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We use the quasinormal modes of a massless scalar perturbation to probe the phase transition of the high dimension charged AdS black holes. As a result, the signature of the critical behavior of this black hole solution is detected [ 1 , 2 ]. As an alternative, we can use also the unstable circular photon orbits around a $RN-AdS_4$ BH in massive gravity to establish a direct link between the null geodesics and the critical behavior thermodynamic of such black hole solution. The analysis shows that the unstable circular radius $r_0$ and the minimal impact parameter $\xi_0$ corresponding to the unstable circular orbits are a good tool to reveal the thermodynamic phase structure [ 3 ].
1 https://arxiv.org/abs/1606.08524
2 https://arxiv.org/abs/1701.00872
3 https://arxiv.org/abs/1902.00557