Abstract
We investigate a new property of retarded Green’s functions using AdS/CFT. The Green's functions are not unique at special points in complex momentum space. This arises because there is no unique incoming mode at the horizon and is similar to the “pole skipping” phenomenon in holographic chaos. Our examples include the bulk scalar field, the bulk Maxwell vector and scalar modes, and the shear mode of gravitational perturbations. In these examples, the special points are always located at \U0001d714★ = –i(2πT) with appropriate values of complex wave number.
Highlights
The Green’s function at the special point is not unique because it depends on the slope δq/δω
The Green’s functions are not unique at special points in complex momentum space. This arises because there is no unique incoming mode at the horizon and is similar to the “poleskipping” phenomenon in holographic chaos
If the sound pole is analytically continued to a pure imaginary q and if it is extended to finite q, it coincides with the special point
Summary
We mostly consider the Schwarzschild-AdSp+2 (SAdSp+2) black hole background:. We consider various perturbations in the black hole background. We consider the scalar field, Maxwell field, and gravitational perturbations. The perturbations are decomposed under the transformation of boundary spatial coordinate xi. The Maxwell perturbations AM are decomposed as scalar mode (diffusive mode): Av , Ax , Ar , vector mode: Ay. For example, the scalar mode transforms as scalar under the transformation. For p = 2, gravitational perturbations are decomposed as scalar mode (sound mode): hvv , hvr , hrr , hvx , hrx , hxx , hyy , vector mode (shear mode): hvy , hry, hxy. The Maxwell vector mode and a generic scalar field do not have a hydrodynamic pole. This is the formalism developed by Kodama and Ishibashi [17]
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