5.6 Janis–Newman–Winicour spacetime

The Janis–Newman–Winicour metric [167] can be brought into the form [335]
( ) γ 2( 2 2 2) 2m-- 2 ----dr2---- r--d𝜗--+-sin-𝜗d-φ--- g = − 1 − γr dt + ( 2m-)γ + ( 2m ) γ− 1 , (116 ) 1 − γr 1 − γr-
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 [338337Jump To The Next Citation Point]. 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 (116View Equation) in the form (69View Equation). The metric coefficient
( ) 12−γ R (r) = r 1 − 2m-- (117 ) γr
has a minimum between the singularity and infinity as long as 1 2 < γ < 1 (see Figure 20View Image). This minimum indicates an unstable light sphere (recall Equation (71View Equation)), as in the Schwarzschild case at r = 3m. All qualitative features of lensing carry over from the Schwarzschild case, i.e., Figures 15View Image, 16View Image, and 17View Image remain qualitatively unchanged. Clearly, the precise shape of the graph of Φ in Figure 15View Image 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.
View Image

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 9View Image). 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.

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