Strong gravitational lensing for the photons coupled to a Weyl tensor in a Kerr black hole spacetime

We present first the equation of motion for the photon coupled to a Weyl tensor in a Kerr black hole spacetime and then study further the corresponding strong gravitational lensing. We find that black hole rotation makes propagation of the coupled photons more complicated, which brings about some new features for physical quantities, including the marginally circular photon orbit, the deflection angle, the observational gravitational lensing variables, and the time delay between two relativistic images. There is a critical value of the coupling parameter for existence of the marginally circular photon orbit outside the event horizon, which depends on the rotation parameter of the black hole and the polarization direction of the photons. As the value of the coupling parameter nears the critical value, we find that the marginally circular photon orbit for the retrograde photon increases with the rotation parameter, which modifies a common feature of the marginally circular photon orbit in a rotating black hole spacetime since it always decreases monotonously with the rotation parameter in the case without Weyl coupling. Modeling the supermassive central object in our Galaxy as a Kerr black hole, we estimate the numerical values of the observables including time delays between the relativistic images in the strong gravitational lensing of the photons coupled to Weyl tensor.

Figure 1

Variety of the critical values ${\alpha }_{c1}$ and ${\alpha }_{c2}$ with the rotation parameter $a$. The marginally circular photon orbit radius $x_{ps}$ exists only in the regime $\alpha <{\alpha }_{c1}$ for PPM and in the regime $\alpha >{\alpha }_{c2}$ for PPL. Here, we set $2M=1$.

Figure 2

Variety of the marginally circular orbit radius of PPM with the coupling parameter $\alpha$ for different $a$. Here, we set $2M=1$.

Figure 3

Variety of the marginally circular orbit radius of photon PPL with the coupling parameter $\alpha$ for different $a$. Here, we set $2M=1$.

Figure 4

Change of the strong deflection limit coefficients $\overline{a}$ and $\overline{b}$ for PPM with the coupling parameter $\alpha$ for different $a$ in a Kerr black hole spacetime. Here, we set $2M=1$.

Figure 5

Change of the strong deflection limit coefficients $\overline{a}$ and $\overline{b}$ for PPL with the coupling parameter $\alpha$ for different $a$ in a Kerr black hole spacetime. Here, we set $2M=1$.

Figure 6

Deflection angles for PPM evaluated at $u=u_{ps}+0.003$ is a function of the coupling parameter $\alpha$ for different $a$ in a Kerr black hole spacetime. Here, we set $2M=1$.

Figure 7

Deflection angles for PPL evaluated at $u=u_{ps}+0.003$ is a function of the coupling parameter $\alpha$ for different $a$ in a Kerr black hole spacetime. Here, we set $2M=1$.

Figure 8

Variety of the innermost relativistic image ${\theta }_{\infty }$ for PPM with the coupling parameter $\alpha$ for different $a$. Here, we set $2M=1$.

Figure 9

Variety of the innermost relativistic image ${\theta }_{\infty }$ for PPL with the coupling parameter $\alpha$ for different $a$. Here, we set $2M=1$.

Figure 10

Variety of the angular separation $s$ for PPM with the coupling parameter $\alpha$ for different $a$. Here, we set $2M=1$.

Figure 11

Variety of the angular separation $s$ for PPL with the coupling parameter $\alpha$ for different $a$. Here, we set $2M=1$.

Figure 12

Variety of the relative magnitudes $r_m$ for PPM with the coupling parameter $\alpha$ for different $a$. Here, we set $2M=1$.

Figure 13

Variety of the relative magnitudes $r_m$ for PPL with the coupling parameter $\alpha$ for different $a$. Here, we set $2M=1$.

Figure 14

Variety of the time delays between the relativistic images for PPM with the coupling parameter $\alpha$ for different $a$ in a Kerr black hole spacetime. Here, we set $2M=1$, $n=2$ and $m=1$.

Figure 15

Variety of the time delays between the relativistic images for PPL with the coupling parameter $\alpha$ for different $a$ in a Kerr black hole spacetime. Here, we set $2M=1$, $n=2$ and $m=1$.