### 5.6 Janis–Newman–Winicour spacetime

The Janis–Newman–Winicour metric [167] can be brought into the form [335]
where and are constants. It is the most general spherically symmetric static and asymptotically
flat solution of Einstein’s field equation coupled to a massless scalar field. For it reduces to the
Schwarzschild solution; in this case the scalar field vanishes. For and , there is a naked
curvature singularity at . Lensing in this spacetime was studied in [338, 337]. The main
motivation was to find out whether the lensing characteristics of such a naked singularity can be
distinguished from lensing by a Schwarschild black hole. The result is that the qualitative features of lensing
remain similar to the Schwarzschild case as long as . However, if drops below , they
become quite different. The reason is easily understood if we write Equation (116) in the form (69). The
metric coefficient
has a minimum between the singularity and infinity as long as (see Figure 20). This minimum
indicates an unstable light sphere (recall Equation (71)), as in the Schwarzschild case at . All
qualitative features of lensing carry over from the Schwarzschild case, i.e., Figures 15, 16, and 17 remain
qualitatively unchanged. Clearly, the precise shape of the graph of in Figure 15 changes if is
changed. The question of how the logarithmic asymptote (“strong-field limit”) depends on is dicussed
in [37]. If drops below , has no longer an extremum, i.e., there is no light sphere. This
implies that only finitely many images are possible [152]. In [337] naked singularities of spherically
symmetric spacetimes are called weakly naked if they are surrounded by a light sphere (cf. [59]). In a
nutshell, weakly naked singularities show the same qualitative lensing features as black holes. A
generalization of this result to spacetimes without spherical symmetry has not been worked
out.