Higher order corrections to deflection angle of massive particles and light rays in plasma media for stationary spacetimes using the Gauss-Bonnet theorem

The purpose of this article is twofold. First, we extend the results presented in [G. Crisnejo and E. Gallo, Phys. Rev. D 97, 124016 (2018)] to stationary spacetimes. Specifically, we show that the Gauss-Bonnet theorem can be applied to describe the deflection angle of light rays in plasma media in stationary spacetimes. Second, by using a correspondence between the motion of light rays in a cold nonmagnetized plasma and relativistic test massive particles we show that this technique is not only powerful to obtain the leading order behavior of the deflection angle of massive/massless particles in the weak field regime but also to obtain higher order corrections. We particularize it to a Kerr background where we compute the deflection angle for test massive particles and light rays propagating in a nonhomogeneous cold plasma by including third order corrections in the mass and spin parameters of the black hole.

Figure 1

The points $S$ and $R$ represent the source and receiver position while ${\gamma }_p$ indicates a light ray emitted by the source and that reaches the observer. $b$ is identified with the impact parameter and $C_{\mathrm{R}}$ is a semicircle of radius R connecting $S$ and $R$.

Figure 2

Different contributions to the total deflection angle Eq. (104) for a lens with parameters $M=4.1\times {10}^6{M}_{\odot }$ and $\widehat{a}=0.6$ in terms of the speed $v$ of the test massive particle. We assume that $b=100M$.

Figure 3

Different contributions to the total deflection angle for a homogeneous plasma with ${\omega }_e=\text{constant}$ (obtained from Eq. (104) by replacing $v^2$ by $1-{\omega }_e^2/{\omega }_{\infty }^2$). We consider a lens with parameters $M=4.1\times {10}^6{M}_{\odot }$, $\widehat{a}=0.6$ and $b=100M$.

Figure 4

Leading order contribution to the deflection angle for a lens with parameters $M=4.1\times {10}^6{M}_{\odot }$ and $\widehat{a}=0.6$ and different plasma density profiles of the form $N(r)={\tilde{N}}_0{r}^{-k}$. We assume that $b=100M$. Here ${\alpha }_{\mathrm{vac}}^{(1)}=4M/b$ and ${\alpha }_{\mathrm{p}}^{(1)}=-\epsilon$ for $k=1$, ${\alpha }_{\mathrm{p}}^{(1)}=-\epsilon \pi /2$ for $k=2$ and ${\alpha }_{\mathrm{p}}^{(1)}=-2\epsilon$ for $k=3$. All of them are particular cases of the expression (131). Here $\epsilon \in [0.01,0.018]$.

Figure 5

Higher order contribution to the deflection angle for a lens with parameters $M=4.1\times {10}^6{M}_{\odot }$ and $\widehat{a}=0.6$. We assume that $b=100M$. Here ${\alpha }_{\mathrm{vac}}^{(2,3)}={\alpha }_{\mathrm{vac}}-{\alpha }_{\mathrm{vac}}^{(1)}$ and ${\alpha }_{\mathrm{p}}^{(2,3)}={\alpha }_{\mathrm{p}}-{\alpha }_{\mathrm{p}}^{(1)}$. $\epsilon \in [0.01,0.018]$. As in Fig. 4 each of them is considered for the different density profiles determined by $k=1$, 2 and 3.