Holographic Einstein Rings of Black Holes in Scalar-Tensor-Vector Gravity

With the help of AdS/CFT correspondence, we analyze the holographic Einstein images via the response function of the complex scalar field as a probe wave on the AdS Schwarzschild scalar-tensor-vector gravity (STVG) black hole (BH). We find that the amplitude of the response function $|\langle O\rangle|$ decreases with the increasing values of the coupling parameter $\alpha$, while it increases with the decreasing values of temperature $T$. The frequency $\omega$ of the wave source also plays a significant role in wave periods, as we increase the values of $\omega$, we find a decrease in periods of waves, which means that the total response function closely depends on the wave source. Further, we investigate the optical appearance of the holographic images of the BH in bulk. We found that the holographic ring always appears with the concentric stripe surrounded when the observer is located at the north pole, and an extremely bright ring appears when the observer is at the position of the photon sphere of the BH. With the change of the observational angle, this ring will change into a luminosity-deformed ring or a bright light spot. The corresponding brightness profiles show that the luminosity of the ring decreases, and the shadow radius increases with increasing values of $\alpha$. The relation between temperature $T$ and the inverse of the horizon $h_{e}$ is discussed, which shows the smaller values at the beginning of the horizon $h_{e}$, and then increases as the horizon radius increases. This effect can be used to distinguish the STVG BH solution from other BH solutions. Moreover, these significant features are also reflected in the Einstein ring and the corresponding brightness profiles. In addition, we compare the results obtained by wave optics and geometric optics, which are aligned well, implying that the holographic scheme adopted in this paper is valid.