5.4 Morris–Thorne wormholes

We consider a spacetime whose metric is of the form (69View Equation) with ef(r)S (r) = 1, i.e.,
g = − e2f(r)dt2 + dr2 + e2f(r)R(r)2(d 𝜗2 + sin2 𝜗dφ2) , (112 )
where r ranges from − ∞ to ∞. We call such a spacetime a Morris–Thorne wormhole (see [228Jump To The Next Citation Point]) if
f(r) − → 0, r−2R (r)2 − → 1, (113 ) r→ ±∞ r→ ±∞
such that the metric (112View Equation) is asymptotically flat for r → − ∞ and for r → ∞.

A particular example of a Morris–Thorne wormhole is the Ellis wormhole [102] where

f(r) = 0, R(r) = √r2--+-a2 (114 )
with a constant a. The Ellis wormhole has an unstable light sphere at r = 0, i.e., at the “neck” of the wormhole. It is easy to see that every Morris–Thorne wormhole must have an unstable light sphere at some radius between r = − ∞ and r = ∞. This has the consequence [152Jump To The Next Citation Point] that every Morris–Thorne wormhole produces an infinite sequence of images of an appropriately placed light source. This infinite sequence corresponds to infinitely many light rays whose limit curve asymptotically spirals towards the unstable light sphere.

Lensing by the Ellis wormhole was discussed in [55]; in this paper the authors identified the region r > 0 with the region r < 0 and they developed a scattering formalism, assuming that observer and light source are in the asymptotic region. Lensing by the Ellis wormhole was also discussed in [271Jump To The Next Citation Point] in terms of the exact lens map. The resulting features are qualitatively very similar to the Schwarzschild case, with the radius values r = − ∞, r = 0, r = ∞ in the wormhole case corresponding to the radius values r = 2m, r = 3m, r = ∞ in the Schwarzschild case. With this correspondence, Figures 15View Image, 16View Image, and 17View Image qualitatively illustrate lensing by the Ellis wormhole. More generally, the same qualitative features occur whenever the metric function R (r) has one minimum and no other extrema, as in Figure 9View Image.

If observer and light source are on the same side of the wormhole’s neck, and if only light rays in the asymptotic region are considered, lensing by a wormhole can be studied in terms of the quasi-Newtonian approximation formalism [181]. However, as wormholes are typically associated with negative energy densities [228229], the usual assumption of the quasi-Newtonian approximation formalism that the mass density is positive cannot be maintained. This observation has raised some interest in lensing by negative masses, in particular in the question of whether negative masses can be detected by their (“microlensing”) effect on the energy flux from sources passing behind them. So far, related calculations [64292] have been done only in the quasi-Newtonian approximation formalism.

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