Can strong gravitational lensing distinguish naked singularities from black holes?

In this paper we study gravitational lensing in the strong field limit from the perspective of cosmic censorship, to investigate whether or not naked singularities, if they exist in nature at all, can be distinguished from black holes. The spacetime we explore from this perspective is the Joshi-Malafarina-Narayan (JMN) metric which represents a spherically symmetric solution to the Einstein field equations with anisotropic pressure and contains a naked singularity at the center. JMN geometry is matched with the Schwarzschild metric to the exterior at a finite radius. This metric was recently shown to be a possible end state of gravitational collapse of a fluid with zero radial pressure. In the presence of the photon sphere, gravitational lensing signature of this spacetime is identical to that of the Schwarzschild black hole with infinitely many relativistic images and Einstein rings, all of them located beyond a certain critical angle from optic axis and the inner relativistic images all clumped together. However, in the absence of the photon sphere, which is the case for a wide range of parameter values in this spacetime, we show that we get finitely many relativistic images and Einstein rings spaced reasonably apart from one another, some of which can be formed inside the critical angle for the corresponding Schwarzschild black hole. This study suggests that the observation of relativistic images and rings might, in principle, allow us to unravel the existence of the naked singularity in the absence of the photon sphere. Also, the results obtained here are in contrast with the earlier investigation on Janis-Newman-Winicour (JNW) naked singularities where it was shown that the radial caustic is always present in the absence the photon sphere, which is not the case with JMN geometry where radial caustic is absent. We also point out the practical difficulties that might be encountered in the observation of the relativistic images and suggest that new dedicated experiments and techniques have to be developed in future for this purpose.

Figure 1

The lens diagram. Positions of the source, lens, observer and the image are given by $S$, $L$, $O$ and $I$, respectively. The distances between lens-source, lens-observer and source-observer are given by $D_{ds}$, $D_d$ and $D_s$, respectively. The angular locations of the source and the image with respect to optic axis are given by $\beta$ and $\theta$. The impact parameter is given by $J$.

Figure 2

$\theta$ vs $x_0$ in Schwarzschild spacetime. The image location $\theta$ (in microarcseconds) is plotted as a function of closest approach $x_0$ (dimensionless) in Schwarzschild spacetime for a galactic supermassive black hole scenario. It is a monotonically increasing function.

Figure 3

$\widehat{\alpha }$ vs $\theta$. Einstein deflection angle $\widehat{\alpha }$ (in radian) is plotted as a function of the image location $\theta$ (in microarcseconds) in Schwarzschild spacetime for a galactic supermassive black hole scenario. $\widehat{\alpha }(\theta )$ is monotonically decreasing. It diverges around $\theta \approx 16.8\mathrm{microarcseconds}$ which corresponds to the photon sphere. The horizontal lines correspond to $3\pi /2$ and $2\pi$ for the dashed-red and thick-purple lines, respectively, which mark the onset of the relativistic deflection of light. Relativistic images and Einstein rings can be seen between the photon sphere and the intersection points of horizontal lines with $\widehat{\alpha }$ curve.

Figure 4

$V_{\mathrm{eff}}$ vs $x$. Plotted here is the effective potential $V_{\mathrm{eff}}$ (dimensionless) for the radial motion for the light rays as a function of $x$ (dimensionless). The panel on the left corresponds to the parameter values $M_0\ge 2/3$. The thick, green vertical line corresponds to the photon sphere, whereas the dashed-red line corresponds to the boundary between the interior naked singularity region and exterior Schwarzschild geometry. The effective potential monotonically goes to zero in the interior region. The panel on the right corresponds to the parameter values $M_0<2/3$, where the photon sphere is absent. The thick, vertical red line is the boundary. The effective potential blows up as we approach the singularity.

Figure 5

$\theta$ vs $x_0$ in JMN. The image location $\theta$ (in microarcseconds) is plotted as a function of closest approach $x_0$ (dimensionless) in JMN spacetime for a galactic central supermassive object scenario in an interior region. It is a monotonically increasing function.

Figure 6

$\widehat{\alpha }$ vs $\theta$ for different $M_0$. Einstein deflection angle $\widehat{\alpha }$ (in radian) in JMN spacetime is plotted against the image location $\theta$ (in microarcseconds). The value of $M_0$ for the curves is shown in the legend. $\widehat{\alpha }$ is a monotonically decreasing function. As we increase $M_0$ the maximum deflection angle continues to increase. The deflection angle will be larger than $2\pi$ and thus relativistic images can form for $M_0>0.475$.

Figure 7

${\mu }_t$ vs $\theta$ for JMN spacetime. Plotted here is the tangential magnification near the relativistic Einstein rings for $M_0=0.63$. $\theta$ is in microarcseconds and the y-axis scale has been multiplied by ${10}^9$ for clarity. Tangential magnification blows up at the location of the Einstein ring and decreases as we go away from it.

Figure 8

${\mu }_r$ vs $\theta$ for JMN spacetime. Plotted here is the radial magnification near the relativistic Einstein rings for $M_0=0.63$. $\theta$ is in microarcseconds and the y-axis scale has been multiplied by ${10}^{12}$ for clarity.

Figure 9

$\mu$ vs $\theta$ for JMN spacetime. Plotted here is the total magnification near the relativistic Einstein rings for $M_0=0.63$. $\theta$ is in microarcseconds and the y-axis scale has been multiplied by ${10}^{21}$ for clarity. Total magnification blows up at the location of the Einstein ring and decreases as we go away from it.