Gravitational lensing from compact bodies: Analytical results for strong and weak deflection limits

We develop a nonperturbative method that yields analytical expressions for the deflection angle of light in a general static and spherically symmetric metric. The method works by introducing into the problem an artificial parameter, called $\delta$, and by performing an expansion in this parameter to a given order. The results obtained are analytical and nonperturbative because they do not correspond to a polynomial expression in the physical parameters. Already to first order in $\delta$ the analytical formulas obtained using our method provide at the same time accurate approximations both at large distances (weak deflection limit) and at distances close to the photon sphere (strong deflection limit). We have applied our technique to different metrics and verified that the error is at most 0.5% for all regimes. We have also proposed an alternative approach which provides simpler formulas, although with larger errors.

Figure 1

The potential $V(z)$ in Eq. (11) with $\mu =3/2$ (solid line). The dashed and dotted lines correspond to quadratic Taylor expansions around $z=0$ and $z=\mu$, respectively.

Figure 2

Left panel: approximate deflection angle for the Schwarzschild metric (Eq. (25)) with $\mu =10$ as a function of $\sigma$. The horizontal dotted line represents the exact value. Right panel: percent error of the approach as a function of $\sigma$.

Figure 3

Same as in Fig. 2 for $\mu =1.001$.

Figure 4

${\sigma }_{\mathrm{PMS}}$ obtained numerically solving the PMS condition (broken line) as a function of $1/\mu$ and the linear approximation $\sigma =1-1/2\mu$ (solid line).

Figure 5

Percent error of Eq. (25) as a function of $\mu$ for the deflection angle in the case of the Schwarzschild metric. Left panel: range $1\le \mu \le 2$; Right panel: range $1\le \mu \le 1000$.

Figure 6

Left panel: deflection angle for the Schwarzschild metric as a function of $\mu$ using different approximations and the exact result. Right panel: percent error of the approximate solutions as a function of $\mu$.

Figure 7

Percent error for the deflection angle in the Schwarzschild metric calculated with Method I (Eq. (25)), with the simplified version of Method I (Eq. (53) and with Method II (Eq. (a8)).

Figure 8

Percent error of the deflection angle in the Reissner-Nordström metric calculated with Method I at different values of $\mu$.

Figure 9

Left panel: deflection angle in the Janis-Newman-Winicour metric using $\nu =1/2$ and $b=1$; the solid line is the numerical result, whereas the dashed line corresponds to our analytical formula. Right panel: percent error for the deflection angle using our analytical formula.