Janis–Newman–Winicour spacetime

5.6 Janis–Newman–Winicour spacetime

The Janis–Newman–Winicour metric [167 ] can be brought into the form [335 ]

where $m$ and $γ$ 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 $γ = 1$ it reduces to the Schwarzschild solution; in this case the scalar field vanishes. For $m > 0$ and $γ ⁄= 1$ , there is a naked curvature singularity at $r = 2m ∕γ$ . 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 $1∕2 < γ < 1$ . However, if $γ$ drops below $1∕2$ , 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 $1 2 < γ < 1$ (see Figure 20

). This minimum indicates an unstable light sphere (recall Equation (71

)), as in the Schwarzschild case at $r = 3m$ . 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 $Φ$ in Figure 15

changes if $γ$ is changed. The question of how the logarithmic asymptote (“strong-field limit”) depends on $γ$ is dicussed in [37 ]. If $γ$ drops below $1∕2$ , $R (r)$ 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.

Figure 20: Metric coefficient $R (r)$ for the Janis–Newman–Winicour metric. For $1 < γ < 1 2$ , $R (r)$ is similar to the Schwarzschild case $γ = 1$ (see Figure 9

). For $1 γ ≤ 2$ , $R(r)$ has no longer a minimum, i.e., there is no longer a light sphere which can be asymptotically approached by light rays.