Lensing in Globally Hyperbolic Spacetimes

3 Lensing in Globally Hyperbolic Spacetimes

In a globally hyperbolic spacetime, considerably stronger statements on qualitative lensing features can be made than in an arbitrary spacetime. This includes, e.g., multiple imaging criteria in terms of cut points or conjugate points, and also applications of Morse theory. The value of these results lies in the fact that they hold in globally hyperbolic spacetimes without symmetries, where lensing cannot be studied by explicitly integrating the lightlike geodesic equation.

The most convenient formal definition of global hyperbolicity is the following. In a spacetime $(ℳ, g)$ , a subset $𝒞$ of $ℳ$ is called a Cauchy surface if every inextendible causal (i.e., timelike or lightlike) curve intersects $𝒞$ exactly once. A spacetime is globally hyperbolic if and only if it admits a Cauchy surface. The name globally hyperbolic refers to the fact that for hyperbolic differential equations, like the wave equation, existence and uniqueness of a global solution is guaranteed for initial data given on a Cauchy surface. For details on globally hyperbolic spacetimes see, e.g., [153 , 25 ]. It was demonstrated by Geroch [132 ] that every gobally hyperbolic spacetime admits a continuous function $t : ℳ − → ℝ$ such that $t−1(t0)$ is a Cauchy surface for every $t0 ∈ ℝ$ . A complete proof of the fact that such a Cauchy time function can be chosen differentiable was given much later by Bernal and Sánchez [26 , 27 ]. The topology of a globally hyperbolic spacetime is determined by the topology of any of its Cauchy surfaces, $ℳ ≃ 𝒞 × ℝ$ . Note, however, that the converse is not true because $𝒞1 × ℝ$ may be homeomorphic (and even diffeomorphic) to $𝒞2 × ℝ$ without $𝒞 1$ being homeomorphic to $𝒞 2$ . For instance, one can construct a globally hyperbolic spacetime with topology $4 ℝ$ that admits a Cauchy surface which is not homeomorphic to $3 ℝ$ [239 ].

In view of applications to lensing the following observation is crucial. If one removes a point, a worldline (timelike curve), or a world tube (region with timelike boundary) from an arbitrary spacetime, the resulting spacetime cannot be globally hyperbolic. Thus, restricting to globally hyperbolic spacetimes excludes all cases where a deflector is treated as non-transparent by cutting its world tube from spacetime (see Figure 24 for an example). Note, however, that this does not mean that globally hyperbolic spacetimes can serve as models only for transparent deflectors. First, a globally hyperbolic spacetime may contain “non-transparent” regions in the sense that a light ray may be trapped in a spatially compact set. Second, the region outside the horizon of a (Schwarzschild, Kerr, $...$ ) black hole is globally hyperbolic.

3.1 Criteria for multiple imaging in globally hyperbolic spacetimes
3.2 Wave fronts in globally hyperbolic spacetimes
3.3 Fermat’s principle and Morse theory in globally hyperbolic spacetimes
3.4 Lensing in asymptotically simple and empty spacetimes