Effects of the refractive index of the X-ray corona on the emission lines in AGN

X-ray reflection from an accretion disc produces characteristic emission lines allowing us to probe the innermost regions in AGN. We investigate these emission lines under a framework of Riemannian geometrical optics where the corona has a refractive index of $n \neq 1$. The empty space outside is a vacuum with $n = 1$. The Kerr metric is modified to trace the light rays that are bent due to not only the gravity of the black hole, but also the effects of coronal plasma dependent on $n$. The choice of $n$ alters the null geodesics, producing the effect which is analogous to the light deflection. For the corona with $n > 1$, the disc on the far side within the corona covers a larger area on the observer' sky, enhancing the blue wing of the line and producing more flux difference between the blue peak and extended red tail. The inverse effects are seen when $n < 1$. Moreover, the corona with $n > 1$ and $n < 1$ could induce extra shifts in the blue wing ($\Delta g_{max}$) to higher and lower energy, respectively. These effects are more prominent when the inclination angle is $\gtrsim 60^\circ$ and the corona extends to $\gtrsim 5r_g$. To obtain the deviation of the line shift of $\Delta g_{\rm max} \gtrsim 0.01$, the difference between the refractive index of the corona and that of the empty space must be $\Delta n \gtrsim 0.5%$. Finally, the lensing corona can influence the arrival time of photons that may affect the observed variability of these emission lines