Null geodesics, quasinormal modes, and thermodynamic phase transition for charged black holes in asymptotically flat and dS spacetimes

The numerical study indicates that there exists a relation between the quasinormal modes and the Davies point for a black hole. In this paper, we analytically study this relation for the charged Reissner-Nordstr\"{o}m black holes in asymptotically flat and dS spacetimes. In the eikonal limit, the angular velocity $\Omega$ and the Lyapunov exponent $\lambda$ of the photon sphere, respectively, corresponding to the real and imaginary parts of the quasinormal modes are obtained from the null geodesics. Both in asymptotically flat and dS spacetimes, we observe the spiral-like shapes in the complex quasinormal mode plane. However, the starting point of the shapes do not coincide with the Davies point. Nevertheless, we find a new relation that the Davies point exactly meet the maximum of the temperature $T$ in the $T$-$\Omega$ and $T$-$\lambda$ planes. In higher dimensional asymptotically flat spacetime, even there is no the spiral-like shape, such relation still holds. Therefore, we provide a new relation between the black hole thermodynamics and dynamics. Applying this relation, we can test the black hole thermodynamic property by the quasinormal modes.