Strong gravitational lensing by an electrically charged black hole in Eddington-inspired Born-Infeld gravity

We systematically examine the properties of null geodesics around an electrically charged, asymptotically flat black hole in Eddington-inspired Born-Infeld gravity, varying the electric charge of the black hole and the coupling constant in the theory. We find that the radius of the unstable circular orbit for a massless particle decreases with the coupling constant, if the value of the electrical charge is fixed. Additionally, we consider the strong gravitational lensing around such a black hole. We show that the deflection angle, the position angle of the relativistic images, and the magnification due to the light bending in strong gravitational field are quite sensitive to the parameters determining the black hole solution. Thus, through the accurate observations associated with the strong gravitational lensing, it might be possible to reveal the gravitational theory in a strong field regime.

Figure 1

Effective potential $V(r)$ in EiBI with $Q/M=0.5$ and $\kappa /M^2=6$. The solid, dotted, and broken lines correspond to the cases for $b=2M$, $b_c$, and $7M$, respectively. The position of $R_{\mathrm{UCO}}$ in EiBI is denoted by the vertical solid line, while that in general relativity is denoted by the vertical dot-dashed line for reference. The massless particle with $b=7M$ is scattered by the black hole at $r=r_0$, which is denoted by the filled circle.

Figure 2

Radius of the unstable circular orbit $R_{\mathrm{UCO}}$ around the electrically charged black hole in EIBI as a function of the electrical charge $Q/M$ with the fixed value of the coupling constant $\kappa /M^2$ (left panel) and as a function of $\kappa /M^2$ with fixed value of $Q/M$ (right panel). In the figure, the thick-solid line corresponds to the result in general relativity, i.e., for the Reissner-Nordström spacetime in the left panel and for the Schwarzschild spacetime in the right panel.

Figure 3

Critical impact parameter $b_c$ in EiBI as a function of the electrical charge $Q/M$ with the fixed value of the coupling constant $\kappa /M^2$ (left panel) and as a function of $\kappa /M^2$ with the fixed value of $Q/M$ (right panel). The same as in Fig. 2, the thick-solid line corresponds to the result in general relativity.

Figure 4

Image of scattering orbit of a massless particle, where $\mathrm{\Delta }\phi$ is the deflection angle. The right panel is a magnified view in the vicinity of the scattering position, where $r_0$ is the curtate distance between the particle orbit and the black hole.

Figure 5

For the scattering massless particle, the impact parameter in EiBI is shown as a function of the position of the turn-around point, where the left and right panels correspond to the results for $Q/M=0.5$ and 1.0, respectively. The labels in the figure denote the value of the coupling constant $\kappa /M^2$. For reference, the case in general relativity ($\kappa =0$) is also shown with the thick-solid line.

Figure 6

Deflection angle is shown as a function of the position of the turn-around point. The left and right panels correspond to the results for $Q/M=0.5$ and 1.0, respectively. The labels in the figure denote the value of the coupling constant $\kappa /M^2$ in EiBI, where the case in general relativity ($\kappa =0$) is also shown with the thick-solid line.

Figure 7

Lens diagram, where S, L, O, and I are the positions of the source, lens object, observer, and image, respectively.

Figure 8

The angle of the 1st relativistic image, ${\mathrm{\Theta }}_1^0$, in EiBI as a function of the electrical charge $Q/M$ with the fixed value of the coupling constant $\kappa /M^2$ in the left panel and as a function of $\kappa /M^2$ with the fixed value of $Q/M$ in the right panel. The labels in the figure denote the fixed values of $Q/M$ or $\kappa /M^2$. For reference, the solid-thick line in each panel corresponds to the result in general relativity.

Figure 9

Angular separation $s$ in EiBI as a function of the electric charge $Q/M$ with the fixed value of the coupling constant $\kappa /M^2$ in the left panel and as a function of the value of $\kappa /M^2$ with the fixed value of $Q/M$ in the right panel.

Figure 10

Magnitude of the relative magnification ${\mathcal{R}}_m$ as a function of the electric charge $Q/M$ with the fixed value of the coupling constant $\kappa /M^2$ in the left panel and as a function of the value of $\kappa /M^2$ with the fixed value of $Q/M$ in the right panel.