Pole-skipping and hydrodynamic analysis in Lifshitz, AdS$_2$ and Rindler geometries

The "pole-skipping" phenomenon reflects that the retarded Green's function is not unique at a pole-skipping point in momentum space $(\omega,k)$. We explore the universality of the pole-skipping in different geometries. In holography, near horizon analysis of the bulk equation of motion is a simpler way to derive a pole-skipping point and we use this method in Lifshitz, AdS$_2$ and Rindler geometries. We also study the complex hydrodynamic analyses and find that the dispersion relations in terms of dimensionless variables $\frac{\omega}{2\pi T}$ and $\frac{\vert k\vert}{2\pi T}$ pass through pole-skipping points $(\frac{\omega_n}{2\pi T}, \frac{\vert k_n\vert}{2\pi T}$) at small $\omega$ and $k$ in Lifshitz background. We verify that the position of the pole-skipping points does not depend on the standard quantization or alternative quantization in the boundary theory in AdS$_2\times\mathbb{R}^{d-1}$ geometry. In Rindler geometry, we cannot find the corresponding Green's function to calculate pole-skipping points because it is difficult to impose the boundary condition. However we can obtain "special points" near horizon where bulk equations of motion have two incoming solutions. These "special points" correspond to nonunique of the Green's function in physical meaning from the perspective of holography.