}}^{\ensuremath{-}\frac{1}{2}}$ it is shown that these surfaces constitute the junction of an infinite number of regions and that the complete manifold may be represented in a causal way by a finite number of regions.

Effects of a nonvanishing cosmological constant on the spherically symmetric vacuum manifold

We demonstrate that there exist unstable, spacelike, circular orbits in the Schwarzschild field for all radii in the range $0lrl3m$ The conditions under which spacelike trajectories bend toward or away from the source of the field are derived for the entire $r\ensuremath{-}\ensuremath{\varphi}$ plane We show that any nonradial spacelike geodesic with turning point less than $2m$ will appear spacelike over the entire $u\ensuremath{-}v$ plane The scattering and capture cross sections for a particle on a spacelike trajectory are evaluated Finally, we suggest that there are compelling reasons for rejecting the usual assumption of a global past-future relation in the extended Schwarzschild manifold

Behavior of spacelike geodesics in the extended Schwarzschild manifold

A local interior-exterior distinction is formulated for matching across a hypersurface, and detailed examples are given for spherical dust flows. While the "inside" vs "outside" character is not, in general, globally preserved, the distinction is unambiguous in an $\mathcal{R}$ region, and does afford clarification of certain boundary surfaces and surface densities discussed in the literature.

Some remarks on surfaces of discontinuity in general relativity

IT has been argued1 that even though a stable model of a white hole requires essentially zero delay between the big bang and the inception of expansion of the white hole, one can (at least in principle) still observe certain types of white hole, most likely those which never expand beyond their ‘Schwarzschild spheres’ into our space, and which have, in fact, always been visible. Here we report the main results of an analysis of the optical properties of such white holes, the complete details of which will be published elsewhere.

Kayll Lake

Papers

Effects of a nonvanishing cosmological constant on the spherically symmetric vacuum manifold

Behavior of spacelike geodesics in the extended Schwarzschild manifold

Some remarks on surfaces of discontinuity in general relativity

The present appearance of white holes