Analytical formulas for gravitational lensing: Higher order calculation

We extend to higher order a recently published method for calculating the deflection angle of light in a general static and spherically symmetric metric. We have tested our method on the metric of Schwarzschild and Reissner-Nordström black holes, on the metric of a charged black hole coupled to Born-Infeld electrodynamics and on the metric of Weyl gravity. Since our method is geometrically convergent, as proved in our previous work, our analytical formulas obtained working to fourth order are sufficient to reach errors of few percents even in proximity of the photon sphere.

Figure 1

Error of the deflection angle as a function of $r_0$ for the Schwartzchild metric with $GM=1$. The dotted vertical line shows the location of the photon sphere. The right plot is a blow up of the left plot. (color online)

Figure 2

Same as in Fig. 1 for the RN metric with $GM=1$ and $Q=1$. (color online)

Figure 3

Error for the deflection angle for the Born-Infeld metric with $GM=1$, $Q=1/2$, $b=1$, and $r_0=3$. The horizontal axis specifies the order of the Taylor approximation to the potential. The notation ${\mathrm{PMS}}_x$ indicates the order to which our method is applied. (color online)

Figure 4

Right plot: deflection angle for the Weyl metric with $\beta =\gamma =1$ for $r_0$ close to the photon sphere; Left plot: $|\Delta {\varphi }_{{\mathrm{PMS}}_4}-\Delta {\varphi }_{\mathrm{numerical}}|$. (color online)

Figure 5

Left plot: deflection angle for the Weyl metric with $\beta =1$ and $\gamma =-{\pi }^2/128$; Right plot: percentage error corresponding to the approximate solution in the left plot.