Analytical formulas for gravitational lensing

In this paper we discuss a new method which can be used to obtain arbitrarily accurate analytical expressions for the deflection angle of light propagating in a given metric. Our method works by mapping the integral into a rapidly convergent series and provides extremely accurate approximations already to first order. We have derived a general first order formula for a generic spherically symmetric static metric tensor and we have tested it in four different cases.

Figure 1

Deflection angle as a function of $r_0$ assuming $GM=1$.

Figure 2

Deflection angle as a function of $r_0$ assuming $GM=1$ close to the photon sphere.

Figure 3

Magnification as a function of the impact parameter assuming $a=1000$ and $GM=1$.

Figure 4

Deflection angle for the Reissner-Nordström metric as a function of $r_0$ assuming $GM=1$ and $Q=1/2$.

Figure 5

Ratio between the approximate first order formula and the exact (numerical) result in the Janis-Newman-Winicour metric for different values of $\nu$ and assuming $b=1$. The potential is expanded to order $z^3$.

Figure 6

Deflection angle for the Einstein-Born-Infeld problem as a function of $r_0$ assuming $b=M=1$ and $Q=1/2$. The potential in our formula is expanded to order $z^4$.