2.1 Light cone and exact lens map

In an arbitrary spacetime (ℳ, g), what an observer at an event pO can see is determined by the lightlike geodesics that issue from pO into the past. Their union gives the past light cone of pO. This is the central geometric object for lensing from the spacetime perspective. For a point source with worldline γS, each past-oriented lightlike geodesic λ from pO to γS gives rise to an image of γS on the observer’s sky. One should view any such λ as the central ray of a thin bundle that is focused by the observer’s eye lens onto the observer’s retina (or by a telescope onto a photographic plate). Hence, the intersection of the past light cone with the world-line of a point source (or with the world-tube of an extended source) determines the visual appearance of the latter on the observer’s sky.

In mathematical terms, the observer’s sky or celestial sphere 𝒮O can be viewed as the set of all lightlike directions at pO. Every such direction defines a unique (up to parametrization) lightlike geodesic through pO, so 𝒮O may also be viewed as a subset of the space of all lightlike geodesics in (ℳ, g) (cf. [210Jump To The Next Citation Point]). One may choose at pO a future-pointing vector UO with g(UO,UO ) = − 1, to be interpreted as the 4-velocity of the observer. This allows identifying the observer’s sky 𝒮O with a subset of the tangent space TpO ℳ,

{ | } 𝒮O ≃ w ∈ TpOℳ |g(w, w) = 0 and g(w,UO ) = 1 . (1 )
If UO is changed, this representation changes according to the standard aberration formula of special relativity. By definition of the exponential map exp, every affinely parametrized geodesic s ↦→ λ(s) satisfies ( ) λ(s) = exp s˙λ(0). Thus, the past light cone of pO is the image of the map
(s,w ) ↦− → exp(sw ), (2 )
which is defined on a subset of ]0,∞ [×𝒮O. If we restrict to values of s sufficiently close to 0, the map (2View Equation) is an embedding, i.e., this truncated light cone is an embedded submanifold; this follows from the well-known fact that exp maps a neighborhood of the origin, in each tangent space, diffeomorphically into the manifold. However, if we extend the map (2View Equation) to larger values of s, it is in general neither injective nor an immersion; it may form folds, cusps, and other forms of caustics, or transverse self-intersections. This observation is of crucial importance in view of lensing. There are some lensing phenomena, such as multiple imaging and image distortion of (point) sources into (1-dimensional) rings, which can occur only if the light cone fails to be an embedded submanifold (see Section 2.8). Such lensing phenomena are summarized under the name strong lensing effects. As long as the light cone is an embedded submanifold, the effects exerted by the gravitational field on the apparent shape and on the apparent brightness of light sources are called weak lensing effects. For examples of light cones with caustics and/or transverse self-intersections, see Figures 12View Image, 24View Image, and 25View Image. These pictures show light cones in spacetimes with symmetries, so their structure is rather regular. A realistic model of our own light cone, in the real world, would have to take into account numerous irregularly distributed inhomogeneities (“clumps”) that bend light rays in their neighborhood. Ellis, Bassett, and Dunsby [99Jump To The Next Citation Point] estimate that such a light cone would have at least 1022 caustics which are hierarchically structured in a way that reminds of fractals.

For calculations it is recommendable to introduce coordinates on the observer’s past light cone. This can be done by choosing an orthonormal tetrad (e0,e1,e2,e3) with e0 = − UO at the observation event pO. This parametrizes the points of the observer’s celestial sphere by spherical coordinates (Ψ, Θ),

w = sinΘ cos Ψe1 + sinΘ sin Ψe2 + cosΘe3 + e0. (3 )
In this representation, map (2View Equation) maps each (s,Ψ, Θ ) to a spacetime point. Letting the observation event float along the observer’s worldline, parametrized by proper time τ, gives a map that assigns to each (s,Ψ, Θ, τ) a spacetime point. In terms of coordinates 0 1 2 3 x = (x ,x ,x ,x ) on the spacetime manifold, this map is of the form
xi = F i(s,Ψ,Θ, τ), i = 0,1,2,3. (4 )
It can be viewed as a map from the world as it appears to the observer (via optical observations) to the world as it is. The observational coordinates (s,Ψ, Θ,τ ) were introduced by Ellis [98] (see [100] for a detailed discussion). They are particularly useful in cosmology but can be introduced for any observer in any spacetime. It is useful to consider observables, such as distance measures (see Section 2.4) or the ellipticity that describes image distortion (see Section 2.5) as functions of the observational coordinates. Some observables, e.g., the redshift and the luminosity distance, are not determined by the spacetime geometry and the observer alone, but also depend on the 4-velocities of the light sources. If a vector field U with g(U,U ) = − 1 has been fixed, one may restrict to an observer and to light sources which are integral curves of U. The above-mentioned observables, like redshift and luminosity distance, are then uniquely determined as functions of the observational coordinates. In applications to cosmology one chooses U as tracing the mean flow of luminous matter (“Hubble flow”) or as the rest system of the cosmic background radiation; present observations are compatible with the assumption that these two distinguished observer fields coincide [32].

Writing map (4View Equation) explicitly requires solving the lightlike geodesic equation. This is usually done, using standard index notation, in the Lagrangian formalism, with the Lagrangian 1 i j ℒ = 2gij(x)x˙˙x, or in the Hamiltonian formalism, with the Hamiltonian ℋ = 12gij(x )pipj. A non-trivial example where the solutions can be explicitly written in terms of elementary functions is the string spacetime of Section 5.10. Somewhat more general, although still very special, is the situation that the lightlike geodesic equation admits three independent constants of motion in addition to the obvious one ij g (x )pipj = 0. If, for any pair of the four constants of motion, the Poisson bracket vanishes (“complete integrability”), the lightlike geodesic equation can be reduced to first-order form, i.e., the light cone can be written in terms of integrals over the metric coefficients. This is true, e.g., in spherically symmetric and static spacetimes (see Section 4.3).

Having parametrized the past light cone of the observation event pO in terms of (s, w), or more specifically in terms of (s,Ψ, Θ), one may set up an exact lens map. This exact lens map is analogous to the lens map of the quasi-Newtonian approximation formalism, as far as possible, but it is valid in an arbitrary spacetime without approximation. In the quasi-Newtonian formalism for thin lenses at rest, the lens map assigns to each point in the lens plane a point in the source plane (see, e.g., [298Jump To The Next Citation Point275Jump To The Next Citation Point343Jump To The Next Citation Point]). When working in an arbitrary spacetime without approximations, the observer’s sky 𝒮O is an obvious substitute for the lens plane. As a substitute for the source plane we choose a 3-dimensional submanifold 𝒯 with a prescribed ruling by timelike curves. We assume that 𝒯 is globally of the form 𝒬 × â„, where the points of the 2-manifold 𝒬 label the timelike curves by which 𝒯 is ruled. These timelike curves are to be interpreted as the worldlines of light sources. We call any such 𝒯 a source surface. In a nutshell, choosing a source surface means choosing a two-parameter family of light sources.

The exact lens map is a map from 𝒮O to 𝒬. It is defined by following, for each w ∈ 𝒮O, the past-pointing geodesic with initial vector w until it meets 𝒯 and then projecting to 𝒬 (see Figure 1View Image). In other words, the exact lens map says, for each point on the observer’s celestial sphere, which of the chosen light sources is seen at this point. Clearly, non-invertibility of the lens map indicates multiple imaging. What one chooses for 𝒯 depends on the situation. In applications to cosmology, one may choose galaxies at a fixed redshift z = zS around the observer. In a spherically-symmetric and static spacetime one may choose static light sources at a fixed radius value r = rS. Also, the surface of an extended light source is a possible choice for 𝒯.

View Image

Figure 1: Illustration of the exact lens map. p O is the chosen observation event, 𝒯 is the chosen source surface. 𝒯 is a hypersurface ruled by timelike curves (worldlines of light sources) which are labeled by the points of a 2-dimensional manifold 𝒬. The lens map is defined on the observer’s celestial sphere 𝒮O, given by Equation (1View Equation), and takes values in 𝒬. For each w ∈ 𝒮O, one follows the lightlike geodesic with this initial direction until it meets 𝒯 and then projects to 𝒬. For illustrating the exact lens map, it is an instructive exercise to intersect the light cones of Figures 12View Image, 24View Image, 25View Image, and 29View Image with various source surfaces 𝒯.

The exact lens map was introduced by Frittelli and Newman [122Jump To The Next Citation Point] and further discussed in [9190]. The following global aspects of the exact lens map were investigated in [270Jump To The Next Citation Point]. First, in general the lens map is not defined on all of 𝒮O because not all past-oriented lightlike geodesics that start at pO necessarily meet 𝒯. Second, in general the lens map is multi-valued because a lightlike geodesic might meet 𝒯 several times. Third, the lens map need not be differentiable and not even continuous because a lightlike geodesic might meet 𝒯 tangentially. In [270Jump To The Next Citation Point], the notion of a simple lensing neighborhood is introduced which translates the statement that a deflector is transparent into precise mathematical language. It is shown that the lens map is globally well-defined and differentiable if the source surface is the boundary of such a simple lensing neighborhood, and that for each light source that does not meet the caustic of the observer’s past light cone the number of images is finite and odd. This result applies, as a special case, to asymptotically simple and empty spacetimes (see Section 3.4).

For expressing the exact lens map in coordinate language, it is recommendable to choose coordinates (x0,x1,x2, x3) such that the source surface 𝒯 is given by the equation x3 = x3 S, with a constant 3 x S, and that the worldlines of the light sources are 0 x-lines. In this situation the remaining coordinates 1 x and 2 x label the light sources and the exact lens map takes the form

(Ψ, Θ ) ↦− → (x1,x2). (5 )
It is given by eliminating the two variables s and 0 x from the four equations (4View Equation) with 3 3 F (s,Ψ, Θ,τ ) = x S and fixed τ. This is the way in which the lens map was written in the original paper by Frittelli and Newman; see Equation (6) in [122Jump To The Next Citation Point]. (They used complex coordinates -- (η,η ) for the observer’s celestial sphere that are related to our spherical coordinates (Ψ, Θ ) by stereographic projection.) In this explicit coordinate version, the exact lens map can be succesfully applied, in particular, to spherically symmetric and static spacetimes, with 0 x = t, 1 x = φ, 2 x = 𝜗, and 3 x = r (see Section 4.3 and the Schwarzschild example in Section 5.1). The exact lens map can also be used for testing the reliability of approximation techniques. In [183] the authors find that the standard quasi-Newtonian approximation formalism may lead to significant errors for lensing configurations with two lenses.
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