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

Abstract

The "pole-skipping" phenomenon reflects that retarded Green's function is not unique at a "special point" in momentum space (ω, k). We calculate the lower-half ω-plane pole-skipping points not only of tensor and Maxwell fields in the anisotropic systems near Lifshitz points, but also of axion field in the Lifshitz black holes with hyperscaling violating factor. The frequencies of these special points are always located at negative integer Matsubara frequencies ω n = −i2πT n (n = 1, 2 . . . ). Compared to the systems studied before, the momentums k n of pole-skipping points found in the background of the anisotropic systems near Lifshitz points are located in the complex k-plane rather than just at the real k n or imaginary k n . We study the complex hydrodynamic analyses in these two backgrounds. We find that the dispersion relations in terms of dimensionless variables ω 2πT and |k| 2πT pass through pole-skipping points (w n , |k n |) at small frequency whether momentums are real, imaginary or complex.

Analogue of the pole-skipping phenomenon in acoustic black holes

“…We consider the backgrounds of embedding (2 + 1)−dimensional acoustic black holes into Minkowski spacetime in section 2, Schwarzschild spacetime in section 3, and AdS-Schwarzschild spacetime in section 4, respectively. We show that the frequencies of the lower-half plane of pole-skipping points in all cases of the acoustic black hole are consistent with the imaginary part of the frequencies of QNMs in section 5, which implies that they may have the similar physical meaning as it was discussed in gravitational black holes [14].…”

“…We cannot obtain poles-skipping points by solving Green's function directly without the boundary condition. Nevertheless, we have obtained the "pole-skipping" points by solving the bulk equations of motion near the horizon in Rindler geometry [14]. The procedure developed in [14] also applies to the acoustic black holes.…”

“…The locations of special points (ω n , k n ) can be easily extracted from the determinant of the In Ref. [6,13,14], the authors show that dispersion relation, which arises from the pole of the retarded Green's function (QNMs), passes through the lower half-plane pole-skipping points. We want to see if the relation between the pole-skipping points and the hydrodynamic dispersion relation still holds for acoustic black holes.…”

“…), where w = ω 2πT . These special points have been found in BTZ black hole [6], Schwarzschild-AdS spacetime [7], 2D CFT [8], a holographic system with the chiral anomaly [9], a holographic system at finite chemical potential [10], hyperbolic space [11], the large q limit of SYK chain [12], anisotropic plasma [13], Lifshitz, and Rindler geometries [14]. However, the physical interpretation of the pole-skipping phenomenon remains elusive.…”

“…In [14], we obtain the "pole-skipping" points in Rindler geometry and reveal their universality for different spacetime backgrounds. There should be universal observations; one can only use near horizon analysis to obtain "pole-skipping" points, not depending on the UV property of the Green's function.…”