Light on curved backgrounds

We consider the motion of light on different spacetime manifolds by calculating the deflection angle, lensing properties and by probing into the possibility of bound states. The metrics in which we examine the light motion include, among other items, a general relativistic dark matter metric, a dirty black hole, and a worm hole metric, the last two inspired by noncommutative geometry. The lensing in a holographic screen metric is discussed in detail. We study also the bending of light around naked singularities like, e.g., the Janis-Newman-Winicour metric and include other cases. A generic property of light behavior in these exotic metrics is pointed out. For the standard metric like the Schwarzschild and Schwarzschild-de Sitter cases, we improve the accuracy of the lensing results for the weak and strong regimes.

Figure 1

Behavior of the “photon potential” proportional to $\tilde{V}$ for the Reissner-Nordström naked singularity. As explained in the text in the narrow range of $\alpha$ there is a possibility to trap photons inside a well.

Figure 2

Behavior of the exact solution (32) (solid line) versus the weak field approximations (33) (dashed line), and (24) in [29] with $q=0$ (dotted line).

Figure 3

Plot of the exact solution (32) (solid line), Bozza’s approximation (dotted line), and the approximation (37) (dashed line).

Figure 4

Ratio of the exact deflection angle (32) and the approximate one (37) as a function of the closest approach distance.

Figure 5

Behavior of the rescaled “photon potential” proportional $\tilde{V}$ for the JNW metric (naked singularity) for different values of the parameter $\gamma$. As long as $\gamma <1/2$ the metric displays a local maximum at the rescaled position bigger than 1. Since the nonremovable singular point at $x=1$ cannot be crossed, light can be trapped, in principle, between the singular point and the hump.

Figure 6

The rescaled “photon potential” proportional to $\tilde{V}$ for the holographic screen metric. In case of a naked singularity, photons can be trapped in the local minimum.

Figure 7

Plot of the function $z_{+}(x)$ in the extreme case (solid line), for $m=3$ (dashed line), and $m=6$ (dotted line). Note that the size of the event horizon increases as the value of the rescaled mass parameter $m$ increases. This figure refers to the holographic screen metric.

Figure 8

The rescaled “photon potential” proportional to $\tilde{V}$ for the holographic screen metric showing the photon unstable orbit (maximum) for a large $m$.

Figure 9

The rescaled photon potential proportional to $\tilde{V}$ for the holographic screen metric displaying the naked singularity case where a narrow range of the parameter $m$ leads to a local minimum allowing bound states of photons.

Figure 10

Plot of the function $z_{+}(x)$ for the cases $\alpha =2.18$ (solid line), $\alpha =2.20$ (dotted line), and $\alpha =2.3$ (dashed line). Note that the size of the size of the throat increases as the value of the parameter $\alpha$ increases. This figure refers to the wormhole metric.

Figure 11

Plot of the weak field approximation (69) as a function of the closest distance of approach for $\alpha =2.2$ (solid line), $\alpha =3.2$ (dotted line), $\alpha =5.2$ (long dashed line).

Figure 12

Plot of (20) using the dirty black hole metric for the nonxtremal case (solid line, $\alpha =4$), extremal case (dotted line, $\alpha ={\alpha }_0$), and the dirty minigravastar (dash dotted and long dashed lines for $\alpha =2$ and $\alpha =1$, respectively). It is interesting to observe that in the case of a minigravastar light will form bound states only when the parameter $\alpha$ varies in a certain interval.